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[ "Simultaneous optogenetic manipulation and calcium imaging in freely moving C. elegans", "Simultaneous optogenetic manipulation and calcium imaging in freely moving C. elegans" ]	[ "Frederick B Shipley \nLewis Sigler Institute for Integrative Genomics\nPrinceton University\nPrincetonNJUSA\n", "Christopher M Clark \nDepartment of Neurobiology\nUniversity of Massachusetts Medical School\nWorcesterMAUSA\n\nDepartment of Neurobiology\nStanford University\nUSA\n\nMedical School\nInstitute for Integrative Genomics\nUniversity of Massachusetts\n364 Plantation Street01605WorcesterMAUSA\n\nPrinceton University\n170 Carl Icahn Laboratory08544PrincetonNJUSA\n", "Mark J Alkema [email protected] \nDepartment of Neurobiology\nUniversity of Massachusetts Medical School\nWorcesterMAUSA\n\nDepartment of Neurobiology\nStanford University\nUSA\n\nMedical School\nInstitute for Integrative Genomics\nUniversity of Massachusetts\n364 Plantation Street01605WorcesterMAUSA\n\nPrinceton University\n170 Carl Icahn Laboratory08544PrincetonNJUSA\n", "Andrew M Leifer [email protected] \nLewis Sigler Institute for Integrative Genomics\nPrinceton University\nPrincetonNJUSA\n", "Luis De Lecea ", "Mark J Alkema ", "Andrew M Leifer ", "Lewis-Sigler " ]	[ "Lewis Sigler Institute for Integrative Genomics\nPrinceton University\nPrincetonNJUSA", "Department of Neurobiology\nUniversity of Massachusetts Medical School\nWorcesterMAUSA", "Department of Neurobiology\nStanford University\nUSA", "Medical School\nInstitute for Integrative Genomics\nUniversity of Massachusetts\n364 Plantation Street01605WorcesterMAUSA", "Princeton University\n170 Carl Icahn Laboratory08544PrincetonNJUSA", "Department of Neurobiology\nUniversity of Massachusetts Medical School\nWorcesterMAUSA", "Department of Neurobiology\nStanford University\nUSA", "Medical School\nInstitute for Integrative Genomics\nUniversity of Massachusetts\n364 Plantation Street01605WorcesterMAUSA", "Princeton University\n170 Carl Icahn Laboratory08544PrincetonNJUSA", "Lewis Sigler Institute for Integrative Genomics\nPrinceton University\nPrincetonNJUSA" ]	[]	Understanding how an organism's nervous system transforms sensory input into behavioral outputs requires recording and manipulating its neural activity during unrestrained behavior. Here we present an instrument to simultaneously monitor and manipulate neural activity while observing behavior in a freely moving animal, the nematode Caenorhabditis elegans. Neural activity is recorded optically from cells expressing a calcium indicator, GCaMP3. Neural activity is manipulated optically by illuminating targeted neurons expressing the optogenetic protein Channelrhodopsin. Real-time computer vision software tracks the animal's behavior and identifies the location of targeted neurons in the nematode as it crawls. Patterned illumination from a DMD is used to selectively illuminate subsets of neurons for either calcium imaging or optogenetic stimulation. Real-time computer vision software constantly updates the illumination pattern in response to the worm's movement and thereby allows for independent optical recording or activation of different neurons in the worm as it moves freely. We use the instrument to directly observe the relationship between sensory neuron activation, interneuron dynamics and locomotion in the worm's mechanosensory circuit. We record and compare calcium transients in the backward locomotion command interneurons AVA, in response to optical activation of the anterior mechanosensory neurons ALM, AVM or both.	10.3389/fncir.2014.00028	null	1,984,780	1311.6406	40ad458e4be1f7013369d48faaa3361d325504d6	Simultaneous optogenetic manipulation and calcium imaging in freely moving C. elegans published: 24 March 2014 Frederick B Shipley Lewis Sigler Institute for Integrative Genomics Princeton University PrincetonNJUSA Christopher M Clark Department of Neurobiology University of Massachusetts Medical School WorcesterMAUSA Department of Neurobiology Stanford University USA Medical School Institute for Integrative Genomics University of Massachusetts 364 Plantation Street01605WorcesterMAUSA Princeton University 170 Carl Icahn Laboratory08544PrincetonNJUSA Mark J Alkema [email protected] Department of Neurobiology University of Massachusetts Medical School WorcesterMAUSA Department of Neurobiology Stanford University USA Medical School Institute for Integrative Genomics University of Massachusetts 364 Plantation Street01605WorcesterMAUSA Princeton University 170 Carl Icahn Laboratory08544PrincetonNJUSA Andrew M Leifer [email protected] Lewis Sigler Institute for Integrative Genomics Princeton University PrincetonNJUSA Luis De Lecea Mark J Alkema Andrew M Leifer Lewis-Sigler Simultaneous optogenetic manipulation and calcium imaging in freely moving C. elegans published: 24 March 201410.3389/fncir.2014.00028METHODS ARTICLE Edited by: Reviewed by: Wolfgang Driever, University of Freiburg, Germany David Prober, California Institute of Technology, USA Correspondence:optogeneticscalcium imagingsensorimotor transformationmechanosensationbehavior Understanding how an organism's nervous system transforms sensory input into behavioral outputs requires recording and manipulating its neural activity during unrestrained behavior. Here we present an instrument to simultaneously monitor and manipulate neural activity while observing behavior in a freely moving animal, the nematode Caenorhabditis elegans. Neural activity is recorded optically from cells expressing a calcium indicator, GCaMP3. Neural activity is manipulated optically by illuminating targeted neurons expressing the optogenetic protein Channelrhodopsin. Real-time computer vision software tracks the animal's behavior and identifies the location of targeted neurons in the nematode as it crawls. Patterned illumination from a DMD is used to selectively illuminate subsets of neurons for either calcium imaging or optogenetic stimulation. Real-time computer vision software constantly updates the illumination pattern in response to the worm's movement and thereby allows for independent optical recording or activation of different neurons in the worm as it moves freely. We use the instrument to directly observe the relationship between sensory neuron activation, interneuron dynamics and locomotion in the worm's mechanosensory circuit. We record and compare calcium transients in the backward locomotion command interneurons AVA, in response to optical activation of the anterior mechanosensory neurons ALM, AVM or both. INTRODUCTION Understanding the neural basis of behavior is a fundamental goal of neuroscience. In many cases, however, the normal operation of neural circuits can be studied only in freely behaving animals. Previous neurophysiology experiments in behaving animals employed one of two approaches: Either probes, such as electrodes or optical fibers, were surgically implanted into an animal and recorded via tether or wireless backpack (Wilson and McNaughton, 1993;Lee et al., 2006;Szuts et al., 2011); or animals were head-fixed so that they were sufficiently immobile for electrical recording or microscopy Harvey et al. (2009) ;Seelig et al. (2010). The nematode Caenorhabditis elegans, due to its small size and optical transparency allows for a third approach, whereby an external and non-invasive tracking microscope keeps pace with the worm's motion to image the worm brain while allowing the animal to roam freely without restraint. Moreover, the worm's small nervous system of 302 neurons, its genetic tractability and its known connectome make it well suited to an investigation of the neural basis of behavior. Early recordings of neural activity from freely moving C. elegans used the genetically encoded calcium indicator Cameleon and required the experimenter to manually adjust a microscope stage to keep the worm centered under a microscope objective (Clark et al., 2007). Subsequently, automated tracking systems were developed that used computer vision (Ben Arous et al., 2010) or analog methods (Faumont et al., 2011) to track the worm's body motion automatically by adjusting a motorized stage. For many of these systems, intracellular calcium transients can be recorded while also observing the worm's behavior. Similar systems have been employed to measure neural activity in zebrafish larvae, another small optically transparent organism (Naumann et al., 2010;Muto et al., 2013). These systems have provided a valuable means to correlate activity with behavior and in worms they have elucidated neural coding of temperature during thermotaxis (Clark et al., 2007) and provided insights into neural dynamics correlated with transitions between forward and backward locomotion (Kawano et al., 2011;Piggott et al., 2011). Optogenetics allows for optically stimulating or inhibiting neurons that express light activated proteins, like Channelrhodopsin. C. elegans was the first organism to have its behavior manipulated optogenetically (Nagel et al., 2005). Early experiments relied on genetic specificity for targeting their stimulus. For example, optogenetics was first used to study the mechanosensory circuit in C. elegans, but only through simultaneous stimulation of all touch receptor neurons, because promoters specific to each neuron are unavailable. Patterned illumination overcomes this limitation by combining genetic specificity with optical targeting. By delivering light to only targeted cells or tissues, neurons can be illuminated individually provided that there is a sufficiently sparse expression pattern. Previously we developed an optogenetic illumination system that allows perturbations of neural activity with high spatial and temporal resolution in an unrestrained worm, enabling us to control locomotion and behavior in real time (CoLBeRT) in C. elegans (Leifer et al., 2011). The CoLBeRT system and others like it (Stirman et al., 2011) have been instrumental in defining neural coding of several behaviors in C. elegans including chemotaxis (Kocabas et al., 2012), nociception (Husson et al., 2012) and the escape response (Donnelly et al., 2013). We sought to combine these capabilities and thus simultaneously manipulate and monitor neural activity while also observing behavior. Here we present an instrument that can perturb a neural circuit and immediately observe its effects both on behavior and the activity of other neurons in the circuit. This instrument integrates the functionality of the CoLBeRT tool with that of a previously developed calcium imaging instrument (Leifer, 2012) to enable simultaneous manipulation and monitoring of neural activity in sparsely labeled neural circuits. We used this combined system to investigate the sensorimotor transformation between mechanosensory stimulus, interneuron activity and behavior in the C. elegans escape response circuit. Using this tool we are able to stimulate individual mechanosensory neurons and directly observe the effect on a downstream command interneuron and on the worm's behavior, an experiment that would not be possible with previous techniques. The integration of optogenetics, calcium imaging and behavioral analysis allows us to dissect neural circuit dynamics and correlate neural activity with behavior. To our knowledge, this is the first instrument, in any organism, to allow simultaneous non-invasive manipulation and monitoring of neural activity in an unrestrained freely moving animal. RESULTS EXPERIMENTAL SETUP C. elegans crawls freely on agarose on a motorized x-y translation stage under dark-field near-infrared (NIR) illumination, Figure 1. A high-speed behavior camera records the worm's position and orientation and real-time computer vision software extracts the outline of the worm and uses it to identify the worm's head, tail and centerline and the expected location of targeted neurons. At the heart of the instrument lies a DMD that generates patterned illumination targeted to individual neurons. Every ∼13 ms (75 frames per second) the DMD adjust its mirrors to reflect blue and yellow laser-light only onto targeted neurons. For calcium imaging, the DMD reflects blue and yellow light onto neurons co-expressing the calcium indicator GCaMP3 and a calcium-insensitive fluorescent reference, mCherry. A dedicated imaging path simultaneously records two images of green and red fluorescence from GCaMP3 and mCherry, respectively. To stimulate neurons, the DMD adjusts its mirrors to also reflect laser light onto other cells expressing ChR2. To accurately illuminate targeted neurons and not nearby neighbors, it is important to rapidly update the DMD's illumination pattern in response to the worm's change of position. The latency between a change in worm position and a corresponding change in illumination output is an important metric of the system's capabilities and represents the cumulative delay from a train of processes, including the time required to: expose one frame of the behavior camera, read out a frame from the camera and load it into software, perform computer vision analysis to extract the worm's position and orientation, send instructions to the DMD and finally adjust the DMD's mirrors. To measure this cumulative latency, we used an additional high speed camera to simultaneously optically record the image that is the input to our instrument and the position of the DMD's mirrors which is the system's output. Instead of an actual worm we used a computer monitor to display a stationary image of a worm that we could then manipulate precisely. In response to a step-wise translation of the image of the worm on our computer monitor (the input), the system updated the DMD mirrors (the output) in 28.6 ± 3 ms (mean ± standard deviation, n = 3 trials, measured with a camera recording 198 frames per second). The low-latency observed in the system is due in part to the high speed open-source MindControl computer-vision software package, written in C, which rapidly performs computer vision analysis. We have updated this software with new features for timed stimuli, improved tracking and better stability. Source code is available online at http://github.com/leiferlab/mindcontrol. TARGETED ILLUMINATION AND FLUORESCENT IMAGING IN MOVING WORMS To first validate the instrument, we sought to independently illuminate neurons in a moving worm. We generated a transgenic worm expressing GFP in the six soft-touch mechanosensory neurons, the ALM pair (ALML, ALMR), AVM, PVM, and the PLM pair (PLML, PLMR) and in the command interneuron pair AVA (AVAL and AVAR). Additionally, the promotor for AVA also expresses in nearby pharyngeal neurons (I1, I4, M4, NSM). By adjusting the illumination pattern, we illuminated the cell bodies of either the ALM neuron pair, the neuron AVM, or the AVA neuron pair along with nearby pharyngeal neurons, and we also illuminated different combinations thereof. We recorded fluorescent images as the worm crawled freely. Although these three targets were as close as 50 µm from one another, we observed no instances of errant illumination, see Supplementary Movie 1. OPTOGENETIC STIMULATION OF SINGLE TOUCH RECEPTOR CELL TYPES AND SIMULTANEOUS Ca 2+ IMAGING Calcium levels of AVA have been shown to increase during spontaneous bouts of backward locomotion (Chronis et al., 2007;Guo et al., 2009;Ben Arous et al., 2010;Kawano et al., 2011) and during mechanosensory evoked reversals (Leifer, 2012). Three mechanosensory neurons in the worm's anterior, the ALM pair and the neuron AVM, mediate the reversal response to soft touch (Chalfie and Sulston, 1981). Previously we showed that a brief optogenetical stimulation to either the neuron AVM or the neuron pair ALM was sufficient to induce a reversal (Leifer et al., 2011). During mechanical stimulation, however, all three of the anterior touch neurons are presumed to be activated. We asked whether the interneuron AVA exhibits elevated calcium transients during reversals caused by the stimulation of only a single mechanosensory neuron type. Moreover, we sought to investigate how the interneuron AVA integrates signals from ALM and AVM and how those signals drive the animal's behavioral response. We generated a transgenic line expressing ChR2 in the mechanosensory neurons and GCaMP3 and mCherry in the interneuron pair AVA. We stimulated either ALM, AVM or both neuron types Figure 2A by illuminating them for 2.7 s (n = 14, 12, 13 trials, respectively) in worms that were undergoing forward locomotion. We recorded calcium transients in AVA for at least 15 s before, during and after stimulus. In these experiments the DMD adjusts a subset of its mirrors such that blue and yellow light continuously illuminates GCaMP3 and mCherry in AVA. Only during stimulation does the DMD transiently turn on other mirrors to illuminate neurons ALM and AVM which express ChR2. In this manner, AVA illumination is controlled independently of ALM and AVM illumination. For all worms that reversed in response to stimulus, we observed elevated calcium levels in AVA immediately following stimulation of either ALM, AVM or both. Figure 2B and FIGURE 2 \| (A) ChR2 is expressed in six soft-touch mechanosensory neurons, including ALM and AVM (dark blue). Calcium indicator GCaMP3 and fluorescent reference mCherry are expressed in command interneuron AVA (striped green and red). Light blue shaded regions indicate areas of illumination. "a" is anterior, "p" is posterior, "d" is dorsal, and "v" is ventral. (B) Intracellular calcium dynamics of AVA (green line) are measured before during and after optogenetic stimulation of the ALM touch neuron (2.7 s stimulation, blue shaded region). The velocity of the worms body bending waves is shown in gray. Supplementary Movie 2 show an example trace of AVA calcium activity in response to ALM stimulation. All traces for the combined ALM and AVM stimulation are shown in Supplementary Figure 1. To reject the null hypothesis that our apparent calcium signals arise from instrument noise or motion artifact, we compared the calcium transients observed from GCaMP3 with apparent calcium transients observed from control worms that instead expressed calcium-insensitive GFP (n = 5 trials), see Figure 3A, (green dashed line) and Supplementary Figure 2. During reversals, AVA's activity as measured with GCaMP3, was more than one standard deviation above the mean greater than the apparent activity observed in calcium-insensitive GFP worms. This suggests that the observed signals reflect underlying calcium dynamics rather than motion artifacts or noise. When averaging across trials, neither AVA's response amplitude Figure 3A, nor the worm's reversal velocity Figure 3B were observed to vary significantly whether ALM or AVM or both were stimulated. The fraction of worms that responded to stimulus by reversing also did not vary significantly with stimulus but did depend, as expected, on the presence of the ChR2 cofactor all-trans retinal Figure 4B. Interestingly, however, the amplitude of AVA's activity inversely correlates with the worm's velocity immediately post stimulus (Pearsons correlation coefficient R = −0.83), irrespective of stimulus Figure 4A. Thus our data indicate that AVA's activity directly correlates with the worm's velocity and not with whether ALM, AVM or both are stimulated. MICROSCOPY The optogenetic stimulation and calcium imaging system was built around a modified Nikon Eclipse Ti-U inverted microscope that contained two stacked filter cube turrets. All imaging was conducted using a 10x, numerical aperture (NA) 0.45, Plano Apo objective. Worms were imaged on 6 cm NGM agarose plates on a Ludl BioPrecision2 XY motorized stage controlled by a MAC 6000 stage-controller. Custom real-time computer vision software kept the worm centered in the field of view via an automated feedback loop. BEHAVIORAL IMAGING Worm behavior was imaged on a CMOS camera (Basler aca2000-300 km) under dark-field illumination in the near-infrared (NIR). Dark-field illumination was chosen to provide high contrast between the worm and the agarose plate. NIR illumination was chosen to avoid cross talk with simultaneous fluorescence imaging and also to minimized inadvertent ChR2 activation. An NIR filter (Semrock #FF01-795/150), was mounted in the illumination path of a Nikon halogen lamp. Dark-field illumination was obtained by using a Ph3 phase ring to create an annular pattern of illumination. A custom filter cube (Semrock #FF670-SDi01) in the microscopes upper filter cube turret reflected the behavior imaging path out of the microscope through what would normally be its epifluorescence illumination pathway. From there, a telescope composed of two achromat doublet plano-convex lenses was used to form an image on the camera. A long-pass filter (Omega Optical #3RD710LP) prevented stray light from entering the camera. The Basler camera transferred images to a desktop computer via a PCI Express x16 10-tap Full Camera Link framegrabber (BitFlow Karbon). Images were recorded via the MindControl software discussed below. SPATIO-TEMPORALLY PATTERNED LASER ILLUMINATION To stimulate ChR2 and to illuminate GCaMP3, we used a 473-nm diode-pumped solid state laser (DPSS) (CNI Laser MBL-III-473, 150-mW maximum power, OptoEngine). Similarly, to illuminate mCherry we used a 561-nm DPSS laser (Sapphire 561-150 CW CDRH, 150-mW maximum power, Coherent). Laser illumination entered the microscope from the bottom filter cube turret. The beams from the 473-nm and 561-nm lasers were first aligned to a common beam path by a series of mirrors and a dichroic mirror (Semrock #FF518-Di01). The combined beam path was then expanded using a telescope of two plano-convex achromat doublet lenses and reflected by a 2-inch diameter mirror onto a 1024 × 768 element DMD (Texas Instruments DLP, Discovery 4100 BD VIS 0.7-inch XGA, Digital Light Innovations). The patterned light from the DMD was imaged onto the sample via an achromat doublet that served as a tube lens, and a custom filter cube (Chroma #59022bs dichroic, Semrock #FF01-523/610 emission filter) in the microscopes lower filter cube turret. Light intensity measured at the sample was 2 mW mm −2 of 473-nm light and 1 mW * mm −2 of 561-nm light. The micromirrors of the system are imaged to the sample using a 10x objective such that each independent mirror element corresponds to a distinct square of illumination at the sample with an area of 1.1 µm 2 . To independently target nearby neurons in a moving worm, however, the neurons must be spatially separated by a sufficient distance. The minimum distance between targets depends on a variety of factors including the worms velocity, the system's latency and the ability of the system to infer the location of targeted neurons from the worm's outline, which can be susceptible to inhomogeneous compression and expansion of the worm body. We have shown the ability to illuminate two targeted neurons (ALM and AVM) spaced as close as 50 µm apart in a freely moving worm. Closer distances may be possible, however, especially in the dorsalventral dimension which is less susceptible to compression or expansion. FLUORESCENCE IMAGING Red and green channel fluorescent images were recorded simultaneously on a Hamamatsu Orca Flash 4.0 siCMOS camera at 30 fps, with 33 ms exposure. To simultaneously image mCherry and GCaMP3 side-by-side we used a DV2 two-channel imager from Photometrics containing a custom filter set (Chroma #565dcxr dichroic, Semrock #FF01-609/54-25 red emission filter, Semrock #FF01-520/35-25 green emission filter). Fluorescent images were captured using HCImage software (Hamamatsu) running on a Dell Precision T7600 computer running Windows 7 with two Intel Xeon Quad Core 3.3 GHz processors and 49 GB of RAM. REAL-TIME COMPUTER VISION SOFTWARE An improved version of the MindControl software (Leifer et al., 2011), written in C, was used to generate patterned illumination and perform real-time feedback of the stage. The software was rewritten for 64-bit Windows 7. Additionally new features were added to give the user more options for performing timed stimulations and to give the user more control over stage feedback parameters. The overall stability of the software was also improved. The MindControl software ran on a second Dell Precision T7600 computer running Windows 7 with two Intel Xeon Quad Core 3.3 GHz processors and 16 GB of RAM. Source code is released under the GNU General Public License and is available for download on GitHub at https://github.com/leiferlab/ mindcontrol. MEASURING LATENCY To measure the latency of the system, we we displayed a stationary image of a worm on an LCD computer monitor and imaged it with the system's behavior camera. A second high speed CMOS camera (Basler aca2000-300 km) and framegrabber (BitFlow Karbon) captured video with 1 ms exposure that simultaneously showed the LCD monitor, the DMD and a stopwatch (iPhone 5 s, Apple Computers). The MindControl software was run with the static worm as an input at the same settings as if we were conducting an optogenetic stimulus experiment. The latency was measured by counting the number of recorded frames between a stepwise translation of the static worm image on the monitor and the corresponding translation of the pattern of the DMD mirrors. The average framerate of the camera was calculated using the stopwatch. FLUORESCENCE IMAGING ANALYSIS Neural activity is reported as normalized deviations from baseline of the ratio between GCaMP3 and mCherry fluorescence, R R 0 = R−R 0 R 0 . The baseline R 0 is defined as the mean of R during a time window at least 15 s in duration beginning with the start of the recording and ending with onset of optogenetic stimulation. We have chosen to report fluorescence intensity as a frame-by-frame ratio of green fluorescence from GCaMP3 to red fluorescence from mCherry so as to better account for artifacts from the animal's motion. The calcium insensitive mCherry serves as a built-in-control to compensate for slight changes in focus or from motion blur. The ratio R = I GCaMP3 −B green I mCherry −B red . The intensities I GCaMP3 and I mCherry were measured as the median pixel intensity in the green and red channels, respectively, of the 40% brightest pixels of a circular region of interest (ROI) centered on the maximal intensity of a Gaussian smoothed image of the neuron pair AVA. The local background B green and B red in the green and red channels, respectively, were measured as the median pixel intensity in an annulus around the neuron. The ROI was selected in each frame by custom MATLAB scripts and confirmed by the user. The ROI for each neuron was a circle of radius 8 pixels and the background was an annulus of inner radius 20 pixels and an outer radius of 22 pixels. Images captured at 2048 × 2048 pixel resolution were binned to 1024 × 1024 pixels before analysis. The image scale was 0.62 µm/pixel during analysis. Custom MATLAB scripts were used to calculated the R/R 0 for each frame, and the timeseries was smoothed with a low-pass Gaussian filter (σ = 5 frames). Transgenic animals expressing GFP instead of GCaMP3 were used to estimate the amount of observed calcium signals that could be attributed to motion artifact or instrument noise. GFP's fluorescence is insensitive to calcium activity, therefore signals of apparent calcium activity observed from GFP can be attributed to instrument noise or motion artifact. Animals expressing GFP were imaged under the same conditions as the GCaMP3 animals, however their reversals were spontaneous rather than induced. The apparent AVA activity of GFP controls is reported using the same R/R 0 metric except that GFP is used in place of GCaMP3. OPTOGENETIC-INDUCED BEHAVIOR EXPERIMENTS Worms were grown on NGM agar plates seeded with 250 µL of OP50 E. Coli mixed with 1 µL of 1 mM all-trans retinal in ethanol solution. Plates were seeded on day 0, worms were transferred to seeded plates on day 1 and imaging was performed on day 2. Spatially distinct regions of the worms body were defined to illuminate the cell bodies of selected neurons. The AVA region corresponded to 90% of the body width and 10% of the body length, centered 10% of the way from the anterior tip of the worm. The AVM region corresponded to 50% of the body width and 13% of the body length, centered 35.5% from the anterior tip, and 35% from the ventral edge. The ALM region was the same size, but centered 42.5% of the way from the anterior tip, and 35% from the lateral edge. Regions were selected to avoid overlap of processes, however the illumination region of AVA includes a small portion (estimated to be less then 20%) of the ALM process. To mitigate suspected worm-to-worm variability in ChR2 expression and retinal uptake, we selected worms for our experiment that reversed in response to whole head illumination but did not reverse or pause in response to illuminating the small portion of the ALM process near AVA. Previously we had observed that illuminating small areas of neuronal processes alone usually caused little effect (Leifer et al., 2011). Approximately three quarters of worms tested responded with reversals to a brief Frontiers in Neural Circuits www.frontiersin.org March 2014 \| Volume 8 \| Article 28 \| 5 whole head illumination, and of those, 39 of 53 worms had no noticeable response to the onset of process illumination and were thus deemed suitable for experimentation. Whole head illumination was performed manually on a fluorescent dissection scope (Nikon SMZ-1500) while process illumination was performed on the main instrument by illuminating the AVA region. For imaging, worms were washed in M9 solution, transferred to approximately 1.5-mm thick NGM agarose plates and covered with mineral oil to improve contrast under dark-field illumination. Imaging began after allowing animals to acclimate for 5 min. Optogenetic stimulation was delivered after >15 s of AVA imaging. Each stimulus constitutes a separate trial. For most experiments animals were stimulated once and immediately discarded. Some worms were stimulated twice with a 3 min rest in between. No worm was stimulated more than twice. Trials where the worm underwent spontaneous reversals prior to stimuli, or prolonged bouts of multiple distinct reversals in response to the stimuli were excluded from analysis. Occasionally during an experiment the real-time computer vision software transiently fails to correctly segment the worm's head and tail. This often occurs when the worm touches itself during the deepest ventral bend portion of an omega turn or when the worm encounters a bubble or other visual artifact in the agarose plate. These periods of segmentation failure are shown as gaps in the behavioral trace, as visible in Supplementary Figure 1. Behavioral data and calcium activity data were analyzed using custom MATLAB scripts, available at http:://github.com/ leiferlab/dualmag-analysis. Each trial was manually classified as reverser or non-reverser. Trials were classified as non-reversers if the animals's velocity failed to drop below zero in a short time window after stimulus. An empirical model of AVA activity was fit to each calcium trace to extract a characteristic amplitude of calcium activity and to define a time region over which to average the velocity (Supplementary Figure 1). This method was chosen because it allows us to compare velocities and calcium amplitudes in a uniform way across trials even when the response varies from trace to trace or even when the worm fails to reverse. Calcium levels in AVA were assumed to increase and then decrease in a similar manner to the voltage across a capacitor in an RC circuit in response to a voltage square wave, f (x) = ⎧ ⎪ ⎨ ⎪ ⎩ 0 if 0 ≤ t A(1 − e −4t/τ 1 ) if 0 < t ≤ τ 1 Ae −(t−τ 1 )/(τ 2 −τ 1 ) if τ 1 < t ,(1) where the onset of optogenetic stimulus occurs at t = 0, A is the amplitude of neural activity and τ 1 and τ 2 are the timescales for calcium increase and decrease and are allowed to vary between trials. Mean velocity was calculated over the time window from stimulus onset (t = 0) to τ 1 . The parameters A, τ 1 and τ 2 were fit for each trace using least-mean squared and the Nelder-Mead method. DISCUSSION The optical neurophysiology system presented here combines targeted optogenetic stimulation with calcium imaging in a freely behaving unrestrained animal. We used the system to investigate how mechanosensory signals from two neuronal cell types in C. elegans are integrated into a downstream command interneuron and how their combined activity influences behavior. The experiments performed here could not be done using any previously available method. A crucial feature of the current system is its ability to independently deliver light to targeted neurons expressing either ChR2 or GCaMP3. The current system, however, cannot independently image a neuron that co-expresses both GCaMP3 and ChR2 together because the two protein's excitation spectra overlap. Nonetheless, with minor modifications, the current system could be adapted to do so by using recently developed red-shifted ChR2 or RCaMP variants (Zhang et al., 2008;Erbguth et al., 2012;Akerboom et al., 2013) such that their excitation spectra no-longer overlap. A consensus has emerged that there is a distinct need for the development of new methods to manipulate and monitor neural activity in behaving animals at cellular resolution and circuit scale (NIH, 2013). The work here represents an important step toward that goal and is the first instance, in any organism, of noninvasive simultaneous optical manipulation and calcium imaging in an unrestrained animal. AUTHOR CONTRIBUTIONS ACKNOWLEDGMENTS This work was supported by grant GM084491 of the National Institutes of Health to Mark J. Alkema. Thanks to Josh Shaevitz, Dmitri Aranov, and Benjamin Bratton for productive discussions. Thanks to Ashley Linder for help rebuilding the optogenetic stimulation portion of the instrument. SUPPLEMENTARY MATERIAL The Supplementary Material for this article can be found online at: http://www.frontiersin.org/journal/10.3389/fncir. 2014.00028/abstract ). An example trace is shown (n = 5 trials). The y axis has been scaled for comparison to Figure 2B. R/R 0 is calculated as before, but with GFP fluorescence instead of GCaMP3. Because GFP is insensitive to modest changes in calcium levels, fluctuations in the signal observed here is presumably due to motion artifact or instrument noise. FIGURE 1 \| 1Schematic of the illumination and imaging systems.A worm moves freely on a motorized stage under infrared illumination (IR). A digital micromirror device (DMD) reflects blue and yellow laser light onto only targeted neurons. Three separate imaging paths simultaneously image the worm's behavior and record simultaneous red and green fluorescence images. Real-time computer vision software monitors the worm's posture and location and controls the DMD, lasers and stage. The DMD's illumination pattern is continuously updated to illuminate only targeted neurons. FIGURE 3 \| 3(A) Mean of AVA activity and (B) worm velocity during a time window are shown (gray crosses) for trials when the worm reversed in response to optogenetic illumination of ALM (n = 8 trials), AVM (n = 6 trials) or both (n = 9). Mean across trials is shown (red squares). Error bars represent standard error of the mean. The dashed green line shows the mean plus standard deviation of the apparent calcium signal observed during reversals in worms expressing calcium-insensitive GFP instead of GCaMP3 (n = 5 trials). FIGURE 4 \| 4(A) For each trial, the amplitude of AVA activity is plotted against the within-trial mean velocity during a trial-specific time window, (n = 39 trials). Both reversers (shaded points) and non-reversers (open points) are included. (B)The fraction, f , of worms that respond by reversing (reversers) are shown for each stimuli. Number, n, indicates total trials. Control worms grown without the ChR2 co-factor all-trans retinal (ATR (−)) are also shown. Error bars are standard error, σ f = f (1 − f )/n, where f is the fraction of worms that reverse out of a total, n.3. MATERIALS AND METHODS3.1. STRAINSStrainAML1 used in our experiment was made by crossing QW309 (zfIs18) [Pmec-4::ChR2::YFP] with QW625 (zfIs42) [Prig-d::GCaMP3::SL2::mCherry]. Control experiments were performed with QW1075 (zfEx416) [Prig-3::GFP:SL2::mCherry], (zdIs5) [Pmec-4::GFP]. Supplementary Movie 1 \| (Smovie1.mp4) Movie, MPEG-4 (10 MB). A control worm expresses GFP in interneuron type AVA and in the mechanosensory neurons including ALM and AVM ((zfEx416) [Prig3::GFP::SL2::mCherry], (zdIs5) [Pmec-4::GFP]). Different combinations of ALM, AVM, and AVA + pharyngeal neurons are illuminated independently. Arrow indicates location of AVA by nearby pharyngeal neurons I1, I4, M4, and NSM. The text indicates targeted neuron. No fluorescence is observed in non-targeted regions. Supplementary Movie 2 \| (Smovie2.mp4) Movie, MPEG-4 (20 MB). Calcium dynamics of command interneuron type AVA are shown before, during Frontiers in Neural Circuits www.frontiersin.panel shows green fluorescence from GCaMP3 in false color. The upper middle panel shows red fluorescence from mCherry in false color. Orange circle indicates location of command interneuron AVA. The upper right panel shows worm behavior. Green circle indicates head. Red triangle indicates tail. The animals outline is shown in yellow. Blue area shows region of blue laser light illumination. Lower panel shows calcium activity. Stimulus onset occurs at t = 0 s. The trial shown here is the same as that in Figure 2B. Background fluorescence that is visible in the red channel is presumably due to non-specific expresion of mCherry. It is thought that the Sl2 sequence that precedes mCherry acts as a weak non-specific promotor in addition to its desired function as a post-translational splice site. This background fluorescence is accounted for when calculating R/R 0 . Supplementary Figure 1 \| Calcium dynamics of command interneuron AVA (green) and animal velocity (gray) are shown before, during and after a 2.7 s stimulation to mechanosensory neurons ALM and AVM for each trial. All trials that resulted in a reversal are shown (9 animals, one trial per animal). The transgenic animals shown here express GCaMP3 and mCherry in interneuron AVA and ChR2 in the soft-touch mechanosensory neurons (lin-15(n765ts)); (zfIs18) [Pmec-4::ChR2::YFP]; (zfis42) [Prig-3::GCaMP3::SL2::mCherry]. Stimulus onset occurs at t = 0 s (first dashed vertical line) and extends for 2.7 s (blue shaded region). An empirical three-parameter model of AVA activity is fit to each calcium transient (blue dashed curve). The fit parameter τ 1 is shown as vertical dashed black line. The average velocity for each trace, as shown in Figures 3B, 4A, is calculated by averaging the velocity between t = 0 and t = τ 1 (region between vertical black dashed lines.) Supplementary Figure 2 \| GFP control. Green fluorescence from animals expressing GFP instead of GCaMP3 in AVA was observed (green line) during spontaneous reversals (QW1075 (zfEx416) [Prig-3::GFP:SL2::mCherry], (zdIs5) [Pmec-4::GFP] Andrew M. Leifer designed the instrument. Andrew M. Leifer and Frederick B. Shipley built the instrument. Christopher M. Clark and Frederick B. Shipley generated transgenic animals. Andrew M. Leifer, Frederick B. Shipley, Christopher M. Clark, and Mark J. Alkema designed experiments. Frederick B. Shipley conducted all experiments. March 2014 \| Volume 8 \| Article 28 \| 7 Genetically encoded calcium indicators for multi-color neural activity imaging and combination with optogenetics. J Akerboom, N Carreras Caldern, L Tian, S Wabnig, M Prigge, J Tol, 10.3389/fnmol.2013.00002Front. Mol. Neurosci. 62Akerboom, J., Carreras Caldern, N., Tian, L., Wabnig, S., Prigge, M., Tol, J., et al. 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[ "RKKY interaction in graphene", "RKKY interaction in graphene" ]	[ "E Kogan \nDepartment of Physics\nBar-Ilan University\nRamat-Gan 52900Israel\n" ]	[ "Department of Physics\nBar-Ilan University\nRamat-Gan 52900Israel" ]	[]	We consider RKKY interaction between two magnetic impurities in graphene. The consideration is based on the perturbation theory for the thermodynamic potential in the imaginary time representation. We analyze the symmetry of the RKKY interaction on the bipartite lattice at half filling. Our analytical calculation of the interaction is based on direct evaluation of real space spin susceptibility.	10.1103/physrevb.84.115119	[ "https://arxiv.org/pdf/1108.1306v9.pdf" ]	118,830,342	1106.5151	810832ac028b5a9bbf06903943974ba6e69bb2e6	RKKY interaction in graphene 23 Sep 2011 E Kogan Department of Physics Bar-Ilan University Ramat-Gan 52900Israel RKKY interaction in graphene 23 Sep 2011(Dated: April 28, 2013)PACS numbers: 7530Hx;7510Lp We consider RKKY interaction between two magnetic impurities in graphene. The consideration is based on the perturbation theory for the thermodynamic potential in the imaginary time representation. We analyze the symmetry of the RKKY interaction on the bipartite lattice at half filling. Our analytical calculation of the interaction is based on direct evaluation of real space spin susceptibility. INTRODUCTION Since graphene was first isolated experimentally [1], it is in the focus of attention of both theorists and experimentalists. Many physical phenomena, well studied in "traditional" solid state physics look quite different in graphene. In this paper we will talk about the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction, first studied (in a normal metal) more than 60 years ago [2][3][4]. This interaction is the effective exchange between two magnetic impurities in a non-magnetic host, obtained as the second order perturbation with respect to exchange interaction between the magnetic impurity and the itinerant electrons of the host. Quite a few theoretical papers published recently considered RKKY interaction in graphene [5][6][7][8][9][10][11]. Though analysis of the RKKY interaction is simple in principle, calculation of the integrals defining the interaction (whether analytical or numerical) can pose some problems. However, substantial progress was achieved in the field. Our interest in the RKKY interaction in graphene started from learning about the theorem stating that for any half-filled bipartite lattice the exchange interaction between the magnetic adatoms is ferromagnetic, if the adatoms belong to the same sublattice, and antiferromagnetic, if the adatoms belong to different sublattices [8]. Also, in this paper and in the following one [10], in the approximation of the linear dispersion law for the electrons, the RKKY interaction in graphene was calculated analytically. However, the integrals obtained in both papers turned out to be divergent, and the complicated (and to some extent arbitrary) cut-off procedure was implemented to obtain from these integrals the finite results. So we started to look for the procedure which will allow to eliminate this problem. The second reason for our interest was the fact that the theorem, mentioned above, was challenged [12]. The claim was that the proof is based on calculation of the magnetic susceptibility of the free electron gas by the imaginary-time method, demanding later analytic continuation from the imaginary frequencies to the real ones. On the other hand, consideration by the real-time method, presented in Ref. [12] does not support the statement of the theorem. To clarify the situation and get rid of the shortages mentioned above, we decided to analyze the problem of RKKY interaction in graphene from the scratch. RKKY INTERACTION We consider two magnetic impurities at the sites i and j and assume a contact exchange interaction between the electrons and the magnetic impurities. Thus the total Hamiltonian of the system is H T = H + H int = H − JS i ·s i − JS j ·s j ,(1) where H is the Hamiltonian of the electron system, S i is the spins of the impurity and s i is the spin of itinerant electrons at site i. Our consideration is based on the perturbation theory for the thermodynamic potential [13]. The correction to the thermodynamic potential due to interaction is ∆Ω = −T ln S ≡ −T ln tr S · e −H/T /Z , (2) where the S-matrix is given by the equation S = exp − 1/T 0 H int (τ )dτ .(3) Writing down s i in the second quantization representation s i = 1 2 c † iα σ αβ c iβ ,(4) the second order term of the expansion with respect to the interaction is ∆Ω = J 2 T 4 αβγδ S i ·σ αβ S j ·σ γδ(5)1/T 0 1/T 0 dτ 1 dτ 2 T τ c † iα (τ 1 )c iβ (τ 1 )c † jγ (τ 2 )c jδ (τ 2 ) . Notice that we have ignored the terms proportional to S 2 i and S 2 j , because they are irrelevant for our calculation of the effective interaction between the adatoms spins. Leaving aside the question about the spin structure of the two-particle Green's function standing in the r.h.s. of Eq. (5) (for interacting electrons), further on we assume that the electrons are non-interacting. This will allow us to use Wick theorem and present the correlator from Eq. (5) in the form − G βγ (i, j; τ 1 − τ 2 )G δα (j, i; τ 2 − τ 1 ),(6) where G βγ (i, j, τ 1 − τ 2 ) = − T τ c iβ (τ 1 )c † jγ (τ 2 )(7) is the Matsubara Green's function [13]. We can connect G βγ with the Green's function of spinless electron G βγ (i, j, τ 1 − τ 2 ) = −δ βγ T τ c i (τ 1 )c † j (τ 2 ) . (8) Presence of delta-symbols allows to perform summation with respect to spin indices in Eq. (5) αβ S i ·σ αβ S j ·σ βα = S i ·S j ,(9) which gives ∆Ω = −J 2 χ ij S i ·S j ,(10) where χ ij = − 1 4 1/T 0 G(i, j; τ )G(j, i; −τ )dτ(11) is the free electrons static real space spin susceptibility. Thus we obtain H RKKY = −J 2 χ ij S i ·S j ,(12) Eq. (11) was applied to calculation of RKKY interaction in graphene for the first time, to the best of our knowledge, in Ref. [14]. The Green's function can be easily written down using representation of eigenvectors and eigenvalues of the operator H (H − E n ) u n = 0.(13) It is G(i, j; τ ) = n u * n (i)u n (j)e −ξnτ × − (1 − n F (ξ n )) , τ > 0 n F (ξ n ), τ < 0 ,(14) where ξ n = E n −µ, and n F (ξ) = e βξ + 1 −1 is the Fermi distribution function. SYMMETRY OF THE RKKY INTERACTION ON THE HALF-FILLED BIPARTITE LATTICE In this Section we'll consider the Hamiltonian of the free electrons in tight-binding representation H = i,j t ij c † i c j .(15) Bipartite lattice we'll understand in the sense, that all the sites can be divided in two sublattices, and there is only inter-sublattice hopping (no intra-sublattice hopping). Thus the Hamiltonian H in matrix representation is H = 0 T T † 0 ,(16) where T is some matrix N ×M ( the first N sites belong to the sublattice A and the last M sites belong to sublattice B). Consider a matrix of even more general form than (16) H = 0 N ×N B N ×M C M×N 0 M×M ;(17) B and C are some arbitrary matrices. The spectrum of the matrixH can be found from a secular equation −EI N ×N B N ×M C M×N −EI M×M = 0.(18) In Ref. [15] it is proved the following property of the determinant of the block matrix A N ×N B N ×M C M×N D M×M = \|A\| D − CA −1 B ,(19) which is valid, provided \|A\| = 0. For non-zero eigenvalues of the matrixH, we can apply Eq. (19) to the determinant (18) to get E 2 I M×M − CB = 0.(20) Thus the spectrum of the bipartite Hamiltonian is symmetric, that is non-zero eigenvalues of the matrix H are present in pares (E, −E). If we write down Eq. (13) explicitly in a matrix form −E n I T T † −E n I u n = 0,(21) it becomes obvious that un(i) = ±u n (i),(22) where u n is the eigenfunction corresponding to E n and un is the eigenfunction corresponding to −E n , and in the r.h.s. of Eq. (22) there is plus sign if the site i belongs to one sublattice, and there is minus sign if the site belongs to the opposite sublattice. Eq. (22) and the fact that for µ = 0 we have n F (ξm) = 1 − n F (ξ m ), immediately convince us that the terms with non-zero energy in Eq. (14) are pairwise antisymmetric (with respect to simultaneous transformation τ → −τ , i ⇄ j and complex conjugation) for the sites i and j belonging to the same sublattice, and pairwise symmetric for the sites i and j belonging to opposite sublattices. The term (terms) with E = 0 is antisymmetric with respect to the above mentioned transformation, no matter which sublattices the sites belong to. Thus for the sites i and j belonging to the same sublattice G(j, i; −τ ) = −G * (i, j; τ ).(23) For the sites i and j belonging to different sublattices G(j, i; −τ ) = G * (i, j; τ ),(24) provided there are no zero energy states, or we can neglect there contribution to the Green's function. Thus for the case considered, Eq. (11) gives ferromagnetic exchange between magnetic impurities on the same sublattice and antiferromagnetic exchange between impurities on opposite sublattices (under the restriction presented above). ANALYTIC CALCULATION OF THE RKKY INTERACTION IN GRAPHENE In calculations of the RKKY interaction in graphene the n in Eq. (14) turns into a 2 (2π) 2 d 2 p, where a is the carbon-carbon distance. (Actually, there should appear a numerical multiplier, connecting the area of the elementary cell with a 2 , but we decided to discard it, which is equivalent to some numerical renormalization of J.) Also u n (i) = e ip·R i ψ p ,(25) where ψ p is the appropriate component of spinor electron wave-function (depending upon which sublattice the magnetic adatom belongs to) in momentum representation. Further on the integration with respect to d 2 p we'll treat as the integration in the vicinity of two Dirac points K, K ′ and present p = K(K ′ ) + k. The wave function for the momentum around Dirac points K and K ′ has respectively the form ψ ν,K (k) = 1 √ 2 e −iθ k /2 νe iθ k /2 ψ ν,K ′ (k) = 1 √ 2 e iθ k /2 νe −iθ k /2 ,(26) where ν = ±1 corresponds to electron and hole band [16]; the upper line of the spinor refers to the sublattice A and the lower line refers to the sublattice B. We'll assume that T = 0 and the Fermi energy is at the Dirac points; E + (k) and E − (k) would be electron and hole energy. Then Eq. (14) takes the form: for i and j belonging to the same sublattice G AA (i, j; τ > 0) = − 1 2 e iK·Rij + e iK ′ ·Rij a 2 (2π) 2 d 2 ke ik·Rij −E+(k)τ ,(27) and for i and j belonging to different sublattices G AB (i, j; τ > 0) = 1 2 a 2 (2π) 2 d 2 ke −E+(k)τ × e i(K+k)·Rij −iθ k − e i(K ′ +k)·Rij +iθ k .(28) For τ < 0 we should change the sign of the Green's functions and substitute E − for E + . For isotropic dispersion law E(k) = E(k) we can perform the angle integration in Eqs. (27) and (28) to get d 2 ke ik·Rij −E(k)τ = ∞ 0 dkkJ 0 (kR)e −E(k)τ (29) d 2 ke ik·Rij ±iθ k −E(k)τ = e ±iθ R ∞ 0 dkkJ 1 (kR)e −E(k)τ (J 0 and J 1 are the Bessel function of zero and first order respectively, and θ R is the angle between the vectors K− K ′ and R ij ). For the linear dispersion law E ± (k) = ±v F k,(30) using mathematical identity [17] ∞ 0 x n−1 e −px J ν (cx)dx (31) = (−1) n−1 c −ν ∂ n−1 ∂p n−1 p 2 + c 2 − p ν p 2 + c 2 , we can explicitly perform the remaining integration. Calculating integrals (29) we obtain χ AA (R ij ) = a 4 256v F R 3 [1 + cos((K − K ′ )·R ij )](32)χ AB (R ij ) = − 3a 4 256v F R 3 [1 − cos((K − K ′ )·R ij − 2θ R )] .(33) The approach presented above can be easily applied to the bilayer graphene. We'll consider Bernal (Ã − B) stacking. Because the low-energy modes are localized on A andB sites [18], we consider RKKY interaction of the magnetic adatoms siting on top of carbon atom in A and/orB sites. The low-energy modes are characterized by the spectrum E ± (k) = ± k 2 2m(34) and wave functions ψ ν,K (k) = 1 √ 2 e −iθ k νe iθ k ψ ν,K ′ (k) = 1 √ 2 e iθ k νe −iθ k ,(35) where this time the upper line of the spinor refers to the sublattice A and the lower line refers to the sublatticeB [18] (we ignore the trigonal warping). So for the case of bilayer we reproduce Eq. (27) (of course, the result for G AA equally refers to GBB); Eq. (28) is changed to G AB (i, j; τ > 0) = 1 2 a 2 (2π) 2 d 2 ke −E+(k)τ × e i(K+k)·Rij −2iθ k − e i(K ′ +k)·Rij +2iθ k .(36)Calculation of G AA would demand the integral [19] ∞ 0 J 0 (x) exp(−px 2 )xdx = 1 2p exp − 1 4p .(37) After simple calculus we obtain for bilayer graphene χ AA (R ij ) = ma 4 16π 2 R 2 [1 + cos((K − K ′ )·R ij )] . (38) We return to monolayer graphene. The case of magnetic adatom siting on top of carbon atom, certainly does not exhaust all the possibilities for the adatom positions in graphene lattice [11,14]. However, under rather general assumptions the specific position of the adatom can be taken into account by changing in Eq. (14) the product of the components of the spinor wave function ψ p to an appropriate matrix element. Thus, using the results of Ref. [11], for the case of substitutional impurities instead of Eqs. (27) and (28) we obtain G AA (i, j; τ > 0) = − 1 2 e iK·Rij + e iK ′ ·Rij a 4 (2π) 2 d 2 kk 2 e ik·Rij −E+(k)τ G AB (i, j; τ > 0) = 1 2 a 4 (2π) 2 d 2 kk 2 e −E+(k)τ × e i(K+k)·Rij −3iθ k − e i(K ′ −k)·Rij +3iθ k .(39) After simple calculus we obtain χ SASA (R ij ) = X SASA v F R 7 [1 + cos((K − K ′ )·R ij )] and χ SASB (R ij ) = − X SASB v F R 7 [1 − cos((K − K ′ )·R ij − 6θ R )] , where X SASA and X SASB can be easily calculated analytically. DISCUSSION In this Section we would like to compare our results with the previously obtained ones and additionally justify our line of reasoning. The correction to the thermodynamic potential can be also written down using frequency representation [13], which would give χ ij = T 4 n,m u * n (i)u m (i)u n (j)u * m (j) ω 1 iω − ξ n 1 iω − ξ m ,(40) where ω = πT (2l + 1) (l is an integer) is Matsubara frequency. (Eq. (14) was taken into account.) Performing in Eq. (40) summation with respect to Matsubara frequencies we obtain χ ij = 1 4 n,m u * n (i)u m (i)u n (j)u * m (j) n F (ξ m ) − n F (ξ n ) ξ n − ξ m . (41) This is in fact the result obtained originally [2][3][4], by using standard quantum mechanics (off-the-energy shell) perturbation theory. We used the term bipartite lattice, but actually neither the symmetry of spectrum , presented after Eq. (20), nor the symmetry of wave functions presented in Eq. (22) do not require any space periodicity (or any order at all) in the position of the sites. They even do not require that the Hamiltonian will be Hermitian, so they remain, say, in non-Hermitian quantum mechanics. The results of Ref. [12] correspond to Eq. (41) with a small but substantial difference: the terms with ξ n = ξ m are discarded [20], which breaks the symmetry of the RKKY interaction we discussed. We want now to consider a simple toy model to additionally explain that these diagonal terms are relevant and should be where they are. Our arguments will follow the consideration of the magnetism of electron gas in Ref. [21]. Let the spectrum of H consists of pairs of states having the same energy, and H int has non-zero matrix elements only between the states belonging to the same pair. Then the quantum mechanical problem of finding the spectrum of the Hamiltonian H T can be solved exactly, each doublet is split, E (1,2) n = E n ±\|V n,1;n,2 \|. The thermodynamic potential is Ω = n,± Ω 0 (E n ± \|V n,1;n,2 \|) , where Ω 0 (E) is the thermodynamic potential of the isolated level with the energy E. Expanding with respect to interaction we obtain ∆Ω = n ∂ 2 Ω 0 ∂E 2 n \|V n,1;n,2 \| 2 = − n ∂n F (E n ) ∂E n \|V n,1;n,2 \| 2 ,(43) which corresponds to the diagonal terms in Eq. (41). The issue of diagonal terms can be also connected to the difference between the real-and imaginary-time approaches the authors of Ref. [12] emphasize in their paper. Our opinion is that calculation of magnetic susceptibility using real-time method (Kubo formula) gives the adiabatic susceptibility. On the other hand, for the calculation of the RKKY interaction we need the isothermal susceptibility, which is given by the imaginary-time method. Analytical calculations of the RKKY interaction can be done using Eq. (41). In this case instead of Eqs. (32) and (33) we would obtain χ AA (R ij ) = a 4 4π 2 v F R 3 [1 + cos((K − K ′ )·R ij )] ∞ 0 dxxJ 0 (x) ∞ 0 dx ′ x ′ J 0 (x ′ ) 1 x + x ′ (44) χ AB (R ij ) = − a 4 4π 2 v F R 3 [1 − cos((K − K ′ )·R ij − 2θ R )] ∞ 0 dxxJ 1 (x) ∞ 0 dx ′ x ′ J 1 (x ′ ) 1 x + x ′ .(45) Eqs. (44) and (45) are particularly convenient to be compared with the results of Ref. [10]. Using the identity [17] ∞ 0 x ν x + z J ν (cx)dx = πz ν 2 cos νπ [H −ν (cz) − Y −ν (cz)],(46) where H ν (z) is the Struve function and Y ν (z) is the Neumann function, we can present integrals in Eqs. (44) and (45) as π 2 ∞ 0 dxx 2 J 0 (x) Y 0 (x) − H 0 (x) + 2 πx π 2 ∞ 0 dxx 2 J 1 (x)[Y −1 (x) − H −1 (x)].(47) These integrals are similar to those standing in Eqs. (18) and (25) of Ref. [10], but contrary to the latter, our integrals diverge. This is guaranteed by the asymptotics of Struve functions H ν (x) − Y ν (x) → 1 √ πΓ ν + 1 2 x 2 ν−1 + O (x/2) ν−3 .(48) A deficiency of the previous analytic calculations of the RKKY interaction in graphene is, to our mind, not due to them using the frequency representation of the Green's function (though we find the imaginary time representation more convenient for the calculations), but due to them first calculating static spin susceptibility in momentum space χ(q) = νν ′ ,p M ν,ν ′ ,p,q n F [E ν ′ (p + q)] − n F [E ν (p)] E ν (p) − E ν ′ (p + q)(49) (we shouldn't worry here what the matrix element M is) and then calculating χ (R ij ) making a Fourier transformation χ (R ij ) = a 2 (2π) 2 d 2 qχ (q) e iq·R ij .(50) Both integrals turn out to be ultra-violet divergent, and cut-offs should be introduced. We, on the other hand, calculated directly χ in real space representation, thus avoiding the problem of divergence of the integrals completely. There is another problem with calculating the RKKY interaction (in normal metals) which has a long history [22,23]; it arises when we combine the integrals (49) and (50) into a single double integrals. The problem is which integration: with respect to q or with respect to p we should do first. We also avoid this problem completely. The contact exchange interaction we used can be easily justified in the case of s-wave orbital of the magnetic adatom [11]. The case of d-wave orbitals is more complicated. To find the physically meaningful form of Kondo perturbation, it is appropriate to go back to the possible origin of the Kondo model, i.e., the Anderson model. Following seminal paper by Schrieffer [24], let us specify the magnetic impurity as being the S-state ion, say Mn ++ , whose d-shell has the configuration S 5/2 . Since the S-state ion cannot change the orbital angular momentum of a conduction electron, one should use states which transform according to the irreducible representations of the point group of the crystal about the impurity center [24]. Such approach for the case of d-wave orbitals was realized by Zhu et al. [25] (see also Ref. [26]). Considering the magnetic impurity above the center of the honeycomb (plaquette impurity), they started from the classification of the degenerate 3d-orbitals of the magnetic atom with respect to irreducible representations of the symmetry group C 6v and inferred that d z 2 belongs to A 1 , (d xz , d yz ) belong to E 1 and (d x 2 −y 2 , d xy ) belong to E 2 representations. Specifying their approach, we'll take into account hybridization of the d-orbitals of the magnetic impurity with the p z states of the carbon atoms around the plaquette. The selection rules for matrix elements demand that from the states \|i >, where i ∈ P, and P is the set of sites surrounding the plaquette, we'll chose combinations realizing the same representations as above. Thus the hybridization Hamiltonian for the 3d magnetic impurity in terms of the irreducible reps of the system will take the form H hyb = λ,α,i∈P v λ i c † iα f λα + h.c. ,(51) where operators f † (f ) create (annihilate) electrons at the d-orbitals of the magnetic impurity, and index λ enumerates the orbitals d z 2 , d xz , d yz , d x 2 −y 2 , d xy . From Eq. (51), following Ref. [24] under appropriate assumptions we can get the p − d exchange model [11] H pd = − λ,α,β,i,j∈P Jv λ i v λ j * S · σ αβ c † iα c jβ .(52) In Ref. [8], the p − d exchange Hamiltonian (for the so-called coherent case) was previously taken in a very specific form H pd = −J α,β,i,j∈P S · σ αβ c † iα c jβ ,(53) which in fact takes into account only the hybridization between d z 2 and the combination of the p-states on the plaquette, realizing irreducible representation A 1 , that is i∈P \|i > / √ 6. Such specific form led to the conclusion that 1/\|R − R ′ \| 3 term in the RKKY interaction between the plaquette impurities vanishes. When the general form of the hybridization Hamiltonian (51) is taken into account, this conclusion seems to us unjustified. I am grateful to B. Uchoa, J. Bunder, I. Titvinidze, M. Potthoff, and L. Sandratskii for very useful discussions. The work was done during the author's visit to Cavendish Laboratory, Cambridge University and finalized during the author's visit to the I. Institute of Theoretical Physics, Hamburg University. 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[ "Contact induced spin relaxation in Hanle spin precession measurements", "Contact induced spin relaxation in Hanle spin precession measurements" ]	[ "T Maassen \nPhysics of Nanodevices\nZernike Institute for Advanced Materials\nUniversity of Groningen\nGroningenThe Netherlands\n", "I J Vera-Marun \nPhysics of Nanodevices\nZernike Institute for Advanced Materials\nUniversity of Groningen\nGroningenThe Netherlands\n", "M H D Guimarães \nPhysics of Nanodevices\nZernike Institute for Advanced Materials\nUniversity of Groningen\nGroningenThe Netherlands\n", "B J Van Wees \nPhysics of Nanodevices\nZernike Institute for Advanced Materials\nUniversity of Groningen\nGroningenThe Netherlands\n" ]	[ "Physics of Nanodevices\nZernike Institute for Advanced Materials\nUniversity of Groningen\nGroningenThe Netherlands", "Physics of Nanodevices\nZernike Institute for Advanced Materials\nUniversity of Groningen\nGroningenThe Netherlands", "Physics of Nanodevices\nZernike Institute for Advanced Materials\nUniversity of Groningen\nGroningenThe Netherlands", "Physics of Nanodevices\nZernike Institute for Advanced Materials\nUniversity of Groningen\nGroningenThe Netherlands" ]	[]	In the field of spintronics the "conductivity mismatch" problem remains an important issue. Here the difference between the resistance of ferromagnetic electrodes and a (high resistive) transport channel causes injected spins to be backscattered into the leads and to lose their spin information.We study the effect of the resulting contact induced spin relaxation on spin transport, in particular on non-local Hanle precession measurements. As the Hanle line shape is modified by the contact induced effects, the fits to Hanle curves can result in incorrectly determined spin transport properties of the transport channel. We quantify this effect that mimics a decrease of the spin relaxation time of the channel reaching more than 4 orders of magnitude and a minor increase of the diffusion coefficient by less than a factor of 2. Then we compare the results to spin transport measurements on graphene from the literature. We further point out guidelines for a Hanle precession fitting procedure that allows to reliably extract spin transport properties from measurements.	10.1103/physrevb.86.235408	[ "https://arxiv.org/pdf/1210.0093v1.pdf" ]	118,586,635	1210.0093	de3d36c111e85feaf24fb834a7d7900e9d9cee58	Contact induced spin relaxation in Hanle spin precession measurements T Maassen Physics of Nanodevices Zernike Institute for Advanced Materials University of Groningen GroningenThe Netherlands I J Vera-Marun Physics of Nanodevices Zernike Institute for Advanced Materials University of Groningen GroningenThe Netherlands M H D Guimarães Physics of Nanodevices Zernike Institute for Advanced Materials University of Groningen GroningenThe Netherlands B J Van Wees Physics of Nanodevices Zernike Institute for Advanced Materials University of Groningen GroningenThe Netherlands Contact induced spin relaxation in Hanle spin precession measurements (Dated: May 3, 2014)numbers: 7576+j7540Gb7225Dc7280Vp In the field of spintronics the "conductivity mismatch" problem remains an important issue. Here the difference between the resistance of ferromagnetic electrodes and a (high resistive) transport channel causes injected spins to be backscattered into the leads and to lose their spin information.We study the effect of the resulting contact induced spin relaxation on spin transport, in particular on non-local Hanle precession measurements. As the Hanle line shape is modified by the contact induced effects, the fits to Hanle curves can result in incorrectly determined spin transport properties of the transport channel. We quantify this effect that mimics a decrease of the spin relaxation time of the channel reaching more than 4 orders of magnitude and a minor increase of the diffusion coefficient by less than a factor of 2. Then we compare the results to spin transport measurements on graphene from the literature. We further point out guidelines for a Hanle precession fitting procedure that allows to reliably extract spin transport properties from measurements. In the field of spintronics the "conductivity mismatch" problem remains an important issue. Here the difference between the resistance of ferromagnetic electrodes and a (high resistive) transport channel causes injected spins to be backscattered into the leads and to lose their spin information. We study the effect of the resulting contact induced spin relaxation on spin transport, in particular on non-local Hanle precession measurements. As the Hanle line shape is modified by the contact induced effects, the fits to Hanle curves can result in incorrectly determined spin transport properties of the transport channel. We quantify this effect that mimics a decrease of the spin relaxation time of the channel reaching more than 4 orders of magnitude and a minor increase of the diffusion coefficient by less than a factor of 2. Then we compare the results to spin transport measurements on graphene from the literature. We further point out guidelines for a Hanle precession fitting procedure that allows to reliably extract spin transport properties from measurements. I. INTRODUCTION New concepts like the spin transfer torque, the transport of spin information over long distances and the prospect of spin field effect transistors keep spintronics an inspiring field 1,2 . But before new types of spintronic devices can be build we need both materials that efficiently generate spin currents as injector and detector electrodes and materials with long spin relaxation lengths (λ) and times (τ ) to transport the spins with only little losses. While ferromagnetic metals can spin polarize currents and are therefore used to inject and detect spins, semiconductors offer low spin relaxation which make them good candidates to be used as transport channels. One of the main challenges when combining the two types of materials is the "conductivity mismatch" problem 3,4 . As the electrical resistance in the ferromagnetic electrodes is in general lower than in the semiconducting transport channel, the injected spins tend to be reabsorbed by the leads and loose their spin orientation. Graphene being an intermediate between metal and semiconductor systems is a prototype example for the conductivity mismatch, as graphene based devices can be well described following simple spin diffusion theory. Here, the long spin relaxation lengths of several µm measured at room temperature are already promising 5 , but still stay behind the theoretical prospects based on the high mobilities combined with weak spin orbit coupling and low hyperfine interactions 6 . While some research aims to understand the spin relaxation mechanism in graphene [7][8][9][10][11][12] and to understand the influence of the direct environment of the graphene transport channel [13][14][15][16][17][18] the conductivity mismatch can play an important role in the origin of spin relaxation in the measured devices. To prevent this mismatch high resistive barriers between the contacts and the graphene channel are included 4,[19][20][21][22][23] . The most common and reliable way to probe spin transport properties is by performing measurements in the non-local spin-valve geometry 2,5,24 because it enables to separate spin and charge currents, avoiding spurious effects 25 . Popinciuc et al. 19 describe, in agreement with Takahashi and Maekawa 26 , that the measured amplitude of the spin signal in the non-local geometry is strongly reduced for low contact resistances R C 27 . To quantify the effect Popinciuc et al. introduce the so called R-parameter that is defined for a 2-dimensional channel by R = (R C /R sq )W , where R sq is the square resistance and W the width of the diffusive channel. 19,28 In this article we start by summarizing the dependence of the non-local amplitude on the contact resistance discussed in Ref. 19 and then we are going to focus on determining how the Hanle spin precession is influenced by low contact resistances. We discuss that not only the amplitude but also the shape of Hanle precession curves is changed for low values of the R-parameter (corresponding to low contact resistances) and simulate Hanle measurements including contact induced relaxation. We quantify the contacts' influence by fitting the data with the standard Hanle formula without taking contact induced effects into account, assuming R → ∞. Note that fitting with the standard Hanle formula is the common method to analyze experimental spin precession data in almost all published work. The difference of the extracted spin relaxation time τ f it and diffusion coefficient D f it to the parameters used in the simulations is quantified and we compare these results to data obtained on graphene spin-valve devices where a reduction of τ was reported for low contact resistances 20 . Finally, we point out how to extract correctly the spin transport properties from Hanle precession measurements by excluding spurious background effects. V V I µ S x I R C high R C low R C high R C low a) b) c) 0 L L C FIG. 1: (Color online) Sketches of spin diffusion through a diffusive channel with a spin injector and detector separated by a distance L in non-local geometry for (a) high and (b) low contact resistances. The width of the contacts in x-direction, LC , is small compared to L or the spin relaxation length λ. (c) The spin chemical potential µS indicating the spin accumulation below the injector electrode and the exponential decay of the spin signal (red dotted curve). The spin accumulation influenced by the contact induced spin relaxation is denoted by the black solid curve. Fig. 1 (a) presents a sketch of the non-local measurement geometry with an injecting electrode at x = L on the right side, sending a charge current via a resistive barrier 4 into the channel to the right side end and a detecting electrode on the left side (at x = 0), measuring the voltage difference between the contact and the left side end. As the electrodes are ferromagnetic, the injecting electrode induces a spin imbalance that accumulates below the electrode and diffuses away from it in both positive and negative x-direction of the channel (red dotted curve in Fig. 1 (c)). The second ferromagnetic electrode detects the spins at x = 0 and the measured voltage is normalized with the injected current to obtain the nonlocal resistance R nl that is given by 19,26,27 II. CONTACT INDUCED SPIN RELAXATION R nl = ± P 2 R sq λ 2W (2R/λ) 2 exp(−L/λ) (1 + 2R/λ) 2 − exp(−2L/λ)(1) The model leading to this result is based on the onedimensional description of a diffusive channel with an injector and detector on distance L with P the polarization of the contacts and λ = √ Dτ the spin relaxation length in the channel with the diffusion coefficient D. The width of the contacts (L C ) is considered to be negligible compared to L and λ 19 . Also we assume 1 − P 2 ≈ 1 (applicable to graphene devices where P < 30% 20 ) and are considering an infinite homogeneous transport channel. The effect of an inhomogeneous transport channel is discussed elsewhere 16 . The R-parameter is being calculated using the contact resistance of the injector and detector. In case R is not the same for the two electrodes an effective single R-parameter can be calculated using 1/R ef f ≈ (1/R 1 + 1/R 2 )/2 with the R-parameters of the injector and detector R 1 and R 2 (see Appendix A). The meaning of the R-parameter gets clear when it is normalized with the spin relaxation length λ. The normalized value corresponds to the ratio of the contact resistance and the spin resistance of the channel R s ch = R sq λ/W so R/λ = R C /R s ch . Hence, R/λ describes the ratio of spins diffusing through the channel and relaxing, versus those being reabsorbed by the contact, making it a good measure for the influence of the contacts. Eq. (1) shows that the spin signal R nl has a maximum for high contact resistances (R → ∞) and is reduced for low R-values. A significant change is observed for R/λ ≤ 1 ( Fig. 2 (b), inset). On the other hand the amplitude of the signal is reduced with increasing L from a maximum at L = 0. The characteristic length ratio of the system is L/λ. While the effect on the normalized amplitude (R nl /R R→∞ nl with the amplitude without contact induced effects R R→∞ nl ) is smaller for short distances between injector and detector electrode it stays approximately constant for L/λ ≥ 1. Popinciuc et al. discuss in detail the effect of low contact resistances on the measured non-local amplitude but, while included in the model, the effect on the Hanle curve is only discussed qualitatively 19 . In the following we are going to present a quantitative analysis of the influence of low contact resistances on Hanle measurements. We show that the extracted spin transport properties of the transport channel can be limited by the contact induced relaxation and are therefore incorrectly determined when low R measurements are analyzed without considering the influence of the contacts. Fig. 2 (a) shows Hanle precession data that was simulated for different values of R/λ with L/λ = 1 using the model system of Fig. 1 (a) described in Ref. 19. Note that the amplitude of the Hanle curves is normalized at B = 0, which is necessary as the amplitude scales with (R/λ) 2 for R/λ 1 and changes by 5 orders of magnitude between R/λ = 0.001 and R/λ = 10. A significant change in the Hanle shape is visible in Fig. 2 (a), pointing to an effective change of the spin transport properties. The strongest change in the shape is seen between R/λ = 0.01 and R/λ = 1 while the curve shape is saturating for both small and large R/λ values denoting spin transport limited by the contacts or by the properties of the channel, respectively. Fig. 2 (b) shows a similar dataset for L/λ = 10. We also see a change in the Hanle shape but the effect is much weaker for this larger distance of the injector and detector. Remarkably, in both cases the curves stay in the characteristic Hanlelike shape for all R/λ. Therefore it is possible to fit the data using the solutions to the Bloch equations 29,30 that do not take the effect of the low resistive contacts into account (see Appendix B) R/λ = 0.001 R/λ = 0.01 R/λ = 0.1 R/λ = 1 R/λ = 10 a) b) L/λ = 1 0.0 0.1 0.2 B (T) 0 1 0 1 R nl /R B=0 nl B (T) L/λ = 10 R/λ R² R nl /R Rd µ S dt = D∇ 2 µ S − µ S τ + ω L × µ S .(2) Here ω L is the Larmor frequency ω L = gµ B / B, with the gyromagnetic factor g (g-factor, g ≈ 2 for free electrons), the Bohr magneton µ B and the magnetic field B. By fitting simulated data without taking the effects of the contacts into account, we can determine what happens when one fits the data obtained in samples with corresponding R-and L-values in the standard manner. 31 The results from these fits are presented in Fig. 3. Note that while the simulations were performed with D = 0.01 m 2 /s, τ = 100 ps they do not depend on the specific value of D and τ . Hence, we get the same results for different D and τ resulting in the same λ = √ Dτ as the fitting results depend only on the ratios R/λ and L/λ. The graphs show the fitted values τ f it and D f it normalized by the actual values for the channel τ and D as a function of R/λ for different L/λ ( Fig. 3 (a) and (c)) and as a function of L/λ with different R/λ ( Fig. 3 (b) and (d)). While all values converge for high R/λ to the intrinsic values, we see a strong decrease of τ f it and a moderate increase of D f it for small R/λ. Looking at Fig. 3 (a) in more detail we see that the decrease in τ f it is the strongest the shorter the distance L between injector and detector relative to λ. We also see that the values saturate for small values of R/λ as already perceivable in Fig. 2 (a). In this limit the effect of the contacts is maximized. τ f it shows changes of nearly up to five orders of magnitude which means that in a measurement with parameters of R/λ = 0.001 and L/λ = 0.01 we would underestimate τ by a factor of 5 × 10 4 . The length dependence of the effect is more clearly presented in Fig. 3 (b) where the τ f it data is plotted as a function of L/λ for different R/λ. Here we see that while the decrease of τ f it is stronger for shorter distances the effect gets negligible for L/λ ≥ 10. That means that contact induced effects can be circumvented by measuring on a longer distance. This is only limited by the reduced measured amplitude for longer distances L (see Eq. (1)). Fig. 3 (c) and (d) show the same plots for D f it . Also here we see the strongest effect for small R/λ and L/λ and no significant change for R/λ = 100 or L/λ = 100. On the other hand, the values for D f it show a much weaker change than the values of τ f it and the change is directed in the opposite direction than the change of τ f it . Similar to τ f it , the D f it values also seem to saturate for small R/λ and the changes are less than a factor of 2. While most curves presented in Fig. 3 have a smooth shape and a continuous change with L/λ and R/λ, the data for D f it shows for values of L/λ ≤ 0.1 combined with values of R/λ ≤ 1 a non-smooth behavior. This is related to the fact that the diffusion in the channel gets for small L dominated by the contact induced effects for short distances and low contact resistances and the shape of the Hanle curves gets strongly influenced. The spin accumulation has no significant decay between the injector and the detector electrode so the system becomes similar to 3-terminal Hanle precession 32 . As a result, the fits become insensitive to the specific value of D and one can only determine τ 18 (see Appendix B). Therefore we omitted the data for D f it for L/λ < 0.1 in Fig. 3 (c) and (d). Note that in the limit L/λ 1 and R/λ < 10 the values for τ f it saturate as they are dominated by the contact induced effects and can be described by a basic formula related to the back diffusion of the spins into the contact (see Appendix C). Fig. 3 shows clear trends for τ f it /τ and D f it /D as a function of R/λ and L/λ. We are going to discuss in the following how to understand the physics behind the presented results. The sketch in Fig. 1 (a) presents the spin injection and detection for high contact resistances, e.g. due to tunnel barriers between the channel and the contacts. Here the spin diffusion in the channel remains undisturbed and the injected spins diffuse freely through the channel before being detected by the spin sensitive detector. In this way, measurements detect the intrinsic spin transport properties of the channel and the simple exponential decay of the spin signal (red dotted curve in Fig. 1 (c)) is obtained. In the case of low contact resistances the spin transport is influenced both at the injector and at the detector electrodes. When diffusing through the channel the low resistive detector has a high probability of detecting the spins as soon as they are near the contact as it acts as a spin sink ( Fig. 1 (b)). Therefore the effective traveling time is reduced and the measured diffusion coefficient enhanced as D = L 2 /2τ D where τ D is the diffusion time for the length L. At the same time the proximity to the low resistive contacts also causes spins to relax, which reduces the relaxation time. The extra relaxation is depicted by the kink at the detector in the black solid curve in Fig. 1 (c), describing the decay of the in general reduced spin accumulation in the system. Fig. 3 (b) and (d) show a reduction of the contact induced effects for larger L/λ. This can be easily understood by the fact that for a longer distance between the electrodes the ratio of the time the spins stay in the channel compared to in close proximity to the contacts grows resulting in relatively less influence of the contacts on the spin transport. Fig. 4 (a) shows the spin relaxation length λ f it resulting from the fitting results presented in Fig. 3 for L/λ ≥ 0.1. The shape of the λ f it -curve is comparable to the behavior of τ f it . This is due to the fact that λ f it is mainly influenced by the spin relaxation time τ f it with a change of up to a factor 1000 (for L/λ ≥ 0.1). As D f it shows only a change of less than a factor 2 we get a maximum reduction of λ f it by a factor 25. This λ f it value would be used in the analysis of a measurement to calculate the polarization P . If we take the amplitude simulated with Eq. (1) (inset, Fig. 2 (b)) and assume spin transport without contact induced spin relaxation (Eq. (1) for R → ∞) we extract the effective polarization P f it with R nl (λ f it , P f it , R → ∞) = R nl (λ, P, R). The resulting value is up to 500 times re- duced for small values of R/λ compared to the real P value ( Fig. 4 (b)). Note that the largest change in P f it compared with P is observed for long distances. III. DISCUSSION After discussing the effects observed in the simulations let us have a look on measurements on real devices using graphene as the transport channel. In spin transport samples in graphene it is difficult to produce high resistive contacts and to control the quality of the contactgraphene interface. So a data set with similar quality samples with only a change of the contact resistance is difficult to produce. On the other hand in a single device the quality of the contacts is most of the time comparable. Therefore it is relatively easy to check the length dependence of the spin transport properties in this kind of system, assuming similar R-parameters for all electrodes. Two sets of data obtained on two different graphene devices with three different injector-detector distances are presented in the work by Wojtaszek et al. in Fig. 4 (a) and Fig. 5 (a) of the supplementary information of Ref. 33. In both cases a minor increase of D is reported when measuring on a shorter distance and in the first case also a minor decrease of τ , pointing to weak but apparent contact induced relaxation 34 . With R ≥ 3 µm and λ ≈ 5 µm the measurements were also performed in a regime where one would expect this kind of weak contact induced effects as L/λ ≈ R/λ ≈ 1 (see Fig. 3 and they would be able to measure this λ without significant influence of the contacts (see Fig. 4 (a)). Such strong differences of the spin signal magnitude between non-local and local configuration as between Refs. 20 and 22 have also been observed in traditional semiconductors like silicon in the non-local 35 and 3-terminal 32 configuration. IV. GUIDELINES FOR A GOOD AND RELIABLE HANLE FIT In this paper we discuss how Hanle measurements are influenced by contact induced relaxation that can lead to incorrectly determined spin transport properties of the channel. Independently from that, the fitting procedure can also give incorrect results for the spin transport properties when performed incorrectly. In this section we are therefore commenting on typical pitfalls in analyzing Hanle precession data. The fit to a Hanle curve is unambiguous if performed in the right way. Fig. 5 (a) illustrates how a fit can still give wrong results on the example of a Hanle precession fit when assuming a wrong background resistance. The background resistance is represented by the R nl (B → ∞) value and is the fitting baseline. Fig. 5 (a) shows a fit to the central peak of a Hanle curve (without contact induced effects) with a baseline shifted by +5% of the amplitude. The fit results in an increase of τ by > 10% and of D by > 45% and therefore a misestimation of λ by more than 25% compared to the values used to simulate the data. However, when fitting the curve with these values the fit presents itself faulty when including the high field tails of the curve as shown in the inset of Fig. 5 (a). Fitting to high B-field values gives therefore a good indication of the quality of the fit. However, this identification of a bad fit can be partly masked by data noise in combination with anisotropic magneto resistance effects or the out-of-plane tilting of the magnetization of the ferromagnetic electrodes at high field values, adding an additional background resistance 36 . Another indication of a good fit is the fitted curve reproducing the "shoulders" of the measured curve, where R nl has a minimum (for parallel alignment of the injector and detector electrode). This is obviously not accomplised in the presented case ( Fig. 5 (a), inset). The larger the ratio L/λ the more pronounced are the shoulders, so measuring on a longer distance enhances the reliability of the fit. While measuring to high magnetic field values to determine the background resistance is in any case advisable, there is a way to avoid such spurious background effects in a fit. Measuring the spin precession both for parallel and for antiparallel orientation of the electrodes, and subtracting the signals from each other removes most spurious (not spin related) background effects as done in several recent works 16,18,33,37 . By taking the mean of the parallel and the antiparallel measurements, one can also extract the B-field dependent background resistance. Finally, a minor error in a fit can also occur if the magnetic field values are not properly calibrated. The effect of a correction factor for the magnetic field value is the same as the effect of a changed g-factor as ω L ∝ B and ω L ∝ g. A wrong B-field calibration is therefore linearly passed on to τ and 1/D 38 . V. CONCLUSIONS We discuss the effect of low resistance contact induced spin relaxation on Hanle precession data and quantify the misinterpretation of spin transport properties in a transport channel that can arise from this effect. As fitting Hanle curves is a common way to extract spin transport properties we use the model presented in Ref. 19 to simulate Hanle measurements and fit the data using the standard formula, neglecting the contact induced effects. The observed rescaling of the spin relaxation time and the diffusion coefficient only depend on the ratios R/λ and L/λ and the fitting results show that a strongly decreased τ f it by up to nearly five orders of magnitude and a moderately increased D f it by less than a factor of 2 can be observed for small R/λ and L/λ. On the other hand large values for both R/λ or L/λ show a convergence of τ f it and D f it on the undisturbed values τ and D, independent on L or R, respectively. This shows that the spin relaxation induced by the contacts can in principle be avoided when measuring on a longer distance. We then discuss how these values of τ f it and D f it lead to a wrong estimate of the contact polarization before comparing our results for τ f it and D f it qualitatively with measurements on graphene in the literature. The modeled effect of the contacts on spin transport only depends on the resistance of the barrier and not on the type of barrier. Hence, although most contact interfaces used in the non-local geometry to study spin transport in graphene are not truly in the tunneling regime, we can conclude that with the the resistance of the commonly used barriers the effect of back diffusion into the contacts on the spin transport is only minor and the spin transport properties are mainly limited by other effects. 5,7,10,14,16,[18][19][20]33,37,39 While explicitly discussing the effect of low resistive contacts on the non-local geometry, similar effects also play a role for local measurements 40 . We also briefly discussed the guidelines for a good and reliable Hanle fit as an incorrectly performed fit can also lead to misinterpretations of the spin transport properties of a diffusive channel while a correct fit leads to unambiguous results. The discussion in the main text focused on the symmetric case when the injector and the detector contacts have equal R-parameters. Here we address the general case of dissimilar injector and detector contacts and demonstrate an equivalence that allows us to map this general case to the more symmetric one presented above. The general expression for the non-local resistance R nl given by 26 R nl = ± R sq λ 2W exp (−L/λ) 2 i=1    P 2R i λ 1 − P 2    ×    2 i=1   1 + 2R i λ 1 − P 2    − exp (−2L/λ)    −1 (A1) where R 1,2 correspond to the R-parameters of the injector and detector contacts, and the rest of the parameters are the same as those presented in the discussion of Eq. (1) 27 . This equation can be simplified by realizing that for highly spin polarized contacts (P ≈ 1) there is no contact induced spin relaxation, even for low resistance contacts. Therefore if we consider 1 − P 2 ≈ 1 we obtain R nl = ± P 2 R sq λ 2W 2R 1 λ 2R 2 λ exp (−L/λ) 1 + 2R 1 λ 1 + 2R 2 λ − exp(−2L/λ) (A2) which has a similar structure as Eq. (1) 19 . Following simple algebra, we can equate both equations and solve for the R-parameter of Eq. (1), which can be understood as an effective R-parameter R ef f (R 1 , R 2 , L, λ) given by, allowing us to map the case of dissimilar contacts into the symmetric case of equal contacts with R 1,2 = R ef f . One example of such a mapping is shown in Fig. 6 (a) for the representative case of L/λ = 1 and R i /λ = 0.1-10. We observe that when R 1 = R 2 then R ef f ≈ min (R 1 , R 2 ) and also the trivial case of 2R ef f λ = 2R1 λ 2R2 λ + 2R1 λ 2R2 λ 2 − 1 + 2R1 λ + 2R2 λ − e −2L/λ 2R1 λ 2R2 λ (e −2L/λ − 1) 1 + 2R1 λ + 2R2 λ − e −2L/λ(R 1 = R 2 = R ef f . The exact mapping depends on L and on λ, which requires careful application to analyze experimental data. We remark that this issue is absent for the case of 2L/λ ≈ 0, where Eq. (A3) reduces to the simple form, 1 R ef f = 1 2 1 R 1 + 1 R 2 (A4) equivalent to a 3-terminal measurement where both contacts are in a parallel configuration. Although Eq. (A4) is strictly speaking valid only when both contacts are closely spaced, we have observed that it offers a reasonable approximation even at finite separation L between the contacts. In Fig. 6 (b) we compare the resulting R ef f /λ for the extreme case of large separation (2L/λ 1) from Eq. (A3), to the value obtained from the simpler Eq. (A4). Surprisingly, in the experimentally relevant range of intermediate conductivity mismatch R i /λ = 0.3-10, Eq. (A4) deviates from the exact result at infinite separation only by less than 20%. For a strong conductivity mismatch (R i /λ ≤ 0.1) one should apply the exact result of Eq. (A3). following the model presented in Ref. 19 including contact induced spin relaxation and are fitted neglecting the contact induced effects. Fig. 7 shows how well the simulated data can be fitted with a Hanle curve for different values of R/λ and L/λ. The curve for R/λ = 0.1 and L/λ = 3 shows that even for small R/λ-values (corresponding to a contact resistance of R = 100 Ω when R sq = 1 kΩ and W = 1 µm) we get an excellent fit (although with a reduced τ f it and increased D f it ). On the other hand the curve and fit for the combination R/λ = 0.01 and L/λ = 0.1 points out that the fit is not describing the curve properly for very small values of the two parameters. This is especially well visible close to B = 0 ( Fig. 7 (b)) where due to the strong contact induced relaxation we observe a distinct drop of R nl which the fit cannot describe. In Fig. 3 (c) and (d) of the main text it is visible for which sets of parameters the fits do not describe the Hanle curves well, as those are the points that do not show a smooth line shape when plotting D f it as a function of R/λ or L/λ. Appendix C: τ f it in the limit L/λ 1 We can obtain τ f it by performing a standard Hanle fit on simulated data that includes contact induced relaxation. Here we show that we can approximate the value of τ f it for small L using an easy reasoning. In the limit L/λ 1 our system resembles the 3-terminal Hanle geometry 32 as we have two contacts connected to approximately the same point of the transport channel with one of the contacts injecting spins and the other detecting them. At the same time there is the transport channel pointing in two directions away from the injection point. Therefore we get for the spin resistance 1/R * spin = 1/R s ch + 1/R C . If we now take the ratio of the spin resistance including the contact resistance (R * spin ) and R spin = R s ch (R * spin for R C → ∞) we get: = R/λ 1 + R/λ (C1) The spin resistance is proportional to the non-local signal (R spin ∝ R nl ) and for L = 0 and R → ∞ the nonlocal signal is proportional to the spin relaxation length R nl ∝ λ (see Eq. (1)). Therefore we get: R * spin R spin = λ f it λ ≈ τ f it τ (C2) We can use here for both R * spin and R spin the relation R nl ∝ λ. This is obviously valid for R spin and for R * spin ∝ λ f it we have to keep in mind that λ f it is obtained assuming R → ∞ so we have to assume this also here in this analysis. The relation between the ratio of the spin relaxation lengths and the ratio of the spin relaxation times is valid as D f it /D ≈ 1. Hence, we get the result: τ f it τ ≈ R/λ 1 + R/λ 2 . (C3) In the limit L/λ 1 we therefore expect τ f it /τ = (0.98, 0.83, 0.25, 8.3 × 10 −3 , 9.8 × 10 −5 ) for R/λ = (100, 10, 1, 0.1, 0.01) in good agreement with the values in Fig. 3 (b) in the limit L/λ 1. 27 For a negligible spin resistance of the ferromagnetic electrodes compared to the one of the barrier and to the resistance of the transport channel (Rsq) we call the resistance of the interface barrier between the contact electrode and the transport channel "contact resistance" (RC ). The contact resistance consists of the parallel resistance for spin up (R ↑ ) and spin down (R ↓ ). We assume R ↑ ≈ R ↓ as we are discussing polarizations of P < 30% leading to an error in determining the effect of the contact resistance on the spin relaxation of less than 10%. In the case of transparent contacts (i.e. no high resistive barrier), RC is equal to the spin resistance of the ferromagnetic leads R F spin . In this case we also have to replace the spin polarization of the barrier P by the spin polarization of the ferromagnetic leads P F . 28 We are discussing here a 2-dimensional channel like graphene. In case of a 3-dimensional channel R would be R = (RC /ρ)A where A is the cross section and ρ the resistivity of the transport channel. online) Simulated spin precession curves for different values of R/λ with (a) L/λ = 1 and (b) L/λ = 10. For the simulations we use D = 0.01 m 2 /s, τ = 100 ps, W = 1 µm and Rsq = 1 kΩ (representative of graphene devices) with contact resistances between 1 and 10 4 Ω. The amplitude of the curves is normalized for clarity with R nl (B = 0). The inset in panel (b) shows R nl from Eq. (1) as a function of R/λ for L/λ = 1, normalized by R nl (R → ∞) (black solid line) and the asymptote ∝ R 2 (red dashed line). FIG. 3 : 3(Color online) The change in τ f it and D f it fitted for different L/λ as a function of R/λ ((a) and (c)) and for different R/λ as a function of L/λ ((b) and(d)). For small values of L/λ the fits become insensitive to the specific value of the diffusion coefficient, resulting in the non-smooth behavior shown for L/λ = 0.1 in panel (c). Therefore the data for L/λ < 0.1 is not shown in panel (c) and (d). FIG. 4 : 4(Color online) (a) The change in λ f it /λ calculated using τ f it and D f it from Fig. 3. (b) The effective polarization P f it normalized with the actual polarization P . The values are plotted as a function of R/λ for different L/λ. The values for L/λ = 100 and 10 overlap in both panels. ) 33 . 33Han et al. present a study of the dependence of the spin transport properties on the quality of the resistive barrier between the graphene channel and the contacts in Ref.20. They show that between tunneling injection of spins and the injection with transparent contacts the measured spin relaxation time decreases while the diffusion coefficient is increased in agreement with our simulations' results. On the other hand the results for a "pinhole" barrier with intermediate resistance present an intermediate spin relaxation time but also a reduced diffusion coefficient. While the spin relaxation time fits into the expectations for an intermediate contact resistance, the reduced diffusion coefficient cannot be explained by the contact resistance but has to be related to a lower quality sample or other effects. Our model also points to the fact that the recent reported differences between the results for the spin relaxation length, based on the analysis of 4-terminal nonlocal Hanle precession measurements 20 and based on the analysis of the magnitude of spin-valve measurements in local 2-terminal geometry with very high contact resistances (R C > 1 MΩ) 22 cannot be explained by contact induced relaxation. If one would measure with the configuration of Han et al.20 with L = 5.5 µm and R ≈ 200 µm a material with a spin relaxation length of ∼ 100 µm and a spin relaxation time of ∼ 100 ns as reported in Ref.22 one would only see a reduction of the fitted spin relaxation time by a factor of τ f it /τ ≈ 1/3 (seeFig. 3 (a)) leading to a reduced spin relaxation length of λ f it /λ ≈ 1/2 (seeFig. 4 (a)) as one would have L/λ ≈ 0.05 and R/λ ≈ 2. Therefore the standard Hanle analysis would yield λ f it ≈ 50 µm and τ f it ≈ 30 ns but Han et al. report λ f it ≈ 2.5 µm and τ f it ≈ 0.5 ns20 . With λ f it ≈ 2.5 µm Han et al. are in the regime of negligible contact induced relaxation with L/λ ≈ 2 and R/λ ≈ 80, so the difference in the measured λ is not based on contact induced relaxation but has to be related to other effects. Even for a spin relaxation length of λ = 20 µm it would be L/λ ≈ 0.25 and R/λ ≈ 10 for the system of Han et al. FIG. 5 : 5(Color online) (a) The influence of a baseline shift shown by means of a Hanle spin precession curve, shifted +5% of the precession amplitude upwards (black solid curve) and a fit assuming no shift (red dotted curve). The baseline of the fit is therefore at R nl = 0 while the baseline of the data is denoted by the black dashed line. The same Hanle curves on a larger B-field range are shown in the inset. A clear difference is visible for \|B\| > 0.3 T. (b) The change of the diffusion coefficient and the spin relaxation time resulting from data with a baseline shift and fits assuming no baseline shift. The presented data was simulated using D = 0.01 m 2 /s, τ = 100 ps and L = 1 µm. online) Mapping the problem of dissimilar contacts R1 = R2 into the simpler one of identical contacts with a common R ef f . (a) Two-dimensional map of equivalent R ef f /λ as a function of Ri/λ of the contacts for L/λ = 1 from Eq. (A3). (b) Normalized deviation of R ef f /λ obtained from Eq. (A4) relative to the exact result from Eq. (A3) in the limit of L/λ 1. The values are normalized using (R ef f (L/λ = 10) − R ef f (L/λ = 0))/(R ef f (L/λ = 10). FIG. 7 : 7Appendix B: Fitting simulated Hanle curves The research presented in this paper is based on the following concept: Hanle precession curves are simulated (Color online) Two sets of simulated data with the corresponding fits assuming an amplitude of 1 and a baseline at 0. Both sets were simulated for D = 0.01 m 2 /s, τ = 100 ps and the values for L/λ and R/λ shown in the legend. (a) shows the full Hanle curve, (b) zooms in on the part close to B = 0. 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[ "Topological states from topological crystals", "Topological states from topological crystals" ]	[ "Zhida Song ", "Sheng-Jie Huang ", "Yang Qi ", "Chen Fang ", "Michael Hermele " ]	[]	[]	We present a scheme to explicitly construct and classify general topological states jointly protected by an onsite symmetry group and a spatial symmetry group. We show that all these symmetry-protected topological states can be adiabatically deformed into a special class of states we call topological crystals. A topological crystal in, for example, three dimensions is a real-space assembly of finite-sized pieces of topological states in one and two dimensions protected by the local symmetry group alone, arranged in a configuration invariant under the spatial group and glued together such that there is no open edge or end. As a demonstration of principle, we explicitly enumerate all inequivalent topological crystals for noninteracting time-reversal symmetric electronic insulators with spin-orbit coupling and any one of the 230 space groups. This enumeration gives topological crystalline insulators a full classification.	10.1126/sciadv.aax2007	null	96,510,052	1810.02330	bd983d30afdd5d8088f54a93802c7453bf4630ff	Topological states from topological crystals Zhida Song Sheng-Jie Huang Yang Qi Chen Fang Michael Hermele Topological states from topological crystals C O N D E N S E D M A T T E R P H Y S I C S for the Advancement of Science. No claim to original U.S. Government Works. Distributed under a Creative Commons Attribution NonCommercial License 4.0 (CC BY-NC). We present a scheme to explicitly construct and classify general topological states jointly protected by an onsite symmetry group and a spatial symmetry group. We show that all these symmetry-protected topological states can be adiabatically deformed into a special class of states we call topological crystals. A topological crystal in, for example, three dimensions is a real-space assembly of finite-sized pieces of topological states in one and two dimensions protected by the local symmetry group alone, arranged in a configuration invariant under the spatial group and glued together such that there is no open edge or end. As a demonstration of principle, we explicitly enumerate all inequivalent topological crystals for noninteracting time-reversal symmetric electronic insulators with spin-orbit coupling and any one of the 230 space groups. This enumeration gives topological crystalline insulators a full classification. INTRODUCTION Symmetry-protected topological (SPT) phases are gapped many-body ground states that can only be adiabatically deformed into product states of local orbitals by breaking a given symmetry group or by closing the energy gap (1). Typical examples are topological insulators, topological superconductors, and the Haldane spin chain [hereinafter, if not explicitly stated otherwise, the terms "topological insulator" and "strong topological insulator" (STI) will refer to the three-dimensional (3D) topological insulator protected by time-reversal symmetry]. The best-understood SPT phases are those of noninteracting fermions; not long after the discovery of topological band insulators, free-fermion topological phases were completely classified for systems with internal (i.e., nonspatial) symmetries (2)(3)(4)(5). Of course, crystalline symmetries play a central role in solid-state physics, so attention naturally began to turn to topological crystalline insulators (TCIs), which are electronic insulators whose topologically nontrivial nature is protected, in part, by point group or space group symmetry (5)(6)(7)(8). Sparked by the prediction and observation of TCIs in SnTe (7,(9)(10)(11), remarkable theoretical (12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24) and experimental progress (25)(26)(27) has followed over the last few years. Despite these developments, somewhat unexpectedly, a unified picture of the classification of noninteracting electron TCIs has yet to emerge. The primary tool for the classification of free-fermion topological phases with spatial symmetries has been K-theory (3) and equivariant K-theory (28). A number of concrete classification results have been obtained (12,14,16,17,22), but reflecting the complexity of K-theory, there is a paucity of concrete results for 3D (d = 3) insulators with general space group symmetry and time-reversal symmetry. Moreover, it is not understood how or whether electron interactions can be included within K-theory. Therefore, there is a need to develop alternate means to classify TCIs and other crystalline SPT (cSPT) phases jointly protected by internal and spatial symmetries. Ideally, to provide a useful complement to K-theory, these methods will be real space based and physically transparent and allow for interactions to be included. Here, we propose a general method for classifying and constructing cSPT phases, which is then applied to the case of electronic TCIs in all 230 space groups, with time-reversal symmetry and spin-orbit coupling. We also extend these results to include STIs. Our approach is based on recent developments in the seemingly harder problem of classifying interacting cSPT phases (29)(30)(31)(32)(33). The key idea is to first argue that any cSPT phase is adiabatically connected to a real-space crystalline pattern of lower-dimensional topological states, which we refer to as a topological crystal (29,32). One then develops a classification of phases of matter in terms of topological crystals. For bosonic cSPT phases with only space group symmetry, the resulting classification (32) agrees with that obtained in complementary approaches based on tensor network states and gauging crystalline symmetry (30,31). Recently, Shiozaki et al. (34) have discussed how to formulate the topological crystal approach within K-theory via the Atiyah-Hirzebruch spectral sequence. Our approach is related to, but goes beyond, layer constructions of TCIs. Any construction of a TCI in terms of decoupled layers, including the archetype of weak topological insulators as stacks of d = 2 topological insulators, is a topological crystal. However, by comparison to the recent systematic study of layer constructions in (35), we show that in certain nonsymmorphic space groups, there are TCIs that do not have a layer construction but can still be realized as topological crystals. We emphasize that all the topological states implied by symmetry eigenvalues proposed in (19,20) are contained in our classification. The work in (35) shows that only five symmetryeigenvalue-implied TCIs-five weak topological insulators-are not layer constructable. Here, we explicitly construct these weak topological insulators as topological crystals. The results we obtain are related to the recent work of Khalaf et al. (36), who considered TCIs with anomalous surface states (dubbed sTCIs) and proposed a classification for sTCIs with point group and space group symmetry via the surface states of doubled STIs. The TCI classifications produced by our approach, which focuses on the bulk and does not assume anomalous surface states or a description in terms of Dirac fermions, agree with the sTCI classifications in (36). This agreement shows that all the TCIs we classify have anomalous surface states for some surface termination. RESULTS Topological crystals We begin by considering a d = 3 system with symmetry G = G int × G c , where G int is some internal symmetry and G c is either a crystalline site symmetry (i.e., point group) or space group. (The assumption that G is a direct product of G int and G c is not necessary and is only made for simplicity of discussion and because it holds for the electronic TCIs to be later discussed.) We assume the system is in an SPT phase (which could be the trivial phase); that is, below an energy gap, the ground state \|y> is unique and symmetry preserving, and moreover, \|y> is adiabatically connected to a trivial product state if the symmetry G is broken explicitly. Moreover, we restrict to those SPT phases that only remain nontrivial in the presence of crystalline symmetry; that is, \|y> is adiabatically connected to a product state if G c is explicitly broken, even if G int is preserved. To avoid complications associated with gapless boundary states, we consider periodic boundary conditions. To proceed, we identify an asymmetric unit (AU), which is the interior of a region of space that is as large as possible, subject to the condition that no two points in the region are related by a crystalline symmetry. The AU is then copied throughout space using the crystalline symmetry, and we denote the resulting union of nonoverlapping AUs as A. This construction gives 3D space a cell complex structure (see Methods), where the 3-cells are the individual (nonoverlapping) copies of the AU in A. The 2-cells lie on faces where two 3-cells meet, with the property that no two distinct points in the same 2-cells are related by symmetry. Similarly, 1-cells are edges where two or more faces meet, and 0-cells are points where edges meet. The 3-cells are in one-to-one correspondence with elements of G c : arbitrarily choosing one 3-cell to correspond to the identity element, each other 3-cell is the unique image of this one upon acting with g ∈ G c . The 2-skeleton X 2 is the complement of A. An example of this cell structure for the space group P1 is shown in Fig. 1. The work in (32) argued that \|y> is adiabatically connected to a product of a trivial state on A, with a possibly nontrivial state on X 2 [see section VI of (32)]. More precisely, one considers a thickened version of X 2 , with characteristic thickness w, and its complement. For the argument to go through, it is important that w ≫ x, where x is any characteristic correlation or entanglement length of the shortrange entangled state \|y>. If G c is a point group symmetry, then this requires no assumptions because w can be taken sufficiently large. For space group symmetry, w is limited by the unit cell size, and one must make the assumption that, by adding a fine mesh of trivial degrees of freedom, it is possible to make the correlation length x as small as desired. This assumption not only allows the reduction to a topological crystal state but also implies that the correlation length of the topological crystal state itself is much smaller than the unit cell size; this is important because it allows us to associate a well-defined lower-dimensional state with each cell of X 2 . While we believe that this assumption is likely to hold, it is not proven, and strictly speaking, it should be treated as a conjecture. If this conjecture is false, then our approach is simply restricted to those cSPT phases whose correlation length is not bounded below upon adding trivial degrees of freedom. We note that a preliminary version of the idea of reduction to X 2 was discussed in (32); there, unlike in the present work, the idea was not developed into a tool to obtain classifications. The result of this reduction procedure is a topological crystal state. The state on X 2 can be understood by associating a d b -dimensional topological phase with each d b -cell of X 2 , where d b = 0, 1, and 2. These lower-dimensional states are referred to as the "building blocks" of the topological crystal, and d b is the block dimension. The building blocks must be glued together so as to eliminate any gapless modes in the bulk while preserving symmetry; for instance, d b = 2 blocks will generally have gapless edge modes, which must gap out at the 1-cells and 0-cells where the building blocks meet. Whereas crystals are formed by periodically arranged atoms, i.e., zero-dimensional objects, topological crystals are "stacked" from building blocks which are themselves topological states in lower dimensions. TCI classification First, we note that a number of topological invariants are already known that distinguish different phases and should be a part of any classification of TCIs. In particular, these invariants include the weak ℤ 2 invariants (37,38), mirror Chern numbers (ℤ invariant) (39), and ℤ 2 invariants associated with rotation (40), glide reflection (17,18), inversion (21,41,42), rotoreflection (35,36), and screw rotation (35, 36) symmetries. All possible combinations of these invariants that can be realized in TCIs with a layer construction were enumerated in (35). While one could not prove a priori that these seven quantities exhaust all independent topological invariants, in this work, we show they play a special role as a complete list of invariants for TCIs. That is, we find that any two inequivalent TCIs differ by at least one of these invariants. To further apply the tool of topological crystals to the case of electronic TCIs, we consider a system of noninteracting electrons with spin-orbit coupling, with internal symmetries of charge conservation and time reversal; that is, G protectint ¼ Uð1Þ⋊ ℤ T 2 . In addition, we have to specify the action of symmetry on fermionic degrees of freedom; for instance, we have Kramers time reversal T with T 2 = (−1) F , where (−1) F is the fermion parity operator. More generally, some equations in the group G c are also modified by factors of fermion parity, in a manner determined from the d = 3 Dirac Hamiltonian describing relativistic electrons (see the Supplementary Materials). Formally, this amounts to specifying an element w f ∈ H 2 (G, ℤ 2 ); we emphasize that w f is uniquely determined by G in the physical setting we are considering. The next step is to understand what kind of topological crystal states can be placed on X 2 . First, we consider topological crystals built out of d = 2 topological states. There are two kinds of 2-cells, those that coincide with a mirror plane and those that do not. 2-Cells coinciding with a mirror plane can host a d = 2 mirror Chern insulator (MCI) state, which is characterized by a ℤ invariant (39). The MCI state can be understood by diagonalizing the mirror operation s : z → −z, where z is the coordinate along the normal direction of the mirror plane. Because s 2 = (−1) F , one-electron wave functions can be divided into two sectors with mirror eigenvalue ±i. Because sT = Ts, time reversal exchanges these two sectors, which therefore have equal and opposite Chern numbers, leading to a ℤ invariant. Each sector can be understood as a d = 2 fermion system in class A, which has a ℤ classification. We see that the relevant symmetry class is, thus, effectively modified from AII to A on a mirror plane; this modification of symmetry class is familiar from classifications of reflectionsymmetric free-fermion topological phases in momentum space (12). For 2-cells not coinciding with a mirror plane, the symmetry class remains AII, and these cells can host a d = 2 STI (2dTI), which is characterized by a ℤ 2 invariant. We also have to consider the possibility of topological crystals built from the d = 1 and d = 0 states. 1-Cells only host trivial states: The effective symmetry class on a 1-cell can be either AII or A (see the discussion on MCIs above), and in either case, the classification in d = 1 is trivial. On the other hand, there are nontrivial topological crystals built from d b = 0 building blocks, which are atomic insulators formed from patterns of localized filled orbitals. Although there are distinct atomic insulators constituting different quantum phases of matter and although these distinctions may be a source of interesting physics, all these states, being product states of localized orbitals, are, in a sense, topologically trivial. Therefore, we ignore distinctions among atomic insulators in our classification. Formally, this is accomplished by taking a certain quotient (Methods). In the future, our results could be extended to include d b = 0 building blocks, which would facilitate a more direct comparison with K-theory-based approaches, which do include such states. We, thus, see that there are two kinds of TCIs, both built from d b = 2 blocks. We refer to TCIs built from MCI blocks as mirror TCIs (MTCIs), while TCIs built from 2dTI blocks are dubbed ℤ 2 TCIs. Of course, a general TCI can have mixed MTCI and ℤ 2 TCI character, and the classification is a product of MTCI and ℤ 2 TCI classifications. To proceed, we consider the requirement that the building blocks must be glued together to eliminate any gapless modes in the bulk. As shown in Methods, this requirement implies that MTCIs can always be decomposed into decoupled planar MCI layers. ℤ 2 TCIs are not quite as simple. If we consider placing a 2dTI on some subset of the 2-cells of X 2 , then it can be shown these building blocks can be glued together into a topological crystal if and only if every 1-cell is the edge of an even number of 2dTI blocks (Methods). While, sometimes, these states can be decomposed into decoupled 2dTI layers, this is not always true. For example, in space group P4 2 2 1 2 (#94), for which the possible topological crystals are described below, we find a topological crystal that is beyond the scope of layer construction, as shown in Fig. 2B. In this state, the 2-cells decorated with 2dTIs form a complicated yet connected structure. Intuitively, one may lower the two yellow facets at z ¼ 1 2 down to z = 0, such that the 2dTIs form decoupled layers; however, such a process breaks the screw symmetry 4 001 \| 1 2 1 2 1 2 È É . More rigorously, the non-layer constructability can be proved by observing that the topological invariants of this state, specifically its nontrivial weak ℤ 2 invariants, cannot be obtained in any TCI constructed from decoupled 2D layers (35). Having described TCIs in terms of topological crystals, we next use these states to classify TCIs. First, we discuss equivalence relations among topological crystal states and argue that two distinct topological crystals on X 2 are in different phases. Following (29,32), we need to consider an additional equivalence relation, beyond those for the d = 2 phases of matter on the 2-cells. Within an AU and all its copies under symmetry, we create a small bubble of 2dTI and expand the bubble until it joins with the AU boundary; this process can be achieved by adiabatic evolution, preserving symmetry, within a finite time, so any two states related in this way belong to the same phase. (Equivalently, this process can be achieved by acting with a finite depth symmetry-preserving quantum circuit.). The reason we consider a bubble of 2dTI and not something else is that this is the only nontrivial d = 2 state that can exist within the AU, where the only symmetries are charge conservation and time reversal. The source of this equivalence relation is the arbitrary width w of the thickened X 2 space in the dimensional reduction procedure; making w larger corresponds to bringing in additional degrees of freedom from the "bulk." However, in the present case, this equivalence operation has a trivial effect because every 2-cell is joined with two layers of 2dTI, one on each side of the 2-cell. In Methods, we give an example (without time-reversal symmetry) where this equivalence relation has a nontrivial effect. Therefore, any two distinct topological crystal states are in different quantum phases of matter. So, to obtain a classification of TCIs, we need to enumerate possible topological crystals. First, we observe that topological crystals form an Abelian group C under stacking, i.e., upon superposing two different states in the same space. Because MCIs (2dTIs) are characterized by ℤ (ℤ 2 ) invariants, C is a product of ℤ and ℤ 2 factors, with the ℤ factors generated by MTCIs and the ℤ 2 factors generated by ℤ 2 TCIs. Because the MTCIs can be decomposed into decoupled planar layers, there is one ℤ factor for each symmetry-inequivalent set of mirror planes. For any particular crystalline symmetry of interest, the classification C is easily worked out by considering possible colorings of the faces of the AU with the MCI and 2dTI states. To provide a concrete illustration, we here explicitly work out the topological crystals for space group P4 2 2 1 2. P4 2 2 1 2 has a tetragonal lattice and is generated from three translations {1\|100}, {1\|010}, and {1\|001}, a fourfold screw 4 001 \| 1 indicating whether the corresponding e 2 i 's are decorated (=1) or not (=0). The gluing condition can be expressed in matrix form ∑ j A ij n j ¼ 0 mod 2ð1Þ where A ij is defined to be the number of 2-cells (modulo 2) that are symmetry equivalent to e 2 j , for which e 2 i is an edge. For the setting in Fig. 2, one can immediately read out A ¼ 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 B B B B B B @ 1 C C C C C C Að2Þ Solving Eq. 1, we get three independent states that generate all possible topological crystals under stacking: (i) n 1 = n 4 = 1, n 2 = n 3 = 0 ( Topological invariants Now, we turn to the topological invariants characterizing topological crystals. First, all MTCIs are characterized by real-space Chern numbers associated with certain mirror planes, and the mirror Chern numbers in momentum space for each of them are listed in (35). Therefore, we focus on ℤ 2 TCIs. Given a ℤ 2 TCI and its corresponding topological crystal, for each symmetry operation g ∈ G c , we assign a ℤ 2 number d(g). We arbitrarily choose one AU and let r be a point inside, then we set d(g) = 1 [d(g) = 0] if a path connecting r to gr crosses through an odd (even) number of 2dTI 2-cells. It is shown in Methods that (1) d(g) is well defined, independent of the arbitrary choices of AU, r, and the path connecting r to gr, and (2) d(g 1 g 2 ) = d(g 1 ) + d(g 2 ). The latter property implies that d is a homomorphism from G c to ℤ 2 [or, equivalently, an element of H 1 (G c , ℤ 2 )], which means that it is enough to specify d(g) for the generators of G c . d(g) encodes all the ℤ 2 invariants for TCIs listed earlier, by choosing different operations g. For instance, if g is a translation, then d(g) is the corresponding ℤ 2 weak invariant, if g is inversion, then d(g) is the ℤ 2 inversion invariant, and so on. As an example, the topological crystal shown in Fig. 2B has d({1\|100}) = 0, d({1\|010}) = 0, d({1\|001}) = 1, d 4 001 \| 1 2 1 2 1 2 È É À Á ¼ 0, and d({2 110 \|000}) = 0. Taking advantage of the results in (35), we find that these invariants, together with the mirror Chern number, uniquely label all the TCIs in our classification, and moreover, we find all TCIs that are beyond layer construction by comparing with (35) (Supplementary Materials). As suggested by the above discussion of invariants, the classification C of TCIs has a simple relationship with H 1 (G c , ℤ 2 ), which allows us to efficiently compute C and obtain the full TCI classification C for all space groups (Methods); the results are given in Table 2. Moreover, we find that there are 12 groups hosting topological crystals beyond layer construction (Fig. 3); for these non-layerconstructable states, we tabulated their invariants and symmetry-based indicators (19,20) in the Supplementary Materials, completing the mapping from indicators to TCI invariants (35). Table 1 gives the classification of TCIs protected by point group symmetry for the 32 crystallographic point groups in three dimensions. Last, given the classification of TCIs, we obtained simple rules that extend this classification to include STIs. The key fact is that upon stacking two identical STIs together, one can either obtain a trivial state or a nontrivial TCI. Upon identifying the state thus obtained, we obtain a full classification of all topologically nontrivial insulators of noninteracting electrons with time-reversal symmetry and spinorbit coupling. Unified classification of STIs and TCIs We have focused thus far on classifying TCIs, where crystalline symmetry is required to protect a nontrivial cSPT phase. Here, we address a more general problem, namely, the classification of all d = 3 free-electron insulators with time-reversal symmetry, spin-orbit coupling, and arbitrary crystalline point group or space group symmetry. We still ignore distinctions among atomic insulators, so the one new state that must be added as a generator of the classification is the STI, which is, of course, robust even upon breaking crystalline symmetry. First, we assume that the STI is compatible with an arbitrary crystalline symmetry. We expect that this is true, but to our knowledge, it has not been proved rigorously. One argument in favor of this expectation is to note that the STI can be described by a continuum theory of a massive Dirac fermion, which is invariant under arbitrary rigid motions of 3D space. This symmetry can be broken down to an arbitrary space group or point group symmetry, for instance, by adding a periodic potential, which produces a model of an STI with arbitrary space group symmetry. This is not quite a rigorous argument because one has to show that it is possible to regularize the continuum theory in a manner compatible with an arbitrary lattice symmetry. Another argument is to note that any centrosymmetric space group has a ℤ 4 indicator (20), and according to the Fu-Kane formula (37), the root state with z 4 = 1 is an STI. While it was argued that any symmetry indicator can be realized by a band structure, there is no guarantee that the resulting band structure is an insulator (20). Assuming an STI can indeed be found for each centrosymmetric space group, then we need only note that every noncentrosymmetric space group is a subgroup of some centrosymmetric space group, so an STI compatible with the latter is compatible with the former. To show this expectation holds rigorously, a straightforward approach would be to find a small number of space groups that contain all space groups as subgroups and exhibit a model realizing an STI for each of these symmetry groups. Next, we would like to compute the classification C full including both TCIs and STIs. The topological crystal picture tells us that TCIs are a subgroup (i.e., C ⊂ C full ) because stacking two TCIs produces another TCI or a trivial state. It is also true that C full /C ≃ ℤ 2 , because this quotient corresponds to ignoring the distinctions among TCIs, which leaves only aℤ 2 generated by the STI. It follows that \|C full \| = 2 \|C\| when the TCI classification is finite. One might expect C full = C × ℤ 2 , with the ℤ 2 factor generated by the STI, but this is not true in general because stacking two identical STIs can result in a nontrivial TCI. Put another way, C full can be a nontrivial group extension of C by ℤ 2 , and we need to solve this group extension problem. We proceed by choosing a particular STI state and stacking this state with itself to get a state we call (STI) 2 . We know that (STI) 2 has trivial strong index and is, thus, either a nontrivial TCI or a trivial state, that is, (STI) 2 ∈ C. We need to determine the element of C given by (STI) 2 . First, we observe that our choice of STI is arbitrary under stacking with a TCI because this stacking does not change the strong invariant. It is obvious that stacking STI with a ℤ 2 TCI does not affect (STI) 2 , but stacking STI with an MTCI can change the ℤ invariants of (STI) 2 by arbitrary even integers, depending on the choice of MTCI. We, thus, see that the information in (STI) 2 that is independent of the arbitrary choice of STI is precisely the information preserved under the map p : C→C≃H 1 ðG c ; ℤ 2 Þ introduced in Methods. Therefore, (STI) 2 is characterized by a homomorphism from G c → ℤ 2 , namely, p((STI) 2 ) ∈ H 1 (G c , ℤ 2 ). Determining p((STI) 2 ) solves the group extension problem and determines the group C full . Denoting the homomorphism given by p((STI) 2 ) by d : G c → ℤ 2 , it is natural to conjecture that d(g) = 0 when g is a rigid motion preserving the orientation of space (e.g., translations and rotations), and d(g) = 1 when g reverses orientation (e.g., inversion, reflections, and glide reflections). This conjecture is natural because the map d should depend only on the crystalline symmetry G c , and there does not seem to be any other nontrivial map that can be defined in a uniform way for all G c . We can establish this conjecture using results of Khalaf et al. (36), where the authors studied surface theories obtained by stacking two STIs. For a crystalline symmetry G surf c preserved by some surface termination, they considered a mass texture on the boundary satisfying m gr = s g m r , where g∈G surf c , r is a point on the boundary, m r is the Dirac mass, and s g = ±1 keeps track of sign changes in the mass. They showed that s g 1 g 2 = s g 1 s g 2 , i.e., s g defines a homomorphism from G surf c to ℤ 2 . Moreover, they showed that, in the case of stacking two identical STIs, s g = det R g (this result follows from equation 14 of (36) upon taking h ð1Þ g ¼ hð2Þ g , as appropriate for identical STIs). Here, g is the rigid motion {R g \|t g }, where R g is an O(3) matrix and t g is a translation vector. Because det R g = 1 (det R g = −1) for orientation-preserving (orientation-reversing) operations, this result is identical to our conjecture upon identifying that s g = (−1) d(g) . Physically, s g and d(g) should be, thus, identified because the gapless lines on the surface where the mass changes sign are, in the topological crystal picture, precisely the gapless edges of 2-cells touching the surface. The argument is not quite complete because the crystalline symmetry G c cannot generally be preserved by a surface termination. However, we found that specifying the seven types of invariants listed in the Results section uniquely determines a TCI phase (element of C). Therefore, we can take G surf c to be the subgroup of G c associated with each invariant. It is always possible to choose a surface termination preserving such G surf c , so we can run the above argument for each such subgroup. This then determines p((STI) 2 ). This result determines the group structure of C full . There are three cases: (i) If G c contains only orientation-preserving operations, then C full = C × ℤ 2 . (ii) If G c contains orientation-reversing operations but no mirror reflections, and hence C has no ℤ factors, then one of the ℤ 2 factors in C is replaced in C full by a ℤ 4 factor. (iii) If G c contains mirror reflections, then (STI) 2 generates a ℤ factor in C. In this case, C full and C have the same group structure, but in C full , the generator of one of the ℤ factors is an STI state. These rules easily allow one to obtain C full for all the crystallographic point groups and space groups, and the results are given as tables in the Supplementary Materials. Because p((STI) 2 ) is known, it is also straightforward to explicitly construct the topological crystal corresponding to (STI) 2 , up to the arbitrariness in defining STI. We note that recently, (43) used K-theory to obtain classifications for crystallographic point 1 N/A 1 ℤ 2 2 ℤ 2 m ℤ 2/m ℤ × ℤ 2 222 ℤ 2 2 mm2 ℤ 2 mmm ℤ 3 4 ℤ 2 4 ℤ 2 4/m ℤ × ℤ 2 422 ℤ 2 2 4mm ℤ 2 42m ℤ × ℤ 2 4/mmm ℤ 3 3 N/A 3 ℤ 2 32 ℤ 2 3m ℤ 3m ℤ × ℤ 2 6 ℤ 2 6 ℤ 6/m ℤ × ℤ 2 622 ℤ 2 2 6mm ℤ 2 62m ℤ 2 6/mmm ℤ 3 23 N/A m 3 ℤ 432 ℤ 2 43m ℤ m 3m ℤ 2 groups in d = 3. To compare these results with ours for all point groups would, for instance, require extending our results to include topological crystals with d b = 0 bulding blocks. However, for the nine nontrivial point groups where all operations share a common fixed line or fixed plane (C n , C s , and C nv ), states with d b = 0 building blocks are not relevant, and for these point groups, our results agree with those of (43). DISCUSSION Our method provides a unified, real-space perspective for TCIs, complementary to the momentum-space perspective usually taken for free-fermion systems. Real-space constructions have an advantage when electron interactions are considered. For example, because all ℤ 2 TCIs are made from 2dTIs, from the stability of the 2dTI phase against interactions, we immediately know that the classification of ℤ 2 TCIs is not affected by interactions. On the other hand, the ℤ classification of d = 2 MCI states collapses to ℤ 8 in the presence of interactions (44). This implies that the classification of MTCIs does collapse, but the ℤ invariants characterizing MTCIs are robust to interactions modulo 8. One can easily use the idea of topological crystals to classify freeelectron d = 3 insulators with time-reversal but without spin-orbit coupling, that is, with SU(2) spin rotation symmetry. For these systems, time reversal and crystalline symmetry can be taken to act trivially on electron operators, with no factors of fermion parity, and in particular T 2 = 1. It can then be seen that all 2-cells have symmetry class AI, while 1-cells can be in class AI or A. In all these cases, only trivial states are possible, and no free-electron TCIs can occur. This means that any nonzero symmetry-based indicator implies some topological nodes in the bulk, proved by exhaustion in (45). The topological crystal approach developed here can be applied in many other physical settings. For instance, one can classify topological crystalline superconductors, described at the free-fermion level by Bogoliubov-de Gennes Hamiltonians. Moreover, as other works have begun to explore, it is also possible to use topological crystals to classify interacting fermionic cSPT phases. METHODS Cell complex structure A cell complex is a topological space constructed by gluing together points (0-cells) and n-dimensional balls (n-cells). In more detail, to construct a cell complex X, one starts with a set of discrete points X 0 , referred to as the 0-skeleton. Next, one forms the 1-skeleton X 1 by attaching a set of 1-cells to X 0 . To attach a 1-cell, one starts with a closed interval on the real line (whose interior is the 1-cell), and the two endpoints are identified with points in X 0 . The resulting space X 1 is given by attaching the 1-cells to X 0 . The process continues in the natural way; for instance, to attach a 2-cell to X 1 , we start with a 2D disc D with boundary (whose interior is the 2-cell) and identify ∂D with a subset of X 1 using a continuous map from ∂D to X 1 . The n-skeleton X n is given by attaching the n-cells to X n−1 . A more detailed discussion can be found in the book by Hatcher (46). Here, we describe in more detail how 3D space ℝ 3 can be given a cell complex structure upon choosing an AU. An AU is an open subset of ℝ 3 that is as large as possible, subject to the condition that no two points in the AU are related by the action of G c . The choice of AU is not unique. While not strictly necessary, we can always choose an AU such that the boundary of the closure of the AU consists of segments of flat planes; that is, in the case of space group symmetry, the AU can be chosen as the interior of a polyhedron. Once we choose an AU, it and its copies under the action of G c form the 3-cells, which are in one-to-one correspondence with the elements of G c . The union of all the 3-cells is denoted by A, and its complement X 2 = ℝ 3 − A is the 2-skeleton of the cell complex. We choose 2-cells of X 2 satisfying three properties: (1) Each 2-cell is a subset of a face where two 3-cells meet. To be precise, we say that two 3-cells meet at a face when the intersection of their closures is S C I E N C E A D V A N C E S \| R E S E A R C H A R T I C L E S C I E N C E A D V A N C E S \| R E S E A R C H A R T I C L E homeomorphic to a 2-manifold (possibly with boundary), and we define the face where they meet to be this intersection. (2) No two distinct points in the same 2-cell are related under the action of G c . Note that a 2-cell may be a subset of a mirror plane, in which case the mirror symmetry will take every point in the 2-cell to itself. This property ensures that each 2-cell has no spatial symmetries; if there is a mirror symmetry, it acts on the 2-cell effectively as an internal symmetry. (3) The 2-cell structure on X 2 respects the G c symmetry. Precisely, given a 2-cell e 2 and a symmetry operation g ∈ G c , the image g(e 2 ) is also a 2-cell. It is always possible to choose a set of 2-cells satisfying these properties: We started with the set of faces where pairs of 3-cells meet and took their interiors as 2-manifolds. This gives a set of 2-cells for X 2 satisfying properties (1) and (3), but property (2) need not be satisfied. This can be rectified by dividing up the 2-cells until property (2) is satisfied. Letting A 2 be the union of all the 2-cells, the 1-skeleton X 1 is X 2 − A 2 . We choose 1-cells to satisfy three properties very similar to those for 2-cells. A difference from the 2-cell case is that different numbers of 2-cells can meet at a 1-cell; we would like to ensure that the same set of n 2-cells meets everywhere along the extent of a given 1-cell. We therefore modified property (1) as follows: Each 1-cell is a subset of an edge where exactly n 2-cells meet. More precisely, we say that n 2-cells meet at an edge when the intersection of their closures is homeomorphic to a 1-manifold (possibly with boundary), and the edge where they meet is defined to be this intersection. Apart from these n 2-cells, we require the edge to have empty intersection with the closure of any other 2-cell. Properties (2) and (3) are required to hold with the obvious modifications. The 0-cells are just the points where two or more 1-cells meet. Formally, letting A 1 be the union of all 1-cells, the 0-cells are the points of X 1 − A 1 . We illustrate this rather abstract discussion with some examples. First, we considered G c = C i , the point group generated by inversion symmetry. We took the AU to be the half space z > 0, and the 3-cells are then the two half-spaces z > 0 and z < 0. There are two 2-cells, which are z = 0 half planes with y > 0 and y < 0, and two 1-cells, which are z = y = 0 half lines with x > 0 and x < 0. Last, the single 0-cell is the point at the origin. As a second example, we took G c to be space group #1, which consists only of translation symmetry. We set the lattice constant to unity and take the three Bravais lattice basis vectors to be (1,0,0), (0,1,0), and (0,0,1). A natural choice for an AU is simply the interior of a unit cell, i.e., the region 0 < x, y, z < 1. The 3-cells are then the copies of the AU under translation. There are three kinds of 2-cells. One type consists of the xy plane (i.e., z = 0 plane) region with 0 < x, y < 1 and its images under translation, and the other two types are similar but lie in the xz and yz planes. Similarly, there are three kinds of 1-cells, with one type consisting of the x = y = 0 region with 0 < z < 1 and its images under translation. The other two types are similar regions oriented along the x and y axes. Last, the 0-cells are points (n x , n y , n z ), with n x , n y , and n z integers. Formal structure of classification resolved by block dimension Here, we describe the Abelian group structure of TCIs (32) and how taking a certain quotient allows us to ignore distinctions among atomic insulators. We let D d b be the Abelian group classifying insulators whose nontrivial building blocks are dimension d b and below. That is, states classified by D d b can be reduced to a state on X 2 where all n-cells with n > d b host a trivial state. Clearly, d b = 0, 1, and 2, and we have the sequence of subgroups D 0 ⊂ D 1 ⊂ D 2 . D 2 as the classification of all TCIs or, at least, all those that can be classified in terms of topological crystals. The observation that all 1-cells are trivial implies that D 0 = D 1 . Phases in D 0 are atomic insulators, which we wish to exclude from consideration. Although there are distinct atomic insulators constituting different quantum phases of matter, all atomic insulators are, in some sense, topologically trivial. We can eliminate these states by taking the quotient C = D 2 /D 0 , which gives the desired classification of TCIs. Gluing MCI building blocks: Planar decomposition of MTCIs Here, we address the consequences of the gluing conditions for MTCIs, i.e., topological crystals built from MCI building blocks placed on the 2-cells of X 2 . In particular, we show that MTCIs can always be decomposed into decoupled planar MCI states placed on mirror planes. We consider a mirror plane P, and note that we must have P ⊂ X 2 . Therefore, up to a set of measure zero, P is a union of 2-cells of X 2 . We consider a 2-cell e 2 1 ⊂ P and place an MCI state on e 2 1 , whose invariant is some element of ℤ. We want to show that symmetry and gluing along 1-cells imply that every 2-cell in P is an MCI with the same ℤ invariant as the state in e 2 1 . It is enough to show that this holds for a single 2-cell e 2 2 ⊂ P that is adjacent to e 2 1 in P. That is, e 2 1 and e 2 2 meet at a 1-cell e 1 ⊂ P, as illustrated in fig. S1. To proceed, we consider a number of cases. In case (1), there exists an element g ∈ G c that maps e 2 1 to e 2 2 . First, we show that symmetry requires that both 2-cells host an MCI state with the same invariant (this is not a priori obvious; it is conceivable that some symmetry operations could change the sign of the invariant). To begin, we claim that either gs = sg, when g preserves the orientation of the mirror plane, or gs = (−1) F sg, when g reverses the orientation of the mirror plane. If we ignore factors of fermion parity, then gs = sg or, equivalently, gsg −1 = s. To see this, we observe that g ∈ G P , where G P G c is the group of symmetries of the mirror plane. Moreover, s is the only nontrivial element of G P that acts on the mirror plane as the identity rigid motion. The operation gsg −1 also acts on the mirror plane as the identity rigid motion, and s cannot be conjugate to the identity in G P ; therefore, gsg −1 = s. of the (−1) F factor. We choose coordinates so that the mirror plane is the z = 0 plane, and the action of s on the Dirac field Y(r) is given in the Supplementary Materials. We consider a symmetry operation g that takes the mirror plane into itself, with action on the Dirac field g : YðrÞ→M g Yðr′Þ ð 3Þ where r′ ¼ Or þ t →ð4Þ where O is an orthogonal matrix. The requirement that the mirror plane goes into itself under g implies that t z = O zx = O zy = 0. Moreover, because O is an orthogonal matrix, O xz = O yz = 0 and O zz = ±1. We are free to multiply g by inversion and/or translations within the z = 0 plane to make g into a rotation. This can be done because both translations and inversion preserve the orientation of the z = 0 plane. Moreover, translations have no effect on M g , while inversion commutes with s. After doing this, there are two possibilities for g. One possibility is a rotation by q with axis normal to the plane; this operation preserves the orientation of the plane, and we have M g = exp (iqm 3 /2) so that g commutes with s. The other possibility is a C 2 rotation with axis normal to the plane; this operation reverses the orientation of the plane and anticommutes with s. This establishes the claim that gs = sg, when g preserves the orientation of the mirror plane, or gs = (−1) F sg, when g reverses the orientation of the mirror plane. Now we use this claim to show that the MCI states on e 2 1 and e 2 2 have the same ℤ invariant. Consider a one-electron state \|y> supported only on e 2 1 , whose mirror eigenvalue is given by s \| y > = i \| y>. The state g \| y> is supported on e 2 2 and has mirror eigenvalue +i if g preserves the orientation of the plane, and −i if it reverses the orientation of the plane. Because the Chern number of each sector with fixed mirror eigenvalue is preserved when g preserves orientation and reversed when g reverses orientation, it follows that the two MCI states have the same ℤ index. To complete the discussion of case (1), we need to address gluing of the two MCI states at e 1 . It is enough to consider only symmetries that take the set of cells fe 1 ; e 2 1 ; e 2 2 g into itself. There are two subcases. In case (1a), the only relevant symmetry is the mirror reflection itself. In this case, it is obvious that two MCI states with the same invariant can be glued together along e 1 . In case (1b), e 1 is contained within a C 2v axis. To analyze gluing at e 1 , we studied the edge theory at e 1 for MCI states on e 2 1 and e 2 2 . The edge of e 2 1 (e 2 2 ) consists of a pair of counterpropagating fermion modes c R1 and c L1 (c R2 and c L2 ). We assemble these 1D fermions into the four-component field y T = (c R1 c L1 c R2 c L2 ). Denoting by s′ the mirror symmetry exchanging the two 2-cells, we take the symmetries to act by T : y→ðim 2 Þyð5Þ s : y→im 3 t 3 yð6Þ s′ : y→im 3 t 1 yð7Þ where the m i and t i Pauli matrices act in the 4 × 4 matrix space just as in the earlier discussion of the relativistic Dirac Hamiltonian. These symmetries act appropriately on fermions and are compatible with the two 2-cells having the same MCI index. These symmetries allow the mass term y † m 2 t 2 y, which gaps out the fermions on e 1 and thus glues the two 2-cells together. Now, we move on to case (2), where there is no element g ∈ G c that maps e 2 1 into e 2 2 . In this case, symmetry does not determine which state is placed on e 2 2 , but we will see that this is determined by gluing at e 1 . We found it useful to consider the 3-cells that meet at e 2 1 and e 2 2 . These are defined in fig. S2. We considered two subcases. In case (2a), e 2 1↑ ¼ e 2 2↑ . It follows immediately that e 2 1↓ ¼ e 2 2↓ , so there is only a single 3-cell above and below the mirror plane in the region shown in fig. S2. This means that e 2 1 and e 2 2 are the only 2-cells meeting at e 1 , which further implies that the only symmetry taking e 1 into itself is the mirror symmetry. The gluing condition at e 1 is then satisfied if and only if an MCI state is placed in e 2 2 . In case (2b), e 2 1↑ ≠e 2 2↑ , which implies that e 2 1 and e 2 2 are not the only 2-cells meeting at e 1 . The additional 2-cells come in mirror-symmetric pairs above and below the mirror plane. There are two further subcases. In case (2b.i), the only symmetry taking e 1 into itself is the mirror symmetry. In this case, the additional pairs of 2-cells do not coincide with mirror planes. Therefore, for each pair, the only nontrivial possibility is that both 2-cells host a 2dTI state, in which case the 2dTI edges of the pair can be gapped out at e 1 . The relevant edge theory for each pair at e 1 is the same as that discussed in case (1), except with only s ′ and T symmetry (i.e., without s symmetry), and it follows from the discussion there that this edge can be gapped. Therefore, these additional pairs of 2-cells can be effectively eliminated, and the gluing condition again requires us to place an MCI state on e 2 2 . Last, in case (2b.ii), e 1 is contained in a C 3v axis, where the C 3v symmetry is generated by s and a 3-fold rotation C 3 . Here, there are six 2-cells that coincide with the mirror planes meeting at e 1 . These 2-cells come in three pairs, with each pair contained in one of the three mirror planes that intersect e 1 . e 2 1 and e 2 2 constitute one such pair, with the other two pairs obtained from it under the 3-fold rotation. We suppose that an MCI state is placed on e 2 1 and its rotation images, but not on e 2 2 (see fig. S3); we show that it is impossible to gap out the resulting edge theory at e 1 , which imply that, again, an MCI state must be placed on e 2 2 . The edge fermions for the e 2 1 MCI are c R1 and c L1 , with s : c R1 →ic R1 c L1 → Àic L1 &ð8Þ The images of these fermions under C 3 rotation arec R2 ¼ C 3 c R1 C À1 3 and c R3 ¼ C 2 3 c R1 C À2 3 , with identical expressions holding for the leftmoving modes. The generators of the C 3v group satisfy the following relations, acting on a fermion field s 2 ¼ À1ð9ÞC 3 3 ¼ À1ð10ÞðsC 3 Þ 2 ¼ À1ð11Þ Using these relations, we find s : c R2 → Àic R3 c L2 →ic L3 c R3 → À ic R2 c L3 →ic L2 8 > > < > > :ð12Þ Now, focusing only on the mirror reflection symmetry, we can change variables to diagonalize s and find that the c R2 , c R3 , c L2 , and c L3 fermion modes can be gapped out. This leaves the c R1 and c L1 edge of the MCI state on e 2 1 , which cannot be gapped; this establishes the desired result. Gluing condition for ℤ 2 TCIs Here, we consider the gluing condition for ℤ 2 TCIs, i.e., the requirement that there are no gapless modes within the bulk. If we consider placing 2dTI states on a subset of the 2-cells of X 2 , then we will show that the gluing condition can be satisfied if and only if an even number of 2dTI edges meet at each 1-cell. This is illustrated in fig. S4 for the space groups P3 (no. 143) and P4 (no. 75). One direction is trivial to show: If the gluing condition is satisfied, then an even number of 2dTI edges must meet at each 1-cell e 1 because, otherwise, time reversal would forbid e 1 from being gapped. Now, we suppose that we place 2dTI states on 2-cells of X 2 such that an even number of 2dTI edges meets at each 1-cell. We would like to show that each 1-cell can be gapped; thus, the gluing condition is satisfied. We do this by considering the different possible point group symmetries of a 1-cell e 1 , which may impose constraints on gluing of 2dTI edge modes along e 1 . If e 1 has trivial point group symmetry, then the only symmetries are charge conservation and time reversal, and an even number of 2dTI edge modes can always be gapped. If e 1 is contained in a mirror plane, then 2dTI edge modes come in mirror-symmetric pairs above and below the mirror plane, and we have already shown in the above that each such pair can be gapped. Next, we consider the case where e 1 is contained in a C n axis. When n = 2, 2dTI edge modes come in pairs related by C 2 symmetry, and the action of C 2 symmetry on such a pair is identical to that of the s ′ symmetry discussed above (see Eq. 7 and the surrounding discussion). Therefore, these pairs of edge modes can be gapped. When n > 2 is even, edge modes come in groups of n related by C n symmetry, and these can be grouped into n/2 pairs related by C 2 symmetry. We can focus on one such pair and gap it out, then use the C n symmetry to "copy" its mass term to the other n/2 − 1 pairs, which gaps out all the modes respecting the C n symmetry. Last, for C 3 symmetry, edge modes must come in groups of six, with two sets of three edge modes related by symmetry. We can take a pair of edge modes unrelated by symmetry and gap these out and then use the C 3 symmetry to copy the resulting mass term to the other two pairs of edge modes. Similar arguments can be applied when e 1 is contained in a C nv axis. Here, 2-cells that can host 2dTIs lie away from the mirror planes, so that edge modes come in groups of 2n related by C nv symmetry. Viewing C nv as generated by a mirror reflection and C n rotation, we can start by gapping out a pair of neighboring edge modes related by the mirror symmetry and then copying its mass term using the rotational symmetry. Gluing at 0-cells In the analysis of the gluing condition in the above, we started with a set of topological states on 2-cells and considered gluing these states together at 1-cells. In principle, this may not be the end of the story; we needed to consider gluing at 0-cells. That is, we needed to ensure that there are no localized protected gapless states at 0-cells, which would violate the gluing condition. However, there is a simple reason this cannot occur: consider the set of gapped 1-cells that meet at a 0-cell. We can lump these 1-cells together and view them as a single gapped d = 1 system, with the 0-cell as its endpoint. Upon decomposing the spectrum into irreducible representations of any site symmetry at the 0-cell, this d = 1 system divides into sectors that are either in class A or AII. To have a protected gapless state on the 0-cell, the d = 1 system would need to be topologically nontrivial, but class A and AII have a trivial classification in d = 1. Nontrivial example of equivalence operation In the Results section, we discussed an equivalence operation involving creating bubbles of 2dTI within each AU, which turns out to have a trivial effect on the classification of time-reversal symmetric electronic TCIs. To clarify the nature and role of this equivalence operation, here we discuss an example without time-reversal symmetry where it does modify the classification. We consider d = 3 insulators without timereversal symmetry with point group 1. Ignoring the inversion symmetry, this is a system in symmetry class A. We choose the 2-skeleton X 2 as the z = 0 plane, the 1-skeleton X 1 as the z = y = 0 line, and the 0-skeleton X 0 as the point at the origin, as shown in Fig. 4A. Phases constructed by decorating X 0 with d b = 0 building blocks are atomic insulators, which are excluded from consideration; thus, we need only to consider X 1 and X 2 . The only internal symmetry on X 1 and X 2 is the charge conservation, which protects a ℤ-classification in d = 2, corresponding to Chern insulators, and a trivial classification in d = 1. Thus, to obtain the TCI classification, we only need to consider topological crystals obtained by decorating X 2 with Chern insulators. First, we consider the gluing condition on X 2 . X 2 decompose into two 2-cells: e 2 1 : z ¼ 0; x > 0; e 2 2 : z ¼ 0; x < 0. We decorate e 2 1 with a Chern insulator with Chern number C (∈ℤ) and e 2 2 with a symmetric copy of this Chern insulator under inversion. Since chirality remains unchanged under inversion, this copy has the identical Chern number C, and the chiral boundary states of e 2 1 and e 2 2 cancel each other on the 1-skeleton X 1 . Therefore, the gluing condition allows a ℤ classification on X 2 . We now consider the equivalence operation. We create a bubble of Chern insulator with Chern number 1 in the z > 0 region. Because x > 0 and e 2 2 : z ¼ 0; x < 0. (B) The equivalence operation. When a bubble made of 2D Chern insulator with Chern number 1 is created in z > 0, another Chern bubble with Chern number −1 in z < 0 will be created because of the inversion symmetry. The arrows represent the orientation of the Chern number. We enlarge these two bubbles until they join with the z = 0 plane; then, the Chern number of the state on z = 0 will be subducted by 2. of the inversion symmetry, another bubble will be created in the z < 0 region as shown in Fig. 4B. We enlarge the two bubbles respecting the inversion symmetry, until the bottom surface of the z > 0 bubble joins with X 2 and the rest of the bubble moves to infinity. At the same time, the top surface of the z < 0 bubble also joins with X 2 . Therefore, after this operation, the total Chern number on X 2 is increased by 2, which implies that the classification is reduced from ℤ to ℤ 2 . This class A ℤ 2 TCI has been discussed in (21,47,48). Topological crystals, topological invariants, and H 1 (G, ℤ 2 ) In this section, we give a more detailed discussion of the ℤ 2 -valued function of invariants d(g) characterizing a topological crystal. On the basis of this discussion, we established a connection between the classification C of TCIs and H 1 (G c , ℤ 2 ), which allows C to be computed with the aid of standard computer algebra tools such as GAP (49). In particular, we show that C and H 1 (G c , ℤ 2 ) have the same number of generators. In the text, we defined d(g) only for a ℤ 2 TCI. It will be useful for our present purposes to give a definition valid for an arbitrary topological crystal. First, we introduce the notion of a ℤ 2 coloring of X 2 , which is given by associating a ℤ 2 number to each 2-cell of X 2 so that each 2-cell is either colored and assigned 1, or empty and assigned 0. These ℤ 2 numbers must be assigned to respect the crystalline symmetry and satisfy a gluing condition, namely, for each 1-cell, an even number of the 2-cells meeting there must be colored. ℤ 2 colorings can be added using the ℤ 2 addition law, and this makes ℤ 2 colorings into a group that we denote byC [we remark that ℤ 2 colorings can be viewed as elements of the homology group H 2 (X 2 ; ℤ 2 ) satisfying a symmetry condition, but we will not make use of this here.] Next, we observe that there is a map from topological crystals (elements of C) toC. We denote this map by p : C→C. Empty cells of the topological crystal map to empty cells in the ℤ 2 coloring. Cells decorated with 2dTI map to colored cells. Cells decorated with an MCI state map to colored cells when the mirror Chern number is odd and to empty cells when it is even (we used the convention that the smallest possible mirror Chern number is 1; some authors use a definition of mirror Chern number that is twice our definition). Last, given a ℤ 2 coloring of X 2 , we define d(g) as in the Results section. That is, we arbitrarily choose one AU, let r be a point inside, and define d(g) = 1 if a path connecting r to gr crosses an odd number of colored 2-cells, while d(g) = 0 if the path crosses an even number of colored 2-cells. The path should be chosen to avoid 0-cells and 1-cells but is otherwise an arbitrary continuous path. We now establish some properties of d(g) quoted in the Results section. First, we show that d(g) is independent of the path chosen to connect r to gr and is, thus, a well-defined function mapping G c to ℤ 2 . Any two such paths are related by a finite number of moves, where a segment of the path is passed through a 1-cell. The gluing condition says that an even number of colored 2-cells meet at every 1-cell, so these moves do not affect d(g). At this stage, we have not yet shown that d is independent of the arbitrary choice of AU. Next, we show that d is a homomorphism from G c to ℤ 2 , that is, d(g 1 g 2 ) = d(g 1 ) + d(g 2 ). We compute d(g 1 g 2 ) by considering a path from r to g 1 g 2 r that first goes from r to g 1 r and then goes from g 1 r to g 1 g 2 r. The number of colored 2-cells modulo 2 crossed by the first segment is d(g 1 ) by definition. The second segment is related by symmetry to a path joining r to g 2 r, so d(g 2 ) is the number of colored 2-cells (modulo 2) crossed by the second segment. Therefore, d(g 1 g 2 ) = d(g 1 ) + d(g 2 ). Last, we show that d does not depend on the arbitrary choice of AU. Let d(g) be the function defined by choosing an AU with a point r inside, and let d ′ (g) be the function defined by choosing a different AU, which contains a point g 0 r for some g 0 ∈ G c . Then, d ′ (g) is the number of colored 2-cells (modulo 2) crossed by a path connecting g 0 r to gg 0 r. By symmetry, this number is the same as for a path joining r to g À1 0 gg 0 r, which shows that d′ðgÞ ¼ dðg À1 0 gg 0 Þ. But this implies that d ′ (g) = d(g) because d is a homomorphism and ℤ 2 is Abelian. Our construction of d gives a map D :C→H 1 ðG c ; ℤ 2 Þ, where H 1 (G c , ℤ 2 ) is viewed as the group of homomorphisms from G c to ℤ 2 . D is an isomorphism, soC≃H 1 ðG c ; ℤ 2 Þ. It is easy to see that D is injective. To see that D is surjective, we need to show that given d : G c → ℤ 2 , we can construct a corresponding ℤ 2 coloring. Upon arbitrarily choosing an AU, the 3-cells of ℝ 3 are in one-to-one correspondence with elements of G c . We then color each 3-cell with the ℤ 2 number d(g). Given a 2-cell, let g 1 and g 2 label the two 3-cells that meet at the 2-cell. We then color the 2-cell with the ℤ 2 number d(g 1 ) + d(g 2 ). The resulting assignment of ℤ 2 numbers to 2-cells is clearly symmetric and satisfies the gluing condition and is, thus, a ℤ 2 coloring of X 2 . By construction, D maps this ℤ 2 coloring to d. Now that we have shown thatC≃H 1 ðG c ; ℤ 2 Þ, we would like to show that C andC have the same number of generators. First, we observed that C = C MCI × C 2dTI , where C MCI is the classification of MTCIs (i.e., topological crystals built from MCI states) and C 2dTI is the classification of ℤ 2 TCIs. C MCI is a product of ℤ factors, and C 2dTI is a product of ℤ 2 factors. We introduce a similar decompositioñ C ¼C MCI ÂC 2dTI , whereC MCI is defined to be the subgroup of ℤ 2 colorings whose colored 2-cells lie in mirror planes, andC 2dTI is the subgroup of ℤ 2 colorings where all 2-cells lying in mirror planes are empty. ClearlyC,C MCI , andC 2dTI are all products of ℤ 2 factors. To prove thatC ¼C MCI ÂC 2dTI , it is enough to show that an arbitrary ℤ 2 coloring c ∈C can be written uniquely as c ¼ c m c m for some c m ∈C MCI and c m ∈C 2dTI . Given c ∈C, we consider a 1-cell e 1 contained in a mirror plane. n 2-cells meet at e 1 , two of which lie in the mirror plane, and n − 2 of which lie outside the mirror plane. The n − 2 2-cells outside the mirror plane can be grouped into pairs related by mirror reflection, so that the two 2-cells in each pair are either both colored or both empty. It follows that the two 2-cells contained in the mirror plane are also either both colored or both empty. Therefore, we define a new ℤ 2 coloring c m ∈C MCI by starting with c and replacing all colored 2-cells not lying in mirror planes with empty cells. Similarly, if we replace all the colored 2-cells within mirror planes with empty cells, then we obtain c m ∈C 2dTI . It is obvious that c ¼ c m c m and that c m and c m are unique. Using the above discussion, we showed that the map p : C→C gives a one-to-one correspondence between generators of C andC, so the two groups have the same number of generators. First, restricting p to C 2dTI gives an isomorphism between C 2dTI andC 2dTI , so these subgroups clearly have the same number of generators. Second, we can take each ℤ factor of C MCI to be generated from a topological crystal obtained by decorating the 2-cells of a mirror plane, as well as all symmetry-equivalent mirror planes, with an MCI state of unit Chern number. The above discussion implies thatC MCI is generated by ℤ 2 colorings obtained by coloring all the 2-cells of set of symmetryequivalent mirror planes, and these generators are images of the C MCI generators under p, giving a one-to-one correspondence between generators of C MCI andC 2dTI . These results make it a simple matter to compute the TCI classification C. First, we compute H 1 (G c , ℤ 2 ); this can be done using GAP (49). Then, we know that the number of ℤ factors in C is n M , the number of symmetry-inequivalent sets of mirror planes. We then obtain C from H 1 (G c , ℤ 2 ) by replacing n M of the ℤ 2 factors with ℤ factors. For a G c crystallographic point group or space group, n M can be obtained immediately from information tabulated in the International Tables for Crystallography (50). The results of this procedure are presented in Table 1 for crystalline point groups, and in Table 2 for space groups. As discussed in the Introduction, we note that Khalaf et al. (36) have also obtained the results in these tables via a mathematically equivalent procedure. While useful, we emphasize that this largely automated procedure is not a substitute for explicit real-space construction of topological crystals for a given symmetry group of interest, as described in Results. The latter procedure not only results in the same group structure but also provides additional physical insight and a starting point for further analysis, by giving an explicit real-space construction of each of the TCI phases classified. We also obtained the results in Table 2 by automating the explicit real-space constructions. SUPPLEMENTARY MATERIALS Supplementary material for this article is available at http://advances.sciencemag.org/cgi/ content/full/5/12/eaax2007/DC1 Fig. S1. Illustration of the 2-cells e 2 1 and e 2 2 and the 1-cell e 1 used to discuss the effect of gluing conditions on MTCIs. Fig. S2. The 3-cells e 3 1↑ and e 3 1↓ (e 3 2↑ and e 3 2↓ ) meet at the 2-cell e 2 1 (e 2 2 ). Fig. S3. Cross section through e 1 and the 2-cells that meet at e 1 , in case (2b.ii), where e 1 is contained in a C 3v axis. Fig. S4. Illustration of the effects of the gluing condition for ℤ 2 TCIs for two representative space groups. Table S1. Topological crystals beyond layer construction. Table S2. Classification C full of STIs and TCIs with time-reversal symmetry and spin-orbital coupling for all point groups. Table S3. Classification C full of STIs and TCIs with time-reversal symmetry and spin-orbital coupling for all space groups. Fig. 1 . 1Cell complex structure for space group P1 (#2). (A) The AU 0 < x, y < 1 and 0 < z < 1 2 . (B) The symmetry-inequivalent 2-cells (colored faces), 1-cells (bold lines), and 0-cells (red dots). The other cells can be obtained from these by acting with symmetry operations. Fig. 2 . 2\|000} (here, the lattice constants are set to 1). The AU can be chosen as the region 0 < x; y; z < 1 2 . The 2-cells and 1-cells are given, respectively, by e 2 i¼1;2;3;4 and e 1 i¼1;2;3;4;5;6 , as shown inFig. 2A, and their images under symmetry. There are no mirror planes, so each inequivalent 2-cell can be decorated with a 2dTI state, and possible configurations are described by four ℤ 2 numbers, n i = 1Topological crystals in space group P4 2 2 1 2 (#94). (A) The symmetryinequivalent 2-cells ðe 2 i¼1;2;3;4 Þ and 1-cells ðe 1 i¼1;2;3;4;5;6 Þ are represented by colored faces and bold lines, respectively. Here, the lattice constants are set to 1, the unit cell is given by 0 ≤ x, y, z < 1, and the AU is given by 0 < x; y; z< 1 2 . (B to D) The three independent ℤ 2 topological crystal generators, where only 2-cells decorated with 2dTIs are shown. (C) and (D) are layer constructions, whereas (B) is not. Fig. 2B); (ii) n 1 = n 2 = 1, n 3 = n 4 = 0 (Fig. 2C); and (iii) n 3 = n 4 = 1, n 1 = n 2 = 0 (Fig. 2D). While states (ii) and (iii) are obviously layer constructions, state (i) is not layer constructible, as discussed above. Fig. 3 . 3Illustrations of the topological crystals beyond layer construction. (A) to (L) show these states for space groups Pnn2 (#34), Pnnn (#48), P4 2 (#77), P4 2 /n (#86), P4 2 22 (#93), P4 2 2 1 2 (#94), P4 2 cm (#102), P4n2 (#118), P4 2 /nnm (#134), Pn3 (#201), P4 2 32 (#208), and Pn3m (#224), respectively. Inequivalent 2-cells are represented by different colors. The topological invariants of these topological crystals and the coordinates of the plotted 2-cells can be found in the Supplementary Materials. Fig. 4 . 4Illustration of the effects of the equivalence operation in point group 1 without time-reversal symmetry. (A) The 2-cells: e 2 1 : z ¼ 0; Table 1 . 1Classifications C of TCIs for noninteracting electrons with time-reversal symmetry, spin-orbit coupling, and crystalline point group symmetry. N/A denotes a trivial classification. Full classifications including strong topological insulators can be easily obtained from this table by using the group extension rules given in Results and are also provided as a table in the Supplementary Materials.Point group TCI classification Table 2 . 2TCI classifications C for noninteracting electrons with time-reversal symmetry and spin-orbit coupling, given for all space groups. N/A denotes a trivial classification. Full classifications including strong topological insulators can be easily obtained from this table by using the group extension rules given in Results and are also provided as a table in the Supplementary Materials.1 ℤ 3 2 41 ℤ 3 2 81 ℤ 3 2 121 ℤ Â ℤ 2 2 161 ℤ 2 201 ℤ 2 2 2 ℤ 4 2 42 ℤ 2 Â ℤ 2 2 82 ℤ 2 2 122 ℤ 2 2 162 ℤ Â ℤ 2 2 202 ℤ 3 ℤ 4 2 43 ℤ 2 2 83 ℤ 2 Â ℤ 2 2 123 ℤ 5 163 ℤ 2 2 203 ℤ 2 4 ℤ 3 2 44 ℤ 2 × ℤ 2 84 ℤ Â ℤ 2 2 124 ℤ Â ℤ 3 2 164 ℤ Â ℤ 2 2 204 ℤ × ℤ 2 5 ℤ 3 2 45 ℤ 3 2 85 ℤ 3 2 125 ℤ Â ℤ 3 2 165 ℤ 2 2 205 ℤ 2 6 ℤ 2 Â ℤ 2 2 46 ℤ Â ℤ 2 2 86 ℤ 3 2 126 ℤ 3 2 166 ℤ Â ℤ 2 2 206 ℤ 2 2 7 ℤ 3 2 47 ℤ 6 87 ℤ Â ℤ 2 2 127 ℤ 3 × ℤ 2 167 ℤ 2 2 207 ℤ 2 2 8 ℤ Â ℤ 2 2 48 ℤ 4 2 88 ℤ 2 2 128 ℤ Â ℤ 2 2 168 ℤ 2 2 208 ℤ 2 2 9 ℤ 2 2 49 ℤ Â ℤ 4 2 89 ℤ 4 2 129 ℤ 2 Â ℤ 2 2 169 ℤ 2 209 ℤ 2 10 ℤ 2 Â ℤ 3 2 50 ℤ 4 2 90 ℤ 3 2 130 ℤ 3 2 170 ℤ 2 210 ℤ 2 11 ℤ Â ℤ 3 2 51 ℤ 3 Â ℤ 2 2 91 ℤ 3 2 131 ℤ 3 × ℤ 2 171 ℤ 2 2 211 ℤ 2 2 12 ℤ Â ℤ 3 2 52 ℤ 3 2 92 ℤ 2 2 132 ℤ 2 Â ℤ 2 2 172 ℤ 2 2 212 ℤ 2 13 ℤ 4 2 53 ℤ Â ℤ 3 2 93 ℤ 4 2 133 ℤ 3 2 173 ℤ 2 213 ℤ 2 14 ℤ 3 2 54 ℤ 4 2 94 ℤ 3 2 134 ℤ Â ℤ 3 2 174 ℤ 2 214 ℤ 2 2 15 ℤ 3 2 55 ℤ 2 Â ℤ 2 2 95 ℤ 3 2 135 ℤ Â ℤ 2 2 175 ℤ 2 × ℤ 2 215 ℤ × ℤ 2 16 ℤ 5 2 56 ℤ 3 2 96 ℤ 2 2 136 ℤ 2 × ℤ 2 176 ℤ × ℤ 2 216 ℤ 17 ℤ 4 2 57 ℤ Â ℤ 3 2 97 ℤ 3 2 137 ℤ Â ℤ 2 2 177 ℤ 3 2 217 ℤ × ℤ 2 18 ℤ 3 2 58 ℤ Â ℤ 2 2 98 ℤ 3 2 138 ℤ Â ℤ 2 2 178 ℤ 2 2 218 ℤ 2 19 ℤ 2 2 59 ℤ 2 Â ℤ 2 2 99 ℤ 3 × ℤ 2 139 ℤ 3 × ℤ 2 179 ℤ 2 2 219 ℤ 2 20 ℤ 3 2 60 ℤ 3 2 100 ℤ Â ℤ 2 2 140 ℤ 2 Â ℤ 2 2 180 ℤ 3 2 220 ℤ 2 21 ℤ 4 2 61 ℤ 3 2 101 ℤ Â ℤ 2 2 141 ℤ Â ℤ 2 2 181 ℤ 3 2 221 ℤ 3 22 ℤ 4 2 62 ℤ Â ℤ 2 2 102 ℤ Â ℤ 2 2 142 ℤ 3 2 182 ℤ 2 2 222 ℤ 2 2 23 ℤ 3 2 63 ℤ 2 Â ℤ 2 2 103 ℤ 3 2 143 ℤ 2 183 ℤ 2 × ℤ 2 223 ℤ × ℤ 2 24 ℤ 3 2 64 ℤ Â ℤ 3 2 104 ℤ 2 2 144 ℤ 2 184 ℤ 2 2 224 ℤ Â ℤ 2 2 25 ℤ 4 × ℤ 2 65 ℤ 4 × ℤ 2 105 ℤ 2 × ℤ 2 145 ℤ 2 185 ℤ × ℤ 2 225 ℤ 2 26 ℤ 2 Â ℤ 2 2 66 ℤ Â ℤ 3 2 106 ℤ 2 2 146 ℤ 2 186 ℤ × ℤ 2 226 ℤ × ℤ 2 27 ℤ 4 2 67 ℤ 2 Â ℤ 3 2 107 ℤ 2 × ℤ 2 147 ℤ 2 2 187 ℤ 3 227 ℤ × ℤ 2 28 ℤ Â ℤ 3 2 68 ℤ 4 2 108 ℤ Â ℤ 2 2 148 ℤ 2 2 188 ℤ × ℤ 2 228 ℤ 2 2 29 ℤ 3 2 69 ℤ 3 Â ℤ 2 2 109 ℤ × ℤ 2 149 ℤ 2 2 189 ℤ 3 229 ℤ 2 × ℤ 2 30 ℤ 3 2 70 ℤ 3 2 110 ℤ 2 2 150 ℤ 2 2 190 ℤ × ℤ 2 230 ℤ 2 2 31 ℤ Â ℤ 2 2 71 ℤ 3 × ℤ 2 111 ℤ Â ℤ 3 2 151 ℤ 2 2 191 ℤ 4 32 ℤ 3 2 72 ℤ Â ℤ 3 2 112 ℤ 3 2 152 ℤ 2 2 192 ℤ Â ℤ 2 2 33 ℤ 2 2 73 ℤ 4 2 113 ℤ Â ℤ 2 2 153 ℤ 2 2 193 ℤ 2 × ℤ 2 34 ℤ 3 2 74 ℤ 2 Â ℤ 2 2 114 ℤ 2 2 154 ℤ 2 2 194 ℤ 2 × ℤ 2 35 ℤ 2 Â ℤ 2 2 75 ℤ 3 2 115 ℤ 2 Â ℤ 2 2 155 ℤ 2 2 195 ℤ 2 continued on next page The relativistic Dirac Hamiltonian as discussed in the Supplementary Materials allows us to determine the presence or absence36 ℤ Â ℤ 2 2 76 ℤ 2 2 116 ℤ 3 2 156 ℤ × ℤ 2 196 N/A 37 ℤ 3 2 77 ℤ 3 2 117 ℤ 3 2 157 ℤ × ℤ 2 197 ℤ 2 38 ℤ 3 × ℤ 2 78 ℤ 2 2 118 ℤ 3 2 158 ℤ 2 198 N/A 39 ℤ Â ℤ 3 2 79 ℤ 2 2 119 ℤ Â ℤ 2 2 159 ℤ 2 199 ℤ 2 40 ℤ Â ℤ 2 2 80 ℤ 2 2 120 ℤ 3 2 160 ℤ × ℤ 2 200 ℤ 2 Song et al., Sci. 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[ "Stabilization of STEP electrolyses in lithium--free molten carbonates", "Stabilization of STEP electrolyses in lithium--free molten carbonates" ]	[ "Stuart Licht \nDepartment of Chemistry\nGeorge Washington University\n20052WashingtonDCUSA\n" ]	[ "Department of Chemistry\nGeorge Washington University\n20052WashingtonDCUSA" ]	[]	This communication reports on effective electrolyses in lithium--free molten carbonates. Processes that utilize solar thermal energy to drive efficient electrolyses are termed Solar Thermal Electrochemical Processes (STEP). Lithium--free molten carbonates, such as a sodium--potassium carbonate eutectic using an iridium anode, or a calcium--sodium--potassium carbonate eutectic using a nickel anode, can provide an effective medium for STEP electrolyses. Such electrolyses are useful in STEP carbon capture, and the production of staples including STEP fuel, iron, and cement. Recently, a series of carbon dioxide splitting and electrochemical syntheses in molten carbonates were introduced. 1--5 Unlike, the traditional synthesis of lime (from limestone) or iron (from hematite or magnetite) these staples are produced without emission of CO 2 . Additionally in molten carbontaes, carbon dioxide can be directly split to oxygen and carbon (at temperatures to ~800°C), or to carbon monoxide (at temperature above 800C°). Each of these electrolyses is endothermic, and requires less energy at higher temperature. Processes that utilize solar thermal energy to provide this energy and drive efficient electrolyses are termed Solar Thermal Electrochemical Processes (STEP). 6--7 Large natural reserves of lithium, sodium and potassium carbonates, Li 2 CO 3 , Na 2 CO 3 and K 2 CO 3 exist. Li 2 CO 3 melts are significantly more conductive than Na 2 CO 3 or K 2 CO 3 melts. 5 This high conductivity of lithium carbonate sustains high rates of electrolytic synthetic production (A/cm 2 ) at low overpotential. 5 However, the sodium and potassium salts are more prevalent and therefore a lithium--free molten carbonate could present economic advantages.5We have explored the general energetics of molten carbonate electrolysis, 1 and previously calculated the thermochemical energetics of lithium carbonate compared to sodium or potassium carbonate electrolyses as a function of temperature. 2 Here, effective experimental electrolysis is demonstrated in such lithium--free carbonate melts. Li 2 CO 3 has a lower melting point (mp 723°C), than Na 2 CO 3 (mp 851°C) or K 2 CO 3 (mp 891°C), but a mix of the salts has lower melting point. We have previously explored effective electrolyses in both the pure Li 2 CO 3 melt, and a Li 0.90 Na 0.62 K 0.48 CO 3 melt. Mixed alkali carbonate melting points can be low, including 399°C for this eutectic Li 0.90 Na 0.62 K 0.48 CO 3 mix, and 695°C for the Na 1.23 K 0.77 CO 3 eutectic salts. The addition of calcium carbonate can decrease the melting point of a carbonate mix. CaCO 3 , as aragonite, decomposes at 825 °C, and as calcite melts at 1339 °C. 8 A variety of molten carbonates have been characterized with, and without, added calcium carbonates. 9--At 750°C in molten, pure lithium carbonate (Li 2 CO 3 , Alfa Aeasar, 99%) electrolysis was conducted with a nickel anode as the oxygen electrode (pure Ni 200 McMaster 9707K59) and a coiled steel wire (14 gauge) as the cathode. Experimental details of this and related molten lithium carbonate cells have been extensively detailed. 3,6 A thin nickel oxide overlayer forms on the anode, which is subsequently highly stable towards oxygen evolution. 3 The electrolysis evolves oxygen from the anode and forms a thick black graphite layer on the cathode. The carbon is readily removed from the cathode, by extracting, cooling and uncoiling the cathode wire (and the carbon then falls off the wire). The cathode product is characterized by x--ray powder diffraction (XRD powder diffraction data were collected on a Rigaku Miniflex diffractometer) and analyzed as graphite with the Jade software package. 14 The cathode wire can be used repeatedly, and does not exhibit signs of corrosion and remains the same diameter during repeated, and/or extended, electrolyses. The lithium 750°C molten carbonate cell sustains electrolysis potentials of less then 2 V at anodic current densities in excess of 100 mA cm 2 , and this sustained electrolysis potential decreases to less than 1.0 V when high concentrations of Li 2 O, such as 3.7 molal, are added to the electrolyte. 3 When calcium carbonate is added to the molten lithium carbonate, the electrolysis products are not only oxygen and reduced carbon, but also lime (calcium oxide) in accord with: below 800°C: CaCO 3 → CaO↓ + C + O 2 (1) above 800°C: CaCO 3 →CaO↓ +CO+1/2O 2 (2)	null	[ "https://export.arxiv.org/pdf/1209.3512v1.pdf" ]	97,151,285	1209.3512	cedd164778f39cf1692c4c8cca85ce045f5fe406	Stabilization of STEP electrolyses in lithium--free molten carbonates Stuart Licht Department of Chemistry George Washington University 20052WashingtonDCUSA Stabilization of STEP electrolyses in lithium--free molten carbonates 1 This communication reports on effective electrolyses in lithium--free molten carbonates. Processes that utilize solar thermal energy to drive efficient electrolyses are termed Solar Thermal Electrochemical Processes (STEP). Lithium--free molten carbonates, such as a sodium--potassium carbonate eutectic using an iridium anode, or a calcium--sodium--potassium carbonate eutectic using a nickel anode, can provide an effective medium for STEP electrolyses. Such electrolyses are useful in STEP carbon capture, and the production of staples including STEP fuel, iron, and cement. Recently, a series of carbon dioxide splitting and electrochemical syntheses in molten carbonates were introduced. 1--5 Unlike, the traditional synthesis of lime (from limestone) or iron (from hematite or magnetite) these staples are produced without emission of CO 2 . Additionally in molten carbontaes, carbon dioxide can be directly split to oxygen and carbon (at temperatures to ~800°C), or to carbon monoxide (at temperature above 800C°). Each of these electrolyses is endothermic, and requires less energy at higher temperature. Processes that utilize solar thermal energy to provide this energy and drive efficient electrolyses are termed Solar Thermal Electrochemical Processes (STEP). 6--7 Large natural reserves of lithium, sodium and potassium carbonates, Li 2 CO 3 , Na 2 CO 3 and K 2 CO 3 exist. Li 2 CO 3 melts are significantly more conductive than Na 2 CO 3 or K 2 CO 3 melts. 5 This high conductivity of lithium carbonate sustains high rates of electrolytic synthetic production (A/cm 2 ) at low overpotential. 5 However, the sodium and potassium salts are more prevalent and therefore a lithium--free molten carbonate could present economic advantages.5We have explored the general energetics of molten carbonate electrolysis, 1 and previously calculated the thermochemical energetics of lithium carbonate compared to sodium or potassium carbonate electrolyses as a function of temperature. 2 Here, effective experimental electrolysis is demonstrated in such lithium--free carbonate melts. Li 2 CO 3 has a lower melting point (mp 723°C), than Na 2 CO 3 (mp 851°C) or K 2 CO 3 (mp 891°C), but a mix of the salts has lower melting point. We have previously explored effective electrolyses in both the pure Li 2 CO 3 melt, and a Li 0.90 Na 0.62 K 0.48 CO 3 melt. Mixed alkali carbonate melting points can be low, including 399°C for this eutectic Li 0.90 Na 0.62 K 0.48 CO 3 mix, and 695°C for the Na 1.23 K 0.77 CO 3 eutectic salts. The addition of calcium carbonate can decrease the melting point of a carbonate mix. CaCO 3 , as aragonite, decomposes at 825 °C, and as calcite melts at 1339 °C. 8 A variety of molten carbonates have been characterized with, and without, added calcium carbonates. 9--At 750°C in molten, pure lithium carbonate (Li 2 CO 3 , Alfa Aeasar, 99%) electrolysis was conducted with a nickel anode as the oxygen electrode (pure Ni 200 McMaster 9707K59) and a coiled steel wire (14 gauge) as the cathode. Experimental details of this and related molten lithium carbonate cells have been extensively detailed. 3,6 A thin nickel oxide overlayer forms on the anode, which is subsequently highly stable towards oxygen evolution. 3 The electrolysis evolves oxygen from the anode and forms a thick black graphite layer on the cathode. The carbon is readily removed from the cathode, by extracting, cooling and uncoiling the cathode wire (and the carbon then falls off the wire). The cathode product is characterized by x--ray powder diffraction (XRD powder diffraction data were collected on a Rigaku Miniflex diffractometer) and analyzed as graphite with the Jade software package. 14 The cathode wire can be used repeatedly, and does not exhibit signs of corrosion and remains the same diameter during repeated, and/or extended, electrolyses. The lithium 750°C molten carbonate cell sustains electrolysis potentials of less then 2 V at anodic current densities in excess of 100 mA cm 2 , and this sustained electrolysis potential decreases to less than 1.0 V when high concentrations of Li 2 O, such as 3.7 molal, are added to the electrolyte. 3 When calcium carbonate is added to the molten lithium carbonate, the electrolysis products are not only oxygen and reduced carbon, but also lime (calcium oxide) in accord with: below 800°C: CaCO 3 → CaO↓ + C + O 2 (1) above 800°C: CaCO 3 →CaO↓ +CO+1/2O 2 (2) We have explored the general energetics of molten carbonate electrolysis, 1 and previously calculated the thermochemical energetics of lithium carbonate compared to sodium or potassium carbonate electrolyses as a function of temperature. 2 Here, effective experimental electrolysis is demonstrated in such lithium--free carbonate melts. Li 2 CO 3 has a lower melting point (mp 723°C), than Na 2 CO 3 (mp 851°C) or K 2 CO 3 (mp 891°C), but a mix of the salts has lower melting point. We have previously explored effective electrolyses in both the pure Li 2 CO 3 melt, and a Li 0.90 Na 0.62 K 0.48 CO 3 melt. Mixed alkali carbonate melting points can be low, including 399°C for this eutectic Li 0.90 Na 0.62 K 0.48 CO 3 mix, and 695°C for the Na 1.23 K 0.77 CO 3 eutectic salts. The addition of calcium carbonate can decrease the melting point of a carbonate mix. CaCO 3 , as aragonite, decomposes at 825 °C, and as calcite melts at 1339 °C. 8 A variety of molten carbonates have been characterized with, and without, added calcium carbonates. 9--At 750°C in molten, pure lithium carbonate (Li 2 CO 3 , Alfa Aeasar, 99%) electrolysis was conducted with a nickel anode as the oxygen electrode (pure Ni 200 McMaster 9707K59) and a coiled steel wire (14 gauge) as the cathode. Experimental details of this and related molten lithium carbonate cells have been extensively detailed. 3,6 A thin nickel oxide overlayer forms on the anode, which is subsequently highly stable towards oxygen evolution. 3 The electrolysis evolves oxygen from the anode and forms a thick black graphite layer on the cathode. The carbon is readily removed from the cathode, by extracting, cooling and uncoiling the cathode wire (and the carbon then falls off the wire). The cathode product is characterized by x--ray powder diffraction (XRD powder diffraction data were collected on a Rigaku Miniflex diffractometer) and analyzed as graphite with the Jade software package. 14 The cathode wire can be used repeatedly, and does not exhibit signs of corrosion and remains the same diameter during repeated, and/or extended, electrolyses. The lithium 750°C molten carbonate cell sustains electrolysis potentials of less then 2 V at anodic current densities in excess of 100 mA cm 2 , and this sustained electrolysis potential decreases to less than 1.0 V when high concentrations of Li 2 O, such as 3.7 molal, are added to the electrolyte. 3 When calcium carbonate is added to the molten lithium carbonate, the electrolysis products are not only oxygen and reduced carbon, but also lime (calcium oxide) in accord with: below 800°C: CaCO 3 → CaO↓ + C + O 2 (1) above 800°C: CaCO 3 →CaO↓ +CO+1/2O 2 (2) CaO is highly insoluble in molten Li 2 CO 3 more dense, and precipitates during the electrolysis. As highly soluble limestone (CaCO 3 ) is continuously fed into the chamber during the electrolysis, lime, carbon and oxygen are produced and the electrolyte is unchanged. 6 Unlike calcium oxide, lithium oxide, Li 2 O, is highly soluble in lithium carbonate. The solubility increases from 9 molal Li 2 O in 750°C Li 2 CO 3 to over 1 molal Li 2 O at 950°C. During electrolysis in Li 2 CO 3 , Li 2 O does not precipitate, but reacts with CO 2 to continuously reform the Li 2 CO 3 and leave the electrolyte unchanged, such as in the net of equations 3 and 4: below 800°C: Li 2 CO 3 → Li 2 O + C + O 2 (3) Li 2 O + CO 2 →Li 2 CO 3 (4) net: CO 2 → C + O 2(5) The carbon dioxide electrolysis potentials further decrease to less than 1.2 V (without added Li 2 O) and less than 0.9 V (with added Li 2 O) when iron, rather than carbon becomes the electrolysis product at the cathode (that is when 3.5m ferric oxide is also added to the molten electrolyte). Cathodic overpotentials increase the iron formation electrolysis potential to above 1V, but higher temperature, or higher concentration of ferric oxide decrease the electrolysis potential (to below 0.6 V for a 10 m Fe(III) dissolved in molten Li 2 CO 3 electrolyte at 950°C. 3 These observe experimental electrolysis potential trends are generally consistent with our thermodynamic electrolysis potential trends as calculated from the free energy of formation of the constituents. Hence the calculated electrolysis potentials of the iron oxide splitting are less than that of the carbon dioxide splitting, Both decrease with increasing temperature, and the Nernst concentration corrected iron oxide splitting potential decreases with increasing ferric concentration. 1,3,15,16 For the STEP iron process, we had previously determined that Fe 2 O 3 dissolves in lithium carbonate as the lithiated salt via: 2 Fe 2 O 3 + Li 2 O → 2LiFeO 2(6) In the process of electrolysis to form iron metal, Li 2 O is liberated: 2LiFeO 2 → 2Fe + Li 2 O + 3/2O 2 (7) The liberated Li 2 O provides a path for the continued dissolution of Fe 2 O 3 , and overall the iron electrolysis process occurs as the combination of eqs 6 and 7: net: Fe 2 O 3 → 2Fe + 3/2O 2(8) We have also demonstrated that stable carbon, calcium oxide, and iron production also occurs in a diminished lithium concentration carbonate melt. In these cases, a significant fraction of the lithium carbonate is replaced by sodium and potassium carbonates. These electrolyses have been demonstrated in the Li 0.90 Na 0.62 K 0.48 CO 3 melt over a wide range of temperature and concentration conditions. 5,6 Silicates and aluminates occur in common iron ore and limestone deposits. We have also demonstrated that stable electrolysis occurs in molten electrolytes in which a fraction of the lithium carbonate has been replaced by silicate or aluminate salts. 3 In this communication, unlike the lithium carbonate melt, under similar conditions, but when the electrolysis is conducted instead in a Na 0.23 K 0.77 CO 3 melt (without lithium carbonate and without added oxides) stable carbon dioxide splitting is not observed. Carbon may form, but tends to falls from the cathode due to excessive corrosion. Specifically, a 30 cm 2 surface area nickel foil anode is used as the oxygen electrode (pure Ni 200 McMaster 9707K59) and a 7.0 cm 2 surface area cathode is prepared by coiling a 18.3 cm long, 1.2 mm diameter steel wire. Anode and cathode are immersed in 50g of molten Na 0.23 K 0.77 CO 3 at 750°C. Electrolysis is conducted at a constant current of 0.5A for five hours, requiring an electrolysis potential of 2.5+0.1 V. In addition to not exhibiting a carbon product, the cathode wire also exhibits signs of corrosion during the (Na 0.23 K 0.77 CO 3 electrolysis, decreasing in diameter from 1.22 to 1.04 mm, as measured subsequent to the five hour electrolysis). While highly stable in lithium carbonate electrolytes, nickel anodes tend to corrode into molten sodium and potassium carbonate mix, observable as a green coloration developing in the electrolyte during extended electrolyses. During the electrolysis, nickel oxide dissolves into the electrolyte as evidenced by x-ray powder diffraction of the electrolyte, subsequent to the electrolysis, which exhibits nickel oxide, and of the cathode product which contains nickel metal (presumably as reduced from the dissolved nickel oxide in the electrolyte). We have shown that iridium is a stable anode material in lithium carbonate melts. Iridium as an oxygen electrode exhibits a similar, low overpotential, and sustains similar high (A/cm 2 ) anodic current densities as those observed at nickel anodes. 3 In addition, we report here that iridium is more stable than nickel in a lithium--free Na 1.23 K 0.77 CO 3 melt. The prepared iridium anode had a surface area of 3.7 cm 2 and consists of a coiled 22.3 cm long, 0.53 mm diameter iridium wire (0.50 mm, 99.8%, Alfa Aeasar), and is immersed with the 7.0 cm 2 area coiled steel wire cathode. The anode exhibits no change in thickness at 750°C during the course of repeated five hour electrolyses at a constant current 0.5 A electrolysis, and the electrolysis potential is stable between 2.3 to 2.5 V during the electrolysis. Added calcium salts can decrease metal solubility and corrosion in molten sodium and potassium salt electrolytes. In molten CaCl 2 it was previously found that NiO solubility decreased with up to 4 mol% CaO concentration, and Ni coated with NiO had much higher stability during anodic polarization. 17 Molten carbonate electrolytic synthesis operates in the reverse mode of molten carbonate fuel cells (MCFC); where rather than fuel injection with electricity as a product, electrical energy is supplied and energetic chemical products are generated. MCFC systems have been studied in greater depth than carbonate electrolysis systems. Ni is a useful cell or electrode candidate material in MCFCs, but slowly degrades via a soluble nickel oxide overlayer. Cassir et al. report the of addition of 10% CaCO 3 to 650 °C Li 1.04 Na 0.96 CaCO 3 is useful to decrease the solubility of NiO from 150 to 100 µmolal in the carbonate mix. 18 A more cost effective solution to the corrosivity of the sodium--potassium STEP carbonate melt (than the use of iridium electrodes) is found by the addition of calcium carbonate or barium salts to the sodium-potassium, lithium--free, carbonate melt (the addition of calcium carbonate is demonstrated here). The addition of calcium carbonate can decrease the melting point of a carbonate mix. The sodium/lithium carbonate mix, Li 1.07 Na 0.93 CO 3 , has a melting point of 499°C, but decreases to below 450°C if 2 to 10 mol% equimolar CaCO 3 and BaCO 3 is added. 19 In addition to the sodium--potassium carbonate electrolytes, electrolyses are also conducted here in calcium--sodium--potassium electrolytes ranging up to a calcium fraction of Ca 0.27 Na 0.70 K 0.75 . Electrodes used are presented in Figure 1. A nickel oxygen anode appears to be fully stable during extended (five hour) 0.5 A electrolyses at 750°C in this melt, using the 30 cm 2 nickel foil anode and a 7.0 cm 2 steel wire cathode, and the electrolysis proceeds at between 1.9 to 2.2V. Unlike the electrolyses conducted in the calcium free (sodium--potassium) carbonate melt, carbon forms and remains on the cathode during electrolysis, and the steel cathode remains the same diameter, as measured subsequent to the electrolysis. As shown subsequent to the electrolyses, in the cathode photographs at the bottom of Figure 1, electrolyses conducted in either Ca 0.16 Na 1.03 K 0.65 or Ca 0.27 Na 0.70 K 0.75 electrolytes exhibit a thick carbon product on the cathode, while this is not the case following electrolysis without calcium carbonate in Na 1.23 K 0.77 CO 3 . The electrolysis potential and subsequent cathode product, during a repeat of the Ca 0.27 Na 0.70 K 0.75 electrolysis, but at a constant electrolysis current of 1A, rather than 0.5A, is presented in Figure 2. The observed increased the stability of the electrolysis in the lithium free (sodium--potassium) carbonate electrolyte containing calcium carbonate may be related to the low solubility of calcium oxide, compared to sodium oxide in these electrolytes. Whereas we find that sodium oxide, Na 2 O, is fully miscible in molten sodium--potassium carbonates. Calcium oxide is nearly insoluble (solubility less than 0.1 molal) from 750 to 950°C, in Na 1.23 K 0.77 CO 3 or Ca 0.27 Na 0.70 K 0.75 . In general the strength of an oxide in melts to donate an electron pair (Lewis basicity) decreases in the order: K 2 O > Na 2 O > Li 2 O > BaO > CaO > MgO > Fe 2 O 3 > Al 2 O 3 > TiO 2 > B 2 O 3 > SiO 2 > P 2 O 5 . 20 The lower basicity, combined with the lower concentration, of the oxide in the calcium containing carbonate electrolyte contribute to the observed decrease in corrosivity and improved electrolysis. Figure 1 . 1Top: Cathode (top left) and anode (top right) prior to 0.5 A, 5 hour lithium--free electrolyses at 750°C with increasing calcium carbonate concentration. The cathode is placed inside the anode, which are both immersed in the molten electrolyte. Bottom: Cathodes after electrolysis in lithium--free molten carbonates. Electrolytes used were respectively: Na 1.23 K 0.77 CO 3 (lower left cathode), Ca 0.16 Na 1.03 K 0.65 (lower middle cathode), and Ca 0.27 Na 0.70 K 0.75 (lower right cathode). Figure 2 . 2Time variation of the electrolysis potential during a five hour electrolysis at 1A in Ca 0.27 Na 0.70 K 0.75 at 750°C. Inset: cathode subsequent to the electrolysis. Acknowledgement. The United States National Science Foundation and the George Washington University provided partial funding for this study. Doctoral graduate student Ulyana Cubeta, who is no longer pursuing this project, participated in the collection of this data. These results will be included in a broader publication related to this topic. STEP (solar thermal electrochemical photo) generation of energetic molecules: A solar chemical process to end anthropogenic global warming. S Licht, J. Phys. Chem., C. 113Licht, S. STEP (solar thermal electrochemical photo) generation of energetic molecules: A solar chemical process to end anthropogenic global warming. J. Phys. Chem., C 2009, 113, 16283--16292. A New Solar Carbon Capture Process: Solar Thermal Electrochemical Photo (STEP) Carbon Capture. 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[ "Resonance structures in coupled two-component φ 4 model", "Resonance structures in coupled two-component φ 4 model" ]	[ "A Halavanau \nDepartment of Theoretical Physics and Astrophysics\nBSU\nMinskBelarus\n", "T Romanczukiewicz \nInstitute of Physics\nJagiellonian University\nKrakowPoland\n", "Ya Shnir \nDepartment of Theoretical Physics and Astrophysics\nBSU\nMinskBelarus\n\nInstitute of Physics\nCarl von Ossietzky University\nOldenburgGermany\n" ]	[ "Department of Theoretical Physics and Astrophysics\nBSU\nMinskBelarus", "Institute of Physics\nJagiellonian University\nKrakowPoland", "Department of Theoretical Physics and Astrophysics\nBSU\nMinskBelarus", "Institute of Physics\nCarl von Ossietzky University\nOldenburgGermany" ]	[]	We present a numerical study of the process of the kink-antikink collisions in the coupled onedimensional two-component φ 4 model. Our results reveal two different soliton solutions which represent double kink configuration and kink-non-topological soliton (lump) bound state. Collision of these solitons leads to very reach resonance structure which is related to reversible energy exchange between the kinks, non-topological solitons and the internal vibrational modes. Various channels of the collisions are discussed, it is shown there is a new type of self-similar fractal structure which appears in the collisions of the relativistic kinks, there the width of the resonance windows increases with the increase of the impact velocity. An analytical approximation scheme is discussed in the limit of the perturbative coupling between the sectors. Considering the spectrum of linear fluctuations around the solitons we found that the double kink configuration is unstable if the coupling constant between the sectors is negative. arXiv:1206.4471v1 [hep-th]	10.1103/physrevd.86.085027	[ "https://arxiv.org/pdf/1206.4471v1.pdf" ]	118,713,130	1206.4471	be7d88cca5e8fd65f9e697fee3c123606b141619	Resonance structures in coupled two-component φ 4 model A Halavanau Department of Theoretical Physics and Astrophysics BSU MinskBelarus T Romanczukiewicz Institute of Physics Jagiellonian University KrakowPoland Ya Shnir Department of Theoretical Physics and Astrophysics BSU MinskBelarus Institute of Physics Carl von Ossietzky University OldenburgGermany Resonance structures in coupled two-component φ 4 model We present a numerical study of the process of the kink-antikink collisions in the coupled onedimensional two-component φ 4 model. Our results reveal two different soliton solutions which represent double kink configuration and kink-non-topological soliton (lump) bound state. Collision of these solitons leads to very reach resonance structure which is related to reversible energy exchange between the kinks, non-topological solitons and the internal vibrational modes. Various channels of the collisions are discussed, it is shown there is a new type of self-similar fractal structure which appears in the collisions of the relativistic kinks, there the width of the resonance windows increases with the increase of the impact velocity. An analytical approximation scheme is discussed in the limit of the perturbative coupling between the sectors. Considering the spectrum of linear fluctuations around the solitons we found that the double kink configuration is unstable if the coupling constant between the sectors is negative. arXiv:1206.4471v1 [hep-th] I. INTRODUCTION Since the early 1960s, the soliton solutions in the non-linear field theories have been intensively studied in various frameworks. It is evident these spatially localized non-perturbative configurations play important role in a wide variety of physical systems. The study of the interaction between the solitons and their dynamical properties has attracted a lot of attention in many different contexts. Simplest example of the topological solitons in one dimension is the class of the kink (K) solutions which appears in the model with a potential with two or more degenerated minima. Double well potential corresponds to the nonintegrable φ 4 model. Further extentions of this model are possible, for example one can consider domain walls obtained by embedding of the 1+1-dimensional φ 4 -kink into higher dimensions. This model has a number of applications in condensed matter physics [1], field theory [2,3] and cosmology [4]. Probably one of the most fascinating developments in this direction is the idea that one may regard our universe as a domain wall [5,6] and Big Bang is associated with the collision of the domain walls [7]. Further, in various branches of physics more complex systems involving several coupled scalar fields arise (see, e.q. [8][9][10]). Properties of the solitons in these extended models are rather different from being a trivial extension of the single-component one, in particular the model may support existence of non-topological solitons [8] or it can be related with application of the supersymmetry in nonrelativistic quantum mechanics [11]. However, relatively little is known about the dynamics of the multicomponent configurations in the extended non-integrable models. Dynamical properties of the usual φ 4 kinks, the processes of their scattering, radiation and annihilation have already been discussed in a number of papers, see e.g. [12][13][14][15][16][17][18][19]. Surprisely, it was discovered numerically that there is dynamical fractal structure in the process of the kink-antikink (KK) collision [14,15,18,19]. Depending on the impact velocity, the collision may produce various results, for some range of values of the impact velocity the KK pair form an oscillating n-bouncing window, after collision the pair reflects back to some finite separation and returns to collide (n − 1) times more before escaping to infinity. Also the process of production of kink-antikink pairs in the collision of particle-like states is chaotic [20]. Another interesting phenomenon occurs when the kink configuration is under the influence of an incident wave [21], it was found that the φ 4 kink starts to accelerate towards the incoming wave although the effect depends on the structure of the potential. The explanation of the appearance of the resonant scattering windows in the KK collision is related to the energy exchange between the translational collective mode of the solitons and an internal vibrational mode of the kink [14,15,18,19]. Another mechanism works in the φ 6 model where the resonance windows appear due to energy transfer between collective bound states trapped by the KK pair [22]. Evidently, in the extended multicomponent model coupling to the second component may strongly affect the process of the resonant energy transfer. A natural extension of the simple model φ 4 model in 1+1 dimension is related to consideration of systems with two or more component scalar fields. Several models of that type have been investigated in recent years, some of them are related to consideration of the complex scalar field, a deformation of the linear O(2)-sigma model known as MSTB-model [23][24][25] or with consideration of a two-component non-linear interaction model [8,9,[26][27][28]. In particular, this model supports existence of solutions with one of the component having the kink structure and the second component being a non-topological soliton. Another situation occurs when the system of coupled scalar fields belongs to the bosonic sector of a supersymmetric system (Wess-Zumino model with polynomic superpotential and two Majorana spinor fields) [29], or with application of supersymmetric methods [11] and related reduction of the second-order field equations to the system of corresponding first-order Bogomol'nyi equations [30,31]. The aim of the present paper is to analyse the process of the KK resonant bouncing scattering in the two-component coupled φ 4 model. We assume that both sectors are two replica of the usual φ 4 model, the fields are coupled through the minimal quadratic term and the kink-antikink collisions are taking place in the first sector. Evidently the fine mechanism of the reversible energy transfer in the bouncing of the kink will be modified in such a model because both the vacuum structure and the spectrum of linear oscillations around the solitons will be different from the standard one-component φ 4 theory. This present work is organized in the following way. In Section II we overview the spectrum of soliton solutions of the coupled two-component φ 4 model and discuss the analytical perturbative approximation to these configurations in both sectors. In section III we present our numerical results of the various kink-antikink (KK) collisions and discuss the resonance structures we observed. In section IV we describe the spectral structure of linear pertubations on the background of the soliton solutions of this model and discuss the possible mechanisms of the resonance energy transfer between the internal modes and the kinks. We present our conclusions in section V. II. TWO-COMPONENT COUPLED φ 4 MODEL We consider the system of two coupled copies of the φ 4 fields L = 1 2 ∂ µ φ 1 ∂ µ φ 1 + 1 2 ∂ µ φ 2 ∂ µ φ 2 − 1 2 φ 2 1 − 1 2 − 1 2 φ 2 2 − a 2 2 + κφ 2 1 φ 2 2 .(1) where a is associated with the mass of the field φ 2 and κ is the coupling parameter [38]. Thus, the potential of the model is U (φ 1 , φ 2 ) = 1 2 φ 2 1 − 1 2 + 1 2 φ 2 2 − a 2 2 − κφ 2 1 φ 2 2 (2) where κ is the coupling constant. Hereafter, for the sake of simplicity we set a = 1. Let us consider the vacuum manifold of the model which is defined by the conditions φ 1,t = φ 2,t = φ 1,x = φ 2,x = 0, U (φ 1 , φ 2 ) = min,(3) that is 2φ 1 (φ 2 1 − 1 − κφ 2 2 ) = 0,(4a)2φ 2 (φ 2 2 − 1 − κφ 2 1 ) = 0. (4b) Evidently, the structure of the vacuum depends on the value of the coupling constant κ. In the case when −1 < κ < 1 there are nine stationary points of the potential (2): • one local maximum φ 1 = φ 2 = 0 (U max = 1), • four saddle points φ 1 = 0, φ 2 = ±1 or φ 1 = ±1, φ 2 = 0 (U saddle = 1 2 ), • four minima φ 1 = ±1/ √ 1 − κ, φ 2 = ±1/ √ 1 − κ (U min = − κ 1−κ ) . If κ > 1 the system becomes unbounded from below, that is it is unstable, so we shall not consider this case. In the limiting case κ = −1 the minima of the potential (2) and its saddle points are degenerated while in the regime κ < −1 the minima appear at the four points φ 1 = 0, φ 2 = ±1 or φ 1 = ±1, φ 2 = 0, which previously, when −1 < κ < 1, were associated with the saddle points. Correspondingly, the former minima become the saddle points in this case, see Fig 1. Thus, there are kink soliton solutions in this model which interpolate between one of the vacua at x = −∞ and another one at x = +∞. To find these configurations explicitly we have to solve the Euler-Lagrange equations of motion, which are the following coupled, nonlinear, partial differential equations: Let us consider \|κ\| < 1. Then a particular static kink solutions interpolate between two of four vacua: ∂ tt φ 1 − ∂ xx φ 1 + 2φ 1 (φ 2 1 − 1 − κφ 2 2 ) = 0, ∂ tt φ 2 − ∂ xx φ 2 + 2φ 2 (φ 2 2 − 1 − κφ 2 1 ) = 0.(5)(±1/ √ 1 − κ, ±1/ √ 1 − κ) . So there are 2 4 2 = 12 types of the (anti)kinks in the model, both topological and nontopological. However the set of discrete symmetries of the Lagrangian (1): x → −x, (6a) (φ 1 , φ 2 ) → (−φ 1 , φ 2 ), (6b) (φ 1 , φ 2 ) → (φ 1 , −φ 2 ), (6c) (φ 1 , φ 2 ) → (φ 2 , φ 1 ). (6d) actually reduces the number of significantly different solitons to two. Indeed, the solitons can be classified according to their topological charge Q i = 1 2 √ 1 − κ ∞ −∞ dx ∂φ i ∂x , i = 1, 2.(7) For each component it can have three values {−1, 0, 1}, kink soliton has topological charge +1 while antikink has topological charge −1. Thus, the configuration can be labeled by the topological charges of the components as (n 1 , n 2 ) where n i ∈ {−1, 0, 1}. Let us consider the most symmetric case φ 2 = ±φ 1 = φ. Then the system of coupled equations (5) becomes reduced to equation ∂ tt φ − ∂ xx φ + 2φ((1 − κ)φ 2 − 1) = 0.(8) Evidently, rescaling of the field φ →φ = φ √ 1 − κ yields the familiar equation of motion of the usual one-component φ 4 theory. So the model (1) in this case is reduced to the two separable copies of the φ 4 model. Static solutions (∂ t φ = 0) of the system (5) can therefore be written as φ 1 φ 2 = ± tanh x √ 1 − κ 1 ±1 .(9) The static kink solution interpolates between the vacua −1/ √ 1 − κ and 1/ √ 1 − κ as x increases from −∞ to ∞. Similarly, the antikink solution interpolates between the stable vacua 1/ √ 1 − κ and −1/ √ 1 − κ as x varies from −∞ to ∞. The corresponding 2-component configuration possess non-zero topological charge in both sectors, so there are four kinks which correspond to the configurations with topological charges (1, 1), (−1, 1), (1, −1) and (−1, −1), respectively. We shall refer to this solution as double kink. The perturbative sector of the model consists of small linear perturbations around one of the solutions, thus by analogy with the usual one-component φ 4 model, one can expect the process of collision between the solitons is related to reversible exchange of energy between the translational and vibrational modes of the individual kinks [14,15,18,19]. However in our case there is another possibility related to energy exchange between the sectors, so the dynamics of the kinks and the spectral structure of the perturbations can be rather interesting. Besides the double kink configuration, there is another solution to the model (1) which corresponds to the topological soliton in one of the sectors and non-topological soliton (lump) in another sector. We shall refer to this solution as lump kinks. In such two-component system the field of the topological soliton in one of the sectors interpolates between the vacua ∓1/ √ 1 − κ and ±1/ √ 1 − κ as before, while the non-topological soliton belongs to the sector with topological charge 0, each of these lump solitons can be settled in one of two vacuum states, so there are 8 such configurations labeled by the topological charges (1, 0), (−1, 0), (0, 1) and (0, −1). For example, the static lump kink solution having first component interpolating between the vacua −1/ √ 1 − κ and 1/ √ 1 − κ and the second component being the non-topological soliton about one of the vacua ±1/ √ 1 − κ is presented in Fig. 2. Other static lump (anti)kinks can be obtained from this solution by using the discrete symmetries of the model (6). Qualitatively, one can understand the physical reason of the lump kink solitons existence, if we consider the kink configuration in one of the sectors, in the configuration space (φ 1 , φ 2 ) it tends to evolve from one vacuum to another through the saddle point (φ 1 (0) = 0, φ 2 (0) = 1). Thus, non-topological soliton in the second sector appears. However the resulting configuration does not reach this point exactly because of the field derivatives contributions in the total energy. Note that although the general structure of the solution is similar to the configuration considered in [8], there is no analytical solution of the model (1), the kink component there is rather different from the solution of the usual φ 4 model. In the case under consideration the topological solitons can be treated as "stretched out" φ 4 kinks which are coupled to the lumps in the second sector. These configurations can be constructed numerically as solutions of the of static equations (5) when we impose the boundary conditions φ 1 (0) = 0, φ 1 (∞) = 1/ √ 1 − κ, φ 2 (0) = 0, φ 2 (∞) = 1/ √ 1 − κ.(10) Since the coupling constant \|κ\| < 1, we can construct an analytical approximation of the lump kink solution considering the coupling between the components perturbatively. The initial decoupled static configuration consist of a kink in the first sector of the model (1) and the trivial vacuum solution in the second sector: φ (0) 1 = tanh(x); φ (0) 2 = 1.(11) Corrections to this configuration can be obtained from the series expansion of the fields φ 1 , φ 2 in powers of perturbation constant [21,33] φ 1 = φ (0) 1 + κφ(1)1 + κ 2 φ(2)1 + . . . ; φ 2 = φ (0) 2 + κφ(1)2 + κ 2 φ(2) 2 + . . . . The first-order time-independent corrections to the initial solution (11) then can be obtained from the equations − φ (1) 1 xx + 6 tanh 2 x − 2 φ (1) 1 − 2 tanh x = 0, − φ (1) 2 xx + 4φ (1) 2 − 2 tanh 2 x = 0.(13) Evidently, as x → ∞ the first order correction φ (1) i → 1 2 supposing that on the spacial asymptotic φ (1) i xx → 0. This is in agreement with the vacuum shift due to perturbative coupling in both sectors 1/ √ 1 − κ ≈ 1 + 1 2 κ. Further, we can try to construct an analytical approximation to the first order correction using an expansion in hyperbolic functions φ (1) 1 = ∞ n=0 A n tanh 2n+1 x, φ (1) 2 = ∞ n=0 B n cosh 2n x .(14) Substitution of these series into the first order equations (13) yields φ (1) 1 = 16 15 tanh x − 1 3 tanh 3 x − 4 75 tanh 5 x − 1 75 tanh 7 x + · · · , φ(1)2 = 1 2 − 1 6 cosh 2 x − 1 28 cosh 4 x − 5 728 cosh 6 x − 45 11284 cosh 8 x + · · ·(15) Similar approximation can be applied to the higher order corrections. Note that the perturbative expansion reveals instability of the double kink configuration in the case κ < 0, the system may decay into a pair of lump kinks. Indeed, the Figure 3 shows the masses of kinks as a function of coupling constant κ. Evidently, for negative values of the coupling κ the mass of a double kink is less than the mass of two lump kinks. That means that the double kink is less energetically favorable state than two separated lump kinks. Kinks in one-component φ 4 model are stable, therefore the perturbation which can destroy the double kink configuration must break the symmetry φ 1 = ±φ 2 . For positive κ the double kink is energetically stable. Moreover it is possible that the double kink could be a final state in collisions of two lump kinks with kinks in different sectors, the outcome evidently depends on the parameters of the model, the impact velocity and the coupling κ. Below we discuss various solitons collisions scenarios and classify them. III. KINK-ANTIKINK COLLISIONS IN TWO-COMPONENT φ 4 MODEL A phenomenon known as the two-bounce resonance soliton scattering in the usual one-dimensional φ 4 model was observed by many researchers in numerical simulations [14,15,18,19]. Unfortunately there is no complete analytic description for such process, the model is non-integrable and the dynamics of the solitons is highly complicated. Numerical simulations performed over last 20 years [14,15,18,19] discovered an interesting result: the process of collision between the solitons is of fractal nature, depending on the impact velocity, it may produce various results and for some range of values of the impact velocity the KK pair form a set of bouncing windows. Although semiclassical consideration [15,18,19] could capture the most important features of the process, the analysis entirely rely on extensive numerical computations. It is known that for initial velocities of a kink and an antikink above a critical value v cr = 0.2598 two incident solitons always escape to infinity after collision, with the emission of some radiation, thus the process is similar to quasielastic scattering of two particle-like objects [14,15,18,19]. For the impact velocities below v cr , the kink-antikink pair generically becomes trapped, it annihilates into radiation, however numerical analysis reveals the fractal structure with a sequence of narrow bouncing windows within which the solitons are again able to escape to infinity. The first of two-bounce window opens at the impact velocity v in = 0.189 then the resonance windows become more narrow as the impact velocity increases. If we look on the edge of a two-bounce window, we can observe a narrow sequence of the 3-bounce windows which are characterized by the same relations between the windows widths and the impact velocities (self-similar structure) [18]. These windows are called 3-bounce ones. Furthermore, on the edge of every 3-bounce window we can find the 4-bounce windows structure, etc. However, the fractal dimension of this dynamical structure of the collision of the solitons is restricted, so we refer to this to as a quasi-fractal structure. The accepted explanation of the appearance of these windows is that they are related to reversible exchange of energy between the translational and vibrational modes of the individual kinks. At the initial impact, some kinetic energy is transferred into internal shape modes of the kink and antikink. They then separate and propagate almost independently, but for initial velocities less than v cr they no longer have enough translational energy to escape their mutual attraction, and so they return and collide one more time [15,18]. At this point some of the energy stored in the shape modes can be returned to the translational modes, tipping the energy balance back again and allowing the kink and antikink to escape to infinity, provided that there is an appropriate resonance between the time interval between the two collisions, and the period of the internal modes. More generally, sufficient energy might be returned to the translational modes after three or more kink-antikink collisions, leading to an intricate nested structure of resonance windows. Thus, the resonance condition is that the time T between two consequent collisions should satisfy the equation [15,18] ω s T = δ + 2πn (16) where δ is a phase, and n is an integer [39]. It can be shown that 2π ωs = 5.13 and δ = 3.3(≈ π). The relation (16) can be modulated [37]. This mechanism allows us to construct an effective model of the collision process [15]. The idea is to separate collective coordinates of the kink-antikink system and make use of the Yukawa interaction between the well separated pair mediated by the meson state of mass m. Surprisingly enough, this model works rather well. Our goal is to analyse the process of collision of the soliton configurations in two-component model (1) numerically. The initial data used in our simulations represent widely separated kinks or lump kinks propagating from both sides towards a collision point, the parameters of the collision process are the impact velocity v and the coupling constant κ. To find a numerical solution of the PDE describing the evolution of the system, we used the pseudo-spectral method on a discrete grid containing 2048 nodes with periodic boundary conditions. For the time stepping function we used symplectic (or geometric) integrator of 8th order to ensure that the energy is conserved. The time and the spacial steps are δx = 0.25 and δt = 0.025, respectively, so the numerical errors are of order of (δt) 8 . Evidently, if the sectors of the two-component model (1) are decoupled, the pattern of collision reproduces the chaotic structure we described above. A. Lump kink collisions -same sectors Let us consider collision of two lump kinks (−1, 0) + (1, 0) at κ = 0.5. The numerical simulations reveal rather interesting pattern, for initial velocities of a lump kink and an lump antikink above a critical value v cr = 0.453, an intermediate state which is created in both sectors at the collision center, decays into lumps in the first sector and into kinks in the second one. Then two solitons escape to infinity, with the emission of some radiation, so the process is similar to quasielastic scattering of two φ 4 kinks up to the flip between sectors: (−1, 0) + (1, 0) → (0, −1) + (0, 1). This process is displayed at Fig. 4. For the impact velocities below v cr we the observe the resonance energy exchange between the sectors with a sequences of bouncing windows within which the solitons are again able to escape to infinity, as seen in Fig. 5. The first of two-bounce windows open at the impact velocity v in = 0.423, the consequent structure of the resonance windows is not so regular as in the case of the usual φ 4 model, also some false windows appear. Evidently, this can be related to more complicated structure of the spectrum of excitations, alongside with the internal mode of the kinks there are degrees of freedom associated with excitations of the lumps, also some collective degrees of freedom in the system kink-lump may appear. In all windows we observed the same permutation of the kink and the lump sectors in the final state. For smaller impact velocities and between the windows, the intermediate state which is created in both sectors at the collision center, decays into two oscillons (one in each sector) with some amount of radiation emitted. The final states of the process then are: (1, 0) + (−1, 0) →      (1, 0) + (−1, 0) bounce (I) (0, 1) + (0, −1) flip (II) (0, 0) annihilation into oscillon (III) . However for ultra-relativistic impact velocities (for example at v = 0.950, κ = 0.386, see Fig. 8), we also found another final state (1, 0) + (−1, 0) → (−1, −1) + (1, 1) (IV).(18) In this process relativistic collision of the lumps leads to the production of the kinks in the second sector. They bind to the kinks from the first sector and then the pairs travel as double kinks. In Fig. 7 we present the final vacuum state in the center of collision after a long time φ i (x = 0, t = 150) as a function of the parameters of the process of collision (1, 0) + (−1, 0), the value of the coupling constant κ and the Depending on the parameters of the collision process, we observed various scenario. If the final vacuum state remains the same after collision, the kinks are bounced back (cf. Fig. 8, panel (I), where we plotted positions of the topological zeros for that collision). On the upper left plot in Fig. 7 the bouncing corresponds to the regions colored in blue. The largest of these regions has a shape of a triangle at κ = 0, note that the bouncing is also observed for some set of higher impact velocities and relatively strong values of the coupling κ, both positive and negative. Black regions in Fig. 7 correspond to the final state [−1, 1], in other words the kink propagating on the left towards the spacial asymptotic x → −∞ after the collision now belongs to the sector with topological charge Q = The regions in Fig. 7 which are colored in red, correspond to final state of the configuration in the region between the collision center and the asymptotic x → −∞ with total topological charge equal zero. This is the process of KK annihilation which dominates at relatively small initial velocities (cf. Fig. 8, panel (III)). Finally, we observed that for the relativistic impact velocities and relatively large positive values of the coupling constant κ the process of the production of the double kink is allowed as illustrated in Fig. 8, panel (IV). The corresponding regions belong to the sector with topological charge Q = (1, 1), in Fig. 7 they are colored in yellow. Note that the borders between those regions are very complicated, presumably they are demonstrating some kind of fractal structure. B. Lump kink collisions -different sectors Since the kink is complemented by a non-topological partner in the opposite sector, there is another process of collision between the kinks from different sectors, mediated by the lumps. Numerical simulations reveal that at relatively small velocities v < 0.565 the energy of collision between the kink and the lump is transformed into excitations of the kinks which are erratically oscillating around the collision center it the both sectors, see Fig. 6. Thus the resulting configuration is actually an excited double kink. It will be shown in the next section that the double kink for κ > 0 has at least two oscillational modes. One of them corresponds to the oscillational mode of the φ 4 kink and has the same frequency √ 3. There is also a mode which is responsible for translational oscillations of the kinks in both sectors around the center of mass. Both of these frequencies can be seen in the power spectrum of the field measured in the center of mass. Moreover due to nonlinearities combinations of those frequencies are also visible (cf. Fig. 9 ). For impact velocity v = 0.565 we observed the bouncing of the kinks, a narrow resonance window opens, a sequence of the bouncing windows of decreasing width follows it up to the v cr = 0.584. This resonance structure is much more regular than in the case of the collision of two lump kinks in the same sector, there is no false windows and the dependence of the windows on the impact velocity closely reminds the well known pattern observed in the usual one-component φ 4 theory [15,18]. Evidently we can consider it as undirect evidence of the simple mechanism of the resonance energy exchange between the kink and the lump which involves only the internal mode of the kink, there is no energy exchange between sectors. Above this threshold velocity the lump kinks are passing through the lumps associated with the kinks from the second sector, thus this is a regime of quasi-elastic collision, see Fig. 10. Variations of the parameters of the collision process, the coupling constant κ and the impact velocity v reveal several possible final states: (1, 0) + (0, −1) →      (1, −1) double kink creation (I) (0, −1) + (1, 0) passage (II) (1, 0) + (0, −1) bounce (III)(19) Note that the double kink production is allowed only for κ > 0, we already mentioned that for κ < 0 the double kinks are unstable with respect to decay into the lump kinks. All the processes decribed above are illustrated in Fig. 11 which presents the final state of the collision (1, 0) + (0, 1) associated to the kink on the left hand side from the collision center propagating towards the spacial asymptotic x → −∞ as Φ = [−1, −1] + 2Q. Similar to the Fig. 7, the black regions Φ = [−1, 1] correspond to the quasi-elastic scattering of the kinks (passage), in this case the topological charge of the scattered kink in the final state is Q = (1, 0) (cf. Fig. 11, (II)). Then the regions in Fig. 11 which are colored in blue, correspond to the bouncing of the kinks, then in the final state we have Q = (1, 0) (cf. Fig. 11, regions (III) and (IV)). Note that there is another possibility, as a result of the collision two kinks can merge forming an (excited) double kink. These processes are depicted as whitish, yellow and light blue colored areas on the right side of the plot (Fig. 11, (I)). C. Double kink -lump kink collisions Another interesting solitonic collision process in the two-component model (1) is the double kink -lump kink collision. Let us take the initial state as, for example (−1, −1) + (1, 0). Since the total topological charge in the first sector is 0 we could expect that the outcome of the collision, at least at small impact velocities, will be just annihilation of the lump kink and one of the components of the double kink. However, a single component of the double kink is not a solution of the model (1), so it has to deform into the lump kink then. Indeed, at relatively small impact velocities v < 0.76 we observed such a process (−1, −1) + (1, 0) → (0, −1) + (0, 0), besides radiation and oscillon remains in the first sector as the result of annihilation of the kink-antikink components. It is known that the resonance excitation of the oscillon can produce KK pairs [20], so at higher energies other scenarios are possible. Indeed at κ = 0.5, within the impact velocity range 0.76 < v < 0.82 we observed another resonance structure between 3 channels (−1, −1) + (1, 0) →      (0, −1) + (0, 0) (0, −1) + (0, 1) + (0, −1) (0, −1) + (1, 0) + (−1, 0) (20) The results of numerical simulations are presented in Fig. 13 where we plotted final velocity of the solitons vs the impact velocity. The bottom plot represents zoom in on the smaller subrange of the impact velocities in between, evidently this plot is qualitatively similar to the upper plot. Thus a new type of self-similar fractal structure appears. Note the width of the windows increases with the increasing of the impact velocity, so this structure is inverted with respect to the usual fractal structure observed in the φ 4 model [15,18]. The complicated fractal structure probably is possible due to the fact that the double kink has at least two bound modes which can couple to each other and to the oscillational mode of the lump kink. Since at the collision process the double kink behaves as a bound state of the lump kinks, this type of collision can be interpreted as a three body problem. Note that for κ < 0 the double kink is unstable, however the perturbation due to the lump kink is relatively weak, so the double kink is moving as a single object and it decays into the pair of lump kinks just before or already during the collision. Finally, as the impact velocity increases above the critical value v cr = 0.82, quasi-elastic scattering with flipping of the sectors is observed: (−1, −1) + (1, 0) → (+1, −1) + (−1, 0)(21) thus, the total topological charge in each sector remains the same but the kink component of the lump kink is getting captured into the double-kink state while the second component of the former configuration becomes the lump kink. In Fig. 14 we have gathered our results for the various channels of the double kink -lump kink collison (1, 1)+(−1, 0). Here we present the values of the fields at the center of collision, evidently the spectrum of the final states is much richer than in the case of the lump kink collisions we discussed above. The colors which represent the final state in Fig. 14, are similar to the palette we used in the Figs. 7,11 above. Note that since the double kink and the lump kink has different masses, the center of collision is not a center of mass of the system, so the dynamics becomes rather involved. For the sake of completeness in Fig. 15 we also presented various examples of the evolution of the topological zeros, associated with the position of the solitons in the process of the double kink-lump kink collision. IV. SPECTRAL STRUCTURE OF LINEAR PERTURBATIONS There is certain similarity between chaotic structures observed in the collision of the kinks in the two-component model (1) and KK resonance bouncing in the usual φ 4 model [15,18]. This analogy can be evidently seen in the collision of the lump kinks, however the chaotic structure of the collision between the double kink and the lump kink is much more complicated. To investigate this analogy in more details we have to make use of the linear expansion around the solitons of the model under consideration assuming perturbations of both components (φ 1 +η 1 e iωt , φ 2 +η 2 e iωt ), here the unperturbed configuration (φ 1 , φ 2 ) corresponds to one of the solutions of the model (1), the double kink or the lump kink. Linearization of the system (5) yields the matrix eigenvalues equation D 2 1 −4κφ 1 φ 2 −4κφ 1 φ 2 D 2 2 η 1 η 2 = ω 2 η 1 η 2(22) where the operators D 2 1 , D 2 2 are Note that in this section we do not consider the coupling constant κ as a perturbation parameter. D 2 1 = − d 2 dx 2 + 6φ 2 1 − 2 − 2κφ 2 2 ,(23a)D 2 1 = − d 2 dx 2 + 6φ 2 2 − 2 − 2κφ 2 1 .(23b) A. Spectral structure of double kink First, let us consider the spectrum of linear oscillations around the double kink solution. In this case φ 2 = ±φ 1 = ± tanh x √ 1−κ and it is possible to diagonalise the system (22) introducing new functions ξ 1 = 1 2 (η 1 + η 2 ) ; ξ 2 = 1 2 (η 1 − η 2 ) .(24) Here the variable ξ 1 describes the motion of the center of mass of the system whereas the variable ξ 2 corresponds to the oscillations of the components about it. Then the system of equations (22) is decoupled and we get − d 2 dx 2 + 6 tanh 2 x − 2 ξ 1 = ω 2 ξ 1 ,(25a)− d 2 dx 2 + 6 + 2κ 1 − κ tanh 2 x − 2 ξ 2 = ω 2 ξ 2 .(25b) Evidently, the first of these equations coincides with the usual modified Peashle-Teller equation that describes the linear excitations around the φ 4 kink, it is well known the spectrum of fluctuations in this case has one translational zero mode ω = 0, the internal mode of the kink ω 2 = 3, which corresponds to the oscillations of the width of the kink, and the continuum modes ω 2 = k 2 + 4: ξ (0) 1 = 1 cosh 2 x ; ξ (1) 1 = sinh x cosh 2 x ; ξ (k) 1 = e ikx (3 tanh 2 x − 3ik tanh x − 1 − k 2 ).(26) The second linearly independent solution is the complex conjugation of the above. In the case under consideration, for example the excitation of the translational mode ξ (0) 1 = 1 2 η (0) 1 + η (0) 2 corresponds to the synchronous translation of the double kink whereas the excitation of the internal mode ξ 1 corresponds to the synchronous wobbling of both components. The meaning of the second mode is different. Indeed, the second equation (25b) is of the same type, so it can be solved by reducing it to the hypergeometric form (see, e.g. [34]). Here, however, we shall follow another route. Let us introduce the ladder operatorsâ † = − d dx + n tanh x;â = d dx + n tanh x,(27) which allows us to transform the equation (25b) to the form â †â + n − 2 ξ 2 = ω 2 ξ 2(28) Here we introduced a new parameter n = − 1 2 + 1 2 25 + 7κ 1 − κ .(29) which is defined as a solution of the algebraic equation n(n + 1) = 6 + 2κ 1 − κ .(30) Evidently, the ground state is defined as the state annihilated by the operatorâ, i.e., aξ 2 = 0.(31) Thus, up to a normalisation factor, the solution to this equation is ξ (0) 2 = 1 cosh n x .(32) Here the integer n labels the internal modes of the double kink system, their number increases with coupling constant since κ(n) = n 2 + n − 6 n 2 + n + 2 . Thus, for −1 < κ < 0 there is only one bound state, for 0 < κ < 3/7 there are two internal modes, for 3/7 < κ < 7/11 there are three internal modes, etc. Indeed, the properties of the modes are specified by the potential U (x) = n(n + 1) tanh 2 x − 2 of the equation (25b), it is getting deeper as the integer n increases. The number of bound states trapped by this potential then also increases. Note that the frequency of the lowest mode is ω 2 = n − 2, thus if n < 2, or equivalently, if κ < 0, there is a negative mode in the spectrum and the lump kink becomes unstable with respect to perturbations. If the coupling constant κ remains positive, the lump kink is stable and excitation of this mode ξ Our numerical studies confirmed this conclusion. An example of the decay of the double kink at κ = −0.1 is displayed in Fig. 16, we can see that the excitation of the lowest translational mode ξ The vibrational mode of the double kink has the following form (up to a normalisation factor): ξ (1) 2 = sinh x cosh n x .(34) The corresponding frequency depends on integer n as ω = 3(n − 1). Note, that action of the creation operator on the ground state (32) yields:â † ξ (0) 2 = 2n sinh x cosh n+1 x .(35) Evidently, this is a (n + 1)-th bound state. So in order to obtain the n-th bound state, the creation operator should act on the (n − 1)-th mode. Indeed, the algebra of the ladder operators (27) is [â,â † ] = 2n cosh 2 x .(36) Therefore â †â + n − 2 â † =â † â †â + n − 2 + 2n − 2n tanh 2 x =â † â †â + n − 2 + p , where p = 2n + 1 and we introduce new ladder operators of degree (n − 1) (cf. Eq. (27)) is an eigenfunction of the operator of degree n (28) and the corresponding eigenvalue is ω 2 , the action of the raising operatorâ transforms it to the eigenfunction of the operator of degree (n − 1) with eigenvalue ω 2 + p. a † = − d dx + (n − 1) tanh x ,â = d dx + (n − 1) tanh x .(38) B. Spectral structure of lump kink Investigation of the spectrum of the linear lump kinks fluctuations is a bit more complicated problem because, unlike in the double kink case, there is no analytical solutions for the classical solutions φ 1 and φ 2 . Thus, our analysis relies on numerical methods, we have to find solutions to the system of equations (22) using shooting in two dimensions or applying the spectral Chebyshev method. Note the the system of the linearized equations (22) can be diagonalised on the spacial asymptotics where the fields approach the vacuum value: φ 1 = φ 2 = 1/ √ 1 − κ as x → ±∞. Introducing the same variables ξ 1 , ξ 2 (24), we obtain then decoupled system of equations − d 2 dx 2 + 4 ξ 1 = ω 2 ξ 1 , (39a) − d 2 dx 2 + 4 1 + κ 1 − κ ξ 2 = ω 2 ξ 2 .(39b) Thus, there are 2 different excitations about the vacuum in our model (1) with 2 different masses m 1 = 2 and m 2 = 2 1+κ 1−κ , they are degenerated only in the special decoupled case κ = 0. One of the interesting manifestation of the presence of two different excitations in the spectrum of linear fluctuations in our model is related with behavior of the lump kink under the influence of an incident wave. It is known in the usual one-component φ 4 theory the kink starts to accelerate in the direction of the incoming wave [35,36] with acceleration proportional to the 4-th power of amplitude of incident wave. This effect is known as the negative radiation pressure. In our model \|φ 1 \| = \|φ 2 \| symmetry restores exactly the φ 4 model so within this sector of theory the same phenomenon takes place. However in our system there is also a different possibility. Because of the difference in masses of small perturbation around the vacuum another mechanism of the negative radiation pressure (proportional to the square of the amplitude of the wave) can be observed [26]. In the case under consideration numerical simulations confirm that if the frequency of the incoming wave is ω 2 = m 2 1 +k 2 the kink accelerates towards the source of the radiation, however if the frequency of the incoming wave is ω 2 = m 2 2 + k 2 the result is inverted, i.e. the positive radiation pressure is observed. Thus, the masses of the bound states of the lump kink are restricted by the lowest of the asymptotic masses m 1 and m 2 . We found the solution of the eigenvalue problem numerically using the spectral Chebyshev method. The results are presented in Figs. 17,18. Evidently, the internal mode of the kink component is antisymmetric while the internal mode of the lump component is symmetric. Considering the dependency of the frequency of the internal mode on the coupling constant κ we found that there is no second internal mode in the system, the second mass threshold is decreasing faster than the frequency of the oscillation mode (cf. Fig. 19). V. CONCLUSIONS Motivated by the recent interest in investigation of the remarkable resonance structures in the parameter space of the kink-antikink system, we have studied dynamical properties of the soliton solution of the two-component coupled φ 4 system. There are two different types of the solitons, the double kink and the lump kink, in both cases our numerical simulations of the collisions between the various types of the solitons reveal some chaotic resonance behavior which however, is rather different from the usual pattern of the KK collision in the one-component φ 4 model. Considering collision of two lump kinks in the same sector below some critical velocity we observed the resonance energy exchange between the sectors with a sequences of bouncing windows similar with the quasi-fractal dynamics observed in the usual φ 4 model. However the structure of the resonance windows in the former case is not so regular, also some false windows appear. At the collisions of the lump kinks above the critical velocity we observed flip of the sectors and consequent scattering of the solitons. In the ultrarelativistic limit another process starts to dominate, the collision of the kinks in the first sector is accompanying by the collision of the lumps in the second sector which produces kink-antikink pairs. We also analysed the collision of the lump kinks in the different sectors and the most complicated process of the douple kink -lump kink collision. In the former case the resonance structure is very regular, it is qualitatively similar to the structure which appears in the one-component model. However the double kink -lump kink collision may lead to various results, there are different channels of this process, the most interesting case is related with collisions at relatively high impact velocities, then a new type of self-similar fractal structure appears. These results need more rigorous investigation which is currently in progress, in particular it would be interesting to find an effective theory which could explain such a behavior. Evidently such a complicated behavior is related with the spectrum of excitations which may be excited in the collision of the solitons and affect the mechanism of the energy exchange between the solitons. Investigation of the linear stability of the soliton solutions showed that for certain range of values of the coupling constant, the double kinks are unstable with respect to linear perturbations, they decay into pair of two lump kinks. We also investigated the spectrum of perturbations of the soliton solutions on the two-component model. We found that in the case of the double kink configuration, in addition to the expectable counterparts of the translational and internal modes, there is a tower of bound states whose number depends on the strength of the coupling between the sectors. Evidently, excitation of these states in the process of the collision of the kinks will strongly affect the mechanism of the energy transfer. Considering the spectrum of linear fluctuation about the lump kinks we found there are excitations of 2 different masses, however there is only one internal mode in the system for all range of values of the coupling constant. It remains to systematically analyze the effect of interaction of the soliton solutions of the two-component model with an incoming wave in perturbation theory, it will answer the question if the effect of negative radiation pressure is also presented in this case. As a direction for future work, it would be interesting to construct an effective collective coordinate Lagrangian for the two-component model which will capture the most important degrees of freedom in the soliton collision. We thank Patrick Dorey and Wojtek Zakrzewski for enlightening discussions. Ya.S. is very grateful to Stephane Nonnenmacher for kind hospitality at the SPhT, CEA Saclay where part of this work was done. A.H. gratefully acknowledges support from the organizers of the 52. Cracow School of Theoretical Physics. This work is supported Figure 1 : 1(Color online) Contour plot of the potential (2) for κ = 0.5 < 1 (left panel) and κ = −1.1 < −1 (right panel). Figure 2 : 2Lump kink static configurations (1, 0) is plotted for κ = 0.5. Figure 3 : 3The mass of the two kink types, the lump kink and the double kink, as function of coupling constant κ. Figure 4 : 4Collision of two lump kinks (−1, 0) + (1, 0) → (0, −1) + (0, 1) at κ = 0.5 and initial velocity vin = 0.5. Figure 5 : 5(Color online) Two lump kinks (−1, 0) + (1, 0) collision at κ = 0.5. The plots represent the field values measured at the collision center as function of impact velocity and time. Upper and bottom plots represent the φ1 and φ2 components, respectively.initial velocity v i . Before the collision the initial value of the field at the center was taken as φ 1 (0, 0) = 1√ 1−κ (the only possiblity for such configuration) and φ 2 (0, 0) = − 1 √ 1−κ (one of two possibilities). We will denote the state of that vacuum in square brackets as Φ = √ 1 − κ[φ 1 , φ 2 ] = [1, −1]. The final state of the vacuum in the collision center then can be defined as Φ = [−1, −1] + 2Q, where Q is the topological charge of the kink moving towards asymptotic x → −∞. − [−1, −1]) = (0, 1). Thus, the collision leads to the topological fliping between the sectors (cf.Fig. 8, panel (II)). Figure 6 :Figure 7 :Figure 8 : 678(Color online) Collision of two lump kinks in different sectors mediated by the lumps at κ = 0.5. (Color online) Final state of the two lump kinks collision (1, 0) + (−1, 0). (Color online) Example of topological zeros motion for two lump kinks collision (1, 0) + (−1, 0) in the same sectors. (I) bounce (1, 0) + (−1, 0), (II) flip (0, −1) + (0, 1), (III) annihilation (0, 0), (IV) double kinks creation (−1, −1) + (1, 1). Figure 9 : 9(Color online) Power spectrum of the field in the first sector in the collision center for (1, 0) + (0, −1) → (1, −1) and κ = 0.5 at vin = 0.2. As the result of the collision an excited double kink is created. One can identify the frequency of the usual φ 4 internal mode ω φ 4 = √ 3, another frequency ωt = − κ = 1.129 (see the next section for details) is lowered because of nonlinearities to ω2 = 1.077, other combination of those frequencies. Figure 10 : 10(Color online) Collision of two lump kinks in different sectors mediated by the lumps at κ = 0.5. The plots represent the field values measured at the collision center as function of impact velocity and time. Upper and bottom plots represent the φ1 and φ2 components, respectively. Figure 11 : 11(Color online) Final state of two lump kinks collision (1, 0) + (0, 1) in different sectors. Figure 12 :Figure 13 : 1213(Color online) Example of topological zeros motion of two lump kinks collision (1, 0) + (0, 1) in different sectors. (Color online) Double kink -lump kink (−1, −1) + (1, 0) collision at κ = 0.5. The plots represent the escape velocity of different solitons produced in both sectors after collision as a function of impact velocity. Bottom plot represent the zoomed in region of the upper plot. Figure 14 : 14(Color online) Final state of the double kink with a lump kink collision (1, 1) + (−1, 0). Figure 15 : 15(Color online) Example of topological zeros motion for collision of a double kink with a lump kink (1, 1) + (−1, 0) with final states (I) unstable double kink decays to pair of lump kinks, one is ejected and second annihilates with the other kink (1, 1) + (−1, 0) → (0, 1) + (1, 0) + (−1, 0) → (0, 1) + (0, 0), (II) annihilation in the first sector and passage in the second sector, (III) annihilation in the first sector, the created oscillon is coupled to the remaining kink in the second sector (0, 0) + (0, 1), (IV) bounce (1, 1) + (−1, 0), (V) annihilation in the first sector and bounce in the second sector (0, 1) + (0, 0), (VI) bounce in one sector and passage in the second sector (1, 0) + (−1, 1). the oscillations of the components of the double kink about the center of mass of the system. to decay of the double kink into two lump kinks which go off in contrary directions. On the other hand, the same perturbative distortion Figure 16 : 16(Color online) Decay of the double kink at κ = −0.1. Initial configuration is taken as φ1(x, 0) = u tanh(x + 0.1) and φ2(x, 0) = u tanh(x − 0.1). Excitation of the lowest translational mode ξ (0) 2 leads to decay of the double kink into two lump kinks moving in opposite directions. of the double kink at positive coupling κ = 0.5 and consequent excitation of the wobbling collective mode does not destroy the configuration. Figure 17 : 17Profiles of the lump kink components φ1, φ2 together with the profiles of the corresponding internal modes η1, η2 are presented at κ = 0.5, the frequency of the internal mode is ω = 1.8537.This, while the function ξ (k) 2 Figure 18 : 18Profiles of the lump kink components φ1, φ2 together with the profiles of the corresponding internal modes η1, η2 are presented at κ = −0.5, the frequency of the internal mode is ω = 1.12245. 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Note that usual classical rescaling of the 2-component model in one spacial dimension does not allow us to absorb all the constants into rescaled scalar fields and couplings. Note that usual classical rescaling of the 2-component model in one spacial dimension does not allow us to absorb all the constants into rescaled scalar fields and couplings.	[]
[ "Preprint typeset in JHEP style -HYPER VERSION N = 1 theories of class S k", "Preprint typeset in JHEP style -HYPER VERSION N = 1 theories of class S k" ]	[ "Davide Gaiotto \nPerimeter Institute for Theoretical Physics\nN2L 2Y5WaterlooOntarioCanada\n", "Shlomo S Razamat \nNHETC\n08854Rutgers, PiscatawayNJUSA\n\nDepartment of Physics\n32000Technion, HaifaIsrael\n" ]	[ "Perimeter Institute for Theoretical Physics\nN2L 2Y5WaterlooOntarioCanada", "NHETC\n08854Rutgers, PiscatawayNJUSA", "Department of Physics\n32000Technion, HaifaIsrael" ]	[]	We construct classes of N = 1 superconformal theories elements of which are labeled by punctured Riemann surfaces. Degenerations of the surfaces correspond, in some cases, to weak coupling limits. Different classes are labeled by two integers (N, k). The k = 1 case coincides with A N −1 N = 2 theories of class S and simple examples of theories with k > 1 are Z k orbifolds of some of the A N −1 class S theories. For the space of N = 1 theories to be complete in an appropriate sense we find it necessary to conjecture existence of new N = 1 strongly coupled SCFTs. These SCFTs when coupled to additional matter can be related by dualities to gauge theories. We discuss in detail the A 1 case with k = 2 using the supersymmetric index as our analysis tool. The index of theories in classes with k > 1 can be constructed using eigenfunctions of elliptic quantum mechanical models generalizing the Ruijsenaars-Schneider integrable model. When the elliptic curve of the model degenerates these eigenfunctions become polynomials with coefficients being algebraic expressions in fugacities, generalizing the Macdonald polynomials with rational coefficients appearing when k = 1.	10.1007/jhep07(2015)073	[ "https://arxiv.org/pdf/1503.05159v2.pdf" ]	53,536,332	1503.05159	2f4793615a142ec4a25f35a5782bd9dde718727c	Preprint typeset in JHEP style -HYPER VERSION N = 1 theories of class S k 17 Mar 2015 Davide Gaiotto Perimeter Institute for Theoretical Physics N2L 2Y5WaterlooOntarioCanada Shlomo S Razamat NHETC 08854Rutgers, PiscatawayNJUSA Department of Physics 32000Technion, HaifaIsrael Preprint typeset in JHEP style -HYPER VERSION N = 1 theories of class S k 17 Mar 2015 We construct classes of N = 1 superconformal theories elements of which are labeled by punctured Riemann surfaces. Degenerations of the surfaces correspond, in some cases, to weak coupling limits. Different classes are labeled by two integers (N, k). The k = 1 case coincides with A N −1 N = 2 theories of class S and simple examples of theories with k > 1 are Z k orbifolds of some of the A N −1 class S theories. For the space of N = 1 theories to be complete in an appropriate sense we find it necessary to conjecture existence of new N = 1 strongly coupled SCFTs. These SCFTs when coupled to additional matter can be related by dualities to gauge theories. We discuss in detail the A 1 case with k = 2 using the supersymmetric index as our analysis tool. The index of theories in classes with k > 1 can be constructed using eigenfunctions of elliptic quantum mechanical models generalizing the Ruijsenaars-Schneider integrable model. When the elliptic curve of the model degenerates these eigenfunctions become polynomials with coefficients being algebraic expressions in fugacities, generalizing the Macdonald polynomials with rational coefficients appearing when k = 1. Introduction There are several known large classes of N = 1 superconformal field theories, which often have some kind of geometric interpretation. A classical example is the class of theories associated to dimer models [1,2,3,4], labelled by bi-partite graphs drawn on a torus and a choice of rank N . These are quiver gauge theories with SU (N ) gauge groups and bi-fundamental matter, with a topology and choice of superpotential which is determined by the bipartite graph. They are associated to a string theory setup involving N D3-branes placed at the tip of a singular toric Calabi-Yau cone. The combinatorial rules for associating quivers to bipartite graphs can be extended to more general two-dimensional geometries [5,6,7], possibly including boundaries, such as disks, or higher genus Riemann surfaces. These constructions produce quiver gauge theories including both SU (N ) gauge groups and SU (N ) flavor groups, again with bi-fundamental matter. There is another natural way to produce large classes of N = 1 superconformal field theories with a geometric origin, by a twisted compactification on a Riemann surface of six-dimensional SCFTs. Such constructions generalize the derivation of class S of N = 2 4d SCFTs from the twisted compactifications of the six-dimensional (2, 0) SCFTs on a Riemann surface decorated with punctures [8,9]. The class S construction may produce both standard gauge theories and strongly-interacting SCFTs which lack a known Lagrangian description. Geometric manipulations of the Riemann surface lead to specific manipulations of the associated SCFTs, which allow one to derive S-dualities relating in various ways the standard gauge theories and the strongly interacting SCFTs. The class S construction has an unexpected computational power, allowing for example to compute the superconformal index of all the four-dimensional class S theories, even though they might lack a Lagrangian description [10,11,12,13], and relate S 4 partition functions to 2d CFTs [14,15]. A straightforward extension of the class S construction to N = 1 gauge theories involves alternative twisted compactifications of the (2, 0) SCFTs. The basic strongly-interacting building blocks remain the same as for the N = 2 class S, but they are glued together using N = 1 gauge multiplets rather than N = 2 ones [16,17,18,19]. In this paper we are interested in a more general extension, which involves the twisted compactification of (1, 0) SCFTs. There is a rather large number of known (1, 0) SCFTs, many of which can be built through F-theory constructions [20,21]. In general, they have tensor branches of vacua where they take the appearance of six-dimensional gauge theories, coupled to matter fields which may themselves be irreducible SCFTs. A subset of the (1, 0) SCFTs become standard gauge theories on their tensor branch and often admits a D-brane engineering construction [22,23]. We focus here on the (1, 0) SCFTs T N k associated to N M5 branes sitting at the tip of an A k−1 singularity of M-theory. These theories are somewhat well understood and have several features in common with the (2, 0) theories. Furthermore, RG flows induced by vevs on the Higgs branch of these theories allow one to reach many more (1, 0) SCFTs [24]. It should be possible to extend our analysis to D-and E-type singularities, but we will not do so here. Our main result is the conjectural definition of a new large class S k of N = 1 SCFTs associated to the compactification of the 6d SCFTs T N k on a Riemann surface with punctures. It is important to observe that the reduced amount of supersymmetry should make one very cautious in extending to N = 1 theories the intuition developed with N = 2 class S theories. With that concern in mind, in this paper we will follow a "bottom up" approach: we focus at first on a set of conventional four-dimensional N = 1 theories which play a "data collection" role analogous to the role of linear quiver gauge theories in class S theories. By looking at the properties and S-dualities of these theories we find several pieces of evidence connecting them to their six-dimensional avatars and to a larger conjectural set of N = 1 SCFTs labelled by punctured Riemann surfaces. In the process, we learn new information about the sixdimensional SCFTs and their compactifications. The basic set of "core" N = 1 gauge theories which are central to our analysis belongs squarely to the family of bi-partite quivers: they correspond to bi-partite hexagonal graphs drawn onto a cylinder. From that perspective, our work selects a subset of bi-partite theories which enjoy a larger than usual set of S-dualities and subjects them to a variety of manipulations to embed them into a larger family of non-bipartite N = 1 SCFTs. The supersymmetric index plays an important role in our analysis. In particular, we are able to recast the index of class S k theories as a 2d TFT, built from wave-functions which are eigenfunctions of novel difference operators, which generalize the elliptic RS difference operators used in bootstrapping the index of class S theories. In principle, that opens the possibility to compute the index of any N = 1 SCFT in the class S k , even in the absence of a Lagrangian description. This paper is organized as follows. In section 2 we motivate our choice of "core" gauge theories by some general considerations on brane constructions. In section 3 we discuss our basic theories and dualities and set the stage for building the class S k models. In particular we introduce the the notions of theories corresponding to punctured Riemann surfaces and the basic punctures, maximal and minimal ones. This and following sections will be divided into two parts with the first part detailing general physical arguments and the second part, the index avatar, giving quantitative evidence and details using the supersymmetric index for the particular example of class S 2 A 1 theories. Next, in section 4 we analyze RG flows starting from the basic models triggered by vacuum expectation values for baryonic operators. Such RG flows lead to theories in the IR which naturally correspond to surfaces with one of the minimal punctures removed. However, unlike the class S case here the removal of a minimal puncture leads to new theory and to the notion of discrete charge labels attached to the Riemann surface. We will also start encountering new strongly coupled theories belonging to our putative classes of models. In section 5 we discuss closing maximal punctures by giving vacuum expectation values to certain mesonic operators. Such vevs lead to more general punctures and we in particular will be interested in closing maximal punctures down to minimal ones. One of the outcomes of this analysis will be a derivation of Argyres-Seiberg like duality frames for our basic duality and yet more irreducible strongly coupled SCFTs. In section 6 we will discuss how to introduce surface defects into our theories by triggering flows with space-time dependent vevs. This construction will give us certain difference operators which we expect to characterize completely the wave-functions and the 2d TFT structure of the index. We will explicitly derive the difference operators for A 1 theories of class S 2 . In section 7 we go back to the higher-dimensional perspective: we study the interplay between class S k theories and the five-dimensional gauge theories which arise from circle compactifications of the T N k theories. We will finish in section 8 with an outlook of possible further research directions. Several appendices complement the text with further technical details. The Z k orbifold of N = 2 linear quivers The analysis of class S theories was greatly facilitated by the existence of a core set of class S N = 2 SCFTs which admit both a Lagrangian description and a six-dimensional engineering construction: linear quiver gauge theories of unitary groups. These are theories which can be engineered by configurations of D-branes in IIA string theory, which are then lifted to M-theory to make contact with compactifications of the A N −1 (2, 0) SCFT onto a cylinder geometry. It is thus natural to seek for some class of N = 1 SCFTs which admit a similar brane engineering construction in IIA and may be lifted to M-theory to a cylinder compactification of the T N k (1, 0) SCFTs. The latter arises on the world-volume of N M5 branes in the presence of an A k−1 singularity. We will thus take the standard NS5-D4 brane system used to engineer N = 2 linear quivers in [25] and super-impose it to an A k−1 singularity in IIA string theory. More precisely, the D4 branes extend along directions 01234, the NS5 branes along directions 012356 and the Z k orbifold action rotates in opposite directions the 56 and 78 planes. See Figure 1 As the degrees of freedom of the original N = 2 gauge theory entirely arise from open strings, we can apply a standard orbifold construction to arrive at our candidate N = 1 theories [26]. The analysis implicitly assumes that appropriate B-fields have been turned on so that the orbifold singularity admits a perturbative description. We will come back to this point momentarily. The orbifold procedure can be implemented in a straightforward way at the level of the gauge theory. Schematically, the orbifold group is embedded both into the global symmetries which correspond to the rotations of the internal space-time directions and into the gauge group, and all fields charged non-trivially under the orbifold action are thrown away. We can focus at first on the D4 branes. In the absence of NS5 branes, they will support a five-dimensional N = 1 gauge theory described by a necklace quiver, the result of orbifolding N = 2 5d SYM. The Z k group acts with charges 0, 1, −1 respectively on the real scalar associated to the 9 direction and on the complex scalars associated to the 56 and 78 directions. Figure 1: The brane configuration which engineers the core N = 1 gauge theories. We draw a setup with N = 4 D4 branes (or kN = 16 fractional D4 branes) sitting at the locus of a Z 4 singularity, intersected by 5 NS5 branes. The D4 branes in each segment between consecutive NS5 branes engineer the four necklaces of SU (N ) gauge groups. The 4-4 strings across the NS5 branes engineer the five sets of zig-zag chiral multiplets between the necklaces. The semi-infinite D4 branes can be either associated to the two necklaces of flavor groups or to five-dimensional necklace gauge theories coupled to the four-dimensional quiver theory. The embedding of Z k in the gauge group splits the gauge fields and the neutral real scalar into k separate blocks of vector multiplets, while the complex scalar fields with charge ±1 under the embedding of Z k in the global symmetry will give bi-fundamental hypermultiplets between consecutive nodes of the 5d quiver. See Figure 2. The six-dimensional (1, 0) SCFT T N k , compactified on a circle, is expected to give a UV completion of precisely such necklace quiver theory with gauge group SU (N ) at each node. We will review the properties of such five-dimensional theory in a later section. For now we only need to observe that it has a U (1) 2k global symmetry, which is an Abelian remnant of the SU (k) β × SU (k) γ × U (1) t global symmetry of the 6d SCFT, together with the rotation symmetry of the compact circle. The non-Abelian gauge symmetry in the UV is broken by Wilson lines for the SU (k) β × SU (k) γ global symmetry, which have to be turned on in order to have a perturbative gauge theory description in 5d, rather than an interacting 5d SCFT. In the string theory setup, this corresponds to the B-field which has to be turned on for the orbifold singularity to admit a perturbative description. In order to arrive to our N = 1 gauge theories, we need to include the effect of the NS5 branes intersecting the D4 branes and in particular the orbifold action on the 4 − 4 strings stretched across NS5 branes. From the point of view of the four-dimensional gauge theory, we can start with an N = 2 linear quiver of four-dimensional U (kN ) gauge groups and embed Z k into the combination of SU (2) R and U (1) R-symmetries which preserves an The necklace quiver N N,k in five dimensions. It is a Z k orbifold of the maximally supersymmetric YM, with k SU (N ) nodes. Notice that each link here represents a full bi-fundamental hypermultiplet and each node a full SU (N ) vector multiplet. There is U (1) global symmetry associated with each link, rotating the bifundamental hypermultiplets. There is also a U (1) symmetry, the instanton symmetry, associated with each gauge group node. N = 1 sub-algebra, under which the 4d vector multiplet scalars (not to be confused with the 5d vector multiplets above!) transform with charge 1 and the 4d hypermultiplets with charge −1/2. The a-th N = 2 vector multiplet will give us a necklace N a of N = 1 SU (N ) gauge groups (dropping the overall U (1) which decouple in the IR), with bi-fundamental chiral multiplets running, say, counter-clockwise along the necklace from the (i + 1)-th to the i-th gauge groups in the necklace. Because the N = 2 bi-fundalemental hypermultiplets have charge −1/2 under the R-symmetry group, we need to embed Z k in the gauge groups accordingly, with integral charges at even nodes and half-integral at odd nodes of the original linear quiver. Then each N = 2 bi-fundamental hypermultiplet is projected down to a set of bi-fundamental chiral fields which zig-zags back and forth between the nodes of consecutive necklaces, say from the i-th node of each necklace N a to the i-th node of the next necklace N a+1 and from the i-th node of N a+1 back to the (i + 1)-th of N a . Thus if we start with a N = 2 linear quiver of n U (kN ) gauge groups, with kN flavors at each end, we end up with a N = 1 quiver of kn SU (N ) gauge groups with the topology of a tessellation of a cylinder, with triangular faces associated to cubic super-potential couplings (arising from the N = 1 superpotential coupling vectors and hypers in the original N = 2 theory) and k SU (N ) flavor groups at each end. This is the theory associated to a bi-partite honeycomb graph drawn on the cylinder, with one side of the hexagons aligned with the cylinder's axis. See Figure 3 for an example with k = 4 and four necklaces. This N = 1 gauge theory is our candidate to describe the compactification of the T N k (1, 0) SCFT on the cylinder, decorated with (n + 1) "minimal punctures" each associated to a single transverse M5 brane wrapping two directions of the A k−1 singularity (the M-theory The "honeycomb" bi-partite graph drawn on a cylinder and the corresponding quiver gauge theory. The top and bottom lines are identified. Each cell of the bi-partite graph maps to an SU (N ) gauge group with 3N flavors corresponding to the six edges of the cell. Each node of the bi-partite graph indicates a cubic superpotential term, with sign associated to the color of the node lift of the NS5 branes in the IIA description). In analogy with the class S analysis, we hope to identify that with a compactification on a sphere, with two extra "maximal punctures" playing the role of the cylinder's ends. See Figure 4. Here and in the next sections, we will study these "core" N = 1 gauge theories, they global symmetries, exactly marginal deformations and S-dualities, in order to find manifestations of their conjectural six-dimensional origin. In particular, we aim to find • One exactly marginal deformation parameter for each complex structure modulus of the underlying Riemann surface. • Global symmetries matching the six-dimensional description. • S-dualities which manifest the indistinguishability among minimal punctures and imply a 2d TFT-like associativity structure for the index • RG flows which relate different types of punctures and, in particular, relate maximal and minimal punctures, thus justifying the picture of a sphere as opposed to a cylinder. The count of global symmetries and exactly marginal deformation parameters are closely related. As each gauge group has N f = 3N c and the super-potentials are cubic, the core theories can be thought of as deformations of a free theory. The (3n + 2)k sets of bi-fundamental hypermultiplets have each a U (1) global symmetry and there are 3nk marginal couplings. By the arguments of [27], the theory will have x exactly marginal couplings iff 2k + x of the global symmetries are unbroken by superpotential terms or mixed gauge anomalies. Figure 4: The six-dimensional lift of the brane configuration which engineers the core N = 1 gauge theories. We have N M5 branes sitting at the locus of a Z k singularity and wrapping a cylinder, intersected by 5 transverse M5 branes. The world volume theory of the N M5 branes is the T N k (1, 0) SCFT, with 5 "minimal" defects (left). We expect that the de-coupling the four-dimensional degrees of freedom can be implemented by replacing the semi-infinite ends of the cylinder by maximal punctures on a sphere (right). The non-anomalous Abelian global symmetries of the core theories can be understood graphically as in Figure 5: each symmetry is associated to a straight sequence of chiral multiplets with alternating charges ±1. There are 2k + n + 1 generators which add up to 0. The set of (n + 1) U (1) α global symmetries associated to each of the (n + 1) blocks of chiral multiplets with the same bi-fundalental hypermultiplet ancestor can be thought of arising from the NS5 branes world volume gauge symmetry. We thus associate them to the minimal punctures in the six-dimensional description. The remaining set of U (1) 2k−1 global symmetries can be thought of as the Cartan generators of the SU (k) β ×SU (k) γ ×U (1) t global symmetry of the underlying 6d SCFT. Thus we are left with n exactly marginal couplings, precisely as in the original N = 2 theory. We associate them to the relative positions of the minimal punctures on the cylinder. Basic building blocks In this section we study the core N = 1 theories and their dualities. It is useful to begin by introducing some nomenclature, which helps abstracting the properties of the core N = 1 theories to the expected properties of class S k theories. Each theory we build will be labelled tentatively by a punctured Riemann surface. Some set of "intrinsic" global symmetries will be always present independently of the choice of punctures: we can denote them as G k ≡ U (1) t × U (1) k U (1) β × U (1) k U (1) γ . It will be convenient to associate fugacities to these symmetries to keep track of charges of different fields and operators. We will denote the fugacities for the above mentioned global symmetries as t, Figure 5: A simple set of generators for the Abelian global symmetries of the core theories. Each arrow represents a generator, acting on the chiral multiplets crossed by the arrow with charge ±1, depending on the sign of the crossing. The sum of all generators is 0. The vertical (red) arrows are chosen to generate the U (1) α global symmetries associated to minimal punctures. The SE pointing (green) arrows, quotiented by their diagonal, generate the U (1) k U (1) β "intrinsic" symmetries. The SW pointing (blue) arrows, quotiented by their diagonal, generate the U (1) k U (1) γ "intrinsic" symmetries. Finally, the anti-diagonal combination of blue and green arrow give the generator of the U (1) t intrinsic symmetry. β i with i β i = 1 and γ i with i γ i = 1, respectively. We will also have a non-anomalous R-symmetry, whose fugacity we can indicate as r or, with index computations in mind, √ pq. Each puncture will be associated to some specific set of flavor symmetries and a specific set of 't Hooft anomalies involving these flavor symmetries and the intrinsic flavor symmetries. It will also be associated to a set of canonical chiral operators with prescribed charges. A "maximal" puncture will be associated to an SU (N ) k global symmetry. Maximal punctures will be labelled by a "color" n ∈ Z k and an "orientation" o = ±1, positive or negative, which determine the pattern of 't Hooft anomalies and the charges of the canonical chiral operators. We will have N units of U (1) t SU (N ) 2 anomaly for all SU (N ) groups, N units of U (1) β i+n−o SU (N ) 2 i and −N units of U (1) γ i SU (N ) 2 i . In other words, SU (N ) i has a mixed anomaly with a U (1) symmetry associated to the fugacity tβ i+n−o γ −1 i . The R-symmetry mixed anomaly should be the same as if the SU (N ) i acted on N fundamental and N anti-fundamental free chiral multiplets with R-symmetry 0 (the anomaly, of course, is computed from the fermions in the chiral multiplets, which have charge −1). We also require a set of "mesons", chiral operators M a i+1 a i which transform as funda-mental/antifundamental of SU (N ) i+1 and SU (N ) i . 1 At a positively oriented puncture, the mesons have fugacity tβ i+n γ −1 i . A useful mnemonic rule is that they involve the β fugacity from the SU (N ) i+1 node, and the gamma fugacity from the SU (N ) i node. At a negatively oriented puncture the mesons have fugacity tβ i+n+1 γ −1 i+1 . A useful mnemonic rule is that they involve the β fugacity from the SU (N ) i node, and the gamma fugacity from the SU (N ) i+1 node. Clearly, a cyclic re-definition of the β i , with fixed γ i , will simultaneously shift the color of all the maximal punctures in a theory. A gluing prescription Next, we give a gluing prescription, an operation on four-dimensional theories which is interpreted in six dimensions as replacing two maximal punctures of opposite orientation and the same color with a tube. Consider a positively oriented maximal puncture of color 0 and flavor groups SU (N ) i and a negatively oriented puncture of color 0 and flavor groups SU (N ) i . We will gauge the diagonal combinations SU (N ) g i of SU (N ) i and SU (N ) i−1 . We will also add k blocks of N 2 chiral fields Φ a i a i+1 in the fundamental/antifundamental representation of SU (N ) g i × SU (N ) g i+1 . These fields also couple to the mesonic operators associated to the maximal punctures, through cubic superpotential couplings, W = λ TrM Φ − Tr M Φ . (3.1) The superpotential couplings determine the charges of Φ a i a i+1 under the intrinsic global symmetry G k such that they have fugacities pqt −1 β −1 i γ i . The mixed anomalies at the SU (N ) g i gauge node cancel out and the intrinsic global symmetries remain non-anomalous. It can be easily seen using Leigh-Strassler-like [28,27] arguments that this gauge theory has one exactly marginal coupling which can be continuously tuned to zero. The fact that the exactly marginal coupling can be switched off is related to the fact that for each SU (N ) gauge factor we have matter equivalent to 3N fundamental and 3N anti-fundamental chiral fields and thus the one-loop contribution to the NSVZ beta-function vanishes, and the superpotential couplings are also classically marginal. In particular, we can count the difference between marginal parameters and broken U (1) symmetries [27]. We have 3k marginal couplings, but we break the 2(2k −2) separate intrinsic Figure 6: WRONG α POWERS The free trinion. Dots on the left an on the right represent SU (N ) groups associated to the two differnet maximal punctures. The horizontal lines are the bi-fundamentals Q and the diagonal lines Q. Q i : t 1 2 αγ −1 i SU(N ) i SU(N ) k−i+1 SU(N ) k−i Q i : t 1 2 α −1 β i symmetries of the two theories together with the k symmetries acting on the Φ i down to a diagonal combination of the intrinsic symmetries. Thus we break 3k − 1 symmetries. Each broken symmetry lifts a marginal coupling, leaving only a single exactly marginal one. We expect it to correspond roughly to the length and twist of the newly-created tube in the underlying Riemann surface. The free trinion Having described some generalities of the setup let us turn our attention to concrete and important examples. Our first ingredient is a free theory which we would like to label by a trinion, a sphere with three punctures. Two of the punctures are maximal and one is a "minimal" one. We will refer to this theory as the free trinion. The free trinion is a collection of 2kN 2 free chiral fields, equipped with a specific action of the intrinsic and puncture-related global symmetries. We organize the fields in two sets of k blocks of N 2 chiral fields, denoted respectively as Q a i b i and Q It is convenient to depict the free trinion as a ring with 2k dots connected by arrows. The dots at even/odd places correspond to SU (N ) i / SU (N ) i flavor groups. See Figure 6. We also define the action of the intrinsic G k = U (1) t × U (1) k U (1) β × U (1) k U (1) γ global symmetries, and of an extra U (1) α associated to the minimal puncture. Expressing the charges in terms of fugacities, we associate fugacity t 1 2 β i α −1 to Q a i b i and t 1 2 αγ −1 i to Q a i b i+1 . It is straightforward to compute the 't Hooft anomalies of the free trinion. The U (1)SU (N ) 2 anomalies for each SU (N ) flavor group receive contributions from two blocks of chirals: we have N units of U (1) t SU (N ) 2 anomaly for all SU (N ) groups, N units of U (1) β i SU (N ) 2 i and U (1) β i SU (N ) 2 k−i+1 , −N units of U (1) γ i SU (N ) 2 i and U (1) γ i SU (N ) 2 k−i . In other words, SU (N ) i has a mixed anomaly with a symmetry associated to the fugacity tβ i γ −1 i , while SU (N ) k−i+1 has a mixed anomaly with a symmetry associated to the fugacity tβ i γ −1 i−1 . There are various U (1) 3 anomalies, such as kN 2 units of U (1) 2 α U (1) t anomaly, −N 2 of U (1) α U (1) 2 β i and N 2 of U (1) α U (1) 2 γ i , and some anomalies involving intrinsic symmetries only. Finally, we have chiral operators we can associate with each puncture. For example, M a i a i−1 = Q b i a i−1 Q a i b i are fundamental/antifundamental of consecutive SU (N ) i groups, with fugacity tβ i γ −1 i−1 , while M b i+1 b i = Q b i+1 a i Q a i b i are fundamental/antifundamental of consecutive SU (N ) i groups, with fugacity tβ i γ −1 i . We associate them to the respective maximal punctures. On the other hand, baryon operators B i = det Q a i b i and B i = det Q b i+1 a i can be associated to the minimal puncture. We see that the first maximal puncture, associated to the SU (N ) i , has colour 1 and positive orientation. The second maximal puncture is associated to the groups SU (N ) k−i . It has negative orientation and colour 0. Gluing two trinions All our core theories are produced by gluing together a sequence of n + 1 free trinions. We can focus on the simplest interacting theory built by gluing two trinions. We can take two free trinions, shift the definition of the β i in the second trinion so that it has a positive maximal puncture of color 0 and a negative of color −1, and glue them together as described in the previous subsection. The result is our candidate for a theory with two maximal punctures, of opposite orientation and color 1 and −1, and two minimal punctures. Let us denote this theory byT k and the quiver description of the theory is in Figure 7. Throughout the paper this theory will play an important role and we will refer to it as our basic interacting theory or basic four-punctured sphere. Our crucial claim is that the interacting theory described here enjoys an S-duality property, which corresponds to the exchange of the two minimal punctures. Notice that this is consistent with 't Hooft anomalies, as the global symmetries U (1) α and U (1) δ associated to the two minimal punctures have mixed anomalies with the intrinsic global symmetries only, and the anomalies are identical. Since we have here an exactly marginal coupling, we can switch on in a correlated manner the gauge couplings and the superpotential coupling such that the beta-functions vanish, the superconformal R-charges of the different fields are the free ones. For example the superconformal anomalies a and c are just the ones of this particular collection of free fields, Q Q Q N Figure 7: The interacting superconformal theory. The circles correspond to SU (N ) groups with the the white ones being gauged and colored ones being flavor symmetries. The fat brown line represent the mesonic operator built from the quarks which is charged under the symmetries associated to maximal puncture but is neutral under the symmetry associated to the minimal puncture. Fat blue line represents a quark from which a baryon is built which is charged under the U (1) symmetry associated with the minimal puncture but is a singlet under the symmetry associated with the maximal puncture. a = k 48 (14N 2 c − 9) , c = k 24 (8N 2 c − 3) . (3.2) We will find it convenient to assign R-charge 0 to all Q and Q and R-charge 2 for all Φ. Notice that in our notation we are focusing on a subset of the full global symmetry of the gauge theory. The quarks and anti-quarks which are associated to a given gauge group can be re-assembled into groups of 2N flavors, so that the theory really has SU (2N ) k × U (1) k+1 global symmetry. The U (1) symmetries are associated to fugacities such as t i = tβ i γ −1 i and η 2 = αδ −1 . The S-duality transformation should invert the definition of the latter U (1) symmetry, the anti-diagonal combination of U (1) α and U (1) δ . This structure is obviously similar to what one encounters in N = 2 SQCD, the k = 1 version of our story. As we will see momentarily, this is no coincidence: all our conjectural S-dualities can be related recursively to the N = 2 SQCD S-duality by Seiberg dualities. Our starting point is a theoryT k defined as a necklace of SU (N ) i gauge groups, connected by bi-fundamental chiral fields Φ a i a i+1 and to fundamental fields Q c i a i and Q a i+1 c i , where the c i indices transform under SU (2N ) global symmetry groups and obvious cubic superpotential. We can take the Φ a i a i+1 to have fugacity t −1 i , Q c i a i of fugacity √ t i η, the Q a i+1 c i of fugacity √ t i η −1 . If we apply Seiberg duality formally at the k-th node, we obtain a new theorỹ T k which has a rather simple description: it consists of a shorter necklaceT k−1 where the (k − 1)-th SU (2N ) flavor group has been gauged and coupled to 2N dual quarks and 2N dual anti-quarks. The theory also has a set of 2N × 2N mesons coupled to the latter fields by a cubic superpotential. The dual quarks and anti-quarks are rotated by the SU (2N ) k−1 and SU (2N ) k global symmetries of the original theory. Crucially, the U (1) flavor symmetry assignments work out in such a way that only fields inT k−1 transform under U (1) η . At this point, we can immediately conclude, recursively, that say the supersymmetric index ofT k is invariant under the transformation η → η −1 : the Seiberg duality manipulation leaves the index invariant, and the index ofT k is built from the index ofT k−1 , with η only entering in the latter. AsT 1 is N = 2 SQCD with N f = 2N c , which has an S-duality which acts as η → η −1 on the flavor fugacities, the index ofT 1 is invariant under the transformation η → η −1 . In order to believe a full S-duality statement, we need to make some assumptions about the RG flows associated to the Seiberg duality we employed, as the SU (2N ) gauge node is strongly-coupled. The RG flow of the theoryT k defines a map from the conformal manifold ofT k−1 to the conformal manifold ofT k . By induction, we can assume that the conformal manifold ofT k−1 has two weakly-coupled cusps, where the quiver gauge theory description is good. Near these cusps, the RG flow will map us to weakly-coupledT k quivers, with opposite U (1) η charge assignments. Thus the crucial assumption for the induction to hold is that the connected path between the two cusps in the conformal manifold ofT k−1 maps to a connected path in the conformal manifold ofT k . In analogy with class S, we can ask if other trinions may exist, in particular trinions which carry three maximal punctures. This is of course far from obvious. In the case of class S, a strong piece of evidence came from the existence of Argyres-Seiberg-like dualities [29]. A key step in the analysis was to find operations which "reduce" a maximal puncture to a minimal one, a Higgs branch RG flow which replaces one end of a linear quiver of SU (N ) gauge groups with a "quiver tail" which is associated to a set of minimal punctures only, all related by S-dualities. That suggests the existence of alternative S-duality frames where all minimal punctures are produced by quiver tails, attached to a conjectural SCFT with maximal punctures only. Due to the intricacies of the quivers we study here, there is a bewildering array of possible RG flows one can trigger by a sequence of vevs for chiral operators. Correspondingly, one can modify a maximal puncture to a wide array of "smaller" punctures. Our challenge is to find a sequence which leads to a minimal puncture. We will undertake this challenge in the following sections. The index avatar Let us discuss the basic theories and the dualities in the language of the supersymmetric index setting the nomenclature for the discussion in the following sections. The reader can accustom him/herself with the standard definitions of the index in appendix A. The index is a very precise tool to test dualities. To be very explicit and keep our formulae readable, we will often specialize to the index of A 1 theories of class S 2 . All we will say can be rather straightforwardly extended to more general cases. Let us start from writing down the index of the free trinion. As discussed in previous subsections and depicted in Figure 9 we associate this trinion to a sphere with two maximal punctures and one minimal puncture. The two maximal punctures are of different colors, and we will return to this feature momentarily. The index can be written by collecting together the contributions of the different free field. For example, in the A 1 k = 2 case the index of the free trinion is, Free trinion I ft (u, α, v; β, γ, t) ≡ I uα v (β, γ, t, p, q) ≡ (3.3) Γ e (t 1 2 v ±1 1 u ±1 1 βα −1 )Γ e (t 1 2 v ±1 1 u ±1 2 γ −1 α)Γ e (t 1 2 v ±1 2 u ±1 1 γα)Γ e (t 1 2 v ±1 2 u ±1 2 β −1 α −1 ) . Here the fugacities u and v correspond to SU (2) 2 and SU (2) 2 flavor symmetries associated to the maximal punctures and α is the U (1) fugacity associated to the minimal puncture. Note the index of the trinion is not symmetric under the exchange of the two maximal punctures. This is a reason why one should associate an additional parameter, color, to the maximal punctures. If we set β = 1 there is no distinction between the colors and indeed the index becomes symmetric under exchanging the two maximal punctures. In the k = 2 case we have two colors which are Z 2 valued, 0 and 1. We will also refer to the two punctures in this case as lower and upper ones. We can glue two trinions together and obtain a theory corresponding to two maximal and two minimal punctures, and we will come to details of the gluing at the level of the index momentarily. We can also continue gluing free trinions to obtain theories corresponding to spheres with many minimal punctures and two maximal ones. The S-duality operation exchanging two punctures surely holds at the level of the index, under the usual assumption of that no new U (1) symmetry emerges accidentally in the IR. Let us for a moment discuss implications of the S-duality for general k and N . At the level of the index, we can glue a free trinion to a theory T with a positive maximal puncture of color 0 by the formula: I T (v; β i , γ i , t) → I T +minα (v, α; β i , γ i , t) ≡ u dµ n (u; β i , γ i , t)I ft (v, α, u; β i , γ i , t)I T (v; β i , γ i , t) , (3.4) where I ft is the free trinion index, dµ n (u, β i , γ i , t) the measure which accounts for the gauge fields and Φ fields involved in the gluing and the u contour integral is the projection on gauge-invariant operators. This can be thought as an integral transformation on the index of T , with a kernel which depends on the fugacity α of the new minimal puncture. The basic S-duality we derived in this section implies that if we apply the transformation twice, the result will not depend on the order of the two transformations. In other words, the integral operators associated to different values of the fugacity α commute. It is reasonable to assume that the transformations can be "diagonalized", as in the class S case [30,31], by expanding the free trinion into "wavefunctions" associated to each puncture, I ft (v, α, u; β i , γ i , t) = λ ψ [1] λ (u; β i , γ i , t)φ λ (α; β i , γ i , t)ψ [0] λ (v; β i , γ i , t) (3.5) with ψ [n] λ (u; β i , γ i , t) ≡ ψ λ (u; β i+n , γ i , t) being the wavefunction for positive maximal punctures,ψ [n] λ (u; β i , γ i , t) ≡ψ λ (u; β i+n , γ i , t) being the wavefunction for negative maximal punctures and φ λ (α; β i , γ i , t) being the wave function for minimal punctures, invariant under color shift, φ λ (α; β i+n , γ i , t) = φ λ (α; β i , γ i , t) . (3.6) We take the maximal wavefunctions to be normalized as v dµ n (v; β i , γ i , t)ψ [n] λ (v, β i , γ i , t)ψ [n] λ (v, β i , γ i , t) = δ λλ . (3.7) Then if we glue together n trinions we get an index I ft n (u i , α k , v i , β i , γ i , t) = λ ψ [n] λ (u i , β i , γ i , t) k φ λ (α k , β i , γ i , t) ψ [0] λ (v i , β i , γ i , t) , (3.8) which is explicitly invariant under S-duality. Note that gluing n free trinions we get a theory of two maximal punctures differing by n units of the Z k color. In the rest of this subsection we will specialize to the case of k = 2 A 1 theories. As already mentioned here we have two colors for maximal punctures. The orientation of the puncture corresponds to the ordering of the two SU (2) groups. We will denote ψ(u; β, γ, t) = ψ [0] (u; β, γ, t) , ψ(u; β, γ, t) = ψ [1] (u; β, γ, t) , (3.9) ψ(u † ; β, γ, t) =ψ(u; β, γ, t) , where we define (u 1 , u 2 ) † = (u 2 , u 1 ). Thus the index of the free trinion will be written as, I ft (u, α, v; β, γ, t) = I uα v = λ ψ λ (u 1 , u 2 ) ψ λ (v 1 , v 2 )φ λ (α) . (3.10) The way to determine the functions ψ λ , ψ λ , φ λ , as well as the possible values of the index λ will be determined in section 6. These functions will turn out to be eigenfunctions of certain difference operators and thus we will refer to them as eigenfunctions. We stress that it is a rather non trivial assumption that such a representation of the index exists. We will discuss more of the physical implications of this property later on. We have the following symmetry properties satisfied by the index of the free trinion following from its explicit expression (3.3), I uα v (β, γ, t) = I vα u (β −1 , γ, t) , I uα v (β, γ, t) = I v † α u † (β, γ −1 , t) , (3.11) I uα v (β, γ, t) = I uα −1 v † (γ , β, t) , I uα v (β, γ, t) = I u † α −1 v (γ −1 , β −1 , t) , It is natural to assume that these transformations act on single eigenfunctions and do not act on the labels λ. Under this assumption (3.11) implies that, ψ λ (u; β, γ, t) = ψ λ (u; β −1 , γ, t) = ψ λ (u † ; β, γ −1 , t) = ψ λ (u; γ, β, t) = ψ λ (u † ; γ −1 , β −1 , t) , ψ λ (u; β, γ, t) = ψ λ (u † ; γ , β, t) = ψ λ (u; γ −1 , β −1 , t) , (3.12) φ λ (α, β, γ, t) = φ λ (α, β −1 , γ, t) = φ λ (α, β, γ −1 , t) = φ λ (α −1 , γ, β, t) . We do not have to assume these properties for what follows but assuming them will make some of the considerations simpler and we will state explicitly when such assumption will be made. Moreover, when explicitly computing the eigenfunctions we find that these indeed are satisfied although we will not prove them mathematically. The gauging We gave a prescription for gluing two theories together at maximal punctures of appropriate color and orientation, by adding extra chiral fields and superpotential couplings to the mesons and gauging diagonal combinations of the flavor symmetries. The gauge group in general is SU (N ) k and in case at hand it is SU (2) 2 . Since there are two types of maximal punctures, we can glue theories along upper or lower punctures. In both cases the gauge group is the same. We identify SU (2) i of one theory with SU (2) 3−i of the other one, and we add bifundamental chiral fields of SU (2) i × SU (2) i+1 . The difference between the two gaugings is that when we glue two upper punctures the two bifundamental chirals have charges (+1, +1) and (−1, −1) under U (1) β × U (1) γ , whereas when gluing two lower punctures the charges are (+1, −1) and (−1, +1). For the index this implies that the functions ψ λ and ψ λ are orthonormal under the following measures, ∆(z) = I 2 V 4 Γ e ( pq t (βγ) ±1 z ±1 1 z ±1 2 ) Γ e (z ±2 1 )Γ e (z ±2 2 ) , ∆(z) = I 2 V 4 Γ e ( pq t (β −1 γ) ±1 z ±1 1 z ±1 2 ) Γ e (z ±2 1 )Γ e (z ±2 2 ) . (3.13) That is v dµ 0 (v; β, γ, t)ψ [0] λ (v, β, γ, t)ψ [0] µ (v, β, γ, t) = dz 1 2πiz 1 dz 2 2πiz 2 ∆(z) ψ λ (z)ψ µ (z † ) = δ λµ , v dµ 1 (v; β, γ, t)ψ [1] λ (v, β, γ, t)ψ [1] µ (v, β, γ, t) = dz 1 2πiz 1 dz 2 2πiz 2 ∆(z) ψ λ (z) ψ µ (z † ) = δ λµ . (3.14) From the free trinions we can obtain the sphere with two maximal punctures, either both upper or both lower, and two minimal punctures. The two theories are physically isomorphic and differ only by re-labeling of the flavor symmetries. The index of the theory with two lower punctures is, I uδvα (t, β, γ, p, q) = [(q; q)(p; p)] 2 dz 1 4πiz 1 dz 2 4πiz 2 Γ e ( p q t (βγ) ±1 z ±1 1 z ±1 2 ) Γ e (z ±2 1 )Γ e (z ±2 2 ) × (3.15) Γ e (t Here the first line comes from the gauging with the second and third lines coming from the contributions of the two trinions. This index has the duality symmetry I uδvα = I uαvδ = I vδuα ,(3.16) which we should expect following our discussion in previous subsections. As discussed before this duality follows from a sequence of Seiberg and S-dualities with the relevant mathematical identities proven in [32,33,34]. Using the eigenfunctions this index is given by I uαvδ = λ ψ λ (u 1 , u 2 )ψ λ (v 1 , v 2 )φ λ (α)φ λ (δ) . (3.17) Gluing many trinions together we can obtain theories with arbitrary number of minimal puncture but with only two maximal punctures. Moreover if the number of minimal punctures is even the two maximal punctures are of the same color and if that number is odd they are of different color. To go beyond these constraints we will have to consider RG flows triggered by vacuum expectation values of the theories obtainable from our trinions and we are turning to that task next. Figure 11: The sphere with two maximal and two minimal punctures with the fugacities associated to the matter fields. The white nodes correspond to gauged SU (2) groups with the colored one to flavor SU (2) groups. We suppressed the pq and t fugacities. αγ v 2 α −1 β −1 βγ 1 βγ αγ −1 α −1 β γ −1 δ u 2 u 1 β −1 δ −1 γδ βδ −1 v 1 Closing minimal punctures and discrete charges The six-dimensional SCFTs which we conjecture lie behind our story have global symmetries which we identify with the four-dimensional intrinsic global symmetries G k . The two copies of U (1) k /U (1) actually enhance to SU (k) and in four dimensional in general only the Cartan subgroup is not broken. In the process of compactification from six to four dimensions, one has a choice of curvature for the line bundles associated to these global symmetries. In the standard class S story, there is a single global symmetry corresponding to U (1) t , and one obtains N = 2 theories for a specific choice of line bundles, and more general N = 1 theories for other choices. We will later comment on the orbifolded versions of these N = 1 theories, which should correspond to different choices of U (1) t curvature. In this section, we would like to assess the four-dimensional meaning of different choices of curvature for the U (1) β and U (1) γ bundles. In order to do so, we look at the possibility of "closing" a minimal puncture, by giving a vev to a chiral operator charged under the U (1) α global symmetry associated to the minimal puncture. An assumption we make is that such vevs leave us inside the class of theories we are discussing, that is the theory in the IR can be associated to a certain Riemann surface. An obvious choice for the operator to receive a vev is one of the (anti)baryons built from the N 2 blocks of the corresponding free trinion. There are 2k such operators, Q N i and Q N i , and we could be giving a vev to any of them. We can do so for any of the minimal punctures. Indeed, the S-duality properties we expect from our theories indicate that we should get the same result, up to a relabeling of minimal puncture fugacities, by turning on (anti)baryon vevs in different free trinions. For our purpose, it is particularly instructive to pick the duality frame where the free trinion we are working on is glued to a maximal puncture of some other generic theory T to give a theory T with an extra minimal puncture and a new maximal puncture. The analysis for a free trinion glued to two other theories can be done in a similar manner. It is straightforward to see the effect of the vev. First of all, the vev Higgses the SU (N ) gauge group coupled to the (anti)quarks and identifies it with the SU (N ) flavor group of the (anti)quarks. The vev converts the cubic superpotential coupling involving that set of (antiquarks)quarks to a mass term for one of the Φ fields and for one block of (quarks)antiquarks for the nearby gauge node. We can integrate these away. When giving a vev to the (anti)quark block, we need to re-define all the symmetry generators, including the R-symmetry one, by an appropriate multiple of the generator of U (1) α in such a way that the (anti)quark block is not charged under any of the assignments re-defined symmetries. It is easy to verify that the final theory we obtain after Higgsing has the same 't Hooft anomalies for the re-defined symmetries as the initial theory, as is obvious from anomaly matching. In particular, all the constraints we impose on 't Hooft anomalies of global symmetries associated to the punctures we did not close remain true after the Higgsing. The resulting theory has three fewer marginal couplings: we lost two superpotential couplings involving the lifted Φ field and one gauge coupling. We also lost three blocks of fields and one global symmetry. Thus we expect to have lost an exactly marginal deformation parameter. This is consistent with the intuition that the minimal puncture is gone. The resulting theory depends on the choice of which 2k (anti)baryons got a vev, and it is not equivalent to the theory T . Indeed, all the 2k resulting theories have the same "punctures", which differ from the punctures of T just by the colour of the maximal puncture the original free trinion was glued to. Thus we obtained 2k potential new class S k theories and discovered that class S k theories need to be labelled by extra data besides the choice of Riemann surface and punctures. We aim to identify the extra data with a choice of curvature for the U (1) β × U (1) γ global symmetries of the underlying six-dimensional theory. Tentatively, we would say that closing a minimal puncture by a vev of a baryon charged under U (1) β i or an anti-baryon charged under U (1) γ i adds a unit of curvature for the corresponding six-dimensional global symmetry. If this idea is correct, adding one unit of curvature for each U (1) β should be the same thing as not adding any, as the U (1) β are identified with the Cartan of a 6d SU (k) global symmetry. The same should be true when adding one unit of curvature for each U (1) γ . We will see that this is indeed the case, as long as we refine slightly our notion of "closing a minimal puncture" by adding to the theory some gauge-neutral chiral multiplets coupled linearly to the surviving baryons (if we turned on a baryon vev) or anti-baryons (if we turned on an anti-baryon vev) in the free trinion associated to the minimal puncture we closed. As we will often have to add gauge-neutral chiral fields with linear couplings to some given chiral operator, it is useful to introduce the notion of "flipping" a chiral operator O: an operation which maps a theory with a chiral operator O to a new theory with an extra chiral multiplet φ coupled to O by a superpotential φO. The new theory lacks the operator O, but usually has a new chiral operator φ with opposite charges to O. Let us now specialize to the k = 2 case and still hold N general while considering closing two minimal punctures. We consider a duality frame where the two minimal punctures reside in two free trinions glued to each other and also both of them glued to some general models. That is the two minimal punctures reside on a "tube" connecting otherwise generic Riemann surfaces. If we give vevs to baryons of fugacities (α −1 (4.1) 1 √ tβ) N and (α −1 2 √ tβ −1 ) N , The theory has a vacuum where M = Λ 2 and B =B = 0, which would precisely Higgs the left and right SU (N ) 2 gauge groups and thus bring us back to the standard theory with two fewer minimal punctures, except for the extra baryonic degrees of freedom B,B. Even at the level of the index, one finds that the N f = N node effectively produces a delta function [35] of the fugacities of the left and right SU (N ) z 2 gauge groups, multiplied by the index of the baryons B,B with fugacities (α −1 1 √ tβ −1 ) N and (α −1 2 √ tβ) N . We can modify our definition of how to close a minimal puncture, by both turning on a vev for the baryon with, say, (α −1 1 √ tβ) N and a linear superpotential coupling between the other baryon B with fugacity (α −1 1 √ tβ −1 ) N and a new gauge-neutral chiral field b. The superpotential forces us at the origin of the baryonic branch of the N f = N node and insures the desired Higgsing. It also removes the undesired free baryons after the Higgsing. We can now go back to general k. We consider a sequence of k trinions and close the corresponding punctures by giving a vev to the baryon charged under U (1) β 1 in the first trinion, U (1) β 2 in the second trinion, etc, and flipping all other baryons charged under the U (1) β . Because of S-duality, we could have picked any other permutation σ, turning on at the σ(a)-th trinion a vev for the baryon charged under U (1) βa . The order we chose simplify the analysis considerably, though. After integrating away the chiral fields which receive mass parameters after the vevs, precisely k−1 gauge nodes end up with N f = N flavors. With the help from the superpotential couplings suppressing the corresponding baryons, the mesons for these gauge nodes get vevs, and initiate another set of Higgsing and lifting of pairs of chiral multiplets. This leads to another set of N f = N nodes, etc. At the end of the RG flow cascade, the k punctures have been completely eliminated. Notice that there is a certain degree of correlation between the choice of punctures on a Riemann surface and the possible values for the discrete charges. Starting from a theory T with a maximal puncture of some colour and orientation, we can glue to it a chain of k free trinions and then close the resulting k minimal punctures in several different ways. This produces new theories with the same punctures as T but different discrete charges. The discrete charges, though, will necessarily add to a multiple of k. A k = 2 example We can give some rather explicit examples of this construction for theories in class S 2 . For example, we can start from our basic interacting theoryT 2 built from two trinions, with two minimal punctures and maximal punctures of opposite orientation and color 1 and −1 ≡ 1. If we give a diagonal vev to, say, the block of quarks of fugacity √ tα −1 β in the second trinion, we are left with an SU (N ) gauge theory with 2N flavors: two blocks of N quarks with flavour symmetries SU (N ) 2 andSU (N ) 2 and two blocks of N anti-quarks with flavour symmetries SU (N ) 1 andSU (N ) 1 . We also have neutral chirals in bi-fundamental representations ofSU (N ) 1 ×SU (N ) 2 andSU (N ) 1 × SU (N ) 2 coupled to the (anti)quarks by cubic superpotential couplings and an additional decoupled setSU (N ) 1 × SU (N ) 1 which will enter the definition of the "mesons" at the maximal punctures of color 1 and −1. Finally, we have the single neutral chiral field we use to flip the baryon made out of the quarks of fugacity β −2 , which are charged underSU (N ) 2 . This is one of four theories which we can obtain by closing the same minimal puncture inT 2 in different ways. They all have one minimal puncture and two maximal of opposite orientation and the same color 1, but different discrete charges. They will be important in defining surface defects in section 6. We can also consider theories obtained fromT 2 by closing both minimal punctures in such a way that the total discrete charge does not cancel out. This produces a variety of "charged tubes", i.e. theories which can be associated to a cylinder with two maximal punctures of the same type and some extra discrete charge. Such charged tubes can be glued to a maximal puncture of some other theory T to shift the discrete charges of that theory without changing the type of punctures. We will discuss some examples in appendix B. The index avatar Let us now translate the discussion above to the language of the index. To study RG flows generated by vacuum expectation values at the level of the index one needs to study its analytical properties. Different poles of the index correspond to operators for which a vev can be turned on with residues being indices of the theories flown to in the IR [12]. To study analytical properties of the index we first take an index of a generic theory corresponding to a Riemann surface and glue to it the free trinion. See Figure 12. The index of such a theory is given by I 2 V dz 1 4πiz 1 dz 2 4πiz 2 I(z) Γ e ( pq t (βγ) ±1 z ±1 1 z ±1 2 ) Γ e (z ±2 1 )Γ e (z ±2 2 ) (4.2) Γ e (t 1 2 z ±1 1 u ±1 1 αγ)Γ e (t 1 2 z ±1 1 u ±1 2 α −1 β −1 )Γ e (t 1 2 z ±1 2 u ±1 1 α −1 β)Γ e (t 1 2 z ±1 2 u ±1 2 αγ −1 ) . We depict in Figure 13 the fugacities associated to different fields of the free trinion. A class of poles in the index above occurs whenever the integration contours are pinched while varying the fugacities. We look thus for pinchings of the integration contours. The poles inside and outside the integration contour coming from the free trinion are located at in : z 1 = t 1 2 u ±1 1 αγq n p m , t 1 2 u ±1 2 α −1 β −1 q n p m , (4.3) z 2 = t 1 2 u ±1 1 α −1 βq n p m , t 1 2 u ±1 2 αγ −1 q n p m , out : z 1 = t − 1 2 u ±1 1 α −1 γ −1 q −n p −m , t − 1 2 u ±1 2 αβq −n p −m , z 2 = t − 1 2 u ±1 1 αβ −1 q −n p −m , t − 1 2 u ±1 2 α −1 γq −n p −m . When some of the in poles coincide with the out poles the integration contours are pinched and the index develops poles. Different poles correspond to vevs for some protected operators. In this section we are interested in the case of the baryonic operators obtaining a vev. We consider the pole in α, the fugacity associated to the U (1) α symmetry of a minimal puncture, at α = t 1 2 β −1 . This pole occurs when an operator with weight tβ −2 α −2 , the baryon Q 2 1 , gets a vacuum expectation value. Giving a vev to such an operator Higgses the z 1 gauge group. By turning on the vacuum expectation value we break the U (1) α symmetry, i.e. we u 2 Figure 13: The glued sphere with fugacities. Q 1 : α −1 β −1 Q 1 : αγ Q 2 : αγ −1 u 1 z 1 z 2 Q 2 : α −1 β close the minimal puncture. There are similar poles at α = t 1 2 β corresponding to vev to baryon Q 2 2 , and α = t − 1 2 γ ±1 corresponding to vevs for baryons Q 2 i . Setting α = t 1 2 β −1 the z 1 integral is pinched at z 1 = u ±1 2 and the residue of the index becomes I V Γ e (tu ±1 2 u ±1 1 β −1 γ) dz 2 4πiz 2 I({u 2 , z 2 }) Γ e ( pq t (βγ) −1 u ±1 2 z ±1 2 )Γ e (β 2 z ±1 2 u ±1 1 ) Γ e (z ±2 2 ) . (4.4) Here and in what follows by residue we more precisely mean the following operation, 2I V Res u→u * 1 u 1 u 2 F(u) → Res u→u * F(u) . (4.5) This operation is natural as it removes the decoupled free chiral associated to the Goldstone boson index of which is I −1 V . The factor of 2 appears since the U (1) α charge of the baryonic operator which is getting a vev is 2. In particular, if the general theory of index I({u 2 , z 2 }) is just a free trinion, the residue is the index of an N f = 4 SU (2) SQCD with additional singlet fields and superpotentials. Let us denote the field corresponding to the prefactor in the integral M , which is in bifundamental of SU (2) u 1 × SU (2) u 2 . The fields in the numerator of the integral are a quark Φ in fundamental of SU (2) u 2 and a quark Q in the fundamental of SU (2) u 1 . In I({u 2 , z 2 }) we have the contribution of additional four quarks, Q 1 and Q 1 , and four gauge singlets in a fundamental of SU (2) u 2 which we denote by M 1,2 . The superpotential then is, ΦM Q + ΦQ 1 M 1 ,(4.6) with M 2 being free fields. This theory enjoys an action of large duality group [36,37,38,39,40]. Under our assumptions that the RG flows generated by vacuum expectation values should leave us in our class of theories, the N f = 4 SU (2) gauge theory should be associated to a Riemann surface. In fact it can be only associated to a sphere with one minimal puncture and two maximal punctures of the same type. This is a new, interacting, trinion we discover in our bootstrap procedure. Note that here the two maximal punctures are of same color and thus the theory should be invariant under exchanging the two factors of the associated flavor symmetries. This new trinion is depicted in Figure 14. For general k such a trinion can be obtained by gluing together k free trinions and closing k − 1 minimal punctures in certain way. Let us study what this residue teaches us about the functions ψ λ , ψ λ and φ λ . As we take the residue for a minimal puncture fugacity and flip the other (anti)baryon, the wave function φ λ (α) in the sum is replaced by the insertion of certain functions of the intrinsic fugacities, i.e. C (β,±) λ ≡ Γ e (pqβ ±4 ) Res α→t 1 2 β ±1 φ λ (α) , C (γ,±) λ ≡ Γ e (pqγ ∓4 ) Res α→t − 1 2 γ ±1 φ λ (α) ,(4.7) which we can interpret as the contribution to the index sum of a unit of positive or negative discrete charge for U (1) β or U (1) γ . Notice that it must be true that C (β,+) λ C (β,−) λ = 1 , C (γ,+) λ C (γ,−) λ = 1 ,(4.8) since as we discussed closing two minimal punctures by giving vevs to the two different types of (anti)baryons leaves behind no discrete charge. Applying the residue prescription to a trinion glued to a maximal puncture, we can get a neat integral relation between wavefunctions of different color: C (β,−) λ ψ λ (u 1 , u 2 ) = (4.9) I V Γ e (pqβ −4 )Γ e (tu ±1 2 u ±1 1 β −1 γ) dz 2 4πiz 2 ψ λ (z 2 , u 2 ) Γ e ( pq t (βγ) −1 u ±1 2 z ±1 2 ) Γ e (z ±2 2 ) Γ e (β 2 z ±1 2 u ±1 1 ) . This relation is actually known as an elliptic Fourier transform [41], and can be inverted by a second elliptic Fourier transform. The invertibility of the elliptic Fourier transform, though, is precisely the index avatar of the Seiberg duality relation for an SU (N ) gauge node with N flavors and flipped baryons, which we used to show how opposite discrete charges cancel out. Thus the elliptic inversion formula gives us the same result as directly taking a residue of a free trinion glued to a φ λ wave function: C (β,+) λ ψ λ (u 1 , u 2 ) = (4.10) I V Γ e (pqβ 4 )Γ e (tu ±1 2 u ±1 1 βγ) dz 2 4πiz 2 ψ λ (z 2 , u 2 ) Γ e ( pq t βγ −1 u ±1 2 z ±1 2 ) Γ e (z ±2 2 ) Γ e (β −2 z ±1 2 u ±1 1 ) . Computing residues of the basic four punctured sphere and removing the appropriate singlets we thus get new trinions with indices, I (β,±) = λ C (β,±) λ ψ λ (u)ψ λ (v)φ λ (α) . (4.11) This is the SU (2) N f = 4 SQCD with singlets and superpotential we discussed above. It will be very useful when we derive the difference equations satisfied by ψ λ . We can consider closing the minimal puncture in the new interacting trinion. The theories obtained in this way would correspond to spheres with two maximal punctures of same color and with non vanishing discrete charges. We will discuss briefly these constructions in appendix B. Closing maximal punctures Our next aim is to give evidence for existence of theories corresponding to spheres with maximal punctures only. To do so we will study RG flows triggered by the vev of chiral mesonic operators which are charged only under symmetries associated to a single maximal puncture and the intrinsic symmetries. To have a concrete example, to which we will refer in the discussion below, the generic theory glued to a free trinion in the previous section can be taken to be a sphere with two maximal and many minimal puncture, but the discussion is completely generic and example independent. A 1 k = 2 The analysis of a general case is a bit cumbersome so we choose to start our discussion with A 1 and k = 2 and gradually crank up these parameters. In the class S theories of type A 1 , maximal punctures turn out to be equivalent to minimal punctures, though they appear different in the brane construction, essentially because mesons and baryons are on the same footing in linear quivers of SU (2) gauge groups. For k = 2 maximal and minimal punctures are clearly different. The simplest choice of chiral operators charged under the global symmetries of a maximal puncture are the meson operators Q i Q i with fugacities u ±1 1 u ±1 2 t β γ ±1 . We refer to fugacities and fields as depicted in Figure 13. Turning on vevs for a single meson operator breaks the SU (2) u 1 × SU (2) u 2 flavor symmetry down to an U (1) δ subgroup. Thus we may hope it will result in a minimal puncture. Without loss of generality we can focus at first on giving a vev to the operator Q 1 Q 1 , with fugacity (u 1 u 2 ) −1 t γ β . More specifically, we give a vev to the component of Q 1 with gauge charge 1 under the SU (2) 1 and Q 1 with gauge charge −1. When turning on these vevs, we need to re-define the intrinsic global symmetries and define U (1) δ by appropriate combinations of the old global and gauge symmetries, in such a way that the fields getting a vev are neutral under the new global symmetries. At the level of fugacities, this is accomplished by setting the the SU (2) u 1 × SU (2) u 2 fugacities to (u 1 , u 2 ) = (t As we turn on the vev for these chiral fields, some other fields are lifted by the cubic superpotentials, which become mass terms. The SU (2) z 1 gauge field is Higgsed and only SU (2) z 2 is left to glue the other free trinions to the one we triggered the mesonic vev in. The crucial observation is that the surviving gauge groups only interact with chiral fields which have the same charge under U (1) α and U (1) δ . This is obvious for the fields in the general theory we glued to the free trinion, which are only charged under the diagonal combination of the two, i.e. have fugacities depending only on z 1 = αδ. Among the chiral fields coupled to SU (2) z 2 , the only ones which have different U (1) α and U (1) δ charges have fugacities β γ z ±1 2 ( δ α ) ±1 and are exchanged by permuting α and δ. Other surviving fields have fugacities pq t (αδβγ) ±1 z ±1 2 . The surviving chiral fields which are not charged under SU (2) z 2 have fugacities tα 2 γ 2 , tδ −2 γ 2 , tα −2 β −2 , and tδ 2 β −2 . In order to find a complete symmetry between U (1) α and U (1) δ we need to remove ("flip") the fields with fugacity tδ 2 β −2 and tδ −2 γ 2 , by adding new fields of fugacity pq(tδ 2 β −2 ) −1 and pq/(tδ −2 γ 2 ) with quadratic superpotential couplings, and add chiral fields with fugacity tδ −2 β −2 , tδ 2 γ 2 with appropriate superpotential couplings. Thus we find that by giving vev to a meson and adding some extra neutral chirals linearly coupled to chiral operators of the original theory we arrive to a theory which has an extra explicit symmetry, permuting the unbroken δ fugacity with (any)one of the minimal punctures fugacities. In other words, starting say from a linear quiver built by concatenating free trinions we have produced a quiver gauge theory which can be rightfully labelled by a single maximal puncture and several minimal ones. An alternative perspective is that the surviving fields in the "α" free trinion, together with the new chiral fields, define a "quiver tail" which can be appended to a negative maximal puncture with color 1 in order to convert it to two minimal punctures of fugacities α and δ. Notice that we could have obtained a similar result starting with the second set of mesons at the original maximal puncture. That would have imposed fugacities, say, z 2 = αδ , (u 1 , u 2 ) = (t 1 2 βδ , t 1 2 1 γδ ) pole corresponding to vev for Q 2 Q 2 . This would have produced an a priori different way to reduce the maximal puncture to a minimal one. It turns out that we can ascribe the difference between the two possible ways to map maximal to minimal to a difference in discrete charges. Indeed, we can probe the difference between these two choices further by closing the newly-created minimal puncture by giving vev to baryonic operators of fugacity δ 2 tγ ±2 or t 1 δ 2 β ±2 . At the level of fugacities, if we set, say, δ = t − 1 2 γ in the relations (u 1 , u 2 ) = (t γδ ) = (t, β γ ) due to Weyl symmetry of u 2 . Similarly, setting δ = t − 1 2 β in the first reduction is equivalent to setting δ = √ tγ −1 in the second one. A detailed analysis of the reduction procedure shows that these pairs of ways to completely close the maximal puncture are indeed equivalent, even when keeping track of the neutral chiral fields we added in the process, as long as we remove an additional singlet chiral field with fugacity β 2 γ 2 in the first way to reduce maximal to minimal and γ 2 β 2 in the second way. Thus we conclude that reducing a maximal puncture to a minimal puncture by giving a vev to a meson with fugacity proportional to γ β or to a meson with fugacity proportional to β γ give class S 2 theories with discrete charges which differ by one unit of U (1) β curvature and one unit of U (1) γ curvature. As we have identified a "quiver tail" which can be attached to a maximal puncture to obtain two minimal punctures, it is natural to do the same step which in class S leads to the definition of non-trivial trinion theories: we can conjecture the existence of SCFTs with one puncture of color 0 and two of color 1 (and appropriate choices of discrete charges), with the property that attaching a quiver tail to one puncture of color 1 will produce our basic core theory built from two free trinions. As we have two different versions of the quiver tail, we seem to need at least two distinct SCFTs, with discrete charges differing by one unit of U (1) β curvature and one unit of U (1) γ curvature. It is instructive to look a bit further to the combination of vevs which we expect to produce and close a minimal puncture starting from a maximal one. It corresponds to giving a vev to mesons with fugacities t γ β u −1 1 u −1 2 and t β γ u −1 1 u 2 . This implies a vev for both chiral fields with fugacity u −1 1 z −1 1 √ tαγ and u −1 2 z 1 √ tα −1 β −1 and chiral fields with fugacities u 2 z −1 2 √ tγ −1 α and u −1 1 z 2 √ tα −1 β. If we are working with a standard core theory built from a sequence of trinions, these vevs force us to turn on chiral fields in the next free trinion as well, because of the cubic superpotential couplings of the second and third fields to the Φ field of fugacity pq t βγz −1 1 z 2 : the extremum equations for Φ require us to turn on a vev for the meson of fugacity t 1 βγ z 1 z −1 2 in the next free trinion. Looking at the theory in detail, we find that the original free trinion has been completely eliminated, while the next free trinion is precisely subject to the vev which reduces the maximal puncture to minimal. This is just another manifestation of the S-duality relations which permute the minimal punctures of the theory. Although we do not have an intrinsic way to compare discrete charges between theories with different types of punctures, we can use some symmetry considerations to set up some useful conventions. Let's declare that the core theories have zero discrete charges. We have several different ways to produce a theory with a single maximal puncture and several minimal ones: a minimal puncture can be produces starting from a maximal puncture of either colour, which can be reduced in two ways. We saw that the two ways of reducing a puncture of color 0 differ by one unit of U (1) β curvature and one unit of U (1) γ curvature. In a similar fashion, the two ways of reducing a puncture of color 0 differ by one unit of U (1) β curvature and minus one unit of U (1) γ curvature. Finally, we saw that reducing a puncture of color 0 and then closing the resulting minimal puncture in a specific way is the same as reducing a puncture of color 1 in a theory with one fewer minimal punctures. Thus it is natural to pick the following symmetric convention to compare the discrete charges of theories before and after reducing maximal punctures: giving a vev to a meson with fugacity proportional to γ ±1 β ∓1 in a color 0 puncture adds ∓1/2 unit of β charge and ±1/2 of γ charge, while giving a vev to a meson with fugacity proportional to γ ±1 β ±1 in a color 1 puncture adds ±1/2 unit of β charge and ±1/2 of γ charge. A 1 general k The general k case for N = 2 is not much harder to analyze, except that now both orientation and color matter. We will proceed by analogy with our k = 2 analysis, and give a prescription to reduce a maximal puncture to minimal. Our prescription will be to give a vev to mesons of fugacity u i u −1 i+1 tβ i γ −1 i . We can give a vev to up to k − 1 of them, say for i = 1, · · · , k − 1. Thus we give a vev to chiral fields of fugacity u i z −1 i √ tβ i α −1 and z i u −1 i+1 √ tγ −1 i α and Higgs k−1 SU (2) groups, leaving only SU (2) k . The chiral field vevs enter the cubic superpotential couplings of the Φ fields between SU (2) i and SU (2) i+1 for i = 1, · · · , k − 2. They force us to also give a vev to the mesons in the nearby trinion, with fugacities z i z −1 i+1 tβ i+1 γ −1 i for i = 1, · · · , k − 2. In turn, these fugacities Higgs k − 2 SU (2) i gauge fields in the next column, for i = 1, · · · , k − 2, but also force us to turn on k − 3 mesons at the next trinion, etcetera. The result is that one end of our quiver of (k − 1)k SU (2) gauge groups is modified to take a triangular shape, with columns of k − 1, k − 2, · · · , 1 SU (2) gauge groups. See Figure 15 for illustration. Figure 15: Example of closing a maximal puncture in a linear quiver with k = 3. Giving a vev to the two mesons denoted by dashed brown lines in the left column the SU (2) 3 group denoted by filled green dots is broken down to U (1). These vevs also Higgs two of the three gauge groups in the second column. The vevs for the mesons through superpotential interactions generate vevs for a meson in the second column, again denoted by dashed brown lines. This meson Higgses another SU (2) gauge group. We end up with a "triangle" of unbroken gauge groups denoted by white dots. We can parameterize the z i and u i in terms of a parameter δ, so that in general the z i are proportional to (αδ) −1 and the u i proportional to δ −1 . Thus in order to identify a symmetry exchanging α and δ we need first to make sure that the fields charged under SU (2) k satisfy such a symmetry. Then we can try to impose the symmetry on neutral fields by adding extra neutral chirals. The relevant fields have fugacities u ±1 k z ±1 k √ tβ k α −1 and z ±1 k u ±1 1 √ tγ −1 k α. Half of these receive masses by the meson vevs, the other half are u −1 k z ±1 k √ tβ k α −1 and z ±1 k u 1 √ tγ −1 k α. They are symmetric if we set δ −2 = u 1 u k γ −1 k β −1 k . Notice that we have set u 2 = u 1 tβ 1 γ −1 1 , etcetera. Thus u 1 u k = u 2 1 t k−1 k−1 i=1 β i γ −1 i and thus δ −1 = u 1 t (k−1)/2 β −1 k . Next we can focus on the lack of symmetry exchanging the α and δ symmetries of neutral fields. The trinion fields have fugacities u ±1 i z ±1 i √ tβ i α −1 and z ±1 i u ±1 i+1 √ tγ −1 i α, but the ones which do not get vevs nor masses and are not eaten by the Higgs mechanism are the ones with fugacities u −1 i z ±1 i √ tβ i α −1 and z ±1 i u i+1 √ tγ −1 i α. Half of the fields have fugacities proportional to α 2 , i.e. tβ 2 i α −2 and tγ −2 i α 2 , and are remnants of the baryons. The other half has α-independent fugacities u −2 i and u 2 i+1 proportional to δ ±2 . If we remove them through linear couplings to new neutral chirals, and replace them with chirals of fugacities tβ 2 i δ −2 and tγ −2 i δ 2 and appropriate superpotential couplings, we arrive at a theory symmetric in α and δ. For the consistency of the picture with further closing the minimal puncture we might need to decouple additional singlet fields charged only under the intrinsic symmetry; we will not analyze this here. Thus we learned how to convert a maximal puncture to a minimal puncture, in k different ways. As for k = 2, these different procedures leave one with different amounts of discrete charges on the surface. For general k we can consider giving vevs to different combinations of mesons closing a maximal puncture down to a puncture Λ with symmetry U (1) ⊂ G Λ ⊂ SU (2) k . On the quiver such choices are classified by carving out multiple triangular wedges from the tail. We will not embark on tail classification here though it is a very interesting problem to discuss. See Figure 16 for illustration. We can call punctures accessible from maximal punctures by RG flows triggered by vevs of mesons "regular" punctures in analogy with class S. There might be other types of punctures one would want to consider but that goes beyond the analysis of this paper. Figure 16: Example of closing a maximal puncture in a linear quiver with k = 3 down to a nonminimal one. We give a vacuum expectation value to a single meson. This breaks the flavor group to SU (2) 2 × U (1) and Higgses one of the gauge groups in the next column. This, next to maximal, puncture together with the maximal and minimal ones are the only "regular" punctures of the A 1 class S 3 . This puncture comes in several varieties depending on the color of the maximal puncture we start with and the particular meson we choose to trigger the flow. A N −1 and general k Let us now analyze the reduction of maximal punctures in A N −1 class S k theories. The basic idea is the same as in A 1 case but now we have many more mesons and thus the details are more involved. For general A N −1 even in class S there is a variety of possible punctures labelled by Young tableaux [8] and to get from a maximal one to minimal one has to turn on vevs for many mesons. We begin our most general A N −1 class S k discussion by prescribing a vev for mesons of fugacity u (a) i+1 (u (a) ) −1 i tβ i γ −1 i . We can give a vev to up to (N − 1)(k − 1) of them, say for i = 1, · · · , k − 1 and a = 1, · · · , N − 1. Thus we give a vev to chiral fields of fugacity z (a) i (u (a) i ) −1 √ tβ i α −1 and u (a) i+1 (z (a) i ) −1 √ tγ −1 i α and Higgs k − 1 SU (N ) groups, leaving only SU (N ) k . These vevs are not yet sufficient for our purposes, as they leave N − 1 U (1) symmetries unbroken. We will thus also turn on an extra set of N − 2 mesons, with fugacities u (a+1) 1 (u (a) k ) −1 tβ k γ −1 k , with a = 1, · · · , N − 2. Thus we give a vev to chiral fields of fugacity z (a) k (u (a) k ) −1 √ tβ k α −1 and u (a+1) 1 (z (a) k ) −1 √ tγ −1 k α with a = 1, · · · , N − 2. These vevs Higgs SU (N ) k to SU (2) k . The vevs enter the cubic superpotentials and force us to turn on also vevs for certain mesons in the next free trinion. Namely, we need mesons with fugacities z (a) i+1 (z (a) i ) −1 tβ i+1 γ −1 i for i = 1, · · · , k − 2 and a = 1, · · · , N − 1, z (a+1) 1 (z (a) k ) −1 tβ 1 γ −1 k and z (a) k (z (a) k−1 ) −1 tβ k γ −1 k−1 with a = 1, · · · , N − 2. These vevs will Higgs the next column of SU (N ) i gauge groups to nothing, except for the last two, Higgsed again to SU (2). These meson vacuum expectation values are implemented by vacuum expectation values for chiral fields in the next trinion of fugacities z (a) i+1 (y (a) i ) −1 √ tγ −1 i α and y (a) i (z (a) i ) −1 √ tβ i+1 α −1 for i = 1, · · · , k−2 and a = 1, · · · , N −1, as well as z (a+1) 1 (y (a) k ) −1 √ tγ −1 k α, y (a) k (z (a) k ) −1 √ tβ 1 α −1 , z (a) k (y (a) k−1 ) −1 √ tγ −1 k−1 α and y (a) k−1 (z (a) k−1 ) −1 √ tβ k α −1 with a = 1, · · · , N − 2. Here the y k ) −1 √ tβ k α −1 and u (N ) 1 (z (a) k ) −1 √ t/γ k α for a = N − 1, N . In terms of the SU (2) fugacityz, such thatz 2 = z (N −1) k (z (N ) k ) −1 , we can write z (N −1) k =z z (N −1) k z (N ) k and z (N ) k = z −1 z (N −1) k z (N ) k , and the flavor fugacities as (z (N −1) k z (N ) k ) 1 2 (u (N ) k ) −1 √ tβ k α −1 and u (N ) 1 (z (N −1) k z (N ) k ) − 1 2 √ tγ −1 k α. The flavor fugacities have a ratio u (N ) 1 u (N ) k (z (N −1) k z (N ) k γ k β k α −2 ) −1 which we want to identify with α N /δ N . Thus we set δ −N = u (N ) 1 u (N ) k (z (N −1) k z (N ) k γ k β k ) −1 α 2−N . As u (a) i+1 = u (a) i t −1 β −1 i γ i , we have u (a) k = u (a) 1 t 1−k β k γ −1 k and u (N ) k = u (N ) 1 t (N −1)(k−1) (β k γ −1 k ) 1−N . Also, (u (N −1) k u (N ) k ) −1 = z (a) k = (u (1) 1 u (N ) 1 ) −1 (t −1 γ k α −1 ) 2−N , and thus δ −N = u (N ) 1 (u (1) 1 ) −1 t (N −1)(k−1)+ 1 2 (N −2) β −N k . With this choice of δ, we have defined our global symmetries in such a way that U (1) α and U (1) δ act in the same way on chiral fields which carry gauge charge. We will have a bunch of gauge-neutral fields charged under both symmetries and by appropriate flips the resulting theory can be made to be fully symmetric under exchanging the U (1) α and U (1) δ symmetries. We have learned how to convert a maximal puncture into a minimal one. The index avatar Let us discuss the above at the level of the index. As usual we specialize to the A 1 k = 2 case. The discussion above makes it clear that one should be able to produce φ λ , up to a normalization factor, by taking a residue of either ψ λ or ψ λ at appropriate values of the fugacities. When we defined φ λ , we have normalized in such a way that the free trinion would have a simple expansion of the form (3.10). In a 2d TFT language, it would be more natural to introduce structure constants C λ and write the index associated to a Riemann surface of genus g with n punctures p a and charges q i in the schematic form I (pa),(q i ) = λ C 2g−2+n λ i C i λ q i a ψ pa λ . (5.1) Thus if we use a convention where core theories have charge 0, we should write φ λ = C λ ψ m λ , with ψ m λ being a properly normalized minimal puncture wave-function. Then with the symmetric charge assignments discussed above, we can write C (β,−) λ C (γ,+) λ ψ m λ (α) = (5.2) Γ e (pqβ −2 γ 2 ) Γ e (tα −2 β −2 )Γ e (tα 2 γ 2 ) Γ e (tα 2 β −2 )Γ e (tα −2 γ 2 ) Res (u 1 ,u 2 )→(t 1 2 γ α , t 1 2 α β ) ψ λ (u 1 , u 2 ) . We can obtain three other similar relations involving the other way to reduce ψ λ and the two other ways to reduce ψ λ , with other prefectures of the form C (β,±) λ C (γ,±) λ . Note that (5.2) together with (4.7) and (4.8) completely determine C λ and C (β/γ,±) λ . We will give explicit expressions for these in appendix B. We can also describe at the level of the index the quiver tail which converts a maximal puncture to two minimal punctures. We start from the general expression for the index of a theory glued to a free trinion, set (u 1 , u 2 ) = (t I = I V Γ e (tα 2 γ 2 )Γ e (tδ −2 γ 2 )Γ e (tα −2 β −2 )Γ e (tδ 2 β −2 ) (5.3) dz 2 4πiz 2 I({αδ, z 2 } † ) Γ e ( pq t (αδβγ) ±1 z ±1 2 ) Γ e (z ±2 2 ) Γ e ( β γ z ±1 2 ( δ α ) ±1 ) . The above manipulations imply a concrete relation between the wavefunctions corresponding to maximal punctures and to minimal ones. For the functions ψ λ , ψ λ , and ψ m λ (5.3) implies that, The relation (5.4) can be used to write the index of our core example of the four-punctured sphere in a very suggestive form. We note that C λ C (β,−) λ C (γ,+) λ ψ m λ (α)ψ m λ (δ) = (5.4) Γ e (pqβ −2 γ 2 )Γ e (tα 2 γ 2 )Γ e (tδ 2 γ 2 )Γ e (tα −2 β −2 )Γ e (tδ −2 β −2 ) I V dz 2 4πiz 2 ψ λ (z 2 , αδ) Γ e ( pq t (αδβγ) ±1 z ±1 2 ) Γ e (z ±2 2 ) Γ e ( β γ z ±1 2 ( δ α ) ±1 ) .I uαvδ = λ C 2 λ ψ λ (u)ψ λ (v)ψ m λ (α)ψ m λ (δ) = (5.5) Γ e (pq γ 2 β 2 )Γ e (tα 2 γ 2 )Γ e (tδ 2 γ 2 )Γ e (tβ −2 α −2 )Γ e (tβ −2 δ −2 ) × I V dz 2 4πiz 2 I {z 2 ,αδ} uv Γ e ( pq t (αδβγ) ±1 z ±1 2 ) Γ e (z ±2 2 ) Γ e ( β γ z ±1 2 ( δ α ) ±1 ) , (5.6) where I h uv = λ C λ C (β,+) λ C (γ,−) λ ψ λ (u)ψ λ (v) ψ λ (h) . (5.7) This index can be interpreted as the index of a new strongly interacting trinion with two lower maximal punctures and one upper puncture and discrete charges (1/2, −1/2). The second equality in (5.5) can be interpreted as the computation of the index in a dual description of the core theory, involving a quiver tail attached to that strongly-interacting SCFT associated to a sphere with three maximal punctures. We can invert the integral in (5.5) using the Spiridonov-Warnaar inversion formula, aka elliptic Fourier transform [41], to obtain explicitly the index of the strongly coupled theory without using the eigenfunction. This is the same inversion procedure used in [11] to obtain the index of the T 3 theory. We will propose an additional, related, duality which connects this interacting trinion to a different Lagrangian theory in appendix D: SU (4) SQCD with four fundamental flavors and two flavors in antisymmetric representation supplemented with gauge singlets and a superpotential. This conjectural, strongly interacting trinion theory can be used as a building block together with the free trinion (with or without closed minimal punctures) to assemble class S 2 theories labelled by a generic Riemann surfaces of genus g with arbitrary numbers of maximal and minimal punctures and arbitrary discrete charges. Discrete charges for U (1) t In our investigations we found that for the space of theories to be closed under gluings and RG flows we have to consider the discrete curvatures for U (1) β and U (1) γ . However we did not have to incorporate such curvatures for U (1) t . Nevertheless, we can consider turning on these curvatures too as was done for k = 1, class S, in [17]. Let us here briefly outline how to apply this generalization. To generalize our story we first should allow two types of theories. These are two copies of theories we discussed till now but with R-symmetry of the two types of theories related as R + = R − + 2q t − , R − = R + + 2q t + . (5.8) The intrinsic and puncture symmetries of the two theories are related as U (1) t × SU (k) β × SU (k) γ → U (1) t −1 × SU (k) β −1 × SU (k) γ −1 , (5.9) SU (N ) k u → SU (N ) k u † . To trinions of type + we associate a new Z valued charge +1 and to trinions of type − we associate charge −1. We glue two trinions of the same type with the appropriate measure we discussed till now, i.e. introducing bi-fundamental fields Φ and coupling them to the mesons associated to the gauged maximal puncture through superpotential. We glue two trinions of different types along a maximal puncture without introducing any extra fields but turning on a quartic superpotential which is a product of the mesons associated to the gauged puncture from the two glued theories. At each gauge node we have here N f = 2N . See Figure 18 for an example. A way to derive the map of symmetries (5.9) is to study anomaly free gluings of two free trinions of different type as depicted in Figure 18. If we consider only one type of theories as we did till now the new Z valued charge will have a fixed value, which we can take to be ±(g − 2 + s) with s being the total number of punctures and the sign determined by which class of trinions we use. However when we glue theories of the two types together and start triggering RG flows closing minimal punctures arbitrary values of the new Z valued charge can be achieved. It would be interesting to develop this generalization in more detail. We make some comments on the index of theories with U (1) t discrete charges in appendix E. Surface defects In this section we will use the strategy of [12] in order to produce difference operators which act diagonally on wavefunctions, associated to BPS surface defects produced by a "vortex construction", i.e. RG flows initiated by position-dependent vevs of chiral operators. Consider two theories T U V and T IR connected by an RG flow initiated by a constant vev of a chiral operator of charge 1 under some U (1) α global symmetry. If we couple the theory T U V to a "vortex", i.e. background connection for the U (1) α global symmetry, with n units of flux concentrated near the origin of the plane, we can give the chiral operator a vev which is constant at infinity, but has a zero of order n near the origin. An RG flow far to the IR will leave us with a surface defect in T IR . Figure 18: An example of a theory with flux for U (1) t . The A 1 k = 2 sphere with two maximal and two minimal punctures built from two trinions of opposite type with the fugacities associated to the matter fields. The white nodes correspond to gauged SU (2) groups with the colored one to flavor SU (2) groups. Note that in our conventions the quarks in the trinion on the left have R-charge zero and on the right R-charge one. Each gauge node has four flavors. We turn on superpotentials for the two "diamond" paths in the quiver. t 1 2 β −1 δ −1 v 2 pq t αγ pq t α −1 β pq t α −1 β −1 u 2 u 1 v 1 t 1 2 γ −1 δ t 1 2 βδ −1 At the level of the index, the vortex construction of surface defect is very simple. The index of T U V has a pole at some value α 0 of the fugacity of U (1) α whose residue is the index of T IR . The pole is accompanied to an infinite family of poles located at α 0 q n , whose residues give the index of T IR in the presence of the vortex defect of order n. Let us take a general theory in class S k and attach to it a sequence of k free trinions to produce a new theory with k extra minimal punctures and the same discrete charges. We know that we can recover the original theory by turning on a baryonic vev for each new minimal puncture, as long as we pick k baryons with distinct U (1) β i charges or anti-baryons with distinct U (1) γ i charges. Thus we can produce 2k classes of interesting surface defects S β i ,n and S γ i ,n by making the vev of one of the k baryons position dependent, or one of the k anti-baryons. Of course, we could also state the construction in term of the interacting trinions built from a sequence of k free trinions by closing k − 1 minimal punctures. This will save us a bit of work below. The index avatar The surfaces defects are very useful in the index considerations since they allow to fix the functions ψ λ (u) by specifying them as eigenfunctions of certain difference operators. As usual, we specialize here to k = 2 and N = 2. The index of the interacting trinion is given by, I (β,−) = Γ e (pqβ −4 )Γ e (tγβ −1 v ±1 1 v ±1 2 )Γ e (t 1 2 βα −1 u ±1 1 v ±1 2 )Γ e (t 1 2 γ −1 αu ±1 2 v ±1 2 ) × I V dz 4πiz Γ e ( pq tβγ v ±1 2 z ±1 ) Γ e (z ±2 ) Γ e (β 2 v ±1 1 z ±1 )Γ e ( t 1 2 βα u ±1 2 z ±1 )Γ e (t 1 2 γαu ±1 1 z ±1 ) . (6.1) We remind the reader that this theory was obtained by closing a minimal puncture of a sphere with two minimal and two maximal punctures by giving a vev to a baryon which set δ = t 1 2 β −1 . The symmetry between u and v is not manifest here, it follows from the duality property of the basic four punctured sphere. We glue this trinion to a generic theory by gauging one of the maximal punctures. The resulting index is I = I 2 V du 1 4πiu 1 du 2 4πiu 2 Γ e ( pq t ( β γ ) ±1 u ±1 1 u ±1 2 ) Γ e (u ±2 1 )Γ e (u ±2 2 ) I 0 (u † )I (β,−) (u, v, α) . (6.2) This index has many interestng poles. If one computes the residue at α = t 1 2 β we will erase Figure 19: Gluing the intercting trinion to a general theory. the minimal puncture and obtain precisely the index of the generic theory we glued the trinion to. As an operatorial statement we say that computing the residue at α = t 1 2 β amounts to acting with identity operator on I 0 . Next we can consider poles in α which have additional powers of q. We claim that this index has, among many others, a pole when α = t The pole is produced by pinching the integration contours between poles of the integrand. The poles arise from the Γ e functions associated to the chiral multiplets which receive a vev. In particular, we have a collision of poles of Γ e (t 1 2 βα −1 u ±1 1 v ±1 2 ) and obtain a finite residue at 2 2 Setting α = t 1 2 βq 1 2 and u1 = q ± 1 2 v ±1 2 in (6.1) one flavor becomes massive and we can evaluate the integral in terms of the Seiberg dual mesons [42]. With the particular charges and generic fugacities for global symmetries appearing in (6.1) one of the mesons contributes zero. However tuning u2 = q ± 1 2 v ±1 1 another meson contributes a pole which cancels the zero. u 1 = q ± 1 2 v ±1 2 , u 2 = q ± 1 2 v ±1 1 . (6.3) Computing the residue is tedious, but straightforward. If we define T (v 1 , v 2 ; β, γ, t) = θ( tv −1 1 v −1 2 q ( γ β ) ±1 ; p)θ( tβv 1 γv 2 ; p)θ( tβ 3 γv 2 v 1 ; p) θ(v 2 1 ; p)θ(v 2 2 ; p) , (6.4) the residue is computed by acting with the following operator on I 0 , S (β,−) (0,1) · f (v 1 , v 2 ) = a,b=±1 T (v a 1 , v b 2 ; β, γ, t)f (q a 2 v 1 , q b 2 v 2 ) . (6.5) Since the theories we consider enjoy S-duality we can act with the difference operator on any of the maximal punctures with the same outcome [12]. The functions ψ λ are eigenfunctions of S (β,−) (0,1) . This operator introduces a surface defect into the N = 1 theories of A 1 class S 2 . We can start from the trinion with opposite β discrete charge and close a minimal puncture with t 1 2 β −1 , or start with one of the trinions with γ discrete charge and close the minimal punctures appropriately. The difference operators one obtains are all related to the above, S (γ,−) (0,1) · f (v 1 , v 2 ) = a,b=±1 T (v a 1 , v b 2 ; γ, β, t)f (q a 2 v 1 , q b 2 v 2 ) , (6.6) S (β,+) (0,1) · f (v 1 , v 2 ) = a,b=±1 T (v a 2 , v b 1 ; β −1 , γ −1 , t)f (q a 2 v 1 , q b 2 v 2 ) , S (γ,+) (0,1) · f (v 1 , v 2 ) = a,b=±1 T (v a 2 , v b 1 ; γ −1 , β −1 , t)f (q a 2 v 1 , q b 2 v 2 ) . The functions ψ λ should be simultaneous eigenfunctions of all these operators and indeed it can be checked that these operators do commute. We can now in principle compute more difference operators introducing more general surface defects by computing other residues of the index above, with higher n. It is a priori straighforward but tedious exercise and we refrain from doing it here. We leave a systematic study of the difference operators and wavefunctions with generic fugacities to future work. Here we will take some degeneration limits which simplify the analysis considerably and allow us to write down some simple, explicit formulae. Although for a general N = 1 theory the limit p → 0 of the index may not be well defined, or useful, the indices of class S k theories with the choice of fugacities used in this paper appear to have a reasonable p → 0 limit, akin to the Macdonald limit of the index of N = 2 gauge theories. In this Macdonald-like limit, the eigenfunctions of this difference operator orthonormal under the vector multiplet measure become (experimentally) polynomials up to an universal pre-factor. The measure under which the polynomial part of the eigenfunction is orthogonal is ∆ k,N (z) = k =1 N a=1 ((z a ) ±2 ; q) N a,b=1 (t β −1 γ z a (z b +1 ) −1 ; q) ,(6.7) with k = 1, N = 2, and β 1 = β, γ 1 = γ. This is a generalization of the A N −1 Macdonald measure. We will discuss a straightforward algorithm to compute the eigenfucntions by diagonalizing the difference operators in Macdonald limit in appendix B. Let us here quote the results if we further set β, γ = 1 and take the Hall-Littlewood-like limit p, q = 0. For the first several wavefunctions we find ψ (0) = 1 (1 − t z ±1 1 z ±1 2 ) 2 , (6.8) ψ (1) ± = 1 (1 − t z ±1 1 z ±1 2 ) 2 ((z 1 + z −1 1 ) ± (z 2 + z −1 2 )), ψ (2) 0 = 1 (1 − t z ±1 1 z ±1 2 ) 2 ((z 2 1 + z −2 1 ) − (z 2 2 + z −2 2 )), ψ (2) ± = 1 (1 − t z ±1 1 z ±1 2 ) 2 ×   − ± √ 2 − t 2 + t z 4 1 + 1 2 (t 2 − 1) z 1 2 − ± √ 2 − t 2 + t z 2 4 + 1 2 (t 2 − 1) z 2 2 ± 2 − t 2 − t + z 1 + 1 z 1 z 2 + 1 z 2 . The hat on the functions reminds us that these functions are not normalized to be orthonormal. Note that the coefficients of the polynomials here are algebraic expressions, roots of polynomial equations, as opposed to rational expressions in the N = 2 case. The N = 1 theories are in this sense irrational though algebraic. The indices are single valued sums over roots of algebraic equations. We denote the normalized functions ψ λ (z) = χ λ (z) (1 − t z ±1 1 z ±1 2 ) 2 ,(6.9) where χ λ (z) is a polynomial. The HL index of the free trinion is then given by t) . I zyα = 1 1 − tz ±1 1 z ±1 2 1 1 − ty ±1 1 y ±1 2 1 − t 2 1 − tα ±2 2 µ χ µ (z 1 , z 2 ) χ µ (y 1 , y 2 ) χ µ (t 1 2 α, t 1 2 α −1 ) χ µ (1, (6.10) -41 -As we will see in appendix B the index of the interacting trinion with three maximal punctures becomes I zyx = 1 1 − tz ±1 1 z ±1 2 (1 − t 2 ) 2 1 − ty ±1 1 y ±1 2 1 1 − tx ±1 1 x ±1 2 2 µ χ µ (z 1 , z 2 ) χ µ (y 1 , y 2 ) χ µ (x 1 , x 2 ) χ µ (1, t) . (6.11) When β, γ = 1 there is no difference between the colors of punctures and we are blind to the different discrete charges. In appendix B we will write the general expressions for the index of any theory in class S 2 when p = 0. Five dimensional interpretation In this section we will re-examine our four-dimensional discussion in the language of boundary conditions and interfaces for a N = 1 five-dimensional gauge theory N N,k , the necklace quiver formed by k SU (N ) gauge groups. This model is the world-volume theory of D4 branes sitting at an A k−1 singularity and conjecturally arising from the compactification on a circle of the six-dimensional SCFTs which inspire our work. Our intuitive picture is that a Riemann surface with s semi-infinite tubes labels interfaces between s copies of N N,k defined on halfspaces. Our first task is to review the properties and definitions of boundary conditions and interfaces for five-dimensional N = 1 gauge theories Generalities A five-dimensional N = 1 gauge theory is labelled by a gauge group G, a flavor group F and a quaternionic representation of G × F which specifies the hypermultiplet content. The hypermultiplets can be given real masses associated to the Cartan sub-algebra of F . The gauge couplings of G can be identified as real masses associated to "instanton" global symmetries U (1) I whose conserved currents are of the schematic form * TrF ∧ F . We can denote the full global symmetry group asF = F × U (1) I . The theory is further labelled by a choice five-dimensional Chern-Simons couplings κ, which may be integral or half-integral depending on the amount of matter fields. Although the gauge theories may have strongly-coupled UV completions, in the IR they are free. That implies that the gauge groups can be treated as very weakly coupled when describing a boundary condition. A simple class of boundary conditions is labelled by two pieces of data: the subgroup G ∂ of the gauge symmetry preserved at the boundary and a choice of boundary condition for the hypermultiplets. Somewhat more general boundary conditions are possible, which involve a generalization of the Nahm pole which occurs in the maximally symmetric case. We will come back to these later in the section. The boundary condition for the hypermultiplets can be strongly coupled. If the hypermultiplets pseudoreal representation is the sum of two conjugate representations, a general construction is available, which starts from some free boundary conditions and adds superpotential couplings to extra four-dimensional N = 1 degrees of freedom living at the boundary. If we denote the two halves of the hypermultiplets as X and Y , we can start from a Y = 0, ∂ ⊥ X = 0 boundary condition and add a linear superpotential coupling W = XO X ,(7.1) to an operator O X in a boundary theory B X . Alternatively, we can start from a X = 0, ∂ ⊥ Y = 0 boundary condition and add a linear superpotential coupling W = Y O Y ,(7.2) to an operator O Y in a boundary theory B Y . As long as we focus on F-terms only, the two constructions are essentially equivalent: B Y can be obtained from B X by "flipping" O X , i.e. by introducing a new chiral field φ with superpotential coupling φO X . Then O Y ≡ φ. A particular case is that the X = 0, ∂ ⊥ Y = 0 can be obtained from Y = 0, ∂ ⊥ X = 0 by adding a chiral with Xφ coupling, and vice versa. Boundary conditions for a five-dimensional gauge theory may have various anomalies. The boundary cubic gauge anomaly receives three contributions: • The bulk Chern-Simons coupling • The boundary theory • The boundary condition for the bulk hypermultiplet: Y = 0, ∂ ⊥ X = 0 contributes half of the anomaly of a boundary chiral field with the same charge as X. 3 This half is the reason the bulk CS coupling may sometimes be half-integral The total gauge anomaly for G ∂ must cancel out. We also have various sources of 't Hooft anomalies. This includes cubic and mixed anomalies arising from the boundary theory, the boundary condition for the hypermultiplets and the bulk CS couplings. The R-symmetry anomaly also receives contributions from the boundary conditions for the gauge fields, which is half of what a four-dimensional G ∂ multiplet would give. A review of N N,k The necklace quiver theory N N,k has a U (1) 2k global symmetry: rotations of the bi-fundamental hypermultiplets and instanton symmetries. See figure 2. A natural way to parameterize these symmetries follows from the UV realization of the gauge theory as a web of fivebranes drawn on a cylinder, with k infinite NS5 branes and N circular D5 branes, see Figure 20. The transverse position of the i-th top half-infinite NS5 brane is the mass parameter for a U (1)β i symmetry which acts with charge 1 on the X a i+1 a i fields of the i-th hypermultiplet, −1 on the Y a i a i+1 fields, 1/2 on the instanton charge at the (i + 1)-th node and −1/2 on the instanton charge at the i-th node. We define the k − 1 symmetry generators U (1) k U (1) β by quotienting by the diagonal symmetry U (1) t . Similarly, the transverse position of the i-th bottom half-infinite NS5 brane is the mass parameter for a U (1)γ i symmetry which acts with charge −1 on the X a i+1 a i fields of the i-th hypermultiplet, 1 on the Y a i a i+1 fields, 1/2 on the instanton charge at the (i + 1)-th node and −1/2 on the instanton charge at the i-th node. We define the k − 1 symmetry generators U (1) k U (1) γ by quotienting by the diagonal symmetry U (1) t . Finally, we need to pick a U (1) p symmetry generator whose mass parameter is associated to the sum of all gauge couplings, i.e. the size of the cylinder. We pick it to act on the instanton charge at the first node. In terms of fugacities, the hypermultiplet fields X a i+1 a i have fugacities tβ i γ −1 i , and the CS coupling at the i-th node is β i−1 γ i−1 β i γ i for i = 1, p β k γ k β 1 γ 1 otherwise. In the following we will encounter variants of this symmetry labeling, where the U (1) β i and U (1) γ i are permuted among themselves by permutations σ and τ respectively, and the U (1) p symmetry generator act on the i-th node. We can denote that as N σ,τ,i N,k . We will also often consider boundary conditions which break U (1) p . In that case, we can refer to N σ,τ, * N,k . In the low energy quiver gauge theory, the mass parameters for these global symmetries are constrained by the requirement that the gauge couplings should be positive. The UV description in terms of fivebranes suggest that symmetry enhancements should occur when some semi-infinite branes are brought together. A maximal case is when all semi-infinite branes are brought together, giving rise to a SU (k) β × SU (k) γ symmetry enhancement. Indeed, as the branes live on a cylinder there are k distinct ways to reach such a symmetry enhancement, as one bring the branes together in the order i, i + 1, · · · , i − 1. The parametrization of N σ,τ,i N,k is adapted to that order. We should also remember the six-dimensional UV completion of the five-dimensional quiver gauge theory, in terms of a circle compactification of the (1, 0) SCFT corresponding to N M5 branes in an A k−1 singularity. Then U (1) p becomes the KK momentum and SU (k) β × SU (k) γ × U (1) t become six-dimensional global symmetries. Then the five-dimensional mass parameters are lifted to the inverse radius of the compactification circle and to Wilson lines for the six-dimensional global symmetries. Intuitively, we may hope to find "duality walls" in the five-dimensional gauge theory, which express the invariance of the UV theory under permutation of two consecutive symmetries U (1) β i and U (1) β i+1 or U (1) γ i and U (1) γ i+1 . More precisely, such a wall would arise as the low energy limit of a Janus configuration in the UV, where the mass parameters for these symmetries are brought across each other as we move along the fifth direction. The notion of duality walls for five-dimensional gauge theories is explored in depth in a separate publication [43]. Here we mention them because the domain walls which are associated to spheres with two punctures and non-trivial discrete charges will turn out to coincide, amazingly, with the duality walls for the N σ,τ,i N,k theories associated to permutations of the U (1) β i and U (1) γ i 5d mass parameters, i.e. of the 6d flavor Wilson lines. This provides a powerful check of our conjectural interpretation of discrete charges as curvature charges for the six-dimensional U (1) β i and U (1) γ i global symmetries. Boundary conditions from maximal punctures Consider any of our four-dimensional gauge theories, with a positive puncture of colour 0. We will build from it a U (1) p -breaking right boundary condition for N N,k , i.e. a boundary condition for N * N,k The anomalies and superpotential couplings match nicely. We can deform the X = 0 boundary conditions by coupling the Y a i a i+1 to the mesons M and on the right to N * N,k . The fact that the interface breaks U (1) p is consistent with the six-dimensional interpretation of the system as an infinite tube with a minimal puncture with a specific location on the cylinder. Upon closing the minimal puncture, though, we will find that U (1) p is restored. Closing minimal punctures and duality walls Next, we close the minimal puncture in the interface between N The vev Higgses the two gauge groups together, so that they form a single SU (N ) 1 gauge group stretching across the interface. The vev also couples the bifundamentals of fugacities √ tγ −1 1 α −1 = tβ 1 /γ 1 linearly to the five-dimensional hypermultiplets on the right hand side of the interface, and √ tγ −1 k α −1 = tβ 1 /γ k on the left hand side, converting the corresponding X = 0 to Y = 0 boundary conditions. The match between the global symmetries across the new interface works almost exactly as before, except for a crucial difference: the U (1) p symmetry acting on the SU (N ) 1 instanton charge both on the left and on the right of the interface is now unbroken! This is consistent with the picture that the minimal puncture has been erased. We have obtained an interface between N (2,3,··· ),(1,2,··· ),1 N,k and N 1 N,k . It turns out that such an interface can be actually decomposed into k − 1 simpler interfaces, each corresponding to a simple duality operation. The rightmost interface is between N The simple interface associated to a permutation of β 1 and β 2 lets all five-dimensional gauge groups go through the interface, except for SU (N ) L 2 and SU (N ) R 2 , which are coupled by a bi-fundamental chiral field Q of fugacity β 2 /β 1 . In turn, det Q is coupled to a gauge-neutral chiral operator b by a linear superpotential W = b det Q. The hypermultiplets also just go through the interface, except the ones charged under SU (N ) L,R 2 . On the right, we set to zero the X R 2 field of fugacity tβ 2 /γ 2 and the Y R 1 field of fugacity t −1 β −1 1 γ 1 . On the left, we set to zero the Y L 2 field of fugacity t −1 β −1 1 γ 2 and the X L 1 field of fugacity tβ 2 γ −1 1 . We introduce superpotential couplings QY L 1 X R 1 + QX L 2 Y R 2 . This gives an interface between N (2,1,3,4,··· ), (1,2,··· ),i N,k and N i N,k for every i except when i = 2. We would like to interpret it as the duality interface for permuting β 1 and β 2 . In a similar manner, we could consider an interface between theories N (1,2,··· ), (2,1,3,4,··· ),i N,k and N i N,k for every i except when i = 2, involving a bi-fundamental chiral fieldQ of fugacity γ 2 /γ 1 going in the opposite direction. We would need superpotential couplingsQX L 1 Y R 1 + QY L 2 X R 2 . We would like to interpret it as the duality interface for permuting γ 1 and γ 2 . In order to test our interpretation, we should check that these interfaces fuse in a manner consistent with the expected properties of the permutation group of the β i and the γ i . The first test is obvious: permuting β 1 and β 2 twice should give the identity, and thus concatenating two copies of the corresponding duality interface should give a trivial interface. This works indeed as expected: the SU (N ) 2 gauge theory in between the interfaces gives at low energy a four-dimensional SU (N ) gauge group with N f = N , as the boundary conditions at the interfaces eliminate completely the bulk hypermultiplets in the interval, and only the two sets of Q chiral multiplets remain, together with the baryonic couplings. We know that such a theory in the IR reduces to the QQ mesons with a non-zero vev, which Higgses the five-dimensional gauge theories on the two sides of the fused interface together. A second test is that permuting β 1 and β 2 should commute with permuting γ 1 and γ 2 . It turns out that this follows from a neat application of Seiberg duality. The SU (N ) 2 gauge theory in between the interfaces now has N f = 2N : the Q andQ give 2N "quarks", while the bulk hypers in the interval give 2N "antiquarks". Seiberg duality at that node exchanges the setups which arise from fusing the interfaces in two different orders. It is even easier to check that the interface for permuting of β 1 and β 2 commutes with, say, the interface for commuting γ 2 and γ 3 . The σ 12 σ 23 σ 12 = σ 23 σ 12 σ 23 relations in the permutation group of the β i can also be checked. If we concatenate the corresponding triplets of interfaces, we get interfaces which support four-dimensional gauge theories which are related by the Seiberg duality of a SU (N ) theory with N f = 2N , up to a slight mismatch in the superpotential couplings which may perhaps explained away by operator mixing. We leave a more detailed analysis to [43]. Finally, we can comment on the relation between different ways of closing a minimal puncture and "curvature charges" in the six-dimensional puncture. By closing a minimal puncture with chirals of fugacities proportional to β i , we obtain an interface between N (2,3,··· ,1),(1,2,··· ),i N,k and N i N,k . The final order of the β j is independent of our choice, but the interfaces glue the U (1) p action on the two sides in different ways. We can try to match the interface with the behaviour of an Abelian SU (k) connection on an infinite tube, asymptotically flat at the two ends, but with curvature (1, 0, 0, · · · ) in between. The curvature will force the Wilson line parameter in the first eigenline to vary as m 1 → m 1 + 1/R as we go from right to left, thus passing across all other Wilson line parameters and back to the initial position on the circle. This seems to roughly match what we see in the interface. Punctures and orbifold Nahm poles It is natural to identify a maximal puncture with a D X Dirichlet boundary condition for the five-dimensional gauge theory, setting to zero the Y a i a i+1 half of the hypermultiplets: If we map a 4d theory T with a maximal puncture to a boundary condition B by coupling it to the 5d theory, we can recover the 4d theory T from the 5d theory on a segment, with boundary condition B at one end and Dirichlet at the other end. The surviving half X a i+1 a i of the hypermultiplets provide the expected "mesons" and the boundary condition produces the desired 't Hooft anomalies at the boundary. The five-dimensional perspective is useful in describing other types of punctures as well. For example, in class S general punctures are associated to a variant of Dirichlet boundary conditions, the Nahm pole boundary conditions. In this section we generalize that notion to the necklace quiver theories N N,k . The boundary conditions we are after are boundary conditions for BPS equations which describe field configurations of the 5d theory which preserve four-dimensional super-Poincare invariance. There are D-term and F-term equations. The F-term equations set the complex moment maps to be zero and require the hypermultiplet vevs to be covariantly constant under a complexified gauge connection of the form D 5 = D 5 − Φ, where Φ is the scalar super partner of the gauge bosons. The D-term equations set D 5 Φ to be equal to the real moment map for the hypermultiplets. For the necklace quiver, we can write schematically X i Y i = Y i+1 X i+1 , D 5 X i = Φ i+1 X i − X i Φ i , D 5 Y i = Φ i Y i − Y i Φ i+1 , (7.3) D 5 Φ i+1 = X i X † i − Y † i Y i − X † i+1 X i+1 + Y i+1 Y † i+1 . We are interested in deformations of the D X boundary conditions, which still set the Y i fields to zero but enforce a singular behavior of the X i , Φ i fields, X i ∼ − B i x 5 , Φ i ∼ − A i x 5 , (7.4) with B i = A i+1 B i − B i A i , A i+1 = B i B † i − B † i+1 B i+1 . (7.5) We can call these boundary conditions orbifold Nahm poles. Notice that we can organize the A i and B i matrixes into two kN × kN matrices t 3 and t + , with t 3 block diagonal of blocks A i and t + with blocks B i under the diagonal. The t + and t 3 give an SU (2) embedding ρ into SU (kN ), with the property that t 3 commutes with a diagonal matrix Ω with N eigenvalues equal to 1, N equal to e 2πi/k , etc. while t + has charge 1 under the action of Ω. If we start from D X and we give a vev to the "mesons", i.e. the boundary values of the X i , such that the kN × kN matrix M with blocks X i under the diagonal is nilpotent, it is natural to expect the boundary condition to flow in the IR to the orbifold Nahm pole labelled by the su(2) embedding ρ in SU (kN ) associated to M . The vevs we used to fully close a maximal puncture are a perfect example of this setup: the "chain" of k(N −1) meson vevs engineers a nilpotent matrix M with a single, large Jordan block. Discussion In this paper we have given a basic description of some the properties of class S k theories. Starting from core theories which we associate to spheres with two maximal and a bunch of minimal punctures we discovered that in order to build classes of theories closed under gaugings and RG flows triggered by vevs for a very particular set of chiral operators we should consider theories which are naturally associated to spheres with more general combinations of punctures. Moreover, the theories should be labelled not just by a punctured Riemann surface, but also by a collection of discrete charges. Some of the theories which followed from our considerations are strongly coupled SCFTs. Some of these can be thought of as IR fixed points of a Lagrangian theory but some lack such a description. Using these SCFTs we in principle now can construct theories corresponding to Riemann surfaces of arbitrary genus. Our analysis is rather incomplete in many respects and leaves room for many new insights to be uncovered. We can supplement our analysis with a very partial list of interesting questions and open problems which we would like to see addressed in the near future. Comparison with bipartite theories Although we have focussed on quivers drawn on a cylinder and compactifications of the sixdimensional theories on surfaces of genus 0, we could have also readily defined core theories associated to tori with kn minimal punctures, by gluing together the two maximal punctures of our standard core theories. It would be interesting to explore in full the relation between such genus 1 class S k theories and the standard bipartite quiver gauge theories associated to toric Calabi Yau singularities. See appendix C for some examples. Classification of punctures We have discussed at first two types of punctures, maximal and minimal, the former coming in k varieties. We have argued that more general "regular" punctures may be defined by turning vevs of collections of mesons at a maximal puncture, and proposed a five-dimensional classification in terms of su(2) embeddings in SU (kN ) commuting with a Z k subgroup. The duality walls we encountered in five dimensions also suggest that one may define maximal or other regular punctures labelled by general permutations of the β i and γ i as well. A natural open problem is to complete a systematic classification of "regular" punctures and the corresponding quiver tails. Precision study of the spectrum We have given a prescription to compute, at least in principle, the index for any of the theories in A N −1 class S k . It would be interesting to actually extract from the indices information about the operators of the putatively new strongly coupled SCFTs. For example, their marginal and relevant deformations [18]. It would be also interesting to find a systematic way to determine their conformal anomalies. Extension to other (1, 0) SCFTs In this paper we discussed theories obtained, conjecturally, by reducing the T N k (1, 0) SCFTs associated to N M 5 branes on A k singularity. It would be interesting to extend the discussion to other types of (1, 0) SCFTs. Some (1, 0) SCFTs can be obtained from RG flows initiated by Higgs branch vevs in the T N k theories. It may be possible to track the four-dimensional image of these RG flows, perhaps by giving vevs to chiral operators which are charged under intrinsic symmetries only. It should be possible to extend our work to other 4d gauge groups and 6d SCFTs by considering "core" theories defined by brane systems enriched by orientifold planes. Reduction to three dimensions We can consider reducing theories of class S k to three dimensions. When we have a description of the theory in four dimensions in terms of a conformal Lagrangian, in three dimensions the description will not in general be conformal. Moreover, the reduction on a circle will produce additional superpotentials involving monopole operators [44]. Such superpotentials break explicitly symmetries in three dimensions which are anomalous in four dimensions. Keeping track of such superpotentials is crucial to have dualities work for the compactified theories. One interesting aspect of the dimensional reduction in class S 1 was that the reduced theories possess a mirror description which was always Lagrangian, a star-shaped quiver with arms associated to the punctures of the original theory [45]. The difference operators and wave-functions [46] had a simple interpretation in the language of domain walls interpolating between two S-duality frames of N = 4 SYM with SU (N ) gauge groups [47]. The eigenvalue equation was associated to the S-duality relation between a Wilson line a 't Hooft line on the two sides of the duality wall [12,48,49]. We do not know how much of this will generalize to class S k . The difference operators do appear to be related to 't Hooft line operators in a necklace quiver. It may be that the 3d limit of the wave-functions is still related to domain walls interpolating between different duality frames of N = 2 four-dimensional necklace quivers. It would be interesting to explore this idea and perhaps build universal mirrors for the compactification of class S k theories. Surface defects We have discussed RG flows introducing surface defects into our theories. It would be interesting to study properties of these defects in more detail. They appear to give a broad generalization of the elliptic RS difference operators encountered in class S theories. Holography Class S k theories can be in principle studied in large N limit. For example one can consider theories corresponding to Riemann surfaces without punctures to avoid proliferation of flavor symmetries. Such AdS 5 backgrounds were recently considered in [50,51] generalizing some of the k = 1 results discussed in [52]. It would be interesting to study the relations between holography and our results in more detail. Geometrization of Seiberg dualities Starting from quiver theories with bifundamental matter and employing different types of dualities one can in principle generate gauge theory with matter in more intricate, tensor, representations. See appendix D for example. It would be interesting to understand whether class S k can serve as a natural setup to systematize the diverse variety of Seiberg dualities. Quantum mechanical models It will be interesting to study in more detail the quantum mechanical models for which our wavefunctions are eigenfunctions. Here we have discussed in some detail the case of A 1 and k = 2 and the generalization to higher k and N though rather straightforward might be interesting. One can also try and compute other partition functions, such as lens index [53], for class S k theories. In terms of eigenfunctions while the supersymmetric index is related to symmetric functions (and polynomials) the lens index in general is a natural generalization to non-symmetric functions [54,55]. The lens index should also provide a window into subtleties with global properties of the gauge and flavor groups [56]. Here j i are the Cartans of the SO(4) ∼ SU (2) 1 × SU (2) 2 isometry of S 3 and r is the U (1) r R-symmetry. The charges q a correspond to U (1) global flavor symmetries with the set of these symmetries denoted by F. The supersymmetric index of a free chiral field of R-charge R is given by I χ = ∞ i,j=0 1 − p 1− R 2 +i q 1− R 2 +j u −1 1 − p R 2 +i q R 2 +j u = Γ((pq) R 2 u; p, q) ≡ Γ e ((pq) R 2 u) . (A.2) For the sake of brevity In this paper Γ e (z) will stand for the elliptic Gamma function Γ(z; p, q) implicitly defined above. When an SU (N ) z flvor symmetry of a theory with index I(z) is gauged, the index of the gauge theory is given by, I = (q; q) N −1 (p; p) N −1 N ! N −1 i=1 dz i 2πiz i N i =j Γ e (z i /z j ) I(z) , (A.3) with (z; q) ≡ ∞ i=0 (1 − zq i ) , (A.4) being the q-Pochhammer symbol. The index of a free vector field is given by, I V = (p; p)(q; q) . (A.5) We will also encounter theta functions which we will define as θ(z; q) = (z; q)(qz −1 ; q) . (A.6) An important property of the supersymmetric index is that it is independent of the continuous parameters of the theory, such as marginal couplings, and also is independent of the RG scale. Thus it can be computed for example in the UV using a non-conformal description and be the same as the superconformal index of the IR CFT. We will adopt the standard convention that "ambiguous" powers in the arguments of a function denote product over all the possibilities. For example the index of the N = 2 hypermultiplet is given by, Γ e (t 1 2 z ±1 ) = Γ e (t 1 2 z)Γ e (t 1 2 z −1 ) . (A.7) B. Eigenfunctions for A 1 class S 2 We will discuss here explicit algorithm to construct eigenfunctions for A 1 theories of class S 2 in the p = 0 limit. The simplifying factor here is that the eigenfunctions turn out to be proportional to polynomials. For p = 0, as also is the case for class S 1 , the computation is much more involved. However, in principle a perturbative computation in p around p = 0 solution can be set up. See for example [59]. For p = 0 the computation is rather straightforward. The equation we have to solve is S (β,−) (0,1) · ψ(u) = E ψ(u) . (B.1) Here S (β,−) (0,1) was explicitly given in section 6.1. A way to proceed is to make an anzats, ψ(u) = K max (u)χ(u) , (B.2) with K max (u; β, γ) = 1 (t( β γ ) ±1 u ±1 1 u ±1 2 ; q) . (B.3) Then we assume χ(u) to be a polynomial in u i with general coefficients of maximal degree n and symmetric in u i → u −1 i . Since our indices are invariant under u → −u we also can assume that χ(u) has well defined parity. We will have then for polynomials of order n order 1 4 n 2 free parameters, and we also have the eigenvalue as a parameter. Expanding the eigenvalue equation in u i and demanding it to hold for any monomial term will give us a system of nonlinear equations for these coefficients. The non-linearity comes because the eigenvalue can multiply other coefficients. We then should solve these system of algebraic equations which in general has a finite set of solutions. Let us illustrate this explicitly for the lowest orders. Assuming the polynomial is of degree zero we get that S (β,−) (0,1) · 1 − E ∝ E − 1 − β 4 t 4 + β 2 γ 2 t 2 + β 2 t 2 γ 2 , (B.4) and thus the order zero polynomial is an eigenfunction given that, E = 1 + β 4 t 4 − β 2 γ 2 t 2 − β 2 t 2 γ 2 . (B.5) The first eigenfunction is then given by, ψ 0 (u) = K max (u) (t 2 ; q)(t 2 β γ ±2 ; q) . (B.6) We have normalized it to have unit norm under the gluing measure. At next order we make an ansatz, χ(u) = u 1 + 1 u 1 + H(u 2 + 1 u 2 ) . (B.7) Then we compute taking q → 0 to avoid horrendous expressions lim q→0 q 1 2 (S (β,−) (0,1) − q − 1 2 E ) · χ(u) u 0 1 = (B.8) − u 4 2 − 1 H t 2 β 4 γ 2 − β 2 − γ 2 − γ 2 (E − 1) + βγt β 2 γ 2 t 2 − 1 γ 2 u 2 = 0 , lim q→0 q 1 2 (S (β,−) (0,1) − q − 1 2 E ) · χ(u) u 3 1 = (B.9) u 2 2 − 1 βHt t 2 − β 2 γ 2 − γ E + β 2 γ 2 t 2 − 1 γ = 0 . Here E = q − 1 2 E . This system of equations can be reduced to a quadratic equation in one of the two variables H or E and thus has two solutions. Ultimately we obtain, H = (B.10) ∓ 4β 4 γ 4 (1 − t 2 β −2 γ −2 ) (1 − t 2 β 2 γ 2 ) + t 2 (1 − β 2 γ 2 ) 2 (β 2 + γ 2 ) 2 + t(1 − β 2 γ 2 ) β 2 + γ 2 2β 3 γ 3 (1 − t 2 β −2 γ −2 ) . We should further normalize χ(u) to have norm one. We note that this expression is algebraic in fugacities. Taking for simplicity β, γ = 1 we obtain that here H = ±1 recovering the result advertised in section 6.1. Note also that with this value for H all the symmetry properties (3.12) are satisfied. We can continue to derive eigenfunctions in this manner. Going to higher orders we get higher order polynomial equations which do not have closed form solutions, but we can solve them perturbatively in the fugacities. The self-adjointness of the difference operator Let us check that the basic difference operator we computed is self adjoint under the gauging measure. That is for two sufficiently nice behaving functions f (u) and g(u) we have, du 1 u 1 du 2 u 2 Γ e ( pq t ( β γ ) ±1 u ±1 1 u ±1 2 ) Γ e (u ±2 1 )Γ e (u ±2 2 ) f (u)S (β,−) (0,1) (u † )g(u † ) = (B.11) du 1 u 1 du 2 u 2 Γ e ( pq t ( β γ ) ±1 u ±1 1 u ±1 2 ) Γ e (u ±2 1 )Γ e (u ±2 2 ) (S (β,−) (0,1) (u)f (u))g(u † ) . We note that, Γ e ( pq t ( β γ ) ±1 u ±1 1 u ±1 2 ) Γ e (u ±2 1 )Γ e (u ±2 2 ) u i →q 1 2 u i = θ(q −1 u −2 1 ; p) θ(u 2 1 ; p) θ(q −1 u −2 2 ; p) θ(u 2 2 ; p) θ( t q ( β γ ) ±1 1 u 1 u 2 ; p) θ(t( β γ ) ±1 u 1 u 2 ; p) Γ e ( pq t ( β γ ) ±1 u ±1 1 u ±1 2 ) Γ e (u ±2 1 )Γ e (u ±2 2 ) . (B.12) Then we have that Γ e ( pq t ( β γ ) ±1 u ±1 1 u ±1 2 ) Γ e (u ±2 1 )Γ e (u ±2 2 ) T (u −1 2 , u −1 1 )g(q − 1 2 u 2 , q − 1 2 u 1 )f (u 1 , u 2 ) u i →q 1 2 u i = (B.13) Γ e ( pq t ( β γ ) ±1 u ±1 1 u ±1 2 ) Γ e (u ±2 1 )Γ e (u ±2 2 ) T (u 1 , u 2 )g(u 2 , u 1 )f (q This implies that under change of coordinates {u i } → {q 1 2 u i } the first term, out of four, on the left hand side of (B.11) in the expansion of the difference operator maps exactly to the fourth term on the right hand side. This can be repeated for the other three terms and thus implies that the difference operator is self-adjoint. The index of generic class S 2 A 1 theory Let us here write down the index of a generic theory residing in A 1 class S 2 . We will specialize to the case p = 0 where we can write very explicitly expressions for the index. The index is given in terms of the following building blocks. The eigenfunctions ψ λ (u) ≡ K max (u)χ λ (u) are associated to the maximal punctures. From (4.8),(4.7), and (5.2) we deduce that C (γ,+) λ ≡ Φ λ (β, γ) = (t 2 γ 2 β ±2 ; q) (t 2 γ −2 β ±2 ; q) 1 2 χ λ (t, γ β ; β, γ) χ λ (tγ 2 , 1 βγ ; β, γ) . (B.19) The index of a theory corresponding to genus g surface with m u upper maximal punctures, m d maximal lower punctures, m m minimal punctures, charge β under U (1) β discrete symmetry, and charge γ under U (1) γ discrerte symmetry is given by, m d i=1 K max (u i ; β, γ) mu j=1 K max (u j ; β −1 , γ) For example the index of the free trinion is given by β ; β, γ) χ λ ( γ β , t; β, γ)χ λ (γβ, tβ −2 ; β, γ) I uα v = (t 2 β ±2 γ ±2 ; q) 1 2 (t( β γ ) ±1 u ±1 1 u ±1 2 ; q)(t(βγ) ±1 v ±1 1 v ±1 2 ; q)(tβ ±2 α −2 ; q)(tγ ±2 α 2 ; q) × λ χ λ (u; β, γ)χ λ (v; β −1 , γ)χ λ (t . (B.21) Note that if we specialize fugacities for different symmetries, i.e. ignore/break corresponding symmetries the indices simplify. For example, taking γ = β, and thus identifying the two corresponding U (1) symmetries, the difference between 1 and 2 is gone and we label the theories by Riemann surface and one integer. Taking β = 1 and thus neglecting the U (1) β symmetry there is no difference between upper and lower punctures and also 1 is not a meaningful number any more. Let us mention here that setting both β = γ = 1 one can derive from (5.4) the following factorization property, χ λ (t 1 2 α, t 1 2 α −1 )χ λ (t 1 2 δ, t 1 2 δ −1 ) χ λ (1, t) = χ λ (αδ −1 , αδ) . (B.22) Which in particular implies that switching off β and γ we cannot distinguish in the index maximal punctures from pairs of minimal ones. The eigenfunctions satisfy many properties which guarantee them to be consistent with our considerations. For instance we can check that χ λ ( γ β , t; β, γ) χ λ (γβ, tβ −2 ; β, γ) ; β, γ) = χ λ (tγ 2 , 1 βγ ; β, γ) χ λ (t, γ β ; β, γ) which implies that reducing maximal puncture in two different ways to minimal punctures discussed in section 5.1 gives the same result. Sphere with two maximal punctures We can start from the (β, −) interacting trinion of section 4.2 and close the minimal puncture. There are here only two bayons available for which we can turn on vevs. These correspond to either δ = t 1 2 β −1 or δ = t − 1 2 γ −1 as can be seen from (6.1). In both cases the gauge group is Higgsed and we get a collection of chiral fields. In the former case the index becomes (in Macdonald limit for simplicity), is given by a collection of chiral fields coupled through a superpotential. This theory has − 3 2 units of U (1) β discrete charge and 1 2 unit of U (1) γ discrete charge. The index of this theory in Macdonald limit is I (β,− 3 2 ),(γ,+ 1 2 ) (u, δ, ) = (B.27) (β 4 ; q)( β 2 γ 2 ; q)(q; q) ( t δ u ±1 1 ; q)( tδ βγ u ±1 2 ; q)( β 2 δ u ±1 1 ; q)( δβ γ u ±1 2 ; q)( t β 2 2 ; q)(tγ 2 2 ; q)( t 2 γ 2 β 2 ; q) × dz 4πiz (z ±2 ; q)(t 3 2 γ z ±1 ; q) (t 1 2 1 βδ u ±1 2 z ±1 ; q)(t 1 2 γδu ±1 1 z ±1 ; q)( β 2 t 1 2 γ z ±1 ; q) = (β 4 ; q)( β 2 γ 2 ; q)(tβγδ u ±1 2 ; q) (tβ −1 γ −1 δ u ±1 2 ; q)( β γ ( δ ) ±1 u ±1 2 ; q)(β 2 (δ ) ±1 u ±1 1 ; q)(t γ β u ±1 1 u ±1 2 ; q)( t β 2 δ 2 ; q)( t β 2 2 ; q)(tγ 2 δ 2 ; q)(tγ 2 2 ; q) . Note that the result is explicitly invariant under exchanging the two minimal punctures. We can attach this theory to a trinion with opposite discrete charges and three maximal punctures of the same color to obtain yet another duality frame for the basic four punctured sphere. For the eigenfunctions the above implies the following relation, 2 ) (u † , , δ) (u ±2 1 ; q)(u ±2 2 ; q)(t( γ β ) ±1 u ±1 1 u ±1 2 ; q)ψ λ (u) . C λ Φ λ (β, γ) Thus we can glue our new trinion to an interacting trinion with appropriate discrete charges to obtain yet another Argyres-Seiberg like frame for our basic interacting four-punctured sphere. One might consider closing punctures in the new trinion. We can further close the maximal puncture in two different ways to obtain a theory corresponding to sphere with three maximal punctures. We also can close a minimal puncture in four different ways and obtain a sphere with one maximal and one minimal punctures. C. Fun with tori Let us discuss here several simple examples of theories corresponding to torus with one minimal puncture or no puncture in the k = 2 A 1 case. Torus with one minimal puncture can be obtained for example by gluing together the two maximal punctures of the trinion with two maximal punctures of same color and a minimal puncture with one of the discrete charges, i , being ±1; the theory we obtained by closing a minimal puncture in our core example of an interacting theory. Since here we have a Lagrangian, depicted in Figure 21, we can observe that the theory has one-dimensional conformal manifold. Giving a vev to one of the chiral fields as depicted in Figure 22 we Higgs two of the gauge groups and obtain a theory which we can associate to torus with no punctures but with two units of one of the discrete charges in class S 2 . This theory has an alternative interpretation as A 1 theory of class S 1 corresponding to torus with two punctures (with additional singlet fields flipping the quadratic operators built from the two adjoints). This theory is also know as Y p=1,q=1 (modulo the extra superpotential terms) in the Y p,q nomenclature [60]. We have an additional gauge singlet field coupled to one of the bifundamental chirals connecting two nodes without an adjoint field. Another two equivalent ways to obtain a torus with no punctures come from giving vacuum expectation value to bifundamental built from quarks connecting one of the other pairs of gauge groups. This theory has a unit of both β and γ discrete charges. The theory so obtained is the T 1,1 theory or Y p=1,q=0 theory (again with extra singlets and superpotential), SU (2) 2 gauge theory with four bifundamental chirals and a superpotential. See figure 23. Finally we can give a vacuum expectation value to the quadratic singlet built from the adjoint field. This amounts to setting δ = t 1 2 β. This vacuum expectation value Higgses the corresponding group and gives masses to four of the quarks. The remaining theory is the SU (2) 2 with two bi-fundamental flavors and two additional singlets and is associated to torus with no punctures and no discrete charges. See figure 24. D. Class S 2 interpretation of a selfdual SU (4) SYM with tensor matter Let us take our basic interacting four punctured sphere of A 1 class S 2 and perform Seiberg duality on one of the gauge nodes. We have discussed this duality in section 3 and here let t t where this equation can be viewed as definition of Υ λ . We will derive the relation of Υ λ to the eigenfunctions soon. The fact that (D.7) can be written in terms of single sum over eigenfunction is a manifestation of S-duality or in this case the Seiberg duality of [37]. We have the following relation, T (v 1 , v 2 ; β, γ, t) = T (v 2 , v 1 ; β −1 , γ −1 , pq t ) = θ(tv 1 v 2 ( γ β ) ±1 ; p)θ( tβ qγ v 1 v 2 ; p)θ( tβ 3 γ q v 2 v 1 ; p) θ(v 2 1 ; p)θ(v 2 2 ; p) . (E.3) We can derive yet another relation between eigenfunctions. Let us glue as in section 6 an interacting trinion to a general theory but now glue it to the puncture of the opposite type, i.e. glue ψ puncture toψ one without extra fields but with a quartic superpotential. Next, we close the minimal puncture residing on the interacting trinion. Assuming dualities to hold, i.e. that we can freely move around on the Riemann surface different punctures and consider various pair of pants decompositions leading to the same theory, we deduce thatψ and ψ have to be orthonormal under the measure involving only the N = 1 vector multiplets. These two facts translate into the following relation between eigenfunctions, ψ λ (u) = Γ e ( pq t β γ ±1 u ±1 1 u ±1 2 )ψ λ (u) . (E.4) Note that this means thatψ are eigenfunctions of the following operator, S (β,−) (0,1) = Γ e ( pq t β γ ±1 u ±1 1 u ±1 2 ) S (β,−) (0,1) Γ e (t β γ ±1 u ±1 1 u ±1 2 ) . (E.5) We can compute this operator to be . (E.7) Note also that the operators S In the k = 1 case discussed in [18] the two types of difference operators turn out to coincide but for higher k one derives two different commuting operators. Figure 2 : 2Figure 2: The necklace quiver N N,k in five dimensions. It is a Z k orbifold of the maximally supersymmetric YM, with k SU (N ) nodes. Notice that each link here represents a full bi-fundamental hypermultiplet and each node a full SU (N ) vector multiplet. There is U (1) global symmetry associated with each link, rotating the bifundamental hypermultiplets. There is also a U (1) symmetry, the instanton symmetry, associated with each gauge group node. Figure 3 : 3Figure 3: The "honeycomb" bi-partite graph drawn on a cylinder and the corresponding quiver gauge theory. The top and bottom lines are identified. Each cell of the bi-partite graph maps to an SU (N ) gauge group with 3N flavors corresponding to the six edges of the cell. Each node of the bi-partite graph indicates a cubic superpotential term, with sign associated to the color of the node a i b i+1 : the former transform as fundamental/anti-fundamental representation of SU (N ) i × SU (N ) k−i+1 and the latter in anti-fundamental/fundamental representation of SU (N ) i × SU (N ) k−i . The sets of k SU (N ) i and k SU (N ) i global symmetries are associated respectively with the two maximal punctures. Figure 8 : 8Seiberg duality of one of the SU (N ) gauge groups in the case of k = 2. The node with a circle is an SU (2N ) gauge group. There is a superpotential term for each triangle in this quiver theory. Figure 9 : 9The sphere with two maximal punctures of different color and one minimal puncture, aka the free trinion. Figure 10 : 10The sphere with two maximal punctures of same kind and two minimal punctures. we end up removing most of the chiral fields in the two corresponding free trinions. The SU (N ) 1 gauge groups to the left of the two trinions, in the middle, and to the right are all Higgsed to a common SU (N ) 1 . The other SU (N ) 2 gauge group which glues the two trinions together survives, but most of the fields charged under it are lifted by mass terms: only bifundamentals connecting it to the left and right SU (N ) 2 gauge groups survive. Thus we have a node with N f = N flavors. The low energy description of such a node is well-known: a set of mesons and baryons constrained by the equation, det M − BB = Λ 2N . Figure 12 : 12Gluing the free trinion to a general theory. Figure 14 : 14The sphere with two maximal punctures of the same kind and one minimal puncture. This corresponds to N f = 4 SU (2) SQCD with some singlet fields and a superpotential. with δ being a fugacity for U (1) δ and the gauge fugacity for SU (2) 1 to z 1 = αδ. for the first type of maximal to minimal reduction we obtain the same result as if we were setting δ = √ tβ −1 in the same type of maximal to minimal reduction (u 1 , u 2 fugacities of the next row of gauge groups. These enforce vevs of mesons at the next trinion with fugacities y ) −1 tβ k γ −1 k−2 for a = 1, · · · , N − 2, thus Higgsing all but three SU (N )s to SU (2). As the vevs propagate along the quiver, the number of SU (2) groups increases linearly. At some point SU (3) groups appear, etc.We can parameterize the fixed z to δ. Thus in order to check for a symmetry between α and δ we only need to focus on the initial trinion.Few fields charged under the surviving SU (2) survive Higgsing and do not get a mass: z and pinch the contour at z 1 = αδ. The residue of the index is given by Figure 17 : 17The sphere with two lower maximal punctures and one upper maximal puncture. This corresponds to the simplest position-dependent vev for the baryon operator. Figure 20 : 20The brane picture for the necklace quiver N N,k . The top and bottom edges are identified forming a cylinder. . The gauge anomaly cancels out because of the balance between fundamental and anti-fundamental fields. The SU (N ) i gauge group has a mixed anomaly associated to the fugacity tβ i−1 γ γ i , which precisely cancels against the contribution from the CS coupling for a right boundary condition. Similarly, a positive puncture of color n can be coupled from the right to N (n+1,n+2,··· ),(1,2,··· ), * N,k . We can use negatively oriented punctures to define left boundary conditions as well. For example, if we have a negatively oriented puncture of color 0, shifting the i indices by one so that mesons have fugacity tβ i γ −1 i and anomalies tβ i γ −1 i−1 , the total anomalyβ i γ i β i−1 γ i−1precisely cancels against the contribution from the CS coupling for a left boundary condition. Thus a negative puncture of color n can be coupled from the left to N a vev to the bifundamental chirals of fugacity √ tβ 1 α which are coupled to the five-dimensional gauge groups SU (N ) L 1 and SU (N ) R 1 . k and we can associate it to the permutation of β 1 and β 2 . The next interface is between N can associate it to permutation of β 1 and β 3 , etc. Φ (β, γ) γ Φ(γ, β) − β m d i=1 χ λ (u i ; β, γ) mu j=1 χ λ (u j ; β −1 , γ) mm k=1 χ m λ (δ k ) . Figure 21 : 21Torus with one puncture and one of the i = ±1. The nodes are SU (2) gauge groups. T (v 1 , v 2 ; β, γ, t) = θ(tv 1 v 2 ( γ β ) ±1 ; p)θ( are not the same and an analogue of the former we did not encounter till now for theories without discrete charges for U (1) t . The functionsψ have to be eigenfunctions of both operators if our assumptions are correct and indeed the two operators are selfadjoint under the same measure and commute, This requirement is for a generic theory. As we will see say in appendix B, in some degenerate examples some mesons may be naively missing, but can be reinstated by adding pairs of gauge-neutral chiral fields with superpotential masses which allow one to integrate them away. One element of each pair play the role of the missing meson and the other is coupled to the rest of the theory by superpotential couplings. That follows from the symmetry between Y = 0, ∂ ⊥ X = 0 and X = 0, ∂ ⊥ Y = 0 and the fact that we can switch from one to the other by a flip AcknowledgmentsWe would like to thank Chris Beem and Brian Willett for useful discussions. The research of SSR was partially supported by "Research in Theoretical High Energy Physics" grant DOE-SC00010008. SSR gratefully acknowledges support from the Martin A. Chooljian and Helen Chooljian membership during his stay at the Institute for Advanced Study, and would like to thank KITP, Santa Barbara, and the Simons Center, Stony Brook, for hospitality and support during different stages of this work. The research of DG was supported by the Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development and Innovation.A. The supersymmetric indexThe supersymmetric index[57,58]counts with signs and weights the protected operators of the theory. It can be defined either as an S 3 × S 1 partition function with supersymmetric boundary conditions on the S 1 or as a trace over the Hilbert space on S 3 . The latter definition takes the following form, I(p, q; u) = Tr H S 3 (−1) F p j 1 +j 2 − 1 2 r q j 1 −j 2 − 1 2 r a∈F u qa a .(A.1)We find that the eigenfunctions satisfy the symmetry properties(3.12), from which and from (4.8) we can deduce that (β 2 γ −2 ; q) (tα 2 β −2 ; q)(tα −2 γ 2 ; q) (tα −2 β −2 ; q)(tα 2 γ 2 ; q) χ λ (t Thus we deduce thatwith the K-factors given by,Finally the structure constant is set from (4.8) and (4.7) to be,(B.24)and in the latter case we obtain,Again, we can close the maximal punctures. Gluing these two-punctured spheres to a general theory does not change the numbers of punctures but shifts the discrete charges. For eigenfunctions this implies that they are "eigenfunctions" also of the following integral operators with well prescribed eigenvalues,One can check that these equations hold for the eigenfunctions that we derived. Thus the functions ψ λ are eigenfunctions of difference operators with eigenvalues related to surface defects, and are eigenvalues of integral operators with eigenvalues being related to discrete charges for intrinsic symmetries.Sphere with one maximal and two minimal puncturesWe can start from the (β, −) interacting trinion of section 4.2 and partially close one of the maximal punctures to obtain a trinion with one maximal and two minimal punctures. As we discussed we can do this in general in two different ways by giving vevs to two different mesons. However, in this case since the theory is not generic and has rather small flavor symmetry only one meson exists, and it corresponds to setting (u 1 , u 2 ) → (t 1 2 γ , t 1 2 β ). This can be clearly seen from the index (6.1). After turning on the vev the gauge part of the theory is SU (2) with three flavors as one flavor acquires mass. Thus, in the Seiberg dual frame it The two gauge groups connected by these quarks are Higgsed to a diagonal combination. The quarks with the vacuum expectation value do not couple through superpotential and thus do not generate mass terms. The resulting theory in the IR is depicted on the right. It is S 1 theory of type A 1 corresponding to torus with two punctures. The three class S 2 U (1) β × U (1) γ × U (1) t symmetries map to the U (1) t symmetry of class S 1 as well to the two U (1) symmetries corresponding to class S 1 punctures. us writing the index in the Seiberg dual frame,.The first line has the singlet mesonic operators appearing after the duality transformation and the last line is the N = 2 block. Note that we cannot here take Macdonald limit of the index of the ingredients. We can perform the S-duality transformation on the N = 2 block exchanging α with δ which will give us the exchange of the two minimal punctures which we discussed in section 3. However, we also can act with the other elements of the S-triality. Denoting, in the index computing the pole at δ = t − 1 2 γ, the two gauge groups connected by these quarks are Higgsed to a diagonal combination. The quarks with the vacuum expectation value do couple through superpotential and thus generate mass terms for two other quarks. The resulting theory in the IR is depicted on the right. It is the conifold, T 1,1 , theory. We have a quartic superpotential involving all the bifundamentals. Moreover there are two additional superpotential terms flipping the mesons built from β 2 and γ 2 fields.the first duality above would exchange b with d but now we will exchange c with b to obtainWe see that we have here the N = 2 block coupled to N = 1 SQCD with four flavors in fundamental representation of SU (4) and 2 flavors in the antisymmetric representation. The symmetry under exchanging the two minimal punctures, α ↔ δ, is manifest. Let us denote the N = 1 theory with the antisymmetric matter by T A . The index of this theory is,The theory T A is actually dual to itself under Seiberg duality[37]. The index of this duality was discussed in[61]. The global non-R symmetry of the theory is as follows,The group SU (2) z 2 rotates the two quarks in the antisymmetric representation of SU (4) gauge group. Under Seiberg duality we have the following map of the symmetries,Alternatively Seiberg duality can be thought of as exchanging (SU (2)) 2 u and (SU (2)) 2 v . We would like to understand this theory in terms of class S 2 . The two symmetries u and v live to two maximal punctures of the same color and the symmetries z 2 and αδ will need to be interpreted. We can write the index using the eigenfunctions asFrom here we deduce thatWe already have encountered a similar relation while closing maximal punctures (5.4),We thus use the elliptic Fourier transform to writeOn the second line we have the index of SU (2) SYM with three flavors which can be exactly evaluated in terms of the dual mesons,Plugging this back we note that some of the fields become massive and we obtain,Writing the index of T A using this expression we finally obtain,The index in the square brackets is that of the theory corresponding to a sphere with three maximal punctures of two different colors and appropriate discrete charges, Γ e (pq γ 2 β 2 )I (z,αδ) uv , we encountered in section 5.1. We thus deduce that the SU (4) SYM with four quark flavors and two antisymmetric tensors is equivalent to certain SU (2) gauging with extra chiral fields of the strongly-coupled trinion with three maximal punctures. We have a singlet field M 0 with fugacities t 2 γ 2 β 2 and R-charge zero, fields q with fugacities γ β (αβγδ) ±1 w ±1 and R-charge one, and fundamental quarks Q with R-charge one and fugacities t −1 β γ w ±1 . Moreover the SCFT has mesons M associated to the maximal puncture we partially gauge. This mesons are in fundamental of the gauge SU (2) and have fugacities t(γβ) ±1 (αδ) ±1 with R-charge zero. The superpotential we turn on is thus,(D.15)Note also that the gauging is non anomalous with all the symmetries at hand. We can now go back further to the four-punctured sphere. The duality frame (D.3) is thenOn the second line we have again SU (2) SYM with three flavors index of which can be evaluated to be,Plugging this back to (D.16) we obtainThis is the Argyres-Seiberg frame for the four-punctured sphere obtained previously in (5.5).E. Comments on index of theories with U (1) t discrete chargeLet us outline here how to compute the index of theories with U (1) t discrete charges. The index for k = 1 case was thoroughly discussed in[18](see also[62,63,64]). The higher k follows similar pattern but there are some new features which we will discuss here. We restrict the explicit discussion as usual to A 1 and class S 2 . First, as we discussed in section 5.2, we have to introduce a second set of our theories with charges under the global symmetries aproppriately flipped and R-charges shifted. 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[ "A structural test for the conformal invariance of the critical 3d Ising model", "A structural test for the conformal invariance of the critical 3d Ising model" ]	[ "Simão Meneses \nCentro de Física das Universidades do Minho e\nDepartamento de Engenharia Física\nFaculdade de Engenharia\nDepartamento de Física e Astronomia\nFaculdade de Ciências\nUniversidade do Porto\n4169-007Porto, PortoPortugal\n", "João Penedones \nInstitute of Physics\nÉcole Polytechnique Fédérale de Lausanne (EPFL)\nRte de la Sorge, BSP 728CH-1015LausanneSwitzerland\n", "Slava Rychkov \nInstitut des HautesÉtudes Scientifiques\nBures-sur-Yvette\nFrance\n\nDépartement de Physique\nEcole Normale Supérieure\nParisFrance\n\nTheoretical Physics Department\nCERN\nGenevaSwitzerland\n", "J M Viana ", "Parente Lopes \nCentro de Física das Universidades do Minho e\nDepartamento de Engenharia Física\nFaculdade de Engenharia\nDepartamento de Física e Astronomia\nFaculdade de Ciências\nUniversidade do Porto\n4169-007Porto, PortoPortugal\n", "Pierre Yvernay \nTheoretical Physics Department\nCERN\nGenevaSwitzerland\n" ]	[ "Centro de Física das Universidades do Minho e\nDepartamento de Engenharia Física\nFaculdade de Engenharia\nDepartamento de Física e Astronomia\nFaculdade de Ciências\nUniversidade do Porto\n4169-007Porto, PortoPortugal", "Institute of Physics\nÉcole Polytechnique Fédérale de Lausanne (EPFL)\nRte de la Sorge, BSP 728CH-1015LausanneSwitzerland", "Institut des HautesÉtudes Scientifiques\nBures-sur-Yvette\nFrance", "Département de Physique\nEcole Normale Supérieure\nParisFrance", "Theoretical Physics Department\nCERN\nGenevaSwitzerland", "Centro de Física das Universidades do Minho e\nDepartamento de Engenharia Física\nFaculdade de Engenharia\nDepartamento de Física e Astronomia\nFaculdade de Ciências\nUniversidade do Porto\n4169-007Porto, PortoPortugal", "Theoretical Physics Department\nCERN\nGenevaSwitzerland" ]	[]	How can a renormalization group fixed point be scale invariant without being conformal?Polchinski (1988)showed that this may happen if the theory contains a virial current -a non-conserved vector operator of dimension exactly (d − 1), whose divergence expresses the trace of the stress tensor. We point out that this scenario can be probed via lattice Monte Carlo simulations, using the critical 3d Ising model as an example. Our results put a lower bound ∆ V > 5.0 on the scaling dimension of the lowest virial current candidate V , well above 2 expected for the true virial current. This implies that the critical 3d Ising model has no virial current, providing a structural explanation for the conformal invariance of the model.	10.1007/jhep04(2019)115	[ "https://arxiv.org/pdf/1802.02319v3.pdf" ]	118,585,656	1802.02319	022ad197547abb074fdb27bef95199eab5917e45	A structural test for the conformal invariance of the critical 3d Ising model v1: February 2018 v2: January 2019 1 Feb 2019 Simão Meneses Centro de Física das Universidades do Minho e Departamento de Engenharia Física Faculdade de Engenharia Departamento de Física e Astronomia Faculdade de Ciências Universidade do Porto 4169-007Porto, PortoPortugal João Penedones Institute of Physics École Polytechnique Fédérale de Lausanne (EPFL) Rte de la Sorge, BSP 728CH-1015LausanneSwitzerland Slava Rychkov Institut des HautesÉtudes Scientifiques Bures-sur-Yvette France Département de Physique Ecole Normale Supérieure ParisFrance Theoretical Physics Department CERN GenevaSwitzerland J M Viana Parente Lopes Centro de Física das Universidades do Minho e Departamento de Engenharia Física Faculdade de Engenharia Departamento de Física e Astronomia Faculdade de Ciências Universidade do Porto 4169-007Porto, PortoPortugal Pierre Yvernay Theoretical Physics Department CERN GenevaSwitzerland A structural test for the conformal invariance of the critical 3d Ising model v1: February 2018 v2: January 2019 1 Feb 2019Dedicated to the memory of Joe Polchinski (1954-2018) How can a renormalization group fixed point be scale invariant without being conformal?Polchinski (1988)showed that this may happen if the theory contains a virial current -a non-conserved vector operator of dimension exactly (d − 1), whose divergence expresses the trace of the stress tensor. We point out that this scenario can be probed via lattice Monte Carlo simulations, using the critical 3d Ising model as an example. Our results put a lower bound ∆ V > 5.0 on the scaling dimension of the lowest virial current candidate V , well above 2 expected for the true virial current. This implies that the critical 3d Ising model has no virial current, providing a structural explanation for the conformal invariance of the model. Introduction It is believed that the critical point of the 3d ferromagnetic Ising model is conformally invariant. One strong piece of evidence is the excellent agreement between the critical exponents extracted from experiments and Monte Carlo simulations and from the conformal bootstrap [1,2]. Conformal invariance has been also checked directly on the lattice, by verifying functional constraints that it imposes on the shape of some correlation functions [3]. 1 In this paper we will provide another lattice test of this property, which is qualitatively different and in a sense more robust. Any field theory coming from a local action, and in particular the 3d Ising model close to or 1 We would also like to point out a related lattice study of conformal invariance in 3d percolation [4]. at the critical temperature, has a local stress tensor operator T µν which is conserved: ∂ µ T µν = 0. The structural property of conformally invariant local theories is that this local stress tensor operator is traceless: T µ µ = 0 . (1.1) Our new test will probe this structural property, unlike previous lattice studies which tested its consequences. The key question is: could the critical 3d Ising model be scale invariant (as befits any critical theory, being a fixed point of a renormalization group flow), but not fully conformally invariant? As was lucidly explained by Polchinski [5], 2 a theory will be scale invariant without being conformal if T µν is not traceless but its trace is a total divergence: T µ µ = ∂ ν W ν ,(1.2) where W µ is a vector operator, called the virial current, which is (a) not conserved and (b) not itself a total derivative. 3 Precisely this mechanism is responsible for scale without conformal invariance of the theory of elasticity, perhaps the simplest physically relevant example of this phenomenon [8]. 4 It's then natural to inquire if Eq. (1.2) can hold in the critical 3d Ising model, and we will show that it cannot. Our argument is based on the following simple observation: any operator W µ which is a candidate to appear in the r.h.s. of (1.2) must have two additional properties. First of all, it should, just as T µν itself, be invariant under the internal symmetry of the model, Z 2 in the case of Ising. In addition, since T µν has canonical scaling dimension d, operator W µ should have dimension d − 1 = 2. For the subsequent discussion, let us define V µ as the lowest Z 2 -even vector operator V µ , which is not a total derivative. If we manage to show that ∆ V > 2, this will imply that the model has no virial current candidates of appropriate dimension, and thus must be conformal. Extending the discussion from d = 3 to the whole family of Z 2 -invariant Wilson-Fisher fixed points for 2 d 4, the dimension of V can be determined exactly in d = 2 and d = 4 (see appendix A). Namely, we have: 5 ∆ V = 14 (2d Ising),(1.3) 2 See also [6] for a review. Concerning the 3d Ising model, see especially section 4.2 of [7]. 3 If Wµ is a total derivative, the stress tensor can be "improved" to be traceless, so that Eq. (1.1) is satisfied for the improved Tµν . 4 It should be noted that this mechanism may be realized with a quirk in gauge theories. Namely it may happen that Eq. (1.2) holds but that the virial current is not a gauge invariant operator (and so is not a physical local operator). For example, this is how the 3d Maxwell theory avoids conformal invariance [9]. ∆ V = 11 (4d free massless scalar). ( 1.4) It also follows from the -expansion that the dimension of V in 4− dimensions will be 11±O( ). 6 Based on these results, one can expect that the dimension of V µ in critical 3d Ising model should be significantly larger than 2. In this paper we will show, using lattice Monte Carlo simulations, that this expectation is correct. Namely, our analysis will imply a numerical lower bound on ∆ V : ∆ V > 5.0 (3d Ising) . (1.5) In particular, this proves that ∆ V > 2, and shows that the 3d Ising model has no candidates for W µ . This rules out the scale without conformal invariance scenario based on (1.2), and thus provides a new test of conformal invariance. The paper is structured as follows. In section 2, we set up the lattice Monte Carlo simulation to measure a one-point function in a cubic lattice with peculiar boundary conditions (motivated in appendices C and D). Section 3 contains our numerical results that lead to (1.5). We conclude with a short discussion of the implications of our result. In appendix A, we compute ∆ V in the 2d Ising model and in the theory of a free massless scalar in d = 4. In appendix B, we summarize the general procedure for matching lattice operators with local operators of the critical field theory. This is well known among the practitioners but we do not know any good pedagogical summary in the literature. Lattice setup We simulate the nearest-neighbor ferromagnetic 3d Ising model on the cubic lattice at the critical temperature. The Hamiltonian is H = −β xy s(x)s(y) , s(x) = ±1. We use the known critical temperature β = β c ≈ 0.2216546 [14,15]. Boundary conditions Our lattice has spatial extent L × L × L sites. We set lattice spacing a = 1. Due to the difficulties of measuring a rather high scaling dimension ∆ V , we will only be able to go up to volumes L = 16. We impose periodic boundary conditions in directions x 1 , x 2 , while at x 3 = 0 and x 3 = L − 1 we impose a mixture of fixed and free boundary conditions. Namely, for x 3 = 0 we impose the fixed s = +1 boundary condition for points with L/4 x 1 < 3L/4, while at x 3 = L − 1 we do the same 6 The coefficient of the O( ) correction term could be computed, but we don't need it. for points with L/2 x 1 < L. The rest of the boundaries at x 3 = 0 and x 3 = L − 1 has free boundary conditions (see Fig. 1). The reasons for such a bizarre choice of boundary conditions will be explained shortly. periodic periodic free s = +1 s = +1 free free x 1 x 2 x 3 0 L 1 Lattice operator We will work with the lattice operator O lat µ = s(x)∇ µ s(x) 3 ν=1 [∇ ν s(x)] 2 ,(2.1) where x is a lattice point and ∇ ν s(x) = s(x +ê ν ) − s(x −ê ν ) is the symmetric lattice derivative in the ν direction. Actually the precise form of the operator is unimportant, the only important thing is that O lat µ is not a total lattice derivative. Matching of the lattice operator with critical point operators Close to the critical point, the lattice operator O lat µ can be expanded into a basis of local operators of the critical theory with well-defined scaling dimensions (see appendix B for a review): O lat µ = i c i O i,µ ,(2.2) where O i is the critical theory operator which has a scaling dimension ∆ i , and c i are some latticedependent constants. Barring accidental cancellations, any lattice measurement related to O lat µ will be dominated by operators of lowest scaling dimensions appearing in the r.h.s. of (2.2). This is because the contribution of an operator of dimension ∆ i will be suppressed by 1/R ∆ i where R is a large distance scale (clearly we have to go to large distances to explore the critical point). Notice that operators in the r.h.s. will have to be vectors, but they don't have to be primaries. So, the total derivative terms involving derivatives of various Z 2 -even scalar operators which exist in the 3d Ising model (see Table 2 in [2]) are expected to appear in the r.h.s. of (2.2). The lowest of these are ∂ µ ε and ∂ µ ε , where ε, ε are the lowest-dimension Z 2 -even scalars, of dimension ∆ ε ≈ 1.41, ∆ ε ≈ 3.83. These derivative operators (especially ∂ µ ε) have rather low dimension. Below we will introduce a trick which will allow us to project them out and focus on more interesting terms. Crucially for us, since O lat µ is not a total derivative, the operator V µ we are interested in will appear in this expansion: V lat µ ⊃ CV µ + . . . . The constant C = O(1) is an unknown, non-universal, lattice quantity, and we will assume C = 0 since there is no reason to expect otherwise. The . . . include various terms which we are not interested in, and we should make sure that those terms do not mask the contribution of V µ . Some of these terms involve operators of higher scaling dimension than V . The presence of those terms is harmless since their effect will be subleading in the large volume limit. More annoying are the total derivative terms involving derivatives of various Z 2 -even scalar operators which exist in the 3d Ising model (see Table 2 in [2]). Some of these have a rather low dimension and would mask V µ unless special care is taken. For example, we expect ∂ µ ε to appear in the r.h.s. of (2.2), where ε is the lowest-dimension Z 2 -even scalar, of dimension ∆ ε ≈ 1.41. Another class of total derivative operators which we expect to appear are ∂ ν T µν , divergences of non-conserved spin-2 Z 2 -even operators. Assuming conformal invariance, the lowest such operator has dimension ∆ T ≈ 5.51 [2]. Divergences of higher spin operators are also expected in principle but will not play a role because of their even higher dimension. In our study we will be able to filter out the contributions of derivatives of scalars (like ∂ µ ε) through the following trick, rendered possible by the periodic boundary conditions. We consider the average value of the x 1 -component of V lat µ integrated along a periodic circle in this direction: I(x 2 , x 3 ) = 1 L L−1 x 1 =0 V lat 1 (x 1 , x 2 , x 3 ) (2.3) Integration kills off the derivatives taken in the direction of integration. As a result this integrated observable in the continuum limit does not couple to derivatives of scalars like ∂ µ ε. On the other hand divergences of spin-2 operators survive this projection, and their integral will contribute to I along with the integral of V µ . 7 We will measure the one-point (1pt) function of I. In infinite volume vector operators would have zero 1pt functions, but in finite volume with appropriate boundary conditions they can be nonzero. In our case we will have 4) with no dependence on x 2 due to the translation invariance in that direction. The scaling of this observable with L will be determined by the smaller of the two dimensions ∆ V , ∆ ∂T = ∆ T + 1: I(x 2 , x 3 ) ≡ Obs(x 3 ) = 1 L ∆ I f x 3 L − 1 + . . . ,(2.∆ I = min(∆ V , ∆ T + 1). (2.5) In this work we will only measure ∆ I , but we will not be able to determine which of the two operators V or ∂T dominates the scaling. Another way to determine ∆ I would be to impose periodic boundary conditions also in the x 3 direction and to study finite size scaling for the 2pt function of I at separation L/2. This observable would scale as 1/L 2∆ I . We tried this strategy and found the signal completely swamped by noise, due to large ∆ I . Using the 1pt function improves the signal-to-noise ratio by a factor L ∆ I and will allow us to perform the measurement. The . . . terms in (2.4) decay with a higher power of L. They originate from the higherdimension operators contributing to V lat µ as well as from corrections to scaling arising from the fact that in finite volume the theory is not exactly at the critical point but is still flowing to it in the renormalization group sense. Because of limited statistics, we will unfortunately be forced to simply neglect both of these corrections in our analysis. The function f (t), 0 < t < 1, parametrizes the observable (2.4) in the infinite-volume limit. This function will be measured in our simulation. To have nonzero f (t), the boundary conditions at x 3 = 0, L − 1 should break the flip symmetry in the x 1 direction: x 1 → L − x 1 , under which I changes sign. This is the case for our boundary conditions in Fig. 1. On the other hand, our boundary condition preserves the above x 1 flip accompanied by the x 3 flip: x 3 → L − x 3 , and a periodic shift of the x 1 direction by L/4. As a consequence, our function f (t) will be odd with respect to t = 1/2, and in particular f (1/2) = 0. 7 To kill all possible total derivatives, one could consider periodic conditions in all directions and to integrate over the whole volume. We do not currently have a concrete proposal implementing this idea. The main difficulty is that the one-point function of a vector operator vanishes on the 3-dimensional torus with periodic boundary conditions. We have experimented with several other flip-breaking boundary conditions, and settled for the one in Fig. 1 because it gives rise to a particularly sizable f (t), thus further improving signalto-noise. See appendix C for a list of other possible boundary conditions, and appendix D for a heuristic procedure to quickly evaluate which boundary condition is expected to work best. While it is not directly related to our computation, we would like to mention here one other instance where boundary conditions were used in lattice field theory to make a 1pt function of a tensor operator nonzero. Namely, in 4d lattice gauge theory, the 1pt function of the off-diagonal stress tensor component T 0x was measured imposing the "shifted" boundary conditions, when the fields are made periodic in the spatial directions, and periodic up to a coordinate shift in the Euclidean time direction [16]. This boundary condition is a particular case of the gluing boundary condition discussed in appendix C. Choice of Monte Carlo algorithm We perform Monte Carlo simulations using the single-spin-flip Metropolis algorithm. The choice of Monte Carlo algorithms plays a crucial role in the efficiency of the simulations. It is well known that the Wolff algorithm [17] is more efficient than the Metropolis algorithm at the critical temperature due to the scaling of the computational effort with the system size. However, even though the smaller critical slowdown exponent favors the Wolff algorithm for large systems, for small ones and for some statistical observables, the Metropolis algorithm may be more efficient. This is what happened in our case. To be more concrete, the standard measure of the simulation efficiency is based on the product of the algorithm execution time (τ CP U ) and the integrated autocorrelation time (τ c ). One reason to prefer the Metropolis algorithm is that in our case it led to very small integrated autocorrelation time of the vector operator sampling (this time scale depends on the statistical observable we are trying to measure). Another important factor for this choice was the role of the boundary conditions. The use of Results We eraging over x 2 ). Since our lattice operator (2.1) has range 3, we only did the measurement for 1 x 3 L − 2. The total number of such decorrelated spin configurations that we generated was 2.4 × 10 12 (resp. 3.5 × 10 13 ) for L = 12 (resp. L = 16). A much smaller number sufficed for L = 8. For N = L 3 /4 spin flips between the two measurements, the integrated autocorrelation time between the subsequent measurements of Obs(x 3 ) was close to 1 for every x 3 . Our simulations were parallelized on a cluster and took a total of about 300 CPU-years. The numerical results of these measurements are given in table 1, and are shown in plots below as a function of t = x 3 /(L − 1). 8 In these plots we show the data multiplied by (L/12) ∆ for various values of ∆. According to (2.4), the curves for different L are supposed to collapse if ∆ = ∆ I . At least this is supposed to happen for sufficiently large L, when contributions from the subleading terms . . . in (2.4) become unimportant. In Fig. 2 we take ∆ = 2, the value needed for a virial current candidate. Clearly the curves show no collapse, ruling out the existence of the virial current. A side remark: as mentioned in the previous section, the function f (t) should be odd with respect to t = 1/2 for our choice of the boundary conditions. This antisymmetry is indeed satisfied within error bars, as can be seen in the figures. 9 To assign an error to our determination of ∆ I , we propose the following heuristic procedure. We vary ∆ around 6 and see when the curves clearly deviate from the collapsing behavior in the interval 0.2 t 0.8, judging by the eye. One way to quickly perform this analysis is to use the Manipulate function of Mathematica. This way we arrive at our confidence interval: ∆ I = 6 ± 1. (3.1) L=8 L=12 L=16 0.0 0.2 0.4 0.6 0.8 1.0 -0.00001 -5. × 10 -6 0 5. × 10 -6 0.00001 x 3 /(L-1} (L/12) Δ Obs(x 3 ) Fig. 3: In this plot ∆ = 6, which is our central value for ∆ V . See Fig. 4 for what the collapse plots look like at the extreme ends of the confidence interval. 10 While the "judging by the eye" procedure may seem subjective and ad hoc, we don't believe a much better statistical procedure can be advocated given our limited amount of data. We have cross-checked our determination of ∆ I by focussing on the three points we get the same answer ∆ I = 6 ± 1. 10 If we omit the L = 8 datapoints from our analysis (e.g. if one is worried that these points are still significantly affected by the subleading . . . corrections in (2.4)), then we get ∆I = 5.5 ± 1.5 using the same procedure. We quote this number only for comparison, as we do not feel that completely discarding the L = 8 points is justified. Discussion and conclusions One goal of this paper was to emphasize that there is a simple and robust way to check the conformal invariance of any critical lattice model, which requires the measurement of the lowest non-derivative vector operator V which is a singlet under all global symmetries. This operator can play the role of the virial current, and potentially cause scale without conformal invariance, but only if its dimension is exactly d − 1. In this paper we considered this strategy in the critical 3d Ising model. Since the dimension of V appears to be large, to carry out our measurement we had to introduce several tricks increasing the efficiency of Monte Carlo simulations. In particular, we had to consider an integrated lattice operator to decouple some uninteresting total derivative terms, and to optimize boundary conditions to maximize the (integrated) 1pt function of V , which was our Monte Carlo target. Further boundary condition optimization is likely possible (see appendix D) and might allow to reduce the error bars in future studies. The main limitation of our approach to measuring ∆ V is that while it decouples total derivatives of scalars, it does not do so for divergences of spin-2 operators. As a result we measure not ∆ V but ∆ I = min(∆ V , ∆ T + 1), where T is the lowest non-conserved Z 2 even spin-2. So, our result ∆ I = 6 ± 1 only implies a lower bound ∆ V 5.0 on the dimension of V . Still, the virial current value ∆ V = 2 is soundly ruled out by this lower bound. This confirms that the 3d Ising model is conformally invariant. Now assuming conformal invariance, we know from the conformal bootstrap that ∆ T ≈ 5.51 [2]. This suggests that our measurement of ∆ I was dominated by ∆ T + 1, while V itself may be much higher. This scenario appears likely also in light of extremely high values of ∆ V in d = 2, 4 reported in the Introduction. In this paper we have not carried out any correction-to-scaling analysis. It would be interesting to repeat the simulation in the Blume-Capel model which is in the same universality class as the Ising model but has a free parameter allowing to drastically reduce corrections to scaling [15]. It would be also interesting to determine or bound the dimension of V for the O(N ) and other models. Finally, we would like to comment on the determination of ∆ V using the conformal bootstrap. The numerical conformal bootstrap has determined scaling dimensions of about 100 operators of the critical 3d Ising model [2]. The operators which have been determined appear in the operator product expansions (OPEs) of σ × σ, ε × ε and σ × ε, where σ and ε are the lowest dimension Z 2 -odd and Z 2 -even scalars. The OPEs σ × σ and ε × ε, being OPEs of identical scalars, contain only operators of even spin. The OPE σ × ε contain only Z 2 -odd operators. The operator V , being a Z 2 -even vector, does not appear in these OPEs, and therefore it has not been so far probed by the conformal bootstrap. In the future, the OPEs σ × σ and ε × ε , where σ and ε are the subleading Z 2 -odd and Z 2 -even scalars, will hopefully be included in the bootstrap analysis. These OPEs contain V and can be used to determine its dimension. Of course, determination of ∆ V using the conformal bootstrap already presupposes that the model is conformally invariant. This has to be distinguished from the lower bound on V obtained in our paper, which is valid independently of conformal invariance, and so allowed us to test this property. Note added. In the first arXiv version of this paper [10] the reader will find an appendix criticizing the argument in [11] for conformal invariance of the critical 3d Ising model. We consider the objections raised there still valid, and the rebuttal [12] unsatisfactory. However, we removed the appendix to keep the focus on the positive results obtained in our own work. Z(q, x, y) = O q ∆ O x 2j O y 2j O , (A.1) where the sum runs over all local operators. The quantum numbers (∆, j,j) are the eigenvalues of the dilatation generator D and two commuting rotation generators J 3 andJ 3 . The latter correspond to the decomposition SO(4) = SU (2) × SU (2) of the rotation group. The partition function can be easily computed using the Fock space structure [19]. We start by introducing the partition function z φ of local operators with a single field φ and arbitrary number of derivatives, z φ (q, x, y) = χ 1,0,0 (q, x, y) − χ 3,0,0 (q, x, y) (A.2) where χ ∆, ,¯ (q, x, y) = q ∆ (1 − qxy)(1 − qy/x)(1 − qx/y)(1 − q/(xy)) j=− x 2j¯ j =−¯ x 2j (A.3) is the long character of a conformal multiplet with primary of dimension ∆ and spin ( ,¯ ). The full partition function can then be written as Z(q, x, y) = exp ∞ k=1 1 k z φ q k , x k , y k . (A.4) Moreover, the partition function restricted to Z 2 even/odd operators is given by Z ± (q, x, y) = 1 2 exp ∞ k=1 1 k z φ q k , x k , y k ± 1 2 exp ∞ k=1 (−1) k k z φ q k , x k , y k . (A.5) We are interested in the character decomposition of the Z 2 even partition function. Expanding the given expression and matching the powers of q and dependence on x, y order by order, we arrive at the following expression: Z + = 1 + 4 n=1 χ short 2+2n,n,n + χ 2,0,0 + χ 4,0,0 + χ 6,0,0 + χ 6,1,1 + χ 7, + 3χ 10,0,0 + χ 10,0,2 + 4χ 10,1,1 + χ 10,1,2 + 2χ 10,1,3 + χ 10,2,0 + χ 10,2,1 + 4χ 10,2,2 + χ 10,2,3 + 2χ 10,3,1 + χ 10,3,2 + 3χ 10,3,3 + χ 11,+ O(q 12 ), where χ short 2+2n,n,n = χ 2+2n,n,n − χ 3+2n,n− 1 2 ,n− 1 2 (A.7) is the character associated with a conserved current of spin 2n. This shows that the vector primary with lowest scaling dimension has ∆ = 11 (blue character). As a consistency check, we have determined ∆ V = 11 using an alternative method. We performed the conformal block decomposition of the four-point function 11 where G ∆,s stands for the conformal block of dimension ∆ and spin s (corresponding to the SO (4) irreducible representation ( s 2 , s 2 )). Again we find the first vector primary at dimension 11. One can also see that the vector primary operator we identified is parity-even. This follows immediately because parity odd vector primary operators cannot appear in the OPE of two scalars (like φ 2 and φ 4 ) in a parity symmetric theory. In addition, it is easy see that the vector operator contains 6 fields φ and 5 derivatives. 13 We also studied the conformal character decomposition 11 We normalized the operators φ 2 and φ 4 to have unit two-point function. 12 We use the standard conformal block as defined in [20,21]. 13 The φ content of each primary can be obtained by studying the partition function φ 2 (x 1 )φ 4 (x 2 )φ 2 (x 3 )φ 4 (x 4 ) = 1 xZ(r, q, x, y) = exp ∞ k=1 r k k z φ q k , x k , y k , (A.10) where r is a fugacity for the number of φ's in each local operator. of the free massless scalar in d = 3. The lightest vector primary still contains 6 fields φ and 5 derivatives, which leads to ∆ V = 8 in d = 3. The conclusion that the lowest Z 2 even vector primary has dimension 11 was reached independently by Marco Meineri [22]. He used a different approach, which also provides an explicit expression for this primary in terms of φ and its derivatives. In d = 4 − , this vector primary operator will get an O( ) anomalous dimension, computable starting from an explicit expression in [22]; this will not be done here. One potential worry could be the recombination of this multiplet with a short multiplet when > 0. However, it is well known (see e.g. [23] for a discussion) that the only multiplets that recombine are the multiplet of φ with the one of φ 3 and the multiplets χ short 2+2n,n,n (conserved currents of spin 2n) with χ 3+2n,n− Notice that in all the above discussion we set ∂ 2 φ = 0 in 4d, eliminating operators involving the letter "∂ 2 φ" from consideration. When we go to (4 − ) dimensions, we will have the equation δ µ 1 [ν 1 δ \|µ 2 \|ν 2 . . . δ \|µ 5 \|ν 5 ] ∂ µ 1 φ ∂ µ 2 ∂ ν 2 φ . . . ∂ µ 5 ∂ ν 5 φ, (A.13) of dimension 14 + O( ). This operator is not a primary [26], so the lowest evanescent vector primary is still somewhere higher. We conclude that the evanescent operators cannot compete with the 11 + O( ) primary that we found above. A.2. Two dimensions Here we discuss spectrum of the 2d Ising model in the Z 2 -even sector. The Ising model contains 2 Z 2 -even Virasoro primaries, 1 with h =h = 0 and with h =h = 1 2 . Their Virasoro characters are given by χ 1 (q,q) = χ 0 (q)χ 0 (q), χ (q,q) = χ 1 2 (q)χ 1 2 (q) . (A. 14) The characters χ 0 and χ 1 2 are given by [27] χ 0 (q) = 1 + q 2 + q 3 + 2q 4 + 2q 5 + 3q 6 + 3q 7 + 5q 8 + 5q 9 + 7q 10 + 8q 11 + 11q 12 + . . . (1 + q + q 2 + q 3 + 2q 4 + 2q 5 + 3q 6 + 4q 7 + 5q 8 + 6q 9 + 8q 10 + 9q 11 + 12q 12 + . . .) These Virasoro characteres can de decomposed into characters X h (q) = q h 1 − q , (A.16) of the global conformal algebra. This gives χ 0 = 1 + X 2 + X 4 + X 6 + 2X 8 + . . . The first vector quasiprimary is obtained by combining X h with Xh with h −h = 1. We see that the minimal choice is h = 15 2 ,h = 13 2 , corresponding to the scaling dimension ∆ = h +h = 14. It is also interesting to find a dimension of the first non-conserved spin-2 quasiprimary, for which we need h −h = 2. This is possible for h = 4,h = 2, which gives ∆ = 6. The vector quasiprimaries can also be found by studying the (global) conformal block decomposition of a four-point function involving two different scalar operators. In 2d Ising, the simplest choice is (with ∆ = 1) and TT (with ∆ = 4). Such correlation functions can be easily computed using the conformal Ward identities. In particular, we obtained A(z,z) = lim w→∞ \|w\| 8 (0, 0) TT (z,z) (1, 1) TT (w,w) = 1 16 1 + (1 − 2z) 2 z 2 (1 − z) 2 2 . (A.19) The conformal block expansion in the z,z → 0 channel is given by A = 1 16 G 1,0 + G 5,G ∆,s (z,z) = k ∆+s (z)k ∆−s (z) + k ∆−s (z)k ∆+s (z) 2 s (1 + δ s,0 ) , k β (z) = (−z) β−9 2 2 F 1 β + 3 2 , β − 3 2 , β, z (shifts in the familiar exponents w.r.t. β/2 due to unequal dimensions of external scalars). This confirms that ∆ V = 14 in the 2d Ising CFT. B. Comments on operator matching Here we collect some well known facts about operator matching between UV theory and its IR fixed point. UV theory may be a lattice spin model, a field theory with cutoff, or a continuum limit field theory. B.1. Matching in the lattice spin model We consider first the lattice spin model case, and will explain the necessary modifications to UV representations. The lattice theory has lattice operators which form multiplets under the lattice symmetry group (cubic group). The critical theory is sometimes called CFT, but here we will avoid using this terminology since we don't want to assume conformal symmetry from the start. The important point is that critical theory correlators are defined at all distances 0 < r < ∞, while correlators of the lattice theory are defined at discrete distances r a. How to recover parameters of the critical theory in a lattice simulation? Two issues complicate this extraction. The first issue is that operators of the lattice theory, naturally given in terms of lattice variables, do not have well-defined scaling dimension, but should be thought of as linear combinations of such operators. The second issue is that the lattice theory, even with couplings finetuned to the second-order phase transition, does not sit precisely at the fixed point, but only flows to it at large distances. Let us consider in turn how these issues manifest themselves. Consider the simplest lattice operator, spin S lat (x). We should expand it in critical theory operators. The appearing terms will have to be, as S lat (x), Z 2 -odd cubic group singlets. The expansion (sometimes referred to as matching) will have the form: S lat (x) = A 1 σ(x) + A 2 ∂ 2 σ(x) + A 3 σ (x) + A 4 ∂ µ R µ + d µνλσ (A 5 ∂ µ ∂ ν ∂ λ ∂ σ σ + A 6 R µνλσ ) + . . . (B.1) There are infinitely many terms but we only wrote the first few representative ones. σ and σ are the first two Z 2 -odd scalars of the critical theory (of dimension ∆ σ ≈ 0.518, ∆ σ ≈ 5.29). Derivatives of these operators with indices contracted so that they are scalars can also appear (∂ 2 σ being shown as a representative case). In addition scalar derivatives of tensor Z 2 operators are also expected to appear, the representative case being the divergence of some Z 2 odd vector On a lattice with spacing a, all coefficients A i in this expansion will be given by A i =Ã i a ∆ i , withÃ i a dimensionless number and ∆ i the dimension of the critical operator multiplied by the corresponding coefficient. On a lattice of unit spacing they will be simply O(1) numbers. With the expansion (B.1), correlators of S lat (x) in the lattice theory, can be matched with sums of correlators of operators in the critical theory. For example, for the 2pt function we have: S lat (x)S lat (y) lattice = A 2 1 σ(x)σ(y) + A 1 A 2 (∂ 2 x + ∂ 2 y ) σ(x)σ(y) + A 2 2 ∂ 2 x ∂ 2 y σ(x)σ(y) + A 2 3 σ (x)σ (y) + A 2 4 ∂ x µ ∂ y ν R µ (x)R ν (y) + . . . (B.2) Here the correlator in the l.h.s. can be measured in a lattice simulation, and by this equation it should be equal to a sum of critical correlators in the r.h.s. Consider for example correlators in infinite volume. The critical theory correlators are expressed in terms of scaling dimensions of the fields. For scalars: O i (x)O i (y) = 1 \|x − y\| 2∆ i , (B.3) where 1 is just a normalization. For a vector operator we would have R µ (x)R ν (y) = δ µν + α(x − y) µ (x − y) ν /\|x − y\| 2 \|x − y\| 2∆ R (B.4) Here the constant α equals −2 in a CFT with R µ a vector primary, but in a scale invariant theory but non-conformal theory it could be different. Also in a non-conformal theory there could be nonzero 2pt functions between operators of unequal scaling dimension which then have to be added to the r.h.s. of (B.2). In any case, according to this discussion, and taking into account the expected size of coefficients A i , the r.h.s. of (B.2) contains a series of terms decaying with the distance as const.(a/r) p i where the powers p i are simply related to scaling dimensions of operators appearing in the r.h.s. of (B.1). We see that only dimensionless ratios of distances enter into this expression. If we go to distances r a, then the lowest power p 1 = 2∆ σ will dominate and the first correction will be suppressed by two more powers of the distance. The terms involving d µνλσ tensor will have nontrivial angular dependence, a sign of rotational symmetry breaking. The leading such term will appear from the crossterm σ∂ µ ∂ ν ∂ λ ∂ σ σ and will be tiny, suppressed by 4 powers of the distance. To complete the just given discussion, we need to address the above-mentioned second issue, taken into account by perturbing the action of the critical theory by irrelevant operators. More precisely, we can describe the system by the action I IRF P + d d x g 1 ε (x) + g 2 ε (x) + g 3 d µνλσ L µνλσ (x) + . . .O lat µ = A 1 V µ + A 2 ∂ µ ε + A 3 ∂ µ T µν + A 4 d µνλσ R νλσ + . . . (B.6) This indicates that the r.h.s. can contain vector critical operators (V µ ), derivatives of scalars (∂ µ ) and divergences of tensors (∂ µ T µν , excluding the stress tensor T µν as it is conserved), as well as 15 The idea of improved lattice actions is to use models that allow to tune to zero the couplings of the first few leading irrelevant operators in (B.5). For example, the Blume-Capel model used in [15] allows to set g1 = 0 thus removing the leading corrections to scaling due to ε . 16 The coefficients Ai are of course not the same as for S lat (x). rotation-invariance breaking terms involving higher-rank tensors contracted with special tensors like d µνλσ , to get objects which transform correctly under the cubic group. In the generic case we expect all A i = O(a ∆ i ) as for S lat (x). In the special case of O lat µ (x) being a total lattice derivative, 17 we will have A 1 = A 4 = 0 and only the terms like A 2 , A 3 could contribute. We emphasize that all we know of the coefficients A i on general grounds is that they are O(a ∆ i ) numbers. There is no simple theoretical way to determine these numbers apart from a lattice simulation. All operators which are allowed by lattice and internal symmetries (and total lattice derivative constraints) will appear in the r.h.s. The problem of determining these coefficients is a "long distance" problem: it has to do with how the microscopic theory approaches the IR fixed point at long distances. B.2. Matching in the lattice field theory It is instructive to consider what changes when we replace the spin model by the latticized φ 4 field theory, defined by the lattice action a 3 x   1 2 3 µ=1 (∇ µ φ(x)) 2 + m 2 φ(x) 2 + λφ(x) 4   . (B.7) where ∇ µ φ(x) = [φ(x + ae µ ) − φ(x − ae µ )]/(2a) is the lattice derivative. For each value of the quartic coupling λ > 0 we can find a value of the mass parameter corresponding to a second-order phase transition. For this value m 2 * (λ) the theory flows at large distances to the critical theory, which does not depend on λ and is actually the same as for the Ising spin model. The operators of the UV theory can be then expanded in critical theory operators. For example, we can write an expansion for φ(x) of the same form (B.1) as for the spin operator S lat (x). The symmetry reasoning which led to this expansion remains the same, and the same operators will appear in the r.h.s. However, the discussion of the size of coefficients A i has to be slightly modified. We say that the φ 4 theory is strongly coupled at the lattice scale if the quartic coupling λ is not small. The appropriate dimensionless condition in 3d is λa 1. 18 The effects of such largish quartic coupling are strongly felt already at the lattice scale (and a fortiori at all longer distance scales). Because of this, the RG flow will converge to the IR fixed point at distances r not much higher than a. The matching coefficients in the strongly coupled latticized φ 4 theory will thus be of the same generic size as for the spin Ising model, i.e. A i = O(a ∆ i ). 17 Total lattice derivatives are local operators that when summed over a region of the lattice, reduce to operators at the boundary of that region. For example, ∇µφ(x) = [φ(x + aeµ) − φ(x − aeµ)]/(2a) is a lattice derivative. 18 Notice that the lattice field φ(x) has dimension 1/2 like a free scalar field in 3d. This implies that the quartic coupling λ has mass dimension 1. If on the other hand the quartic satisfies λa 1, the starting point of RG flow finds itself not far from the gaussian UV fixed point (UVFP). The RG trajectory can then be divided into two parts (see Fig. 6). In this case we say that the UV lattice theory is 'weakly coupled'. The first part of the RG flow happens in the neighbourhood of the UVFP. It corresponds to distances 0 , where 0 = 1/λ a. The second part starts at distances ∼ 0 where the flow transitions from the neighbourhood of free UVFP to the strongly interacting IRFP. IRFP a = 0 a ⌧ 1 a = O(1) UVFP Rotational invariant theories I = I U V F P + d 3 x u 1 : φ 2 (x) : +u 2 : φ 4 (x) : +u 3 d µνρσ : φ∂ µ ∂ ν ∂ ρ ∂ σ φ(x) : + . . . (B.8) where u i =ũ i a ∆ U V F P i −3 withũ i dimensionless. Generically, we expect allũ i = O(1). However, for weakly coupled flows we haveũ 2 ∼ λa ∼ a/ 0 1. Furthermore, because we tuned the mass term to its critical value we also haveũ 1 1. The first term breaking rotational invariance has u 3 =ũ 3 a 2 withũ 3 = O(1). 19 The second part of the flow starts at the scale 0 = 1/λ a. Therefore, the scale 0 plays the role of UV cutoff for the second part of the flow. It is then useful to write u i =ū i ∆ U V F P i −3 0 to define dimensionless couplingsū i with respect to the UV cutoff for the second part of the flow. This givesū 2 = O(1) for the quartic coupling andū 3 ∼ (a/ 0 ) 2 1 for the leading irrelevant 19 The couplings of irrelevant operators that involve more than two powers of φ are also suppressed by the small parameter λa because at λ = 0 the lattice path integral is exactly gaussian. coupling that breaks rotational symmetry. The second part of the flow can then be described using the action (B.5) with dimensionless couplingsg i defined by g i =g i ∆ IRF P i −3 0 . We expect g 1 ∼g 2 ∼ O(1) andg 3 ∼ū 3 ∼ (a/ 0 ) 2 1. We thus see that the second part of RG flow starts with some irrelevant operators in the action having dimensionless couplings much smaller than the other ones. This effect was absent in the spin lattice model case, where all irrelevant operators were expected to be present at the cutoff scale with O(1) coefficients in lattice units. As a consequence, rotation breaking in the IR, already small in the spin model case, will be even further suppressed in the weakly coupled lattice field theory case. Now let us discuss matching of operators, which also happens in two stages. First we expand lattice field theory operators into operators of the UVFP. E.g. we will have φ lat = A 1 φ + A 2 : φ 3 : + . . . (B.9) Coefficients of this expansion have a power series expansion in λ. For example we expect A 1 = 1 + O(λa), while A 2 = O(λa 2 ) . Then we have to expand UVFP operators in IRFP operators. This matching is done at the scale 0 . E.g. we have: ( 0 ) ∆ φ φ = B 1 ( 0 ) ∆σ σ + B 2 ( 0 ) ∆ σ σ + . . . (B.10) Since this matching is done at the scale where the flow is strongly coupled, the coefficients B i cannot be easily predicted and are expected to be O(1). Combining the two matchings, we will get expressions for lattice field theory operators in terms of IRFP operators. C. Possible boundary conditions One can imagine modifying our setup described in the main text, by changing the boundary conditions at x 3 = 0, L − 1. The purpose would be to find boundary conditions which lead to an even larger f (t) and thus improve the signal-to-noise ratio. It makes sense to keep translation invariance in the x 2 direction, so that I(x 2 , x 3 ) is x 2 independent and can be averaged in this direction. As discussed in the main text, we have to break the x 1 flip symmetry. One way to do this is to choose different boundary conditions for different parts of the x 3 = 0, L − 1 boundaries, depending on x 1 . In addition to the free and fixed boundary conditions (b.c.) described in the main text, there are two other imaginable types of b.c. worth discussing. C.1. Gluing b.c. The gluing b.c. changes topology of our manifold, by gluing one part of the boundary to another. For example, one can imagine gluing the gray parts of the x 3 = 0, L − 1 boundaries in Fig. 1, instead of imposing the fixed b.c. there. In practice, gluing is achieved by identifying points pairwise or, equivalently in the large L limit, by creating links joining the points being glued. In the just mentioned example, we would be identifying points (x 1 , x 2 , x 3 = L − 1) with (x 1 + L/4, x 2 , x 3 = 0) (0 x 1 < L/2, 0 x 2 < L) Gluing does not have to preserve order, for example we could have instead chosen to glue the gray parts of the boundaries while simultaneously flipping the x 1 coordinate. Such a reversed gluing would be a different boundary condition. One can even glue parts of the same boundary, e.g. the lower and upper white parts of the Fig. 1 (again, in the direct or the reversed x 1 order). x 3 = 0 boundary in C.2. Changing the strength of boundary interactions We may change the strength of interaction among spins belonging to some part of the boundary to β bdry = β c . Two particularly interesting values of β bdry are as follows. • β bdry = β sp ≈ 0.33302. This fixes β bdry to the value corresponding to the "special" boundary phase transition. Recall that the special transition separates the "ordinary" boundary behavior for which the boundary remains disordered at the critical temperature, from the "extraordinary" one when the boundary is ordered at the critical temperature. The ordinary (extraordinary) behavior is realized at β bdry < β sp (β bdry > β sp ). The β sp for the 3d Ising model given above was determined in [28]. Since the boundary points have fewer neighbors than the bulk points, β bdry = β c belongs to the "ordinary" phase, and this explains why β sp > β c . • β bdry = ∞. This enforces that all spins are equal along a part of the boundary, which is the maximally efficient way to enforce the "extraordinary" boundary behavior. Notice that unlike the fixed boundary condition, the spins can still fluctuate between ±1, but only all at once. This difference may seem minor, but it has the following practical consequence. The fixed b.c. can be used if the simulations are performed using the Metropolis algorithm, as in the main text. On the other hand, if the simulations are performed using cluster algorithms, it leads to lowering the acceptance rate since clusters which touch the boundary cannot be flipped. The β bdry = ∞ boundary condition does not have this difficulty. There are many imaginable combinations of the four boundary condition types which break symmetries of the lattice in a way which makes f (t) nonzero. It is tedious to simulate one by one all possible combinations for the Ising model and see which one gives the largest f (t). It would be nice to have a way to guess a good boundary condition. A heuristic method is described in the next appendix. D. Heuristic optimization of boundary conditions Consider the free massless scalar theory on the cubic lattice, described by the action: H = xy (φ(x) − φ(y)) 2 , φ(x) ∈ R . We consider in this theory a lattice operator V lat µ given by the same equation (2.1) with φ(x) instead of s(x). We make a heuristic hypothesis that one can get an idea about the size of I in the critical Ising model by measuring the same quantity in the free scalar theory on the same cubic lattice. One motivation for this hypothesis is that in d = 4 the two theories are actually identical. We won't attempt to justify this hypothesis any further. It's amusing that empirically it seems to work. Once the b.c. is so heuristically guessed, the actual hard computation will be an honest Monte Carlo simulation in the 3d Ising. To use the heuristic, we have to establish a correspondence between boundary conditions for the two models. This correspondence is as follows: 1. The free b.c. in the Ising corresponds to the Dirichlet b.c. for the free scalar. Indeed, the free b.c. in Ising leads to the "ordinary" boundary behavior, where the order parameter is effectively zero on the boundary [29]. 2. The gluing b.c. in Ising clearly corresponds to the same gluing for the free scalar. 3. β bdry = ∞ for the Ising corresponds to imposing that φ(x) remains constant on this part of the boundary for the scalar. 4. β bdry = β sp for the Ising corresponds to the Neumann (i.e. free) boundary condition for the scalar [29]. 5. The fixed 3d Ising boundary condition can be modeled by adding a constant magnetic field (linear in φ(x) term) on the boundary, pushing the free scalar in the needed direction. We won't give full details on how one actually performs the calculation for the free scalar. This calculation is inexpensive since one is computing a gaussian path integral. One constructs the lattice action, evaluates the Green's function, and finally evaluates the observable. The computation is done numerically and takes only a few seconds for a given boundary condition. The most expensive step is the Green's function evaluation which requires to invert an L 3 × L 3 matrix. After playing with the free scalar, we concluded that the boundary condition in Fig. 1 is particularly promising. Notice that since we have the same fixed b.c. on two parts of the boundary, and since we measure a Z 2 -even observable, for the purpose of the heuristic computation we could replace the fixed boundary condition with β bdry = ∞. Before we discovered the heuristic optimization trick, we tried other boundary conditions in the 3d Ising, but they led to a smaller f (t). We could have just postulated the boundary condition in Fig. 1, but we prefer to play in the open. This is because we have not performed exhaustive optimization. Even better b.c. likely exist, and our heuristic may be helpful to search for them. Fig. 1 : 1The boundary conditions used in our simulation. The x 3 = 0 and x 3 = L − 1 faces have a combination of free (white) and fixed s = +1 (gray) boundary conditions. On the other faces the periodic boundary conditions are imposed. This drawing uses the Byzantine perspective only to improve visibility; the actual geometry is an L × L × L cube with parallel sides. The red dashed line is one possible location of the integrated observable (2.3). fixed boundary conditions requires the imposition of an acceptance probability to flip the clusters touching the boundary (see appendix C). On the other hand, if we replace the fixed b.c. by the β bdry = ∞ conditions (see appendix D) each time a cluster touches the boundary the full boundary will be flipped with a clear increase of τ CP U and without any gain in τ c . These reasons led us to opt for the Metropolis algorithm. Our tests showed that for a system size of L = 16, the Metropolis algorithm was able to produce results with error bars comparable to the Wolff algorithm, being faster by a factor of 10. performed Monte Carlo simulations in the setup described in the previous section, with L = 8, 12, 16. The nature of our boundary conditions, with the shift by L/4, requires to increase L in steps of 4. Our simulations were organized as follows. To generate the next sufficiently decorrelated spin configuration we performed N = L 3 /4 steps of the Metropolis algorithm on spins with randomly chosen positions. The measurement of the observable Obs(x 3 ) in (2.4) was then performed (av- Fig. 2 : 2In this plot ∆ = 2, testing (and ruling out) the virial current existence hypothesis.InFig. 3we show what the same plot looks like if we choose ∆ = 6. In fact this value is our best estimate for ∆ I . The curves show collapse within the error bars for 0.2 t 0.8. We consider that the t values closer to the x 3 = 0, L − 1 boundaries are dominated by boundary effects and exclude them from the analysis. , x 3 = 3 (L = 12) and x 3 = 4 (L = 16), which correspond to three close values of t = x 3 /(L − 1). Neglecting the difference in t, the values of the observable at these three points should scale as const./L ∆ I . That this is indeed roughly the case can be seen in the log-log plot in figure 5. Performing the fit using these three points and their mirror images under t → 1 − t, Fig. 4 : 4Determining a confidence interval for ∆ I . Left: ∆ = 5. Right: ∆ = 7. Fig. 5 : 5Observable for x 3 = 2 (L = 8), x 3 = 3 (L = 12) and x 3 = 4 (L = 16) and for the three mirror points (with a minus sign). The dashed line is the best fit c/L ∆ which gives ∆ = 6.03 as the central value. A. Theoretical expectations for the dimension of V In this appendix, we determine the lowest dimension of a vector primary operator at the Wilson-Fisher fixed point in spacetime dimension d = 2 and d = 4. These exactly solvable cases provide an indication for what to expect in d = 3. A.1. Four dimensions The Wilson-Fisher fixed point in d = 4 describes a free massless scalar field φ satisfying the equation of motion ∂ 2 φ = 0. The operator content of this free CFT can be encoded in the partition function , 2 + χ 9 298,0,0 + χ 8,0,2 + 2χ 8,1,1 + χ 8,2,0 + 2χ 8,2 . . . . So the vector primary of dimension 11 will survive as a vector primary of dimension 11 + O( ) in 4 − dimensions. of motion ∂ 2 φ ∝ φ 3 . So when classifying the local operators in (4 − ) dimensions, it would be double counting to consider operators involving ∂ 2 φ. Operators proportional to the equations of motion are known as "redundant operators"[24]. While such "operators" are useful in formal treatments of renormalized perturbation theory[25], they have correlation functions which are zero except at coincident points, and their dimensions do not correspond to critical exponents measurable e.g. in lattice simulations. So redundant operators do not count as local operators of the critical theory. 14A.1.1. Evanescent operatorsHere we will discuss, and exclude, the possibility, that the lowest primary vector in 4− dimension is not the vector primary of dimension 11 + O( ) discussed above, but a still lower vector primary which is an evanescent operator. Recall that the evanescent operators are those which do not exist in d = 4 but only in d = 4 − , see[26] for a discussion. The evanescent operators arise because of antisymmetrization of indices, which kills an operator in d = 4. Thus, they have to involve a contraction withδ µ 1 [ν 1 δ \|µ 2 \|ν 2 . . . δ \|µ 5 \|ν 5 ] (A.11)which in integer dimensions becomesµ 1 µ 2 ...µ 5 ν 1 ν 2 ...ν 5 . (A.12)Any operator involving this contraction will vanish identically in d = 4, because the index µ runs only over 4 values.The lowest vector operator which vanishes in d = 4 but not in d = 4 − is[26] field theory case later on. For definiteness let us think about the d = 3 Ising model, on a cubic lattice of spacing a (we could specialize to a = 1 without loss of generality). We tune the lattice coupling (temperature for the Ising model) to the second-order phase transition. The lattice theory with so finetuned couplings flows, in the RG sense, at large distances to the IR fixed point (IRFP), which we also call "critical theory". The critical theory has full O(3) invariance, while the lattice theory itself has rotational invariance broken to the cubic subgroup. The critical theory has local operators O i (x) which have well-defined scaling dimensions ∆ i and transform in O(3) R µ (dimension of the lowest such vector in the critical Ising theory is unknown). All the above terms are O(3) scalars, hence cubic singlets. However, since rotational invariance is broken by the lattice, some tensor operators may appear as long as they are multiplied by tensors which are invariant under the cubic group but not the full O(3). The first such tensor is the rank-4 tensor with nonzero components d 1111 = d 2222 = d 3333 = 1, and we show two terms involving this tensor, multiplied by A 5,6 . Z 2 -even irrelevant operators, invariant under the cubic symmetry of the lattice, are present generically. By dimensional analysis, the couplings are given by g j =g j a ∆ j −d whereg j are dimensionless numbers. The expansion (B.2) is still true, but correlators in the r.h.s. should be evaluated in the perturbed theory. Specializing again to the 2pt function, the presence of perturbations will lead to the following effect. In addition to the powers p i occurring in the scale-invariant case there will occur powers p i = p i + ω j where ω j = ∆ j − d are all possible correction-to-scaling exponents, with ∆ j dimensions of irrelevant Z 2 -even operators. The smallest such exponent is ω 1 = ∆ ε − 3 ≈ 0.83. 15 Some of these power law corrections will come with nontrivial angular dependence. This is to be expected, since the lattice theory breaks rotation invariance. The smallest rotational invariance breaking exponent ω 3 ≈ 2.02 is related to the dimension of the lowest Z 2 -even cubic group singlet that is not an O(3) scalar. In the case of 3d Ising, this is the lowest Z 2 -even spin-4 operator L µνλσ contracted with the d µνλσ tensor (while R µνλσ in (B.1) was Z 2 -odd). Matching can also be done for lattice operators transforming in nontrivial representations of the lattice symmetry group, vector being our main case of interest. The (d = 3)-dimensional vector representation is irreducible both under O(3) and under the cubic group. For a generic Z 2 -even lattice vector operator O latµ , some representative terms in its expansion will be:16 Fig. 6 : 6Various RG flows on the critical surface of the latticized φ 4 field theory. All flows with λ > 0 end up in the IRFP because we tuned the mass to its critical value. However, flows that start with λa 1 will first be attracted to the UVFP and from there move to the IRFP. More precisely, if the quartic coupling is parametrically small at the UV scale a, the RG flow will be controlled by the UVFP until the scale 0 = 1/λ. At this scale, the flow transitions to the neighborhood of the IRFP. The flow with λ = 0 corresponds to a quadratic theory which ends in the UVFP once the rotation invariance breaking terms have decayed.In the first part of the flow we can approximate the action of the flowing theory expanding around the UVFP action in perturbations parametrized by normalized operators of the gaussian theory: Table 1 : 1Results of Monte Carlo measurements with statistical errors. This corrects some incorrect statements in the first version of this paper[10] and in[11][12][13]. As pointed out in[12], dimension 7 candidate for V considered in our first version is in fact a total derivative. However, their own dimension 7 candidate for V is also incorrect, being a redundant operator (see note14). The raw data in text form can be found inside the tex file of the arxiv submission.9 The way our measurement is organized, all points for the same L, and in particular the symmetric data points, are correlated with an unknown correlation. Thus once the measurement is finished, we cannot easily take advantage of this antisymmetry to reduce the errors by averaging over the symmetric datapoints. However, that the measured function does come out antisymmetric is a check of our procedure. As a side remark, we note that the "exact critical exponents" discussed in Ref.[13] correspond in fact to redundant operators, making the discussion of that paper of little relevance to the physics of the Ising critical point. AcknowledgementsWe would like to thank Leonardo Giusti SR is supported by Mitsubishi Heavy Industries as an ENS-MHI Chair holder. Calculations of this paper were performed at the CERN Theory cluster. We are grateful to the authors of ALPS[18]whose library was very useful for setting up preliminary simulations. Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents. S El-Showk, M F Paulos, D Poland, S Rychkov, D Simmons-Duffin, & A Vichi, ; ! 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[ "Beyond the Standard Model Lectures at the 2013 European School of High Energy Physics", "Beyond the Standard Model Lectures at the 2013 European School of High Energy Physics" ]	[ "Csaba Csáki \nDepartment of Physics, lepp\nCornell University\n14853Ithacany\n", "Philip Tanedo \nDepartment of Physics & Astronomy\nUniversity of California\n92697Irvineca\n", "M Mulders ", "G Perez " ]	[ "Department of Physics, lepp\nCornell University\n14853Ithacany", "Department of Physics & Astronomy\nUniversity of California\n92697Irvineca" ]	[ "Appearing in the Proceedings of the 2013 European School of High-Energy Physics" ]	We introduce aspects of physics beyond the Standard Model focusing on supersymmetry, extra dimensions, and a composite Higgs as solutions to the Hierarchy problem. Lectures at This document is based on lectures by c.c. on physics beyond the Standard Model at the 2013 European School of High-Energy Physics. We present a pedagogical introduction to supersymmetry, extra dimensions, and composite Higgs. We provide references to useful review literature and refer to those for more complete citations to original papers on these topics. We apologize for any omissions in our citations or choice of topics.	10.5170/cern-2015-004.169	[ "https://arxiv.org/pdf/1602.04228v2.pdf" ]	118,586,558	1602.04228	5cdc95afcf5b67a131d837ab848b143560d293d9	Beyond the Standard Model Lectures at the 2013 European School of High Energy Physics June 2013. June 2013 Csaba Csáki Department of Physics, lepp Cornell University 14853Ithacany Philip Tanedo Department of Physics & Astronomy University of California 92697Irvineca M Mulders G Perez Beyond the Standard Model Lectures at the 2013 European School of High Energy Physics Appearing in the Proceedings of the 2013 European School of High-Energy Physics Parádfürdő, HungaryJune 2013. June 2013 We introduce aspects of physics beyond the Standard Model focusing on supersymmetry, extra dimensions, and a composite Higgs as solutions to the Hierarchy problem. Lectures at This document is based on lectures by c.c. on physics beyond the Standard Model at the 2013 European School of High-Energy Physics. We present a pedagogical introduction to supersymmetry, extra dimensions, and composite Higgs. We provide references to useful review literature and refer to those for more complete citations to original papers on these topics. We apologize for any omissions in our citations or choice of topics. The Hierarchy Problem At loop level, the Higgs mass receives corrections from self interactions, gauge loops, and fermion loops (especially the top quark). Diagrammatically, = + + These loops are quadratically divergent and go like d 4 k (k 2 − m 2 ) −1 ∼ Λ 2 for some cutoff scale Λ. Explicitly, δm 2 H = Λ 2 32π 2 6λ + 1 4 9g 2 + 3g 2 − y 2 t (1.1) If Λ 10 tev (for example, Λ ∼ M Pl ), then the quantum correction to the Higgs mass is much larger than the mass itself, δm 2 H m 2 H . This is the Hierarchy problem: the Higgs mass is quadratically sensitive to any mass scale of new physics. This problem is specific to elementary scalars. Unlike scalars, the quantum corrections to fermion and gauge boson masses are proportional to the particle masses themselves. In this way, small fermion and gauge boson masses are technically natural: the loop corrections are suppressed by the smallness of the tree-level parameter. For fermions this is because of the appearance of a new chiral symmetry in the massless limit. For gauge bosons this is because gauge symmetry is restored in the massless limit. By dimensional analysis, the corrections to these mass parameters cannot be quadratically sensitive to the cutoff, Λ, ∆m e ∼ m e ln Λ m e (1.2) ∆M 2 W ∼ M 2 W ln Λ M W . (1.3) The Hierarchy problem is independent of the renormalization scheme. It is sometimes argued that in dimensional regularization there are no quadratic divergences since the 1/ poles correspond to logarithmic divergences. This is fallacious. The Hierarchy problem isn't about the cancellation of divergences, it is about the separation of the electroweak and uv scales. Any new physics coupled to the Higgs will reintroduce the quadratic dependence on the scale at which the new physics appears. For example, suppose new physics enters at the scale m S by a four-point interaction between the Higgs and an additional complex scalar, ∆L ⊃ λ S \|H\| 2 \|S\| 2 . The contribution to the Higgs mass from a loop of the S particle is δm 2 H = λ S 16π 2 Λ 2 UV − 2m 2 S ln Λ UV m S + (finite) . (1.4) Suppose one chose to ignore the term quadratic in the loop regulator, Λ 2 UV -note that there's no justification to do this-the logarithmically divergent piece (corresponding to the 1/ ) and the finite pieces are proportional to the squared mass scale of the new physics, m 2 S . The regulator Λ UV is not a physical scale, but m 2 S is the scale of new physics. The Higgs mass is quadratically sensitive to this scale, no matter how one chooses to regulate the loop. This quadratic sensitivity is true even if these new states are not directly coupled to the Higgs but only interact with other Standard Model fields. For example, suppose there were a pair of heavy fermions Ψ which are charged under the Standard Model gauge group but don't directly interact with the Higgs. One still expects two loop contributions to the Higgs mass from diagrams such as those in Fig. 1. These contributions are of the form δm 2 H ∼ g 2 16π 2 2 aΛ 2 UV + 48m 2 F ln Λ UV m F + (finite) . (1.5) This is indeed of the same form as (1.4). Note that in this case, the sensitivity to the new scale is softened by a loop factor. The Higgs mass operator \|H\| 2 is a relevant and thus grows in the infrared. From the Wilsonian perspective, the Hierarchy problem is the statement that is is difficult (finely tuned) to choose a renormalization group trajectory that flows to the correct Higgs mass. In summary, the Hierarchy problem is the issue that the Higgs mass m H is sensitive to any high scale in the theory, even if it only indirectly couples to the Standard Model. Thus naïvely one would expect that m H should be on the order of the scale of new physics. In the Wilsonian picture, the Higgs mass is a relevant operator and so its importance grows towards the ir. Indeed, m H is the only relevant operator in the Standard Model. The implication of the Hierarchy problem is that there should to be new physics at the tev scale that eliminates the large loop contributions from above the tev scale 1 . In these lectures we explore some of options for the physics beyond the sm that enforce naturalness. Before going into further detail, here is a brief overview of some of the possibilities for this to happen: • Supersymmetry: relate the elementary scalar Higgs to fermions in such a way that the chiral symmetry protecting the fermion mass is extended to also protect the scalar mass. • Gauge-Higgs unification: relate the the elementary scalar Higgs to an elementary gauge field so that gauge symmetry also protects the Higgs mass. • Technicolor, Higgsless: there is no Higgs boson, just a dynamically generated condensate. • Composite Higgs, warped extra dimensions: There is a Higgs, but it is not elementary. At the tev scale the Higgs "dissolves": it becomes sensitive to large form factors that suppresses corrections. • Pseudo-Goldstone Higgs: The Higgs is a pseudo-Goldstone boson of a spontaneously broken symmetry. This gives some protection against quadratic divergences, usually removing the one-loop contribution. In practice one must still combine with additional mechanisms, such as collective symmetry breaking. • Large extra dimensions: The fundamental Planck scale is actually ∼ tev and only appears much larger because gravity is diluted through its propagation in more directions. Supersymmetry Recall that under an infinitesimal transformation by an 'ordinary' internal symmetry, a quantum field φ transforms as ϕ i → (1 ij + i a T a ij )ϕ j , (2.1) where a is an infinitesimal parameter, T a is the [bosonic] generator of the symmetry, and i, j label the representation of φ with respect to this symmetry. These internal symmetries do not change the spin of φ: bosons remain bosons and fermions remain fermions. Supersymmetry (susy) is a generalization of this 'ordinary' symmetry where generator is now fermionic. Thus a susy transformation changes fermions into bosons and vice versa. Further reading: Wess and Bagger [2] is the canonical reference for the tools of supersymmetry. The text by Terning has a broad overview of susy and its modern applications in particle physics. Additional reviews include [3][4][5]. Key historical papers are collected in [6] and a more personal account is presented in [7]. More formal topics in susy that are beyond the scope of these lectures, but are key tools for model builders, can be found in [8][9][10]. The SUSY algebra The '60s were very successful for classifying hadrons based on Gell-Mann's SU(3) internal symmetry. Physicists then tried to enlarge this group to SU (6) so that it would include SU(3) Gell-Mann × SU(2) spin , (2.2) but they were unable to construct a viable relativistic model. Later this was understood to be a result of the Coleman-Mandula 'no go' theorem which states that one cannot construct a consistent quantum field theory based on a nontrivial combination of internal symmetries with space-time symmetry [11]. The one exception came from Haag, Lopuszanski, and Sohnius: the only nontrivial combination of an internal and spacetime symmetry is to use a graded Lie algebra whose generators are fermionic [12]. Recall that fermionic objects obey anti-commutation relations rather than commutation relations. The main anti-commutation relation for susy is: Q A α , Qα B = 2P µ σ µ αβ δ A B ,(2.3) where the Q and Q are susy generators (supercharges) and P µ is the momentum operator. Here the α andα are Lorentz indices while A, B index the number of supercharges. For completeness, the rest of the algebra is [M µν , M ρσ ] = i(M µν η νρ + M νρ η µσ − M µρ η νσ − M νσ η µρ ) (2.4) [P µ , P ν ] = 0 (2.5) [M µν , P σ ] = i(P µ η νσ − P ν η µσ ) (2.6) [Q A α , M µν ] = (σ µν ) β α Q A β (2.7) [Q A α , P µ ] = 0 (2.8) {Q A α , Q B β } = αβ Z AB . (2.9) The Z AB may appear for N > 1 and are known as central charges. By the Coleman-Mandula theorem, we know that internal symmetry generators commute with the Poincaré generators. For example, the Standard Model gauge group commutes with the momentum, rotation, and boost operators. This carries over to the susy algebra. For an internal symmetry generator T a , [T a , Q α ] = 0. (2.10) This is true with one exception. The susy generators come equipped with their own internal symmetry, called R-symmetry. For N = 1 there exists an automorphism of the supersymmetry algebra, Q α → e it Q α Qα → e −it Qα,(2.11) for some transformation parameter t. This is a U(1) internal symmetry. Applying this symmetry preserves the SUSY algebra. If R is the generator of this U(1), then its action on the susy operators is given by Q α → e −iRt Q α e iRt . (2.12) By comparing the transformation of Q under (2.12), we find the corresponding algebra, [Q α , R] = Q α [Qα, R] = −Qα. (2.13) Note that this means that different components of a susy multiplet have different R charge. For N > 1 the R-symmetry group enlarges to U(N ). Properties of supersymmetric theories Supersymmetric theories obey some key properties: 1. The number of fermionic degrees of freedom equals the number of bosonic degrees of freedom. To see this, first introduce an operator (−) N F such that, (−) N F \|q = + \|q boson − \|q fermion (2.14) where N F is the fermion number operator. Note that (2.15) so that (−) N F and the supercharges anticommute, (−) N F , Q A α = 0. Next consider the operator in (2.3) weighted by (−) N F . When one sums over the states in a representationwhich we write as a trace over the operator-one finds: (2.16) where in the last step we've used the cyclicity of the trace to convert the first term into the second term up to a minus sign. By (2.3) the left-hand side of this equation is simply Tr (−) N F 2σ µ αβ P µ . Note that since Poincaré symmetry is assumed to be unbroken, P µ is identical for each state in a representation. Thus we are left with the conclusion that Tr(−) N F = 0, (2.17) which implies that there is an equal number of fermions and bosons. (−) N F Q A α \|q = −Q A α (−) N F \|qTr (−) N F Q A α , Q Ḃ β = Tr −Q A α (−) N F Q Ḃ β + (−) N F Q Ḃ β Q A α = 0, 2. All states in a supersymmetry multiplet ('supermultiplet' or superfield) have the same mass. This follows from the equivalence of P µ acting on these states. 3. Energy for any state Ψ is positive semi-definite Ψ\|H\|Ψ ≥ 0 and the energy for any vacuum with unbroken susy vanishes exactly, 0\|H\|0 = 0. Classification of supersymmetry representations For the basic case of N = 1 susy there is a single supercharge Q and its conjugate Q. The massless representations of this class of theories are separated into two cases: • (anti-)chiral superfield: contains a complex scalar and a 2-component (Weyl) spinor. • vector superfield: contains a 2-component (Weyl) spinor and a gauge field. These are the only N = 1 representations that do not involve fields with spin greater than 1. Multiplets when there is more supersymmetry. If there are more susy charges, e.g. N = 2, then the smallest representation is the hypermultiplet which contains a 4-component (Dirac) fermion and two complex scalars. For supersymmetric extensions of the sm it is sufficient to focus only on the N = 1 case since this is the only case which admits the observed chiral fermions of the Standard Model. One can compare the number of bosonic and fermionic degrees of freedom in these representations. In the chiral superfield, the complex scalar carries 2 degrees of freedom while the complex Weyl spinor carries 4 degrees of freedom. Recall, however, that fermions only have two helicity states so that in fact only 2 of these fermionic degrees of freedom propagate on-shell. Since one of the key points of using fields to describe physical particles is that we can describe off-shell propagation, we would like to also have supersymmetry hold off-shell. This requires adding two 'dummy' scalar degrees of freedom, which we package in a non-propagating 'auxiliary' complex field F : Field off-shell degrees of freedom on-shell degrees of freedom scalar, φ 2 2 fermion, ψ 4 2 auxiliary, F 2 0 For the vector superfield the Weyl spinor has 4 (2) off-(on-)shell degrees of freedom while the massless gauge boson has 3 (2) off(on-)shell degrees of freedom after identifying gauge equivalent states. As in the chiral superfield, the number of on-shell degrees of freedom match automatically while the number of off-shell degrees of freedom require an additional non-propagating auxiliary field. In this case we introduce a real scalar, D: Field off-shell degrees of freedom on-shell degrees of freedom fermion, ψ 4 2 gauge boson, A µ 3 2 auxiliary, D 1 0 Superspace The most convenient way to describe N = 1 supersymmetric field theories is to use the superspace formalism. Here we understand the supersymmetry transformation generated by Q and Q to be a spacetime transformation in an additional fermionic dimension. To do this, we introduce Weyl spinor superspace coordinates θ α andθα. Superfields are functions of x, θ, andθ and encode all of the off-shell degrees of freedom of a supermultiplet. Weyl spinors and van der Waerden notation. We assume familiarity with two-component Weyl spinors. These are the natural language for fermions in four-dimensions. We use the van der Waerden notation with dotted and undotted indices to distinguish the indices of left-and right-chiral spinors. Readers unfamiliar with this notation may consult [2,13]. The encyclopedic 'two component bible' is a useful reference for full details and as a template for doing calculations [14]. The susy algebra tells us that the effect of a susy transformation with infinitesimal parameters and¯ on a superspace coordinate (x, θ,θ) is (x µ , θ,θ) → (x µ + iθσ µ¯ − i σ µθ , θ + ,θ +¯ ). (2.18) It is useful to define the superspace covariant derivatives, D α = + ∂ ∂θ α + iσ µ ααθα ∂ µ Dα = − ∂ ∂θα − iθ α σ µ αα ∂ µ . (2.19) These are 'covariant derivatives' in that they anticommute with the susy generators 2 . They satisfy {D α , Dβ} = −2i(σ µ ) αβ ∂ µ and {D α , D β } = {Dα, Dβ} = 0 (2.20) By expanding in the fermionic coordinates, a generic superfield F (x, θ,θ) can be written in terms of component fields of different spin that propagate on ordinary spacetime, F (x, θ,θ) = f (x) + θψ(x) +θχ(x) + θ 2 M (x) +θ 2 N (x) + θσ µθ v µ (x) + θ 2θλ (x) +θ 2 θξ + θ 2θ2 D(x). This expansion is exact because higher powers of θ orθ vanish identically because an anticommuting number θ 1 satisfies (θ 1 ) 2 = 0. As a sanity check, we are allowed quadratic terms in θ since it is a Weyl spinor and θ 2 = θ α θ α = αβ θ β θ α = 2θ 1 θ 2 . With modest effort, one can work out the transformation of each component of this general superfield by applying the transformation (2.18), expanding all fields in θ andθ, and matching the coefficients of each term. Some of the terms require massaging by Fierz identities to get to the correct form. Fortunately, the general superfield above is a reducible representation: some of these fields do not transform into one another. We can restrict to irreducible representations by imposing one of the following conditions: The chiral and anti-chiral superfields carry Weyl fermions of left-and right-handed helicity respectively. It is convenient to write all anti-chiral superfields into chiral superfields, for example by swapping the right-handed electron chiral superfield with a left-handed positron superfield. The field content is identical, one is just swapping which is the 'particle' and which is the 'anti-particle.' The linear superfield. The defining condition for this superfield includes a constraint that the vector component is divergence free, ∂ µ V µ = 0. It is thus a natural supersymmetrization of a conserved current. We will not consider linear superfields further in these lectures. Supersymmetric Lagrangians for chiral superfields One can check that because Dα(x µ + iθσ µθ ) = 0, any function of y µ = x µ + iθσ µθ is automatically a chiral superfield (χsf). Indeed, the most compact way of writing the components of a χsf is Φ(y, θ) = ϕ(y) + √ 2θψ(y) + θ 2 F (y). (2.25) Again, we point out that this expansion is exact since higher powers of the Weyl spinor θ vanish by the antisymmetry of its components. Under a susy transformation with parameter , the components of the χsf each transform as δ ϕ(x) = √ 2 ψ(x) (2.26) δψ(x) = i √ 2σ µ¯ ∂ µ ϕ(x) + √ 2 F (x) (2.27) δ F (x) = i √ 2¯ σ µ ∂ µ ψ(x). (2.28) Observe that the auxiliary field transforms into a total spacetime derivative. This is especially nice since a total derivative vanishes in the action and so the highest component of a χsf is a candidate for a susy-invariant term in the Lagrangian. Thus we arrive at our first way of constructing supersymmetric Lagrangian terms: write the F -term of a chiral superfield. To generate interesting interactions we don't want to write the F -terms of our fundamental fields-indeed, these are generally not even gauge invariant. Fortunately, one can check that a product of chiral superfields is itself a chiral superfield. Indeed, a general way of writing a supersymmetry Lagrangian term built out of chiral superfields is L = d 2 θ W (Φ) + h.c., (2.29) where W is a holomorphic function of chiral superfields called the superpotential. Note that the integral over d 2 θ is an ordinary fermionic integral that just picks out the highest component of W . Performing the fermionic integral gives Lagrangian terms L = − ∂ 2 W (ϕ) ∂Φ i ∂Φ j ψ i ψ j − i ∂W (ϕ) ∂Φ i 2 . (2. 30) Observe that the superpotential is evaluated on the scalar components of the superfields, Φ = ϕ. One can check that restricting to renormalizable terms in the Lagrangian limits the mass dimension of the superpotential to [W ] ≤ 3. Cancellation of quadratic divergences. One can check from explicit calculations that the susy formalism ensures the existence of superpartner particles with just the right couplings to cancel quadratic divergences. A more elegant way to see this, however, is to note that the symmetries of superspace itself prevent this. While it is beyond the scope of these lectures, the superpotential is not renormalized perturbatively-see, e.g. [8,16] for details. The holomorphy of W plays a key role in these arguments. The symmetries of the theory enforce the technical naturalness of parameters in W , including scalar masses. Superpotential terms, however, do not include the usual kinetic terms for propagating fields. In fact, one can show that these terms appear in the θ 2θ2 term of the combination Φ † Φ θ 2θ2 = F F * + 1 4 ϕ * ∂ 2 ϕ + 1 4 ∂ 2 ϕ * ϕ − 1 2 ∂ µ ϕ * ∂ µ ϕ + i 2 ∂ µψσ µ ψ − i 2ψσ ∂ µ ψ. (2.31) Two immediate observations are in order: 1. The complex scalar ϕ and Weyl fermion ψ each have their canonical kinetic term. The nonpropagating field, F , does not have any derivative terms: its equation of motion is algebraic and can be solved explicitly. This is precisely what is meant that F is auxiliary. 2. Φ † Φ is not a chiral superfield. In fact, it's a real superfield and the θ 2θ2 component is the auxiliary D field. Indeed, in the same way that the highest component of a χsf transforms into a total derivative, the highest component of a real superfield also transforms into a total derivative and is a candidate term for the Lagrangian. We thus arrive at the second way to write supersymmetric Lagrangian terms: take the D-term of a real superfield. We may write this term as an integral over superspace, d 4 θ Φ † Φ, where d 4 θ = d 2 θ d 2θ . More generally, we may write a generic real function K(Φ, Φ † ) of chiral superfields, Φ and Φ † , whose D term is supersymmetric contribution to the Lagrangian. This is called the Kähler potential. The simplest Kähler potential built out of chiral superfields is precisely (2.31) and includes the necessary kinetic terms for the chiral superfield. One can check that restricting to renormalizable terms in the Lagrangian limits the mass dimension of the Kähler potential to [K] ≤ 2. Combined with the condition that K is real and the observation that chiral superfields are typically not gauge invariant, this usually restricts the Kähler potential to take the canonical form, K = Φ † i Φ i . The most general N = 1 supersymmetric Lagrangian for chiral superfields is thus L = d 4 θ K(Φ, Φ † ) + d 2 θ W (Φ) + h.c. . (2.32) This expression is general, but renormalizability restricts the mass dimensions to be [K] ≤ 2 and [W ] ≤ 3. For theories with more supersymmetry, e.g. N = 2, one must impose additional relations between K and W . Assuming a renormalizable supersymmetric theory of chiral superfields Φ i , we may plug in K = Φ † i Φ i and integrate out the auxiliary fields from (2.32). The result is L = ∂ µ ϕ * i ∂ µ ϕ i + iψ iσ µ ∂ µ ψ i − ∂ 2 W ∂ϕ i ∂ϕ j ψ i ψ j − i ∂W ∂ϕ i 2 . (2.33) Here the superpotential is assumed to be evaluated at its lowest component so that W [Φ i (y, θ)] → W [ϕ i (x)]. Observe that dimension-2 terms in the superpotential link the mass terms of the Weyl fermion and the complex scalar. Further, dimension-3 terms in the superpotential connect Yukawa interactions to quartic scalar couplings. Supersymmetric Lagrangians for vector superfields Until now, however, we have only described supersymmetric theories of complex scalars and fermions packaged as chiral superfields. In order to include the interactions of gauge fields we must write down susy Lagrangians that include vector superfields. Suppose a set of chiral superfields Φ carry a U(1) charge such that Φ(x) → exp(−iΛ)Φ(x). For an ordinary global symmetry this is an overall phase on each component of the chiral superfield. For a gauge symmetry, the transformation parameter is spacetime dependent, Λ = Λ(x). Note, however, that this is now problematic because our definition of a chiral superfield, D α Φ = 0, contains a spacetime derivative. It would appear that the naïve gauge transformation is not consistent with the irreducible susy representations we've written because it does not preserve the chiral superfield condition. This inconsistency is a relic of keeping Λ(x) a function of spacetime rather than a function of the full superspace. We noted above that a function of y µ = x µ +iθσ µθ is a chiral superfield and, further, that a product of chiral superfields is also a chiral superfield. Thus a consistent way to include gauge transformations is to promote Λ(x) to a chiral superfield Λ(y) so that exp(−iΛ(y))Φ(y) is indeed chiral. In this way we see that supersymmetry has 'complexified' the gauge group. Under this complexified gauge transformation, the canonical Kähler potential term that contains the kinetic terms transforms to Φ † Φ → Φ † e −i(Λ−Λ † ) Φ. (2.34) For gauge theories one must modify the Kähler potential to accommodate this factor. This is unsurprising since gauging an ordinary quantum field theory requires one to modify the kinetic terms by promoting derivatives to covariant derivatives which include the gauge field. To correctly gauge a symmetry, we introduce a vector (real) superfield (vsf) V which transforms according to V → V + i(Λ − Λ † ) (2.35) and promote the Kähler potential to K(Φ, Φ † ) = Φ † e V Φ. (2.36) A generic vsf has many components, but many can be eliminated by partially gauge fixing to the Wess-Zumino gauge where V = − θσ µθ V µ (x) + iθ 2θλ (x) − iθ 2 θλ(x) + 1 2 θ 2θ2 D(x). (2.37) here V µ (x) is the gauge field of the local symmetry, λ(x) andλ(x) = λ † (x) are gauginos, and D(x) is the auxiliary field needed to match off-shell fermionic and bosonic degrees of freedom. The two gauginos are the pair of two-component spinors that make up a Majorana four-component spinor. This gauge choice fixes the complex part of the 'complexified' gauge symmetry, leaving the ordinary spacetime (rather than superspace) gauge redundancy that we are familiar with in quantum field theory. We have not yet written a kinetic term for the vector superfield. A useful first step is to construct the chiral superfield, W α = − 1 4 DαDαD α V (2.38) = − iλ α (y) + θ β δ β α D(y) − i 2 (σ µσν ) β α F µν (y) + θ 2 σ µ αα ∂ µλα (y). (2.39) One can see that W α is a chiral superfield because DβW α = 0 from the antisymmetry of the components ofD, (2.20). Observe that unlike Φ, the lowest component is a spin-1/2 field. Further, W contains the usual gauge field strength. Indeed, one can write the supersymmetric Yang Mills kinetic terms for the vector superfield as L SYM = 1 4 W α W α \| 2 θ + h.c. = 1 4 d 2 θW 2 + h.c.. (2.40) One can check that this gives the usual kinetic terms for the gauge field and gauginos as well as an auxiliary term. For completeness, the general form of the field strength superfield for a non-Abelian supersymmetric gauge theory is T a W a α = − 1 4 D˙aDȧe −T a V a D α e T a V a . (2.41) Under a non-Abelian gauge transformation the chiral and vector superfields transform as Φ → e −gT a Λ a Φ (2.42) e T a V a → e T a Λ a † e T a V a e T a Λ a . (2.43) The final form of the renormalizable, gauge-invariant, N = 1 supersymmetric Lagrangian is L = d 4 θΦ † i e gV Φ i + d 2 θ 1 4 W a α W αa + h.c. + d 2 θ (W (Φ) + h.c.) . (2.44) Non-renormalization and the gauge kinetic term. Although W 2 looks like it could be a superpotential term, it is important to treat it separately since it is the kinetic term for the gauge fields. Further the arguments that the superpotential is not renormalized in perturbation theory do not hold for the W 2 term. Indeed, the prefactor of W 2 can be identified with the [holomorphic] gauge coupling, which is only corrected perturbatively at one loop order. One way to see this is to note that for non-Abelian theories, the gauge kinetic term W 2 d 2 θ + h.c. also includes a topological term, F F , which we know is related to anomalies. Another way to see this is the note that the simplest demonstration of non-renormalization of the superpotential makes use of holomorphy and the global symmetries of W : the vector (real) superfield from which W α is built, however, is not holomorphic and its fields cannot carry have the U(1) global symmetries used in the proof. Example: SUSY QED As a simple example, consider the supersymmetric version of quantum electrodynamics, sqed. In ordinary qed we start with a Dirac spinor representing the electron and positron. Since we've seen above that a chiral superfield only contains a Weyl spinor, we require two chiral superfields, Φ ± , which we may interpret to be the electron and positron superfields. Our only two inputs are the electromagnetic coupling e and the electron mass m. The latter suggests a superpotential W (Φ + , Φ − ) = mΦ + Φ − . (2.45) Writing out the resulting Lagrangian in components: L sqed = 1 2 D 2 − 1 4 F µν F µν − iλσ µ ∂ µλ + F * + F + + \|D µ ϕ + \| 2 + iψ + D µσ µ ψ + + F * − F − + \|D µ ϕ − \| 2 + iψ − D µσ µ ψ − − ie √ 2 ϕ +ψ+λ − ϕ −ψ−λ + h.c. + e 2 D \|ϕ + \| 2 − \|ϕ − \| 2 + m (ϕ + F − + ϕ − F + − ψ + ψ − ) + h.c. (2.46) We can write this out explicitly by solving for the auxiliary fields D, F ± . The equations of motion are D = − e 2 \|ϕ + \| 2 − \|ϕ − \| 2 F ± = − mϕ * ∓ . (2.47) Plugging this back into the Lagrangian gives L SQED = i=± \|D µ ϕ i \| 2 + iψ i D µσ µ ψ i − 1 4 F µν F µν − iλσ µ ∂ µλ − m 2 \|φ + \| 2 + \|φ − \| 2 − mψ + ψ − − mψ +ψ− − e 2 8 \|ϕ + \| 2 − \|ϕ − \| 2 2 − ie √ 2 ϕ +ψ+λ − ϕ −ψ−λ + h.c. . (2.48) The first line gives the kinetic terms for the electron ψ − , positron ψ − , selectron (φ − ), spositron (φ + ), photon A µ , and photino λ. The second line gives an equivalent mass to the chiral scalars and fermions. The last line gives vertices that come from the supersymmetrization of the kinetic terms: four-point scalar interactions from the D terms and a three-point Yukawa-like vertex with the 'chiral' scalars and photino. The relation between the gauge group and the four-point scalar interaction plays a central role in how the Higgs fits into susy, as we show below. The MSSM We now focus on the minimal supersymmetric extension of the Standard Model, the mssm. To go from the sm to the mssm, it is sufficient to promote each sm chiral fermion into a chiral superfield and each sm gauge field into a vector superfield. Thus for each sm fermion there is a new propagating scalar sfermion (squarks or sleptons) and for each sm gauge field there is also a propagating gaugino, a fermion in the adjoint representation. As we showed above, off-shell susy also implies non-propagating auxiliary fields. The matter (χsf) content of the mssm is shown in Table 1. It is the same as the sm except that we require two Higgs doublet chiral superfields. This is necessary for the cancellation of the SU(2) 2 L ×U(1) Y and SU(2) L Witten anomalies coming from the Higgs fermions, or Higgsinos. An additional hint that this is necessary comes from the observation that the superpotential is a holomorphic function of the chiral superfields while the Standard Model up-type Yukawa coupling requires the conjugate of the Higgs, H = iσ 2 H * . χsf SU(3) c SU(2) L U(1) Y Q 3 2 1 /6 U 3 1 − 2 /3 D 3 1 1 /3 L 1 2 − 1 /2 E 1 1 −1 H d 1 2 1 /2 H u 1 1 − 1 /2 The most general renormalizable superpotential made with these fields can be split into two terms, W = W (good) + W (bad) , W (good) =y ij u Q i H uŪ j + y ij d Q i H dD + y ij e L i H dĒ j + µH u H d (2.49) W (bad) =λ ijk 1 Q i L jDk + λ ijk 2 L i L jĒk + λ i 3 L i H u + λ ijk 4D iDjŪ k . (2.50) In W (good) one can straight forwardly identify the Standard Model Yukawa couplings which give the sm fermions their masses. Since these are packaged into the superpotential these terms also encode the additional scalar quartic interactions required by supersymmetry. The last term in W (good) is a supersymmetric Higgs mass known as the µ-term. By supersymmetry this term also gives a mass to the Higgsinos, which we require since we do not observe any very light chiral fermions with the quantum numbers of a Higgs. The W (bad) terms, on the other hand, are phenomenologically undesirable. These are renormalizable interactions which violate baryon (B) and/or lepton (L) number and are thus constrained to have very small coefficients. Compare this to the sm where B and L are accidental symmetries: all renormalizable interactions of sm fields allowed by the sm gauge group preserve B and L. Violation of these symmetries only occurs at the non-renormalizable level and are suppressed by what can be a very high scale, e.g. M GUT . We see that in the mssm we must find ways to forbid, or otherwise strongly suppress, the terms in W (bad) . Otherwise one would be faced with dangerous rates for rare processes such as proton decay, p + → e + π 0 orνπ + (or alternately with π replaced with K) as shown in Fig. 2. Observe that this is a tree level process and all of the couplings are completely unsuppressed. A simple way to forbid W (bad) is to impose matter parity, which is a Z 2 symmetry with assignments: Figure 2: Proton decay mediated by squarks. Arrows indicate helicity and should not be confused with the 'charge flow' arrows of Dirac spinors [14]. Tildes indicate superpartners while bars are used to write right-chiral antiparticles into left-chiral fields in the conjugate representation. d u d , s, b λ 4 λ 1 Q L uū Superfield Matter parity quark, lepton χsf P M = −1 Higgs χsf P M = +1 gauge vsf P M = +1. Under these assignments, all terms in W (good) have P M = +1 while all terms in (bad) have P M = −1. One can check that one may write matter parity in terms of baryon and lepton number as P M = (−) 3(B−L) . (2.51) A common variation of this is to impose the above constraint using R-parity, P R = (−) 3(B−L)+2s ,(2.52) where s is the spin of the field. Conservation of matter parity implies conservation of R-parity. This is because the (−) 2s factor always cancels in any interaction term since Lorentz invariance requires that any such term has an even number of fermions. Observe that all sm fields have R-parity +1 while all superpartner fields have R-parity −1. (This is similar to T -parity for Little Higgs models.) The diagrams assocaited with electroweak precision observables carry only sm external states. Since R-parity requires pair-production of superpartners, this means that electroweak precision corrections cannot occur at tree-level and must come from loop diagrams. It is important to understand that R-parity (or matter parity) is an additional symmetry that we impose on top of supersymmetry. R-parity has some important consequences: 1. The lightest R-parity odd particle is stable. This is known as the lightest supersymmetric particle or lsp. If the lsp is an electrically neutral color singlet-as we shall assume-it is a candidate for wimp-like dm. 2. Each superpartner (sparticle) other than the lsp will decay. At the end of any such sequence of decays one is left with an odd number (usually one) of lsps. 3. In collider experiments, the initial state has P R = +1 so that only an even number of sparticles can be produced at a time (e.g. via pair production). At the end of the decay these end up as lsps which manifest themselves as missing energy signals at colliders. For most of this document we postulate that the mssm has exact R-parity conservation-though this is something of an ad-hoc assumption. Supersymmetry breaking Any scalar partners to the sm leptons or quarks with exactly degenerate masses as their sm partner would have been discovered long ago. Thus, the next piece required to construct a realistic mssm is a way to break supersymmetry and split the mass degeneracy between the sm particles and their superpartners. Since we want to keep the desirable ultraviolet behavior of supersymmetry, we assume that susy is a fundamental symmetry of nature which is spontaneously broken. susy is unbroken when the supercharges annihilate the vacuum, Q\|0 = Q\|0 = 0. The susy algebra, {Q, Q} = 2σ µ P µ allows us to write the four-momentum operator as P µ = 1 4σ ν {Q, Q} so that the Hamiltonian is H = P 0 = 1 4 Q 1 Q˙1 + Q˙1Q 1 + Q 2 Q˙2 + Q˙2Q 2 . (2.53) Observing that this expression is positive semi-definite, we see that if susy is unbroken, 0\|H\|0 = 0 if susy is broken, 0\|H\|0 > 0 . The vacuum energy can be read from the scalar potential, V [φ] = V F [φ] + V D [φ] (2.54) V F [φ] = i ∂W ∂φ i 2 = i \|F i \| 2 (2.55) V D [φ] = a 1 2 g 2 i φ † i T a φ i 2 = a 1 2 gD a D a . (2.56) We see that susy breaking corresponds to one of the auxiliary fields, F i or D i , picking up a vacuum expectation value (vev). We refer to the case F i = 0 as F -type susy breaking and the case D = 0 as D-type susy breaking. When an ordinary global symmetry is spontaneously broken due to a field picking up a vev there exists a massless boson in the spectrum of the theory known as the Goldstone boson. In the same way, when susy is broken spontaneously due to a auxiliary field picking up a vev, there exists a massless fermion in the theory known as the Goldstino 3 . The spin of this field is inherited by the spin of the susy generators. Heuristically, the massless Goldstone modes correspond to acting on the vev with the broken generators and promoting the transformation parameters to fields. Since the susy transformation parameter is fermionic, the Goldstone field must also be fermionic. For example, if F = 0, then the transformation of the fermion ψ under the broken (susy) generator is δ ψ = 2 F . (2.57) susy acts as a shift in the fermion, analogously to the shift symmetry of a Goldstone boson under a spontaneously broken global internal symmetry. If there is more than one superfield with a non-zero F term, then δ ψ i = 2 F i (2.58) ψ Goldstone = i F i i F 2 i ψ i . (2.59) Note that we have used the convention that, when there is no ambiguity, F refers to the susy breaking background value, dropping the brackets · · · to avoid clutter. One can further generalize this to include a linear combination of gauginos when there is also D-term susy breaking. When ordinary spontaneously broken internal symmetries are promoted to gauge symmetries, their Goldstone modes are 'eaten' and become the longitudinal polarization of the gauge fields. Similarly, gauging supersymmetry corresponds to writing a theory of supergravity. The gravitino then becomes massive by eating the Goldstino from spontaneous susy breaking. Sum rule for broken SUSY Even when it is spontaneously broken, susy is a strong constraint on the parameters of a theory. One of the most important constraints is the susy sum rule, which relates the traces of the mass matrices of particles of different spins. First consider the mass terms for chiral fermions (ψ) and gauginos (λ): i √ 2g (T a ) i j ϕ iλ aψj − ϕ * λψ − ∂ 2 W ∂ϕ i ∂ϕ j ψ i ψ j + h.c. (2.60) We may write this succinctly as a mass matrix, ψ i λ a F ij √ 2D bi √ 2D aj 0 ψ j λ b ,(2.61) where we use the shorthand notation F ij = ∂F i ∂ϕ j = ∂ 2 W ∂ϕ i ∂ϕ j D ai = ∂D a ∂ϕ i = gϕ * i T a . (2.62) Call this fermion mass matrix m (j=1/2) . Next, the scalar mass matrix (m 2 ) (j=0) ij is obtained by the Hessian of the scalar potential, ∂ 2 V ∂ϕ i ∂ϕ * j ∂ 2 V ∂ϕ i ∂ϕ j ∂ 2 V ∂ϕ * i ∂ϕ * j ∂ 2 V ∂ϕ * i ∂ϕ j = F ij F kj + D i a D aj + D i aj D aF ijk F k + D j a D j a F ijkF k + D ai D aj F ikF jk + D ai D j a + D j ai D a . (2.63) Finally, the gauge boson matrix comes from the kinetic terms i g 2 \|A a µ T aα β φ iα \| 2 = \|A a µ D i a \| 2 ,(2.64) and may thus be written (m 2 ) (j=1) ab = D i a D bi + D ai D i b . (2.65) The traces of the squared mass matrices are, respectively, Tr m (j=1/2) m (j=1/2) † =F ijF ij + 4\|D ai \| 2 (2.66) Tr m (j=0) 2 =2F ijF ij + 2D i a D ai + 2D a D i ai (2.67) Tr m (j=1) 2 =2D ai D i a . (2.68) For convenience, we may define the supertrace, a sum of the squared mass matrices weighted by the number of states, Note that this is a tree-level result that assumes renormalizable interactions 4 . STr m (j) 2 ≡ j Tr (2j + 1)(−) 2j m 2 (2.69) = − 2FF − u\|D ai \| 2 + 2FF + 2D i a D ai + 2D a D i ai + 3 · 2D ai D i a (2.70) =2D a (D a ) i i (2.71) =2D a i q ( a) i (2.72) Note that D a = 0 only for U(1) factors, so (D a ) i i = q i , Soft breaking and the MSSM The sum rule (2.74) is a road block to susy model building. To see why, consider the scalar mass matrix (2.63) applied to squarks. In order to preserve SU(3) c , the squarks should not obtain a vev. This implies that the D-terms vanish, D i a = D color = 0, for squarks. Thus further means that quarks only get their masses from the superpotential. Similarly preserving U(1) EM implies that the D-terms corresponding to the electrically charged SU(2) L directions should also vanish: D ± = D 1,2 = 0. This means that the only D-terms which are allowed to be non-trivial are D 3 and D Y , corresponding to the third generator of SU (2) L and hypercharge. The scalar mass matrix for the up-type quarks is then m 2 2/3 = m 2/3 m † 2/3 + 1 2 gD 3 + 1 6 g D Y 1 ∆ ∆ † m 2/3 m † 2/3 − 2 3 g D Y 1 (2.75) m 2 1/3 = m 1/3 m † 1/3 + − 1 2 gD 3 + 1 6 g D Y 1 ∆ ∆ † m 1/3 m † 1/3 + 1 3 g D Y 1 ,(2.76) where the ∆ and ∆ are the appropriate expressions from (2.63) and m 2/3,1/3 correspond to the quadratic terms in the superpotential that contribute to the quark masses. Charge conservation requires the sum of D terms to vanish, so that at least one D term is less than or equal to zero. For example, suppose that 1 2 gD 3 + 1 6 g D Y ≤ 0. (2.77) Let β be the direction in field space corresponding to the up quark. Then β is an eigenvector of the quark mass matrix m 2/3 with eigenvalue m u . Then (2.77) implies that β † 0 m 2 2/3 β 0 ≤ m 2 u . (2.78) This implies that there exists a squark in the spectrum that has a tree-level mass less than the up quark. Such an object would have been discovered long ago and is ruled out. More generally, the observation that there is at least one negative D-term combined with the form of the squark matrices (2.75) and (2.76) implies that there must exist a squark with mass less than or equal to either m u or m d . Thus even if susy is broken, it appears that any supersymmetric version of the Standard Model is doomed to be ruled out at tree level. In order to get around this restriction, one typically breaks susy in a separate supersymmetry breaking sector ($ $ $ susy) that is not charged under the Standard Model gauge group. This $ $ $ susy sector still obeys a sum rule of the form (2.74) but the spectrum is no longer constrained by observed sm particles. In order for the $ $ $ susy sector to lend masses to the sm superpartners, one assumes the existence of a messenger sector which interacts with both the sm and the $ $ $ susy sectors. The messenger sector transmits the susy-breaking auxiliary field vev to the sm sector and allows the sm superpartners to become massive without violating the sum rule (2.74). Note that this also allows a large degree of agnosticism about the details of the $ $ $ susy sector-as far as the phenomenology of the mssm is concerned, we only need to know about the $ $ $ susy scale and the properties of the messenger sector. There are two standard types of assumptions for the messenger sector depending on how one assumes it couples to the sm: • Gravity mediation: here one assumes that the sm and $ $ $ susy breaking sectors only communicate gravitationally. The details of these interactions fall under the theory of local supersymmetry, or supergravity (sugra), but are typically not necessary for collider phenomenology. • Gauge mediation: The messenger sector contains fields which are charged under the sm gauge group. An alternative way around the $ $ $ susy sum rule is to construct a 'single sector' model based on strong coupling [17,18]. These turn out to be dual to 5D models of susy breaking using tools that we introduce in Section 3 [19]. Often we are only interested in the properties of the Standard Model particles and their superpartners. We can 'integrate out' the details of the messenger sector and parameterize susy breaking into non-renormalizable interactions. As an example, suppose that a superfield, X, breaks supersymmetry by picking up an F -term vev: X = · · · + F θ 2 . X may also have a scalar vev, but this does not break susy. We then parameterize the types of non-renormalizable couplings that are generated when we integrate out the messenger sector. We have four types of terms: 1. Non-holomorphic scalar masses are generated by higher order Kähler potential terms such as d 4 θ X † X M 2 Φ † Φ = F M 2 ϕ * ϕ + (susy preserving terms). (2.79) d 4 θ X + X † M Φ † Φ = F * M d 2 θ Φ † Φ + h.c. + (susy preserving terms). (2.80) We have written the susy-breaking part of (2.80) suggestively to appear as a non-holomorphic superpotential term. Since Φ † only containsθs and not θ, d 2 θ Φ † Φ = ϕ * F ϕ = ϕ * W [ϕ * ]. For renormalizable superpotentials, this can give an A-term of the form (2.82) or a b-term of the form (2.81) below; the latter with a slightly different scaling with F . The mass scale M is required by dimensional analysis and is naturally the scale of the mediator sector that has been integrated out. For gravity mediation M ∼ M Pl while for gauge mediation M ∼ M mess , the mass of the messenger fields. Doing the Grassmann integral and picking the terms that depend on the susy breaking order parameter F gives a mass m 2 = (F/M ) 2 to the scalar ϕ. Note that F has dimension 2 so that this term has the correct mass dimension. 2. Holomorphic scalar masses are generated by a similar higher order Kähler potential term, d 4 θ X † X M 2 Φ 2 + Φ † 2 = F M 2 ϕ 2 + ϕ * 2 + (susy preserving terms). (2.81) These are often called b-terms. One may want to instead write these masses at lower order in F by writing a superpotential term W ⊃ XΦ 2 . This, however, is a renormalizable interaction that does not separate the $ $ $ susy sector from the visible sector-as one can see the mediator mass does not appear explicitly in such a term. Thus W ⊃ XΦ 2 is subject to the susy sum rule and is not the type of soft term we want for the mssm. 3. Holomorphic cubic scalar interactions are generated from the superpotential, This is a gaugino mass on the same order as the scalar mass and the A-term. d 2 θ X M Φ 3 + h.c. = F M ϕ 3 + ϕ * In principle one could also generate tadpole terms for visible sector fields, but we shall ignore this case and assume that all field are expanded about their minimum. These four types of terms are known as soft supersymmetry breaking terms. The key point is that these do not reintroduce any quadratic uv sensitivity in the masses of any scalars. This is clear since above the susy breaking mediation scale M , the theory is supersymmetric and these divergences cancel. It is common to simply parameterize the soft breaking terms of the mssm in the Lagrangian: L soft = − 1 2 M 3 g g + M 2 W W + M 1 B B + h.c. (2.84) − a u QH u ū + a d QH d d + a e LH d ē + h.c. (2.85) − Q † m 2 Q Q − L † m 2 L L − u † m 2 u ū − d † m 2 d d − e † m 2 e ē − m 2 Hu H * u H u − m 2 H d H * d H d (2.86) − (bH u H d + h.c.)) . (2.87) This is simply a reparameterization of the types of soft terms described in (2.79 -2.83), from which one can read off the scaling of each coefficient with respect to F/M . Note that the trilinear soft terms, a u,d,e , and the soft masses m 2 Q,L,u,d,e are 3 × 3 matrices in flavor space. The trilinear terms are in a one-to-one correspondence with the Yukawa matrices except that they represent a coupling between three scalars. In general, the soft masses cause the squarks and sleptons to have different mass eigenstates than the sm fermions. Phenomenologically, we assume that M 1,2,3 , a u,d,e ∼ m susy (2.88) m 2 Q,u,d,L,e,Hu,H d , b ∼ m 2 susy , (2.89) where m susy is between a few hundreds of gev to a tev. This is the range in which generic mssm-like models provide a solution to the Hierarchy problem. R-symmetry, gauginos, supersymmetry breaking. Recall that when an R-symmetry exists, the different components of a superfield carry different R charges. Because the O(θ) component of W α , F µν , is real, it cannot carry an R charge. This means that the lowest component, the gaugino λ, must have non-zero Rcharge. Further, the gaugino mass term (2.83) breaks this symmetry. One will find that R-symmetry plays an important role in many non-perturbative results in susy. Two important results related to susy breaking and gaugino masses are [20,21]. Electroweak symmetry breaking in the MSSM The most important feature of the Standard Model is electroweak symmetry breaking. Recall that this is due to a tachyonic Higgs mass at the origin being balanced by a positive quartic coupling leading to a non-zero vacuum expectation value. In the mssm we have two Higgs doublets, H u = H + u H 0 u H d = H 0 d H − d . (2.90) We have already seen that supersymmetry relates the scalar quartic coupling to the other couplings of the theory. This then constrains the expected Higgs boson mass. To preserve SU(3) c and U(1) EM we assume that no squarks or sleptons pick up vevs. Then the quartic terms in the Higgs potential come from D-terms, (2.56): V D = g 2 4 H † u σ a H u + H † d σ a H d H † u σ a H u + H † d σ a H d + g 2 4 \|H u \| 2 − \|H d \| 2 2 = 1 2 g 2 \|H † u H d \| 2 + 1 8 (g 2 + g 2 ) \|H u \| 2 − \|H d \| 2 2 ,(2.91) where we have simplified the SU(2) L terms using the relation σ a ij σ a k = 2δ i δ jk − δ ij δ k . We see immediately that the Higgs quartic λ coupling goes like the squared electroweak couplings, g 2 and g 2 . This connection between the Higgs sector and the gauge parameters does not exist in the Standard Model In addition to the D-term contribution, there is also the supersymmetric F -term contribution coming from the µ-term in the superpotential. The quadratic contributions to the Higgs potential are, V F = \|µ\| 2 \|H u \| 2 + \|µ\| 2 \|H d \| 2 + · · · (2.92) We have dropped terms proportional to the Yukawa couplings since we assume the scalar partners of the sm fermions do not acquire vevs. On top of this, there are the soft supersymmetry breaking terms. These include soft masses for each Higgs doublet and a 'holomorphic' b-term which is called B µ (or sometimes Bµ), V soft = m 2 Hu \|H u \| 2 + m 2 H d \|H d \| 2 + (B µ H u · H d + h.c.+ u H − d − H 0 u H 0 d . Further, the D-term couplings are real since they are part of a real superfield. The F -term couplings are made real because they are the modulus of a complex parameter. The couplings of the soft terms, on the other hand, carry arbitrary sign and phase. Combining all of these factors, the full Higgs potential is V H = V D + V F + V soft (2.94) = 1 2 g 2 \|H † u H d \| 2 + 1 8 (g 2 + g 2 ) \|H u \| 2 − \|H d \| 2 2 + \|µ\| 2 + m 2 Hu \|H u \| 2 + \|µ\| 2 + m 2 Hu \|H d \| 2 + (B µ H u · H d + h.c.) . (2.95) To simplify this, we can assume that the charged components of the doublets pick up no vev and write everything in terms of only the neutral components (we address the validity of this assumption below): V H = 1 8 (g 2 + g 2 ) \|H 0 u \| 2 − \|H 0 d \| 2 2 + i=u,d \|µ\| 2 + m 2 H i \|H 0 i \| 2 − 2B µ Re(H 0 u H 0 d ). (2.96) Observe that this potential has a direction in field space, \|H 0 u \| 2 = \|H 0 d \| 2 where the D-term quartic vanishes. This is called a D-flat direction and requires caution. In order to break electroweak symmetry, we must destabilize the origin of field space with a tachyonic mass term to force a linear combination of the neutral Higgses to pick up a vev. In the sm destabilization is balanced by the quartic coupling which forces the vev to take a finite value. We see now in the mssm that one has to take special care to make sure that the destabilized direction does not align with the D-flat direction or else the potential isn't bounded from below. In other words, we must impose a negative mass squared in one direction in the Higgs moduli space while making sure that there is a positive definite mass squared along the D-flat direction. This can be written as two conditions: 1. We require exactly one negative eigenvalue in the neutral Higgs mass matrix, (2.98). One way to nevertheless enforce this relation is to impose it as a boundary condition at some high scale and allow the renormalization group flow to differentiate between them. This is actually quite reasonable, since the β-function for these soft masses include terms that go like the squared Yukawa coupling. The two soft masses flow differently due to the large difference in the top and bottom Yukawas. In fact, the up-type Higgs mass parameter shrinks in the ir and it is natural to assume \|µ\| 2 + m 2 Hu −B µ −B µ \|µ\| 2 + m 2 H d = \|µ\| 2 + m 2 Hu \|µ\| 2 + m 2 H d − B 2 µ < 0.(2.m 2 Hu < m 2 H d . (2.99) A convenient choice is m 2 Hu < 0 and m 2 H d > 0. In this way the mssm naturally admits radiative electroweak symmetry breaking where the tachyonic direction at the origin is generated by quantum effects. Since there are many parameters floating around, it use useful to summarize that the following all prefer electroweak symmetry breaking and no runaway directions: • Relatively large B µ • Relatively small µ • Negative m 2 Hu . Be aware that these are only rough guidelines and are neither necessary nor sufficient. It is standard to parameterize the vevs of the two Higgses relative to the sm Higgs vev by introducing an angle, β, H 0 u = v u √ 2 = v √ 2 sin β H 0 d = v d √ 2 = v √ 2 cos β. (2.100) Minimizing the potential, ∂V /∂H 0 u = ∂V /∂H 0 d = 0, one obtains sin 2β = 2B µ 2\|µ\| 2 + m 2 Hu + m 2 H d (2.101) M 2 Z 2 = −\|µ\| 2 + m 2 H d − m 2 Hu tan 2 β tan 2 β − 1 . (2.102) The second relation is especially strange: it connects the supersymmetric µ term to the softbreaking masses, even though these come from totally different sectors of the theory. In other words, unlike the quartic and gauge couplings which are tied together by supersymmetry, these parameters have no reason to have any particular relation with each other. Further, M 2 Z is experimentally measured and much smaller than the typical expectation for either µ or m 2 H u,d , so it appears that there's some cancellation going on. The Higgs sector of the mssm contains the usual cp-even Higgs h, a heavier cp-even Higgs, the Goldstones of electroweak symmetry breaking, an additional pair of charged Higgses H ± , and a cp-odd Higgs A. With a little work, one can show that the cp-even Higgs masses are m 2 h = 1 2 M 2 Z + m 2 A ± (M 2 Z + m 2 A ) 2 − 4m 2 A M 2 Z cos 2 2β , (2.103) where m 2 A = B µ /(sin β cos β). One can further show that this is bounded from above, m h ≤ M Z \|cos 2β\| ≤ M Z . (2.104) Of course, we now know that m h ≈ 125 gev. In fact, even before the lhc it was known from lep that m h 114 gev. While at first glance (2.104) appears to be ruled out experimentally, this is only a tree-level bound. What this is really saying is that one requires large corrections to the quartic self-coupling to pull up the Higgs mass from its tree level value. Due to the size of y t , the main effect comes from top and stop loops. To maximize the quartic coupling, we are pushed towards large values of tan β since this would put most of the Higgs vev in H u and would make the light Higgs be primarily composed of H u . Examining the H 4 u coupling at loop level, consider diagrams of the form: H 0 u H 0 u H 0 u H 0 u t R t L t L t R H 0 u H 0 u H 0 u H 0 u t L,R t L,R Assuming negligible A terms, the result is λ(m t ) = λ SUSY + 2N c y 4 t 16π 2 ln m t 1 m t 2 m 2 t ,(2.105) where λ SUSY comes from the D-term potential and N c is the number of colors. This equation tells us that in order to push the Higgs mass above the tree-level bound of M Z , one must increase m t . The correction is ∆m 2 h = 3 4π 2 v 2 y 4 t sin 2 β ln m t 1 m t 2 m 2 t . (2.106) For further details, we refer to the treatment in [22] or the encyclopedic reference [23]. 2.13 The little hierarchy problem of the MSSM It has been well known since lep that in order to push m h > 114 gev in the mssm, one requires large stop masses, m t ∼ 1 − 1.4 tev. Pushing the stop mass this heavy comes at a cost, unfortunately. The stops contribute not only to the Higgs quartic-which we need to push the Higgs mass up-but also to the soft mass m 2 Hu from loops of the form + The larger one sets m t , the larger the shift in m 2 Hu . Recall, however, the strange cancellation we noted in (2.102). This equation seems to want m 2 Hu ∼ M 2 Z /2. The loop corrections above contribute a shift of the form ∆m 2 Hu = 3y 2 t 4π 2 m 2 t ln Λ UV m t . (2.107) For m t = 1.2 tev and Λ UV = 10 16 this balancing act between m 2 Hu and M 2 Z /2 requires a fine tuning of M 2 Z /2 ∆m 2 Hu ∼ 0.1%. Physically what's happening is that the stop plays a key role in naturalness by canceling the sensitivity to the uv scale. By pushing the stop to be heavier to increase the Higgs quartic, one reintroduces quadratic sensitivity up to the scale of the stop mass. This is known as the little hierarchy problem of the mssm. SUSY breaking versus flavor The soft breaking Lagrangian introduces many new masses, phases, and mixing angles on top of those found in the Standard Model for a total of 124 parameters [24]. Most of this huge parameter space, however, is already excluded from flavor and cp violating processes. Recall that in the sm, there are no tree-level flavor-changing neutral currents (fcnc) and loop-level contributions are suppressed by the gim mechanism. Lepton number violation is similarly strongly suppressed. In the limit where the Yukawa couplings vanish, y → 0, the Standard Model has a U(3) 5 flavor symmetry where each of the five types of matter particles are equivalent. This flavor symmetry is presumably broken at some scale Λ F in such a way that the only imprint of this uv physics at scales well below Λ F are the Yukawa matrices. This flavor scale can be very large so that effects of this flavor breaking go like 1/Λ F and are plausibly very small. In the mssm, one must further check that the flavor breaking dynamics has already 'frozen out' at the susy breaking scale so that the only non-trivial flavor structure in the susy breaking parameters are the Yukawa matrices themselves. This means we would like the mediator scale M to be below the flavor scale, M Λ F . In gravity mediation, however, Λ med = M Pl , and we can no longer guarantee that the susy breaking mediators are insulated from flavor violating dynamics. This leads to strong constraints on the flavor structure of the mssm soft parameters. For example, consider one of the most carefully studied fcnc processes, kaon anti-kaon (K-K) mixing. The quark content of the mesons are K = ds andK =ds. In the sm this process is mediated by diagrams such as There are similar constraints on cp violating and lepton number violating processes (e.g. dipole moments and µ → eγ). This is the SUSY flavor problem: a generic flavor structure for the mssm soft parameters is phenomenologically ruled out. We are led to conclude that the off-diagonal flavor terms must be strongly suppressed to avoid experimental bounds. One way to do this is to suppose an organizing principle in the susy breaking parameters, soft-breaking universality, 1. Soft breaking masses are all universal for all particles at some high scale. This means that m 2 Q ∝ 1 in flavor space, and similarly for each mssm matter multiplet. 2. If a-terms are not flavor-universal, then the Higgs vev induces similar problematic mixings, L a = a u ij Q iŪj H u + a d ij Q iDj H d + a e ij L iĒj H d . (2.110) To avoid this, assume that a I ij is proportional to the Yukawa matrix, a I ij = A I y I ij . (2.111) This way, the rotation that diagonalizes the sm fermions also diagonalizes their scalar partners. 3. To avoid cp violation, assume that all non-trivial phases beyond those in the Standard Model ckm matrix vanish. These are phenomenological principles. Ultimately, one would like to explain why these properties should be true (or at least approximately so). Gauge mediated SUSY breaking One straightforward realization of soft-breaking universality is to have the messenger sector be flavor universal. A natural way to do this is gauge mediation since the sm gauge fields are blind to flavor [25][26][27][28]. See [29] for a review. $ $ $ susy messenger mssm F X = 0 Φ i ,Φ i sm gauge The main idea is that the susy breaking sector has some superfield (or collection of superfields) X which pick up F -term vevs, F X = 0. This generates mass splittings in the messenger sector superfields, Φ i andΦ i . These messengers obey the tree-level susy sum rules discussed above but are not problematic since all of the components can be made heavy. One then assumes that the messengers are charged under the sm gauge group so that the mssm superfields will feel the effects of susy breaking through loops that include the messenger fields. Note that anomaly cancellation of the sm gauge group typically requires the messenger superfields to appear in vector-like pairs, Φ andΦ with opposite sm quantum numbers. The messenger fields generate non-renormalizable operators that connect the mssm and the susy breaking sector without introducing any flavor dependence for the soft masses. Further, because the messenger scale is adjustable, one can always stay in regime where it is parametrically smaller than the flavor scale M Λ F . Recall the estimates in Section 2.11 for the size of the mssm soft terms. For gauge mediation, M is the mass of the messenger sector fields Φ i andΦ i and F is the susy breaking vev, F X . Below M we integrate out the messengers to generate the mssm soft parameters. The simplest realization of this is minimal gauge mediation. Here one assumes only one susy breaking field X and N m mediators, Φ i andΦ i , in the fundamental representation of an SU(5) gut. The superpotential coupling between these sectors is W =ΦXΦ. (2.112) The contribution to the potential is ∂W ∂Φ 2 = \| X \| 2 \|ϕ\| 2 + \| X \| 2 \|φ\| 2 + ϕφ F X (2.113) The messenger masses are m ψ = X (2.114) m 2 ϕ = X 2 ± F X ,(2.115) using the notation where the angle brackets · · · are dropped when it is clear that we are referring to the vev of a field. Observe that the messenger scale is set by the lowest component vev of the susy breaking parameter, M = X. In what follows we make the typical assumption that F/M 2 1. Note that these masses satisfy the susy sum rule. Now let's consider the spectrum arising from this simple set up. The gauginos of the sm gauge group pick up a mass contribution from diagrams of the form F X X ψΦ ψ Φφ ϕ λ λ The X insertion on the ψ Φ line is required to flip the gaugino helicity (recall that arrows on fermion indicate helicity). The F insertion on the ϕ line is required to connect to susy breaking so that this is indeed a mass contribution that is not accessible to the gauge boson. The F vev is also required to flip from a ϕ to aφ so that the scalar of the chiral superfield picks up a sense of chirality as well. Using powerful methods based on holomporphy [30,31], the gaugino mass for the i th gauge factor is M λ i = F M M 2 g 2 i 16π 2 N m = α i 4π N m F M . (2.116) This expression-which one could have guessed from a back-of-the-envelope estimate-turns out to be exact to leading order in F/M 2 . This is a reflection of the powerful renormalization theorems in supersymmetry, see e.g. [32]. One of the concrete predictions of minimal gauge mediation is the relation M λ 1 : M λ 2 : M λ 3 = α 1 : α 2 : α 3 . (2.117) The heaviest superpartners are those which couple to the largest rank gauge group. The scalar partners of the sm matter particles do not directly couple to the messengers. Thus the masses for the squarks and sleptons must be generated at two loop level. There are many diagrams that include loops of both the messenger scalar and fermion: ϕ ψ ϕ ϕ ϕ ψ ϕ ϕ The loops either include a gauge boson or otherwise use the scalar quartic D-term interaction between messengers and sfermions. The result is that the soft scalar masses go like m 2 soft ∼ g 2 16π 2 2 N m F 2 M 2 C i ,(2.118) where C i is the relevant quadratic casimir. Observe that m 2 soft ∼ m 2 λ so that the sfermions which couple to the higher rank gauge factors pick up more mass. Including the various gauge charges and taking the limit α 3 α 2 α 1 gives a prediction for the sfermion spectrum in minimal gauge mediation, Note that since the messengers interact with the sm superfields only through gauge interactions, the holomorphic soft terms (A and B terms) are typically very small in gauge mediation. One important phenomenological consequence of gauge mediation is that the lightest supersymmetric partner (lsp) is not one of the mssm fields but rather the gravitino whose mass is [29], m 3/2 ∼ F √ 3M Pl ∼ √ F 100 TeV 2 2.4 eV. (2.121) Thus the gravitino is much lighter than the electroweak scale, but is also similarly weakly coupled. The relevant couplings at low energies are not gravitational, but rather through the Goldstino component of the gravitino. This coupling is proportional to the susy breaking vev F . Because of R-parity, any supersymmetric partner produced in the mssm will eventually decay into the nextto-lightest superpartner (nlsp). This nlsp must eventually decay into the gravitino lsp since it is the only decay mode available. When √ F 10 6 GeV, the nlsp is so long lived that on collider scales it behaves effectively like the lsp. On the other hand, if √ F 10 6 GeV, the nlsp decays within the detector. This gives a fairly unique signal with displaced photons and missing energy if the nlsp is the bino, B. The µ-B µ problem of gauge mediation Let's return to an issue we addressed earlier when discussing electroweak symmetry breaking. We wrote two relations (2.101 -2.102) satisfied at the minimum of the Higgs potential. We noted the µ-problem associated with (2.102): µ and m 2 H u,d seem to come from different sectors of the theory but must conspire to be roughly the same scale. In principle, since µ is a supersymmetric dimensionful parameter (the only one in the mssm), it could take a value on the order of the Planck mass. We now present a solution to the µ-problem, but we shall see that this solution will cause problems in gauge mediation due to the second relation, (2.101). One way to address this µ-problem is to forbid it in the supersymmetric limit and then assume that it is generated through the susy breaking sector. For example, a global Peccei-Quinn (pq) symmetry, H u →e iα H u (2.122) H d →e iα H d ,(2.123) prohibits the µ term in the superpotential. Gravity, however, is believed to explicitly break global symmetries. Indeed, gravity mediation of susy breaking will produce a µ term. Consider, for example, the higher order Kähler potential term that couples the susy breaking superfield X to the Higgses [35], d 4 θ X † H u · H d M Pl + h.c. (2.124) When X ∼ θ 2 F , one generates an effective µ term of order µ ∼ F/M Pl . This neatly addresses the µ-problem and ties the µ term to the susy breaking masses. The B µ term that is generated comes from d 4 θ X † XH u · H d M 2 Pl (2.125) and thus is of the same order as µ 2 . This is consistent with the observation in (2.101) that B µ , µ, and the soft breaking terms seem to want to be the same order. We remark that this is no longer true in gauge mediation since F 10 11 GeV, the µ and B µ terms generated from gravitational breaking are far too small. This must be addressed separately in such theories. Variations beyond the MSSM The mssm is under pressure from the lhc. For a review of the status after Run I of the lhc, see [36]. There are two main issues: 1. The Higgs mass m h = 125 GeV is hard to achieve in the mssm since it requires a large radiative correction to the tree level upper bound of m h = M Z . 2. There are no signs of superpartners. With the simplest assumption that m q ∼ m g , the lhc pushes the scale of colored superpartners to be over 1.2 tev. This appears to no longer be natural. In this section we present some model-building directions that the lhc data may be suggesting. Additional D-term contributions One simple direction to increase the tree-level Higgs mass is to add extra D-terms to increase the Higgs quartic coupling [37][38][39][40][41]. This requires charging the Higgs under an additional U(1) X gauge group which one must break above the weak scale. This technique is able to indeed push the tree-level Higgs mass up to the observed value, but one is constrained by changes to Higgs decay branching ratios, particularly h → bb [42,43]. The NMSSM At the cost of adding an additional singlet superfield S to the mssm sector, one may solve the µ problem and also raise the Higgs mass by enhancing its quartic coupling [44][45][46][47][48]. The Higgs sector superpotential for this "next-to-minimal" supersymmetric sm (nmssm, see [49,50] for reviews) is W NMSSM = y u H u QŪ + y d H d QD + y e H dĒ + λSH u H d + 1 3 κS 3 . (2.126) The κ term breaks the Peccei-Quinn symmetry, (2.122 -2.123), to a Z 3 . Since S is a gauge singlet, the D-term potential is unchanged from (2.91). Note, however, that there is no longer a µ term in the superpotential, instead the SH u H d coupling has taken its place. Thus the F -term potential differs from that of the mssm, (2.92), and is instead V F,nmssm = λ 2 \|S\| 2 \|H u \| 2 + \|H d \| 2 + λ 2 \|H u H d \| 2 . (2.127) We observe that the combination λ S plays the role of an effective µ term and solves the µproblem. Finally, there are additional soft terms allowed which augment V soft in (2.93), ∆V soft,nmssm = m 2 S \|S\| 2 + λA λ (SH u H d + h.c.) + 1 3 κA κ (S 3 + h.c.). (2.128) The resulting expression for the Higgs mass is approximately m 2 h ≈ M 2 Z cos 2 2β + λ 2 v 2 sin 2 2β − λ 2 v 2 κ 2 (λ − κ sin 2β) 2 + 3m 4 t 4π 2 v 2 ln m t m t + A 2 t m t 1 − A 2 t 12m t . (2.129) This can be larger than the value in the mssm depending on the value of λ. There are limits on the size of λ coming from perturbativity, but lifting the Higgs mass to 125 gev is fine. The singlet S contributes an additional complex scalar to the Higgs sector and an additional neutralino. Natural SUSY The simplest choices for the mssm parameters-those that treat all the flavors universally, as preferred by the flavor problem-are tightly constrained by the non-observation of new physics at the lhc. Because the lhc is a proton-proton collider, it is easy for it to produce colored superpartners such as squarks and gluinos. These, in turn, are expected to show up as events with many jets and missing energy as the heavy colored states decay into the lsp. The fact that no significant excesses have been found pushes one to consider other parts of the large mssm parameter space. Instead of biasing our parameter preferences by simplicity, one may take a different approach and ask what is the minimal sparticle content required for naturalness? In other words, which superpartners are absolutely required to cancel quadratic divergences? Once these are identified, one may decouple the remaining sparticles and check the experimental constraints on the resulting spectrum. The ingredients of a 'minimally' natural mssm spectrum are [51,52] (see [53][54][55][56] for a re-examination from the early lhc run) 1. Light stops. The largest sm contribution to the Higgs quadratic uv sensitivity is the top loop. Naturalness thus requires that its superpartner, the stop, is also accessible to cancel these loops. Since the stop lives in both the U R and Q L superfields, this typically also suggests that the left-handed sbottom is also light. 2. Light Higgsinos. In order to preserve natural electroweak symmetry breaking, µ should be on the order of the electroweak scale. This is the same parameter that determines the Higgsino mass, so the Higgsinos should also be light. 3. Not-too-heavy gluinos. The stop is a scalar particle which is, itself, quadratically uv sensitive at face value. The main contribution to the stop mass comes from gluon loops so that naturalness requires 'not-too-heavy' (∼ 1.5 tev) gluinos to cancel these loops. In other words, the gluino feeds into the Higgs mass at two-loop order since it keeps the light-stop light enough to cancel the Higgs' one-loop uv sensitivity. Figure 3: Heuristic picture of a natural susy spectrum. All other superpartners are assumed to have masses well above the tev scale and decouple. H t R , t L b L g W B 4. Light-ish electroweak-inos (optional). Finally, if one insists on grand unification, the scale of the gluinos imposes a mass spectrum on the electroweak gauginos with M ew-ino < M gluino . As a rough estimate, Majorana gluinos should have mass 2m t while Dirac gluinos should have mass 4m t . 5. All other particles decoupled. All of the other squarks and sleptons are assumed to be well above the tev scale and effectively inaccessible at Run-I of the lhc. These are shown in Fig. 3. The simplest models have a light stop t L which decays either to a top and neutralino/gravitino, t + N , or a bottom and a chargino, b + C. Bounds on these decays depend on the N ( C) mass. The 'stealthy' region near m N = 0 and the 'compressed' region near m N ≈ m t are especially difficult to probe kinematically. R-parity violation One of the main ways to search for 'vanilla' susy signatures is to trigger on the large amount of missing energy (met or E T ) expected from the neutral lsp. Underlying this assumption is R-parity, which forces the lsp to to be stable. Recall that R-parity was something that we embraced because it killed the supersymmetric terms in the superpotential (2.50) that would violate lepton and baryon number and would be severely constrained by experiments, most notably proton decay. If, however, there were another way to suppress these dangerous operators, then perhaps we could avoid the experimental bounds while giving the lsp a way to decay into non-supersymmetric particles. This would allow us to consider models with R-parity violation (rpv) with no missing energy signal [57][58][59][60][61], see [62] for a review. Such models would be immune to the usual met-based susy search strategies. The simplest way to do this is to turn on only the λ 4ŪDD term. This violates baryon number but preserves lepton number so that protons remain stable. Motivated by naturalness, we may now allow the stop to be the lsp since this is no longer a dark matter candidate. The rpv coupling would allow a decay t →bs, which would be hidden in the large qcd di-jet background. One still has to worry about the effects of this rpv coupling on the partners of the light squarks. Phenomenologically, the strictest bounds come from neutron-anti-neutron oscillation and dinucleon decay. Indeed, most of the flavor bounds on the mssm come from the first two generations of sparticles. One interesting model-building tool is to invoke minimal flavor violation, which posits that the flavor structure of the entire mssm is carried by the Yukawa matrices [63]. This then implies that the coefficient of theŪ iDjDk rpv coupling is proportional to a product of Yukawa elements depending on the generations i,j, and k. This gives a natural explanation for why the rpv couplings of the first two generation squarks are much smaller than the stop. Extra Dimensions The original proposal for extra dimensions by Kaluza [64], Klein [65], and later Einstein [66] were attempts to unify electromagnetism with gravitation. Several decades later the development of string theory-originally as a dual theory to explain the Regge trajectories of hadronic physicsled physicists to revisit the idea of compact extra dimensions [67][68][69]. In early models, the nonobservation of an additional spatial direction was explained by requiring the compactification radius to be too small for macroscopic objects. Further reading: Two of the authors' favorite reviews on this subject are [70] and [71]. This lecture is meant to be largely complementary. Additional references include [72][73][74][75][76][77], which focus on different aspects. Kaluza-Klein decomposition The simplest example to begin with is a real scalar field in 5D where the fifth dimension is compactified to a circle of radius R. The details of the compactification do not change the qualitative behavior of the theory at low energies. The Lagrangian is S = d 5 x 1 2 ∂ M φ(x, y)∂ M φ(x, y) = d 5 x 1 2 ∂ µ φ(x, y)∂ µ φ(x, y) − (∂ y φ(x, y)) 2 ,(3.1) where M = 0, · · · , 5 and x 5 = y. Since y is compact, we may identify energy eigenstates by doing a Fourier decomposition in the extra dimension, φ(x, y) = 1 √ 2πR ∞ n=−∞ φ (n) (x) e i n R y . (3.2) Since φ is real, φ (n) † = φ (−n) . Plugging this expansion into the action allows us to use the orthogonality of the Fourier terms to perform the dy integral. This leaves us with an expression for the action that is an integral over only the non-compact dimensions, but written in terms of the kk modes φ (n) (x), S = d 4 x mn dy 1 2πR e i (m+n) R y 1 2 ∂ µ φ (m) (x)∂ µ φ (n) (x) + mn R 2 φ (m) (x)φ (n) (x) (3.3) = 1 2 d 4 x n ∂ µ φ (−n) ∂ µ φ (n) − n 2 R 2 φ (−n) φ (n) (3.4) = d 4 x n>0 ∂ µ φ (n) † ∂ µ φ (n) − n 2 R 2 φ (n) 2 . (3.5) From the 4D point of view, a single 5D scalar becomes a 'Kaluza-Klein (kk) tower' of 4D particles, each with mass n/R. If there were more than one extra dimension, for example if one compactified on an k-dimensional torus with radii R 5 , R 6 , . . ., then the kk tower would have k indices and masses m 2 n 5 ,n 6 ,··· ,n k = m 2 0 + n 2 5 R 2 5 + n 2 6 R 2 6 + · · · + n 2 k R 2 k , (3.6) where m 2 0 is the higher dimensional mass of the field. Gauge fields A more complicated example is a gauge field. We know that gauge fields are associated with vector particles, but in 5D the vector now carries five components, A M . We perform the same kk decomposition for each component M , A M (x, y) = 1 √ 2πR n A (n) M (x) e i n R y . (3.7) Note that this decomposes into a kk tower of 4D vectors, A (n) µ , and a kk tower of 4D scalars, A (n) 5 . Similarly, the field strengths are antisymmetric with respect to indices M and N so that the action is decomposed according to S = d 4 x dy − 1 4 F M N F M N (3.8) = d 4 x dy − 1 4 F µν F µν + 1 2 (∂ µ A 5 − ∂ 5 A µ ) (∂ µ A 5 − ∂ 5 A µ ) (3.9) = d 4 x n − 1 4 F (−n) µν F (n)µν + 1 2 ∂ µ A (−n) 5 − ∂ 5 A (−n) µ ∂ µ A (n) 5 − ∂ 5 A (n)µ . (3.10) This looks complicated because there is an odd mixing between the 4D vector, A (n) µ , and the 4D scalar A (n) 5 . Fortunately, this mixing term can be removed by fixing to 5D axial gauge, A (n) µ → A (n) µ − i n/R ∂ µ A (n) 5 A (n) 5 → 0,(3.11) for n = 0. Note that for n = 0 there's no scalar-vector mixing anyway. The resulting action takes a much nicer form, S = d 4 x − 1 4 F (0) µν 2 + 1 2 ∂ µ A (0) 5 2 + n≥1 2 − 1 4 F (−n) µν F (n)µν + 1 2 n 2 R 2 A (−n) µ A (n)µ . (3.12) The spectrum includes a tower of massive vector particles as well as a massless (zero mode) gauge boson and scalar. Recall the usual expression for the number of degrees of freedom in a massless 4D gauge boson: (4 components in A µ ) − (longitudinal mode) − (gauge redundancy). (3.13) When the gauge boson becomes massive, it picks up a longitudinal mode from eating a scalar by the Goldstone mechanism. This is precisely what has happened to our kk gauge bosons, A (n) µ : they pick up a mass by eating the scalar kk modes, A (n) 5 . In a theory with (4 + n) dimensions, the (4 + n)-component vector A M decomposes into a massless gauge boson, n massless scalars, a tower of massive kk vectors A µ , and a tower of (n − 1) massive kk scalars. One may similarly generalize to spin-2 particles such as the graviton. In (4 + n) dimensions these are represented by an antisymmetric (4 + n) × (4 + n) tensor, g M N =   g µν A µ ϕ   . (3.14) The massless 4D zero modes include the usual 4D graviton, a vector, and a scalar. At the massive level, there is a kk tower of gravitons with (n − 1) gauge fields and [ 1 2 n(n + 1) − n] scalars. Here we observe the graviton and vector eating the required degrees of freedom to become massive. Matching of couplings It is important to notice that the mass dimension of couplings and fields depend on the number of spacetime dimensions. The action is dimensionless, [S] = 0, since it is exponentiated in the partition function. Then, in (4 + n) dimensions, the kinetic term for a boson gives d (4+n) x (∂φ) 2 = −(4 + n) + 2 + 2[φ] = 0 ⇒ [φ] = 1 + n 2 . (3.15) Note that this is consistent with the dimensions in the kk expansion (3.2). The 5D scalar contains the 4D scalars with a prefactor ∼ R −1/2 that has mass dimension 1/2. Similarly, for fermions, [ψ] = 3 2 + n 2 . With this information, dimensions of the Lagrangian couplings can be read off straightforwardly. For example, the 5D gauge field lives in the covariant derivative, D µ = ∂ µ − ig 5 A µ = ∂ µ − i g 5 √ 2πR A (0) µ + · · · . (3.16) We see that [g 5 ] = −1/2 since [∂] = 1 and [A M ] = 3/2. Further, we find an explicit relation between the 5D parameter g 5 and the observed 4D gauge coupling, g 4 = g 5 √ 2πR . (3.17) More generally, in (4 + n) dimensions the 4D coupling is related to the higher dimensional coupling by the volume of the extra dimensional space, g 2 4 = g 2 (4+n) Vol n . (3.18) One can read off the matching of the gravitational coupling by looking at the prefactor of the Ricci term in the action, S (4+n) = −M 2+n (4+n) d 4+n x √ g R (4+n) = −M 2+n (4+n) V n d 4 x √ g (4) R (4) + · · · ,(3.19) where we've written g for the determinant of the metric. From this we identify 4D Planck mass M Pl from the fundamental higher dimensional Planck mass, M (4+n) , M 2 Pl = M 2+n (4+n) V n . (3.20) The higher dimensional Planck mass is a good choice for a fundamental mass scale for the theory, M * = M (4+n) . (3.21) In a (4 + n) dimensional theory where the characteristic mass scale is M * and a compactification radius R. Then dimensional analysis tells us that the higher dimensional gauge couplings, which are dimensionful, characteristically scale like g (4+n) ∼ M −n/2 * . (3.22) Relating this to the 4D couplings with (3.18) and relating M * to the 4D Planck mass with (3.20) gives R ∼ 1 M Pl g (n+2)/n 4 . (3.23) Plugging in the observed sm gauge couplings on the right hand side gives a compactification radius which is far too small to be relevant at colliders-the first kk modes will be near the Planck scale. Branes and Large Extra Dimensions In the mid '90s, developments in string theory led to a new ingredient that renewed interest in extra dimensions that might be accessible at collider scales. The key idea is that branes, solitonic objects which form lower dimensional subspaces, can trap fields. In other words, not all fields have to propagate in all dimensions. This was introduced by Rubakov and Shaposhnikov [78], who showed that instead of a very small radius of compactification, it may be that our observed universe is constrained to live in a (3+1)-dimensional subspace of a higher dimensional spacetime. Terminology. Models that make use of branes to localize fields are known as braneworld models and are distinguished from models where all fields propagate in the extra dimensions, known as universal extra dimensions. In braneworld models, fields which are allowed to propagate in the full space are said to live in the bulk. Allowing the fields to be brane-localized buys us quite a lot. It allows us to separate particle physics from gravity. One can, for example, force the sm fields to be truly four-dimensional objects that are stuck to a (3+1)-dimensional brane. This avoids the bound on the size of the extra dimension in (3.23), since that relied on the sm propagating in the bulk. With this in mind, one could allow the volume of the extra dimensions to actually be quite large. This idea was explored by Arkani-Hamed, Dimopoulos, and Dvali in the add or large extra dimension scenario [79]. If this were feasible, then (3.20) gives a new way to address the Hierarchy problem. The large volume factor allows the fundamental scale of nature to be much smaller than the observed Planck mass, M * M Pl . If, for example, M * ∼ 1 tev, then there is no Hierarchy problem. Gravity appears to be weaker at short distances because its flux is diluted by the extra dimensions. As one accesses scales smaller than R, however, one notices that gravity actually propagates in (4 + n) dimensions. A cartoon of the braneworld scenario is shown in Fig. 4. How large can this extra dimension be? Doing a rough matching and using Vol n = r n in (3.20) gives R = 1 M * M Pl M * 2/n . (3.24) g r a v i t y b r a n e g r a v it o n Figure 4: Cartoon pictures of a (3+1) dimensional brane in a compact 5D space. (left) The brane (red line) as a subspace. Gravity propagates in the entire space 'diluting' its field lines relative to forces localized on the brane. (right) sm processes localized on the brane, now with an additional dimension drawn, emitting a graviton into the bulk. Pushing the fundamental scale to M * ∼ tev requires R = 10 32/n tev −1 = 2 · 10 −17 10 32/n cm, (3.25) using GeV −1 = 2 · 10 −14 cm. We make the important caveat that this is specifically for the ADD model. Considering different numbers of extra dimensions, • n = 1. For a single extra dimension we have R = 10 15 cm, which is roughly the size of the solar system and is quickly ruled out. • n = 2. Two extra dimensions brings us down to R ≈ 0.1 cm, which is barely ruled out by gravitational Cavendish experiments. • n = 3. Three extra dimensions pushes us down to R < 10 −6 cm. How much do we know about gravity at short distances? Surprisingly little, actually. Cavendish experiments (e.g. Eöt-Wash 5 ) test the r −2 law down to 10 −4 m. These set a direct bound on the n = 2 case that R < 37 µm and M * > 1.4 TeV. For larger n one is allowed to have M * = TeV. One might have objected that one cannot say that M * is the fundamental scale while allowing R, itself a dimensionful quantity, float to take on any value. Indeed, in a completely natural theory, one expects R ∼ 1/M * so that R ∼ tev −1 . This is quite different from what we wrote in (3.24). Indeed, what we have done here is swapped the hierarchy in mass scales to a Hierarchy between R and M −1 * . In other words, we have reformulated the Hierarchy problem to a problem of radius stabilization. This is indeed very difficult to solve in add. Nevertheless, we may explore the phenomenological consequences of an add type model at colliders and through astrophysical observations. • The first thing to consider is the production of kk gravitons. 5 The name is a play on the Eötvös experiment by University of Washington researchers. f f G γ, g The kk graviton couples too weakly to interact with the detector so it appears as missing energy. By itself, however, missing energy is difficult to disentangle from, say, neutrino production. Thus it's useful to have a handle for the hardness of the event (more energetic than Z → νν) so one can look for processes that emit a hard photon or gluon. Thus a reasonable search is a jet or photon with missing energy. It is worth noting that this is the same search used for searching for dark matter, which is also typically a massive particle which appears as missing energy. • Alternately, one may search for s-channel virtual graviton exchange in processes like e + e − → ff . One expects a resonance at the kk graviton mass. • Supernovae can cool due to the emission of gravitons. This is similar to the supernovae cooling bounds on axions. The strongest bounds on n = 2 theories push M * 100 TeV. • An additional byproduct of lowering the fundamental gravitational scale is that one may form microscopic black holes at energies kinematically accessible to the lhc and cosmic rays. For E CM > M * black holes are formed with a radius R S ∼ 1 M * M BH M * 1 n+1 . (3.26) the cross section is roughly the geometric value, σ BH ∼ πR 2 S and can be as large as 400 pb. These microscopic black holes decay via Hawking radiation, T H ∼ 1 R S (3.27) with this energy distributed equally to all degrees of freedom, for example 10% going to leptons, 2% going to photons, and 75% going to many jets. Warped extra dimensions We've seen that the framework of large extra dimensions leads to interesting phenomenology, but the add realization leaves the size of the radius unexplained and is therefore not a complete solution to the Hierarchy problem. The Randall-Sundrum (rs) proposal for a warped extra dimension offers a more interesting possibility [80]. The set up differs from add in that the space between the two branes has a non-factorizable metric that depends on the extra space coordinate, z, ds 2 = R z 2 η µν dx µ dx ν − dz 2 . (3.28) This is the metric of anti-de Sitter space (ads) with curvature k = 1/R. There are two branes located at z = R (the uv brane) and z = R > R (the ir brane) that truncate the extra dimension; in this sense the rs background is often described as a 'slice of ads.' We see that 1/R is naturally a fundamental uv scale of the theory. The metric (3.28) warps down the natural physical scale as a function of the position along the extra dimension. In particular, when R R one finds that near z = R , the scales are warped down to much smaller values. Note the different notation from the add case: the size of the extra dimension is R − R ≈ R , while R should be identified with the radius of curvature. To see how this works, suppose that the Higgs is localized to live on the ir brane at z = R . The action on this brane depends on the 4D induced metricĝ µν (note that √ĝ = √ g/ √ g 55 ), S = d 4 x ĝ ∂ µ H∂ ν Hĝ µν − \|H\| 2 − v 2 2 2 (3.29) We assume that the Higgs vev is on the order of the uv scale, v = 1/R, since this is the fundamental 5D scale. Plugging in the metic gives S = d 4 x R R 4 ∂ µ H∂ µ H R R 2 − \|H\| 2 − v 2 2 2 z=R , (3.30) where indices are implicitly raised with respect to the Minkowski metric. Canonically normalizing the kinetic term viaĤ = R R H, (3.31) allows us to write the action in the form, S = d 4 x ∂ µĤ 2 − λ \|Ĥ\| 2 − 1 2 v R R 2 2 , (3.32) where we see that the canonically normalized Higgs picks up a vev that is warped down to the tev scale. One can further imagine that the cutoff for loops contributing to the Higgs mass are similarly warped down to, say, the tev scale. In this way, the warped extra dimension gives a new handle for generating hierarchies. Readers should be skeptical that we're not just hiding the Hierarchy problem in some fine tuning of the ir scale R relative to the fundamental scale R. Indeed, the real solution to the Hierarchy problem requires a mechanism for radius stabilization, which we present below. Note that typically R ≈ M −1 Pl and R ≈ TeV −1 so that R is roughly the size of the extra dimension. A cartoon of this scenario is shown in Fig. 5. In the remainder of this lecture we'll focus on the rs background. In the appendices we present some additional technical results that may be useful for building rs models. Further details of the rs gravitational background are discussed in Appendices A.1 and A.2. Details of bulk matter fields are discussed in Appendices A.3 and A.4 The Planck scale and hierarchy in RS We have seen how the ads curvature can warp mass scales to be much smaller than the fundamental 5D scale 1/R. It is instructive to also check the observed Planck scale. With respect to the fundamental Planck scale M * (ostensibly M * ∼ 1/R), the gravitational action is where the quantities with subscripts are the determinant of the 5D metric and the 5D Ricci scalar, respectively. By performing the dz integral one finds the effective 4D gravitational action, S g = M 3 * R R dz d 4 x √ g (5) R (5) ,(3.S g = M 3 * R R dz R z 3 d 4 x √ g (4) R (4) = M 3 * 1 2 1 − R R 2 d 4 x √ g (4) R (4) . (3.34) We can thus identify the effective 4D Planck mass by reading off the coefficient, M 2 Pl = M 3 * R 2 1 − R R 2 ∼ M 2 * ,(3.35) so that for a large extra dimension, R R, the 4D Planck mass is insensitive to R and is fixed by the 5D Planck mass, M * ∼ 1/R. This is precisely what we have set out to construct: assuming there is a dynamical reason for R R, we are able to warp down masses to the tev scale by forcing particles to localize on the ir brane while simultaneously maintaining that 4D observers will measure a Planck mass that is much heavier. An alternate way of saying this is that the Hierarchy problem is solved because the sm Higgs is peaked towards the ir brane while gravity is peaked towards the uv brane. What we mean by the latter part of this statement is that the graviton zero mode has a bulk profile that is peaked towards the uv brane. Recall that in flat space, zero modes have flat profiles since they carry no momentum in the extra dimension. In rs, the warping of the space also warps the shape of the graviton zero mode towards the uv brane; the weakness of gravity is explained by the smallness of the graviton zero mode profile where the Standard Model particles live. This should be compared to the case of a flat interval where the zero mode wave function decouples as the size of the extra dimension increases. In this case the coupling with the ir brane indeed becomes weaker, but the graviton kk modes become accessible and can spoil the appearance of 4D gravity. In rs the zero mode doesn't decouple and one doesn't need to appeal to a dilution of the gravitational flux into the extra dimensions as in the add model. See [70] for more explicit calculations in this picture. Bulk scalar profiles in RS In the original rs model, only gravity propagates in the bulk and has kk modes. However, it is instructive to derive the kk properties of a bulk scalar. • This serves as a simple template for how to kk reduce more complicated bulk fields, such as the graviton, in a warped background. See, e.g. [70] for the analysis of the graviton. • We anticipate the 'modern' incarnations of the rs where gauge and matter fields are pulled into the bulk. The properties of these fields are detailed in Appendix A and follow this analysis of the bulk scalar. • As mentioned above, the solution to the Hierarchy problem depends on stabilizing the position of the ir brane, z = R , relative to the uv brane, z = R. The standard technique for doing this requires a bulk scalar. Start with a bulk complex scalar Φ(x, z) with a bulk mass parameter m. The bulk action is S = R R dz d 4 x √ g (∂ M Φ) * ∂ M Φ − m 2 Φ * Φ . (3.36) In principle one may have additional brane-localized interactions proportional to δ(z − R ) or δ(z − R). We use M, N to index 5D coordinates while µ, ν only run over 4D coordinates. Varying with respect to Φ * yields an equation of motion −∂ M √ gg M N ∂ N Φ − √ gm 2 Φ = 0. (3.37) In writing this we have dropped an overall surface term that we picked up when integrating by parts. Specializing to the rs metric, this amounts to picking boundary conditions such that Φ * (z)∂ z Φ(z)\| R,R = 0, (3.38) with the appropriate modifications if there are brane-localized terms. We see that we have a choice of Dirichlet and Neumann boundary conditions. We now plug in the Kaluza-Klein decomposition in terms of yet-unknown basis functions f (n) (z) which encode the profile of the n th mode in the extra dimension: Φ(x, z) = 1 √ R ∞ n φ (n) (x)f (n) (z). (3.39) The factor of 1/ √ R is pulled out for explicit dimensional analysis; that is, the profile f (n) (z) is defined to be dimensionless. By assumption, the φ (n) are eigenstates of η µν ∂ µ ∂ ν with eigenvalue −m 2 (n) , the kk mass. We are thus left with a differential equation for f (n) (z), R z 3 m 2 (n) − 3 z R z 3 ∂ z + R z 3 ∂ 2 z − R z 5 m 2 f (n) (z) √ R = 0. (3.40) This is a Sturm-Liouville equation with real eigenvalues and real, orthonormal eigenfunctions, R R dz R R z 3 f (n) (z)f (m) (z) = δ mn . (3.41) Just as we saw in Section 3.1 for a flat extra dimension, this orthonormality relation diagonalizes the kk kinetic terms. One may now solve (3.40) by observing that through suitable redefinitions this is simply a Bessel equation. The result is a general solution for n > 0 of the form f (n) (z) = c 1 z 2 J α (m (n) z) + c 2 z 2 Y α (m (n) z), (3.42) where J, Y are the familiar Bessel functions and α = √ 4 + m 2 R 2 . The integration constants c 1,2 and the spectrum of kk masses m 2 (n) can be found using boundary conditions on each brane and the orthonormality relation (3.41). The states have a discrete spectrum with spacing of approximately R −1 ∼ tev with profiles peaked towards the ir brane. Sturm-Liouville theory. The orthonormality relations for bulk fields in a warped background are results of the Sturm-Liouville form of the equations of motion for these fields. The generic form of such an equation is ∂ z p(x)∂ z f (n) (z) + q(z)f (n) (z) = −λw(z)f (n) (z) , (3.43) where the weight w(z) > 0 and the eigenvalue λ is identified with the squared kk mass, m 2 n . The regular solutions of such an equation satisfy the orthonormality relation dz f (n) (z)f (m) (z)w(z) = δ mn , (3.44) where the integration is over the relevant interval. In the case of the rs model, this range is (R, R ) and one must use the dimensionful profiles, f (n) (z) → f (n) (z)/ √ R so that the integral is dimensionless. As shown in Appendix A, the equations of motion for bulk fields of non-trivial spin will lead to Sturm-Liouville equations with different weight functions. The orthonormality condition for the kk profiles of these fields then differ in the power of (R/z) weighting the overlap integral. We note that the equations of motion for bulk fields of any spin can be massaged into Bessel equations, which are themselves a special case of a Sturm-Liouville equation. For n = 0 the zero mode profile is f (0) (z) = c 1 z 2− √ 4+m 2 R 2 + c 2 z 2+ √ 4+m 2 R 2 . (3.45) We use this result in the Goldberger-Wise mechanism discussed below, but let us remark that the zero mode is neither consistent with Neumann nor Dirichlet boundary conditions and requires brane localized terms to generate boundary conditions that permit a zero mode. The same general procedure can be used to find the profiles of higher spin bulk fields. In Appendices A.3 and A.4 we work through the additional subtleties coming from fermions and gauge bosons. A Standard Model field is associated with the zero mode of a 5D field, where the sm mass is a correction from electroweak symmetry breaking on the zero mass from the kk decomposition. Note that the meaning of the 5D profile is that a 4D particle, even though it is localized and pointlike in the four Minkowski dimensions, is an extended plane wave in the fifth dimension. The boundary conditions imposed by the branes mean that this system is essentially identical to a waveguide in electrodynamics 6 . Radius stabilization We've now shifted the Hierarchy problem to a question of why the ir scale R is so much larger than the uv scale R. In fact, one should think about R as the expectation value of a dynamical degree of freedom, R = r(x, z) , called the radion. This is identified with the 4D scalar arising from the dimensional decomposition of the 5D metric. This isn't surprising since the metric is, of course, the quantity which measures distances. Thus far in our description of the rs framework, the radion is a modulus-it has no potential and could take any value. This is problematic since excitations of this field would be massless and lead to long-range modifications to gravity. It is thus important to find a mechanism that dynamically fixes R ∼ tev −1 to (1) provide a complete solution to the Hierarchy problem and (2) avoid constraints from modifications to gravity. Don't be fooled by coordinate choices. The original rs literature used variables such that the metric explicitly contained an exponential warping ds 2 = e −2ky dx 2 − dy 2 so that an O(10) value of kπR leads to large hierarchies. Do not confuse this variable choice with a solution to the Hierarchy problem-it just shifts the fine tuning into a parameter to which the theory is exponentially sensitive. The reason why the exponential hierarchy is actually physical in rs (with a dynamically stabilized radius) is that fields propagating in the space are redshifted as they 'fall' towards the ir brane in the gravitational well of the ads background. A standard solution in the rs model is the Goldberger-Wise mechanism 7 [82,83], where radion kinetic and potential energy terms conspire against one another to select vacuum with finite R . To do this, we introduce a massive bulk scalar field Φ(x, z) of the type in Section 3.7. We introduce brane-localized potentials for this field which force it to obtain a different vev at each brane, ϕ uv = ϕ ir , ∆L = −λδ(z − R) Φ 2 − ϕ 2 uv 2 − λδ(z − R ) Φ 2 − ϕ 2 ir 2 λ → ∞. (3.46) This causes the scalar to pick up a z-dependent vev that interpolates between ϕ uv and ϕ ir , Φ(x, z) = ϕ(z) ϕ(R) = ϕ uv ϕ(R ) = ϕ ir . (3.47) The general form of ϕ(z) is precisely the zero mode profile in (3.45) since the vev carries zero momentum in the Minkowski directions. One may now consider the terms in the action of Φ(x, z) (evaluated on the vev ϕ(z)) as contributions to the potential for the radion via R = r(x, z) . The kinetic term for Φ(x, z) contributes a potential to r(x, z) that goes like ϕ (z) 2 . 1. This gradient energy is minimized when ϕ(z) has a large distance to interpolate between ϕ uv,ir since larger R allows a smaller slope. 2. On the other hand, the bulk mass for Φ(x, z) gives an energy per unit length in the z-direction when ϕ(z) = 0. Thus the energy from this term is minimized when R is small. By balancing these two effects, one is able to dynamically fix a value for R . A pedagogical derivation of this presented in [72]. The main idea is that for small values of the bulk Φ(x, z) mass, m 2 R −2 , one may write the Φ(x, z) vev as ϕ = c 1 z −ε + c 2 z 4+ε , (3.48) where ε = α − 2 = √ 4 + m 2 R 2 − 2 ≈ m 2 R 2 /4 is small. The coefficients c 1,2 are determined by the boundary conditions (3.47). The potential takes the form V [R ] = ε ϕ 2 uv R + R 3 R 4 (4 + 2ε) ϕ ir − ϕ uv R R ε 2 − εϕ ir R −4 + O R 4 R 8 ,(3.49) where judicious checkers of dimensions will recall that the dimension of the 5D scalar is [ϕ(x, z)] = 3 2 . The minimum of this potential is R = R ϕ uv ϕ ir 1/ε . (3.50) We can generate the Planck-weak hierarchy with 1/ ∼ 20 and ϕ uv /ϕ ir ∼ 10. A key point here is that we may write the radius in terms of a characteristic energy scale, R ∼ 1/µ, and the potential for µ carries terms that go like µ 4 times a polynomial in µ . This is reminiscent of dimensional transmutation and, indeed, we explain below that the rs scenario can be understood as a dual description of strongly coupled 4D dynamics. The above description of the Goldberger-Wise mechanism neglects the effect of the background Φ field on the rs geometry. For example, one may wonder if the rs metric is even compatible with the Φ vev. In order to account for this gravitational backreaction, one must solve the Φ equation of motion combined with the Einstein equation as a function of the metric (discussed in Appendix A.1) in the presence of the Φ vev. This set of coupled second order differential equations is generically very difficult to solve. Fortunately, there exists a 'superpotential 8 ' trick that one may apply to solve the system exactly. This method is described and demonstrated pedagogically for the Goldberger-Wise field in [70,73]. One finds that it is indeed possible to maintain the rs background in the presence of the bulk field necessary to stabilize the radius. Holographic interpretation Gauge/gravity duality is a way to understand the physics of a warped extra dimension as the dual to a strongly coupled 4D theory. Our goal here is to develop the intuition to use and understand the ads/cft dictionary as an interpretational tool. The most rigorous explicit derivations of this duality are often presented in the language of string theory. This idea is presented pedagogically in the language of 4D quantum field theory (rather than string theory) in [75,76,[85][86][87][88][89]. Those interested in presentations that connect to supergravity and string theory may explore [90][91][92][93][94][95], listed roughly in order of increasing formal theory sophistication starting from very little assumed background. We also point out [96] which is an excellent presentation of dualities between 4D supersymmetric gauge theories that are analogous to the gauge/gravity correspondence. We now introduce a way to re-interpret the observables of rs scenario in terms of the dynamics of a purely four-dimensional theory in its non-perturbative regime. The idea is that the symmetries of the bulk ads space enforce the symmetries of a conformal theory in 4D-this latter theory approximates a strongly coupled theory near a fixed point. Combined with the observation that a shift in z causes an overall rescaling of the ads metric (3.28), we can identify slices of constant z as scale transformations of the 4D [approximately] conformal theory. In this way, the 5D ads theory 'geometrizes' the renormalization group flow of the 4D theory. One then interprets the physics on the uv brane as a 4D conformal theory that sets the boundary conditions for the 5D fields. Slices of constant z describe the rg evolution of this theory at lower energies, µ ∼ 1/z. Because the higher-dimensional theory encodes information about the behavior of a lower-dimensional theory on its boundary, this identification is known as the holographic interpretation of warped extra dimensions. This interpretation is sketched in Fig. 6. Sketch of a more formal description We can better motivate the holographic interpretation by appealing to more formal arguments. One of the most powerful developments in theoretical physics over the past two decades is the ads/cft correspondence-more generally, the holographic principle or the gauge/gravity correspondence [85,[97][98][99]. The conjecture states that type iib string theory on ads 5 × S 5 is equivalent to 4D N = 4 superconformal SU(N ) theory on Minkowski space in the large N limit: ads 5 × S 5 ⇐⇒ N = 4 super Yang-Mills. (3.51) The essence of this duality is the observation that a stack of N so-called D3-branes in string theory can be interpreted at low energies in two ways: 1. A solitonic configuration of closed strings which manifests itself as an extended black hole-like object for which ads 5 × S 5 is a solution. 2. Dirichlet boundary conditions for open strings which admit a non-Abelian U (N ) gauge symmetry associated with the N coincident D3-branes. These correspond to the left-and right-hand side of (3.51) and form the basis of the ads/cft correspondence. The key for us is that the ads 5 × S 5 extra dimension 'geometrizes' the renormalization group flow of the strongly coupled theory by relating the position in the extra dimension z with the rg scale µ. An operator O i in the 4D theory has a source j i (x, µ) that satisfies an rg equation µ ∂ ∂µ j i (x, µ) = β i (j j (x, µ), µ). (3.52) The gauge/gravity correspondence identifies this source as the value of a bulk field j i (x, µ) ⇔ Φ i (x, z) at the uv boundary of the ads 5 extra dimension. The profile of Φ i in the extra dimension is associated with the rg flow of j i (x, µ). Each Minkowski slice of ads 5 represents a picture of the 4D theory probed at a different energy scale µ ∼ 1/z. More concretely, the duality gives a prescription by which the correlation functions of one theory are identified with correlation functions of the other. The parameters of these two theories are related by R 4 4 = 4πg 2 N, (3.53) where R is the ads curvature, is the string length, and g is the Yang Mills coupling. Here we see why ads/cft is such a powerful tool. In the limit of small string coupling α ∼ 2 where string theory can be described by classical supergravity, the dual gauge theory is strongly coupled and very 'quantum'. The correlation functions of that theory are non-perturbative and difficult to calculate, whereas the dual description is weakly coupled. The duality gives a handle to calculate observables in theories outside the regime where our usual tools are applicable. What it means to geometrize the RG flow For our purposes, it is only important that we understand the warped extra dimension as the renormalization group flow of a strongly coupled 4D gauge theory. To see how this rg flow is 'geometrized,' we consider the internal symmetries of the two theories. • The isometry of the S 5 space is SO(6) ∼ = SU(4). This is precisely the R-symmetry group of the N = 4 gauge theory. • The isometry of the AdS 5 space is SO(4, 2), which exactly matches the spacetime symmetries of a 4D conformal theory. Since rs only has a slice of the ads space without the S 5 , we expect it to be dual to a conformal theory without supersymmetry. Steps towards formalizing the holographic interpretation of Randall-Sundrum are reviewed in [96]. Armed with this background, we can develop a working understanding of how to interpret rs models as a picture of a strong, four-dimensional dynamics. Observe that in the conformal coordinates that we've chosen, the metric has a manifest scale symmetry z → αz x → αx. (3.54) Consider 4D cross sections perpendicular to the z direction. Moving this cross section to another position z → αz is equivalent to a rescaling of the 4D length scales. Increasing z thus corresponds to a decrease in 4D energy scales. In this way, the ads space gives us a holographic handle on the renormalization group behavior of the strongly coupled theory. 3.9.4 What it means to take a slice of Anti-de Sitter The rs scenario differs from ads 5 due to the presence of the uv and ir branes which truncate the extra dimension. Since flows along the extra dimension correspond to scale transformations, the branes represent scales at which conformal symmetry is broken. The uv brane corresponds to an explicit uv cutoff for the 4D conformal theory. The ir brane sets the scale of the kk modes. We heuristically identified these with bound states of the strongly coupled theory, and so we can identify the ir brane as a scale where conformal symmetry is spontaneously broken, the theory confines, and one finds a spectrum of bound states. Recall that the bound state profiles are localized toward the ir brane; this is an indication that these bound states only exist as one approaches the confinement scale. The picture of the rs 'slice of ads' is thus of a theory which is nearly conformal in the uv that runs slowly under rg flow down to the ir scale where it produces bound state resonances. The sm, and in particular the Higgs, exist on the ir brane and are thus identified with composite states of the strongly coupled theory. In the extra dimensional picture, we argued that the Higgs mass is natural because the uv cutoff was warped down to the tev scale. In the dual theory, the solution to the Hierarchy problem is compositeness (much like in technicolor): the scalar mass is natural because above the confinement scale the scalar disappears and one accesses its strongly coupled constituents. By comparison, a state stuck on the uv brane is identified with an elementary (non-composite) field that couples to the cft. The meaning of 5D calculations At the level presented, it may seem like the ads/cft correspondence is a magic wand for describing strong coupling perturbatively-and indeed, if you have started to believe this, it behooves you to always know the limits of your favorite tools. A 5D calculation includes entire towers of 4D strongly coupled bound states-in what sense are are we doing a perturbation expansion? First of all, we underscore that the ads/cft correspondence assumes the 't Hooft large N limit, where N is the rank of the gauge group [100]. Further, whether in four or five dimensions, a scattering calculation assumes a gap in the particle spectrum. This gap in the 5D mass is translated into a gap in the scaling dimension ∆ of the 4D cft operators. Thus one of the implicit assumptions of a holographic calculation is that the spectrum of the cft has a gap in scaling dimensions. More practically, a scattering process in 5D include 4D fields with large kk masses. We can say definite things about these large kk mass states, but only as long as these questions include a sum over the entire tower. The RS Radion is a Dilaton We have already met the radion as the dynamical field whose vev sets the distance between the uv and ir brane. Excitations of the radion about this vev correspond to fluctuations in the position of the ir brane. From its origin as a part of the 5D dynamical metric, it couples to the trace of the energy-momentum tensor, r Λ ir T µ µ . (3.55) Observe that this is very similar to the coupling of the sm Higgs except that it is scaled by a factor of v Λir and there are additional couplings due to the trace anomaly-for example, a coupling to gluons of the form [83,101] r Λ ir − 1 2 r Λ ir F 1/2 (m t ) α s 8π (G a µν ) 2 , (3.56) where F 1/2 (m t ) = −8m 2 t /m 2 h + · · · is a triangle diagram function, see e.g. (2.17) of [102]. Why should the radion coupling be so similar to the Higgs? Before one stabilizes the radion vev (e.g. as in Section 3.8), the radion is a modulus and has a flat potential. In the holographic 4D dual, the radion corresponds to the Goldstone boson from the spontaneous breaking of conformal symmetry by the confining dynamics at the ir scale. In other words, in the 4D theory, the radion is a dilaton. This is the reason why it is so similar to the sm Higgs: the Higgs is also a dilaton in a simple limit of the Standard Model. In the sm the only dimensionful parameter is that of the Higgs mass, V (H) = λ H † H − v 2 2 2 . (3.57) In the limit when λ → 0, the Standard Model thus enjoys an approximate scale invariance. If we maintain v = 0 while taking λ → 0, that is, we leave the Higgs vev on, then: • Electroweak breaking SU(2)×U(1) →U(1) gives the usual three Goldstone bosons eaten by the W ± and Z • The breaking of scale invariance gives an additional Goldstone boson, which is precisely the Higgs. Indeed, the Higgs couples to the sources of scale invariance breaking: the masses of the fundamental sm particles, h v m fΨ Ψ + M 2 W W µ W µ + · · · . Realistic Randall-Sundrum Models While the original rs model is sometimes used as a template by lhc experiments to put bounds on kk gravitons, most theorists usually refer to rs to mean a more modern variant than the model presented thus far. In the so-called 'realistic' version of Randall-Sundrum, all of the Standard Model fields are allowed to propagate in the bulk [104][105][106]. Doing this allows one to use other features of the rs framework to address other model building issues. For example, pulling the gauge fields into the bulk can help for grand unification, but this typically leads to unacceptably large corrections to the Peskin-Takeuchi S-parameter. One way to control this is to also allow the fermions to live in the bulk. We explain below that the bulk fermions open up a powerful new way to use the rs background to generate the hierarchies in the Yukawa matrix. Solving the Hierarchy problem requires the Higgs to either be stuck on the ir brane or otherwise have a bulk profile that is highly peaked towards it. Allowing the fermions and gauge fields to propagate in the bulk introduces a tower of kk modes for each state. These tend to be peaked towards the ir brane and, as we learned above, are identified with bound states of the strongly coupled holographic dual. The Standard Model matter and gauge content are identified with the zero modes of the bulk fields. These carry zero kk mass and pick up small non-zero masses from their interaction with the Higgs. When boundary conditions permit them, zero mode profiles can have different types of behavior: • Fermion zero modes 9 are either exponentially peaked toward the ir brane or the uv brane. The parameter controlling this behavior is the bulk mass 10 , see (A.46). • Gauge boson zero modes are flat in the extra dimension, though electroweak symmetry breaking on the ir brane distorts this a bit, see (3.63). The holographic interpretation of a Standard Model field with a bulk profile is that the sm state is partially composite. That is to say that it is an admixture of elementary and composite states. This is analogous to the mixing between the ρ meson and the photon in qcd. States whose profiles are peaked towards the uv brane are mostly elementary, states peaked toward the ir brane are mostly composite, and states with flat profiles are an equal admixture. The effective 4D coupling between states depends on the overlap integral of their extra dimensional profiles. This gives a way to understand the hierarchies in the Yukawa matrices, since these are couplings to the Higgs, which is mostly localized on the ir brane [105][106][107][108][109][110][111]. This is a realization of the split fermion scenario 11 [112][113][114][115]. The zero-mode fermions that couple to the Higgs, on the other hand, can be peaked on either brane. We can see that even with O(1) 5D couplings, if the zero-mode fermions are peaked away from the Higgs, the dz overlap integral of their profiles will produce an exponentially small prefactor. We can thus identify heavier quarks as those whose bulk mass parameters cause them to lean towards the Higgs, while the lighter quarks are those whose bulk mass parameters cause them to lean away from the Higgs. Because the 5D couplings can be arbitrary O(1) numbers, this is often called flavor anarchy. This scenario is sketched in Fig. 7. This framework tells us how to search for 'realistic' rs models. Unlike the original rs model, whose main experimental signature were kk gravitons decaying to sm states like leptons, the profiles of our sm fields tell us what we expect realistic rs to produce. The most abundantly produced new states are those with strong coupling, say the kk gluon. Like all of the rs kk states, this is peaked towards the ir brane. The sm field which couples the most to this state are the right-handed tops. This is because we want the tops to have a large Yukawa coupling, and the left-handed top cannot be too peaked on the ir brane or else the bottom quark-part of the same electroweak doublet-would become heavy. These kk gluons are expected to have a mass 3 tev, so we expect these tops to be very boosted. This suggests experimental techniques like jet substructure (see [116][117][118] for reviews). There are additional features that one may add to the rs scenario to make it even more realistic. From the picture above, the electroweak gauge kk modes lean towards the ir brane where the 9 One immediate concern with bulk fermions is that in 5D the basic spinor representation is Dirac rather than Weyl. Thus one does not automatically obtain a chiral spectrum of the type observed in the sm. While heavy kk states indeed appear as Dirac fermions, one may pick boundary conditions for the bulk fermion field that project out the 'wrong chirality' zero-mode state. See Appendix A.3.6. 10 Observe that this is a manifestation of our identification of bulk masses and scaling dimension in Sec. 3.9.5. 11 Note that the use of an extra dimension to explain flavor hierarchies does not require warping. Higgs can cause large mixing with the sm W and Z. This causes large corrections to the Peskin-Takeuchi T parameter which seems to push up the compactification scale, causing a reintroduction of tuning. A second issue is that the third generation sm fermions also have a large overlap with the Higgs and can induce a large Zbb coupling through the neutral Goldstone. This coupling is well measured and would also require some tuning in the couplings. It turns out, however, that imposing custodial symmetry in the bulk can address both of these problems [119,120]. The symmetry is typically gauged and broken on the ir brane so that it is holographically identified with a global symmetry of the 4D theory-just as in the sm. This introduces several new states in the theory, many of which are required to have boundary conditions that prevent zero modes. A sketch of RS flavor Let us assume that the Higgs is effectively ir brane-localized. The effective 4D Yukawa coupling between a left-handed quark doublet and a right-handed quark singlet is given by the O(1) anarchic (non hierarchicial) 5D Yukawa coupling multiplied by the zero-mode fermion profiles evaluated on the ir brane, , y u i j ∼ O(1) ij × Q i u R j . (3.59) Here we have implicitly treated the Higgs boson profile as a δ-function on the ir brane and integrated over the profiles. In the 4D mass eigenstate basis, y t ∼ 1, we can write u R 3 ∼ Q 3 ∼ 1. For a choice of these parameters, one may then use the bottom mass to determine the value of d R 3 . This, in turn, may be used in conjunction with the ckm matrix, V CKM,i≤j ∼ O Q i Q j ,(3.60) to determine the s of lower generations and so forth. One automatically obtains a hierarchical pattern of mixing. Neutrino zero modes, on the other hand, must be highly peaked on the uv brane. In fact, these are typically even more peaked on the uv brane than the Higgs is peaked on the ir brane. In other words, one should no longer treat the Higgs as purely brane localized 12 and rather as a profile which is exponentially small on the uv brane. In this limit, one can treat the right-handed neutrinos as each having a δ-function profile on the uv brane. Even with O(1) anarchic Yukawa couplings, the smallness of the Higgs profile then suppresses the neutrino mass to automatically be small. Further, since each neutrino Yukawa coupling has the same Higgs mass, one finds larger mixing than in the quark sector, as phenomenologically observed. 3.13 Example: the coupling of the Z in RS As a sample calculation, consider the coupling of the Z boson in rs. We first derive the effective 4D (sm) coupling of the Z in terms of the 5D parameters and then calculate the fcnc induced by the zero mode Z. In the sm the Z is, of course, flavor universal and flavor-changing coupling. Indeed, at zeroth order, rs also prevents such a fcnc since the gauge boson zero mode profile is flat and therefore universal. We will see, however, that the correction to the Z profile induces a small fcnc term. Let us first state some results that are derived in the appendix. The localization of the normalized zero mode fermion profile is controlled by the dimensionless parameter c, Ψ (0) c (x, z) = 1 √ R z R 2 z R −c f c P L Ψ (0) c (x),(3.61) where c/R is the fermion bulk mass and P L is the left-chiral projection operator. Right chiral states differ by P L → P R and c → −c. We have also used the rs flavor function characterizing the fermion profile on the ir brane (larger f means larger overlap with the Higgs), f c = 1 − 2c 1 − (R/R ) 1−2c . (3.62) Each sm fermion has a different bulk mass c which according to the size of its sm Yukawa coupling. For simplicity of notation, we will simultaneously use c as the bulk mass parameter and as a flavor index rather than c i . Further, the profile for the zero mode Z boson is h (0) Z (z) = 1 R log R /R 1 − M 2 Z 4 z 2 − 2z 2 log z R ,(3.63) Starting in the canonical 5D basis where the bulk masses (c parameters) are diagonal, the zero mode fermion coupling to the zero mode Z is g 4D Z (0) µ (x)Ψ (0) c (x)γ µ Ψ (0) c (x) + · · · = dz R z 5 g 5D Z (0) M (x, z)Ψ (0) c (x, z)Γ M Ψ (0) (x, z),(3.64) where Γ M = z R γ M , the prefactor coming from the vielbein. Plugging in the profiles gives g cc 4D = g 5D R R dz 1 R z R −2c f 2 c 1 R log R /R 1 + M Z 4 z 2 − 2z 2 log z R ,(3.65) where the cc superscripts index fermion flavor. We write g cc 4D = g sm + g cc fcnc in anticipation that the term in the bracket proportional to M Z is non-universal and will contribute a fcnc. The leading term, on the other hand, gives the usual sm coupling. Performing the dz integral for that term gives g sm = g 5 f 2 c (R ) 2c R R log R /R R 1 − 2c 1 − R R 1−2c = g 5 R log R /R . (3.66) This is indeed flavor-universal since it is independent of c so that upon diagonalization of the zero mode mass matrix with respect to the Yukawa matrices, this contribution remains unchanged. On the other hand, the term proportional to M Z gives a non-universal contribution. Performing a change of variables to y = z/R and performing the dy integral gives g cc fcnc = −g 5 (M Z R ) 2 log R /R 2(3 − 2c) f 2 c , (3.67) where we've dropped a subleading term that doesn't have the log R /R enhancement. Consider, for example, the coupling between a muon and an electron through the zero mode Z. The unitary transformation that diagonalizes the Yukawa mass matrix goes like f i /f j so that g Z 0 µe fcnc = U † g ee U µe ∼ − f e f µ f 2 µ 3 − 2c µ − f 2 e 3 − 2c e (M Z R ) 2 1 2 log R R g sm . (3.68) We can drop the second term since flavor anarchy requires f 2 e f 2 µ . The result is g Z 0 µe fcnc = −g sm (M Z R ) 2 2(3 − 2c µ log R R f µ f e . (3.69) The observation that the coupling is suppressed by (M z R ) 2 is sometimes called the 'RS GIM mechanism.' Note that in order to do a full calculation, one must also include the non-universal contribution from kk Z bosons. These couplings do not have a (M z R ) 2 suppression, but fcnc diagrams with these kk modes are suppressed by the Z (n) mass. The Higgs from Strong Dynamics Further reading: The original phenomenological Lagrangian papers lay the foundation for the general treatment of Goldstone bosons [122,123]. See §19.6 of [124] for a slightly more pedagogical treatment that maintains much of the rigor of [122,123], or Donoghue, et al. for a discussion tied closely to qcd [125]. Very readable discussions can be found in [126,127]. For a rather comprehensive review that emphasizes the role of 'gauge' symmetries, see [128]. For the composite Higgs see [129,130] or the 2012 ictp "School on Strongly Coupled Physics Beyond the Standard Model" [131] for a modern sets of lectures and [132,133] for a phenomenological reviews. See [134,135] for reviews of the little Higgs scenario. Finally, a recent comprehensive review can be found in [136]. For our last topic we explore models where strong dynamics at a scale Λ ∼ 10 tev produces a light, composite Higgs. The solution to the Hierarchy problem is that there is no elementary scalarbeyond Λ one becomes sensitive to the underlying 'partons' that make up the Higgs. Through the holographic principle, we have already discussed many broad features of this paradigm in the context of warped extra dimensions above. One key question to address is the lightness of the Higgs mass. If Λ ∼ 10 tev, how is it that the Higgs appears at 125 gev? By comparison, the strong coupling scale for quantum chromodynamics is Λ qcd ∼ O(300 mev) while most qcd states, such as the ρ meson and proton are at least as heavy as this 13 . Those who are sharp with their meson spectroscopy will quickly observe that there is a counter-example in qcd: the pions are all lighter than Λ qcd , albeit by only an O(1) factor. The reason that the pions can be appreciably lighter than the other qcd states is the well-known story of chiral perturbation theory, a subset of the more general nonlinear Σ model (NLΣM) construction. The pions are the Goldstone bosons of the spontaneously broken SU(2) L ×SU(2) R flavor symmetry coming from chiral rotations of the up and down quarks. Small explicit breaking of this symmetry generates a mass for the pions so that they are pseudo-Goldstone modes. In the composite models that we consider in this section, we assume a similar structure where the Higgs is a pseudo-Goldstone boson of some symmetry for which Λ ≈ 4πf with breaking scale f ≈ 1 tev. We show that the generic composite Higgs set up still requires some tuning between the electroweak scale v and the symmetry breaking scale f . One way to generate this 'little hierarchy' is through the mechanism of collective symmetry breaking.We close this section by drawing connections to models of an extra dimension and by providing a phenomenological taxonomy of composite Higgs models to help clarify nomenclature. Pions as Goldstone bosons Before exploring composite Higgs models in earnest, it is useful to review strong electroweak symmetry breaking in qcd since this gives a concrete example of the effective theory of Goldstone bosons. It is also useful because electroweak symmetry breaking in qcd formed the motivation for technicolor models that have since fallen out of favor-it is useful to see why this is, and how composite Higgs models are different from a revival of technicolor. First, consider the Lagrangian for pure qcd: a theory of vector-like quarks and gluons, where 'vector-like' mean the left-and right-handed quarks come in conjugate representations, L qcd = − 1 4 G a µν G aµν +q(i / D − m)q. (4.1) This is a theory which becomes strongly coupled and confines at low energies, leading to a spectrum of composite states. This makes it a good template for our own explorations into compositeness. We can already guess that at low energies the effective theory is described by Goldstone bosons, the pions. In anticipation, we examine the global symmetries of the theory. We focus only on the three lightest quarks with masses m i Λ qcd . In the chiral limit, m → 0, the physical quarks are Weyl spinors and have an enhanced U(3) L ×U(3) R global flavor symmetry acting separately on the left-and right-handed quarks, q i L → (U L ) i j q j L (4.2) q i R → (U R ) i j q j R . (4.3) One may write the currents for this global symmetry. For compactness we move back to Dirac spinors and write in terms of the vector (U L = U R ) and axial (U L = U † R ) transformations: (j a V ) µ =qγ µ T a q (j a A ) µ =qγ µ γ 5 T a q (4.4) (j V ) µ =qγ µ q (j A ) µ =qγ µ γ 5 q,(4.5) where the T a are the generators of SU(3). We can identify j V with baryon number, which is conserved in qcd, and we note that j A is anomalous so that it is not a good symmetry and we don't expect to see it at low energies 14 . The vectorial SU (3), with current j a V , is precisely the symmetry of Gell-Mann's eightfold way and can be used to classify the light hadrons. What do we make of the axial SU(3), j a A ? Phenomenologically we can observe that the axial SU (3) is not a symmetry of the low energy spectrum, otherwise we would expect a parity doubling of all the 'eightfold way' multiplets. There is one way out: this symmetry must be spontaneously broken. What could possibly enact this breaking in a theory with no Higgs boson? It turns out that qcd itself can do the job! We assume that the axial SU(3) A is broken spontaneously by a quark-anti-quark condensate, qq = q i L q Ri + h.c. = 0 (4.6) in such a way that the vector SU(3) V is preserved. This is the unique combination that preserves Lorentz invariance and breaks SU(3) A . By dimensional analysis, this 'chiral condensate' takes the form q i q j ∼ δ ij Λ 3 qcd . Given that qcd is strongly interacting in the ir, the existence of this nontrivial vacuum condensate should not be surprising and is indeed supported by lattice calculations. However, the exact mechanism by which this condensate forms is non-perturbative and not fully understood. This also gives a robust prediction: we should have eight pseudoscalar Goldstone bosons as light excitations. These are precisely the pions, kaons, and η. Because SU(3) A is only a symmetry in the chiral m → 0 limit, these are not exactly Goldstone bosons as the symmetry is explicitly broken by the quark masses and electromagnetism. However, because this explicit breaking is small relative to Λ qcd , these excitations are still very light m π Λ qcd and are often referred to as pseudo-Goldstone bosons (sometimes pseudo-Nambu-Goldstone bosons, pngb, in the literature). Note that the electroweak group sits inside the qcd flavor symmetry 15 , SU(3) L × SU(3) R × U(1) B ⊃ SU(2) L × U(1) Y .Y = T R3 + B 2 . (4.8) We say that the electroweak group is weakly gauged with respect to low energy qcd. By this we mean that the gauge couplings are perturbative in all energy scales of interest. This weak gauging is a small explicit breaking of the qcd flavor symmetries and accounts for the mass splitting between the π 0 and π ± . 14 What happens to this symmetry at low energies is rather subtle and was known as the 'U(1) problem.' There is a lot more to the story than simply saying that the axial U(1) is anomalous and so does not appear at low energies. One can construct a current out of j A and a Chern-Simons (topological) current that is anomaly-free and spontaneously broken. This current indeed has a Goldstone pole. However, Kogut and Susskind showed that this current is not gauge invariant. There are actually two Goldstone bosons that cancel in any gauge invariant operator [137]. 15 It has to be true that the electroweak gauge group sits in the full qcd global symmetry group in order for some of the quarks to have non-trivial electroweak charges. A farewell to technicolor Because of (4.7), the spontaneous breaking of SU(3) A by the chiral condensate qq also breaks electroweak symmetry. This is an important observation: even if there were no Higgs boson, electroweak symmetry would still be broken and W and Z bosons would still be massive, albeit with a much smaller mass. This mass comes from 'eating' part of the appropriately charged pseudo-Goldstone bosons. We will see this in slightly more detail below. Readers unfamiliar with this story are encouraged to follow the treatment in [138]. The observation that strong dynamics can-and indeed, does in qcd-break electroweak symmetry led to the development of technicolor theories where the sm is extended by a confining sector [139][140][141][142][143]. By the holographic interpretation of extra dimensions, this type of electroweak symmetry breaking is analogous to the rs scenario where a brane-localized Higgs picks up a vev. The large hierarchy between the Planck and electroweak scales is then understood to be a result of dimensional transmutation. The simplest constructions of these models, however, suffer from several issues. These include the requirement for an additional mechanism to generate fermion masses [144,145] and generically large deviations in flavor and electroweak precision observables [145][146][147]. However, the nail in the coffin for most of these models is observation of the Higgs boson at 125 gev, much lighter than the compositeness scale. Such a state-even if it is not the Standard Model Higgs-is very difficult to explain in the context of these models. As such, even though the models we consider here encode strong dynamics, they are completely different from the pre-Higgs technicolor strong dynamics of the past. To repeat: composite Higgs is not technicolor. In technicolor, the strong dynamics generates a techni-condensate Q Q of techniquarks which spontaneously breaks SU(2) L ×U(1) Y . In the composite Higgs models we consider, there is a spontaneous breaking of some symmetry which produces Goldstone bosons but does not itself break electroweak symmetry. Instead, one of the Goldstone bosons develops a non-trivial potential and is identified with the Higgs doublet. Chiral perturbation theory In this section we review the main framework for describing Goldstone bosons of chiral symmetry breaking, known as chiral perturbation theory. Many of the results highlight general principles that appear in any theory of Goldstone bosons, known as nonlinear sigma models. A completely general treatment of spontaneously broken global symmetries is captured in the the so-called Callan-Coleman-Wess-Zumino (ccwz) construction, which we present in Appendix B. The importance of having a Lagrangian theory of Goldstone bosons is clear from the success of sm predictions before the Higgs discovery. Naïvely, one might wonder how we knew so much about the Standard Model before the Higgs discovery-isn't the Higgs a very central piece to the theory? As we saw above, the key feature is actually electroweak symmetry breaking: whether or not there is a Higgs, one always has the Goldstone bosons which are eaten by the W ± and Z to become massive. It is this nonlinear sigma model that pre-Higgs experiments had studied so carefully. The discovery of the Higgs is a statement that the nonlinear sigma model is uv completed into a linear sigma model. Framework We begin with the concrete example of low-energy qcd that we described above. Given that the chiral condensate qq breaks SU(3) A , we proceed to write down the effective theory describing the interaction of the resulting Goldstone bosons. Let us write U 0 to refer to the direction in field   e i a T a = e 2i a T a . (4.10) U (x) → U L U (x)U † R ,(4. We now promote the transformation parameter a to Goldstone fields, a ∼ π a (x). Since a is dimensionless, in order for π a to have canonical scaling dimension we should rescale by the decay constant 16 f . We may understand the physical meaning of f if we recall Fig. 8, since we want to be an angle that parameterizes the position along the vacuum circle: the Goldstone is a periodic variable with period 2πf , so that f is identified with the value of the symmetry breaking vev. The angle is then π(x)/f . We thus promote a → π(x)/f so that we may define the field U (x), U (x) = e i π a (x) f T a U 0 e i π a (x) f T a = e 2i π a (x) f T a . (4.11) We now have an object U (x) which packages the Goldstone fields, π a (x). Note that U (x) transforms linearly under the full SU(3) L ×SU(3) R group, U (x) → U L U (x)U † R , but the fields that actually describe the low energy spectrum are related in a non-trivial way to U (x). How pions transform We can determine the transformation of the pions π a by using the transformation of the linear field U (x). Under the SU(3) V (unbroken) symmetry, U L = U R = U V , we have U (x) → U V U (x)U † V = U V 1 + 2i π a (x) f T a + · · · U † V ,(4.12) where we can see from the first term that π a (x)T a → U V π a (x)T a U † V . In other words, π a (x) transforms linearly under the unbroken symmetry. Note that the higher order terms also obey this by trivially inserting factors of U † V U V = 1. Indeed, we expected this result because we know that Gell-Mann's eightfold way is precisely a realization of SU(3) V , so our pions must transform as octets. Things are not as simple for the broken symmetry, U L = U † R = U A . In this case the transformation is U (x) → U A U (x)U A ≡ e 2i π a (x) f T a . (4.13) In this case the pion does not transform in a nice, linear way 17 . Unlike the above case, there is no sense in which this looks like π a (x)T a → U A π a (x)T a U † A . The best we can do is say that we have moved U 0 to a new point on the vacuum manifold, which we parameterize by an angle 2π a (x)/f . The transformation π a (x) → π a(x) is nonlinear. To leading order, 1 + 2i π a (x) f T a = (1 + ic a T a ) 1 + 2i π a (x) f T a (1 + ic a T a ) (4.14) so that π a (x)T a = π a (x)T a + f c a T a .(4.15) In other words, to leading order the pion shifts π a → f c a . This shift symmetry in the nonlinear realization is precisely why the pion is massless; the only non-trivial pion Lagrangian terms must carry derivatives. Coset space description. In anticipation of the more general ccwz construction, let us restate the above arguments in a more compact way. The symmetry breaking pattern is the coset SU(3) L ×SU(3) R /SU(3) V . Using the notation above, this means that group elements of the full symmetry U L,R can be written as a product of elements of the unbroken group U V and the [left] coset U A ∈ SU(3) L ×SU(3) R /SU(3) V , U L = U A U V U R = U † A U V . (4.16) One can check that this matches the above cases when one sets U A = 1 or U V = 1. The general transformation of the linear packaging of the pions, U (x) = exp (2iπ a (x)T a /f ), is U (x) → U A U V U (x)U † V U A . Lagrangian description Thus far we have found a convenient way to package the Goldstone fields π a (x) into a linear realization of the full SU(3) L ×SU(3) R symmetry. We would like to write down a Lagrangian describing the dynamics of the Goldstones. Our strategy will be to write the lowest order terms in U (x) that are SU(3) L ×SU(3) R invariant and then expand U (x) in Goldstone excitations about U 0 . One can see that many invariants, such as U (x) † U (x), are independent of the Goldstones. In fact, only derivative terms contain the Goldstone fields. This is consistent with our argument that Goldstones must have derivative couplings. The lowest order non-trivial term is L = f 2 4 Tr ∂ µ U † (x) ∂ µ U (x) (4.18) The pre-factor is fixed by expanding U (x) = 1 + 2i π a (x) f T a + · · · and ensuring that the kinetic term for π a (x) is canonically normalized. We have used the normalization that Tr T a T b = 1 2 δ ab . The higher order terms in the expansion of U yield a series of non-renormalizable pion-pion interactions. Next we weakly gauge the electroweak group. Recall that this sits in SU (3) ∂ µ → D µ where D µ U (x) i j = ∂ µ U (x) i j − igW a µ 1 2 (τ a ) i k U (x) k j + ig B µ 1 2 U (x) i k T 3 R k j . (4.19) We have written the SU(2) L generators as 1 2 τ a = 1 2   τ a 0 0 0 0   ⊂ SU(3) L .(4.20) Promoting ∂ µ → D µ in (4.18) yields L = f 2 4 Tr ∂ µ − ig 2 W a µ (x)τ a U (x) 2 + · · · , (4.21) where we leave the similar term with B µ (x) implicit. Electroweak symmetry breaking One may check that (4.21) has terms that are linear in W (x) such as g 2 f W + µ (x)∂ µ π − (x) + h.c. This is precisely a mixing term between the π + (x) and the W + µ (x). In other words, the W has eaten the Goldstone boson to pick up a longitudinal polarization. This is precisely electroweak symmetry breaking at work. Note that similar terms mixing the W 3 µ and B µ with the π 0 . As usual, the masses of the heavy gauge bosons come from the gauge fields acting on the U 0 'vev' part of U (x), the resulting spectrum is ∆L = g 2 f 2 4 W + W − + g 2 + g 2 4 f 2 Z 2 2 . (4.22) The characteristic mass scale is 100 mev, much smaller than the actual W and Z since most of the mass contribution to those fields comes from the Higgs vev. Diagrammatically, we can imagine the mixing as follows: = + Π + Π Π + · · · (4.23) We have parameterized the strong dynamics in terms of a momentum-dependent form factor Π(q 2 ). What the W boson is really coupling to is the SU(2) L current formed from the quarks, Π µ ν = QCD µ ν (4.24) where the W bosons are coupling to quarks which then interact strongly with one another. In other words, iΠ µν (q) = J + µ (q)J − ν (−q) . (4.25) The qcd corrected W propagator ∆ µν (q) from resumming the diagrams in (4.23) is ∆ µν (q) = −i q 2 − g 2 Π(q 2 )/2 Π µν (q) = η µν − q µ q ν q 2 Π(q 2 ). (4.26) The observation that a charged pion has been 'eaten' to make the W massive is the statement that Π µν (q 2 ) has a zero-momentum pole. Indeed, 0\|J + µ \|π − (p) = if π p µ / √ 2. The qcd blobs in (4.23) also encode, however, the effects of heavier resonances and has poles at the masses of these states. In the 'large N ' limit (large number of colors) one may write the current-current correlation function as a sum of resonances [100,149,150], η µν − q µ q ν q 2 Π(q 2 ) = q 2 η µν − q µ q ν n f 2 n q 2 − m 2 n ,(4.27) where the Goldstone pole appears for m 0 = 0. Figure 9: 'Cat diagram' adapted from [151]. Despite the silly appearance, the key point is that the photon couples to the electric current J µ = eΨγ µ Ψ ('ears') formed from interactions with fundamental quarks in the strongly coupled sector. The 'whiskers' are the pseudo-Goldstone external states when expanding the U (x) field in (4.28). The contribution to the charged meson masses come from the 'two whisker' diagram. Electromagnetic mass splitting In addition to the spontaneous chiral symmetry breaking by strong dynamics, the SU(3) L ×SU(3) R group is also broken explicitly from the gauging of U(1) EM ⊂ SU(3) V . The neutral Goldstones (pions, kaons, and the η) are unaffected by this. The charged Goldstones, on the other hand, pick up masses from photon loop diagrams of the form in Fig. 9. These diagrams contribute to an operator that gives a shift in the [pseudo-]Goldstone mass, ∆L ∼ e 2 Tr QU (x) † QU (x) ,(4.28) where Q = 1 3 diag(2, −1, −1) is the matrix of quark electric charges. Since the electromagnetic force does not distinguish between the down and strange quarks, this diagram gives an equal shift to both the charged pions (e.g. ud) and kaons (e.g. us). Since the up and anti-down/strange quark have the same charge, the bound state is more energetic than the neutral mesons and we expect the shift in the mass-squared to be positive [151,152]. Note that the contribution to the charged pion mass is quadratically sensitive to the chiral symmetry breaking scale, though it is also suppressed by the smallness of α EM . Explicit breaking from quark spectrum One can add quark masses that constitute a small (m q Λ qcd ) explicit breaking of the global symmetry and generate small masses to the pseudo-Goldstone bosons. One can write this as a spurion M = diag(m u , m d , m s ) which has the same quantum numbers as U (x). One can add these terms to the effective Lagrangian by forming the appropriate global symmetry group invariant. In particular, we add to the Lagrangian ∆L ∼ Tr [M U (x)] ∼ Tr M π a (x) f T a 2 + · · · (4.29) In the limit where m u = m d and ignoring the electromagnetic splitting above, one may identify the masses for the pions, kaons, and η (different components of π a ) to derive the Gell-Mann-Okubo relation, m 2 η + m 2 π = 4m 2 K . (4.30) NDA: When the theory breaks down Finally, let us note that the effective Lagrangian for pions is non-renormalizable, so we should say something about the cutoff for this theory. At tree-level, the two-to-two scattering of pions with characteristic momentum p goes like p 2 /f 2 from (4.18). Using naïve dimensional analysis (nda) [153][154][155][156], we see that the loop contributions go like ∼ d 4 k p 2 k 2 f 4 1 k 4 ∼ Λ 2 p 2 16π 2 f 4 . ∼ × Λ 2 16π 2 f 2 ,(4.31) where we have used the shift symmetry (the full SU(3) 2 group structure) to tell us that at the numerator of the integrand carries at least two powers of the external momenta. Validity of our loop expansion thus requires that Λ ∼ 4πf ∼ gev, and this is indeed the scale at which additional qcd states appear. Note that this cutoff, based on perturbativity of the 1/f couplings in the chiral Lagrangian, is slightly different from Λ QCD ∼ O(300 mev), which is the scale where α s becomes non-perturbative. Indeed, this uv behavior of the theory of Goldstones is one of the reasons why we expected either the Higgs or something new to be manifest at the lhc: the sm without a Higgs is simply a nonlinear sigma model. By the Goldstone equivalence theorem, the scattering cross section for longitudinal W boson scattering grows linearly with the center of mass energy. In order to maintain unitarity, one requires that either there is a Higgs boson (a linearization of the nonlinear sigma model) or that the theory becomes strongly coupled so that higher order terms can cancel the unphysical behavior. NDA: Characteristic couplings To show the power of nda, let us explore the qualitative behavior of a strongly coupled theory without doing any calculations. The rules of nda boil down to (1) a factor of 1/f for each particle involved and (2) powers of a heavy scale, m ρ , to make up the remaining dimensions [155]. Rule 1 comes from the fact that the Goldstone fields 18 π(x) appear in the Lagrangian pre-packaged as U (x) = exp(iπ(x)/f ). Rule 2 is straight dimensional analysis with respect to a mass scale which one can take to be a uv scale Λ, or more phenomenologically, the mass of the lowest non-Goldstone resonances, m ρ . We use the ρ meson as an example of such a state. Let us parameterize the separation between the heavy mass scale m ρ and the compositeness scale f by g ρ = m ρ /f . We may interpret g ρ as the 'natural' coupling size of the heavy state ρ to the strong sector. For a strong sector field φ, define the dimensionless combinations x = φ f = g ρ φ m ρ y = ∂ m ρ . (4.32) We'd like to build an nda Lagrangian to estimate the size of couplings. We start by writing some dimensionless function L(x, y). In order to obtain the correct mass dimension of a Lagrangian, we further define L 0 (x, y) = m 4 ρ L(x, y). This function is assumed to contain a kinetic term, L 0 (x, y) ⊃ m 4 ρ x 2 y 2 = g 2 ρ O(∂ 2 , φ 2 ). (4.33) 18 This rule of applies even for non-Goldstone fields. For example, baryons appear a factor of 1/f √ Λ, where the √ Λ makes up for the difference between the scalar and fermion mass dimensions [155]. We see that we have to rescale by g −2 ρ to obtain a canonically normalized Lagrangian, L = 1 g 2 ρ L 0 (x, y) = m 4 ρ g 2 ρ L(x, y) = m 2 ρ f 2 L(x, y). (4.34) Let us use this to determine the expected size of a quartic coupling of strong sector fields. This comes from the O(x 4 ) term in the expansion of L(x, y) so that L ⊃ m 2 ρ g 2 ρ g 4 ρ φ 4 m 4 ρ = g 2 ρ φ 4 . (4.35) Thus we expect the quartic coupling of the φ to go like g 2 ρ ∼ m 2 ρ /f 2 , justifying the intepretation of g ρ as a strong sector coupling scale. Composite, pseudo-Goldstone Higgs The main idea for composite pseudo-Goldstone Higgs models is that the Higgs mass parameter is protected against quadratic quantum corrections up to the compositeness scale because it is a pseudo-Goldstone boson. Above the scale of compositeness, it is simply not an elementary scalar. This should be contrasted with the solutions to the Hierarchy problem already discussed: • Supersymmetry: due to the extended spacetime symmetry, there is a cancellation of the quadratic corrections through the introduction of different-spin partners. • Technicolor/Higgs-less: there is no elementary Higgs and electroweak symmetry breaking proceeds through a Fermi condensate. This is now excluded. • Warped extra dimensions: the Higgs itself is a composite state so that above the compositeness scale it no longer behaves like a fundamental scalar. However, there is no explanation for why the Higgs is lighter than the confinement scale. Note, in particular, that the composite Higgs scenario that we're interested in is distinct from technicolor: the pseudo-Goldstone nature of the Higgs is an explanation for why the Higgs mass is so much lighter than the other bound states in the strongly coupled sector. Goldstone bosons, however, behave very differently from the Standard Model Higgs. We saw that Goldstone bosons have derivative couplings owing to their shift symmetry. The Higgs, on the other hand, has Yukawa couplings and the all important electroweak symmetry-breaking potential. Our goal in this section is to see how to construct a theory of Goldstones which can produce a Higgs particle that has all of the required couplings of the sm Higgs. We shall closely follow the discussion in [129] and refer the reader there for further details and references. The framework Start with a large global symmetry group G, analogous to the 'large' SU(3) L × SU(3) R global symmetry of low energy qcd. We will break this symmetry in two ways: 1. We assume that the strong dynamics spontaneously breaks G to a subgroup H global . This is analogous to chiral symmetry breaking in qcd, SU ( 2. In addition to this, we will explicitly break G by weakly gauging a subgroup H gauge which contains the sm electroweak group SU(2) L × U(1) Y . This is analogous to the gauging of U(1) EM . We assume that the sm electroweak group is a subgroup of H = H gauge ∪ H global so that it is gauged and preserved by the strong dynamics. This is shown on the left of Fig. 10. This results in dim H gauge transverse gauge bosons and (dim G − dim H global ) Goldstone bosons. The breaking G → H global also breaks some of the gauge group so that there are a total of (dim H gauge − H) massive gauge bosons and (dim G − dim H global ) − (dim H gauge − dim H) 'uneaten' massless Goldstones. Now we address the white elephant of the Higgs interactions-can we bequeath to our Goldstone bosons the necessary non-derivative interactions to make one of them a realistic Higgs candidate? This is indeed possible through vacuum misalignment, which we illustrate on the right of Fig. 10. The gauging of H gauge gives loop-level corrections to the dynamical symmetry breaking pattern since this is an explicit breaking of the global symmetry. This is analogous to how the gauged U(1) EM splits the masses of the charged and neutral pions through a photon loop. Loops of sm gauge bosons can generate an electroweak symmetry breaking potential for the Higgs. We illustrate this below. One key point here is that since the Higgs potential is generated dynamically through sm gauge interactions, the electroweak scale v is distinct from the G → H symmetry breaking scale f . The 'angle' ξ = v f decoupled) and technicolor, respectively. We note that this parameter is also a source of tuning in realistic composite Higgs models. Once the pseudo-Goldstone Higgs state is given non-derivative interactions, these interactions generically introduce quadratic divergences at loop level which would lead to an expected O(1%) tuning. To avoid this, one needs to introduce a smart way of dealing with these explicit breaking terms called collective symmetry breaking which we discuss below. First, however, we focus on the effects of gauge bosons on the Higgs potential. We have the following constraints for picking a symmetry breaking pattern: 1. The sm electroweak group is a subgroup of the unbroken group, SU(2) L × U(1) Y ⊂ H. In fact, it is better to have the full custodial SU(2) L × SU(2) R ∼ = SO(4) group embedded in H since this will protect against large contributions to the ρ-parameter. 2. There is at least one pseudo-Goldstone boson with the quantum numbers of the sm Higgs. To protect the ρ-parameter, it is better to have a (2, 2) under the custodial group. At this point we have said nothing about the sm fermions. These, too, will have to couple to the strong sector to generate Yukawa couplings with the Higgs. We show below that a reasonable way to do this is to allow the sm fermions to be partially composite, a scheme that we had already seen in the holographic interpretation of the rs scenario. Indeed, extra dimensions provide a natural language to construct composite Higgs models. Minimal Composite Higgs: set up We now consider an explicit example, the minimal composite Higgs model, which was explored in [158,159] using the intuition from the rs framework. Following the guidelines set above, we would like to choose choose H global = SO(4), the custodial group which is the minimal choice to protect the ρ-parameter. However, the SO(4) = SU(2) L × SU(2) R charge assignments don't give the correct U(1) Y charges, as is well known in left-right symmetric models. Thus our 'minimal' choice for H global requires an additional U(1) X so that one may include hypercharge in the unbroken group, H, Y = (T R ) 3 + X. (4.37) We then choose G = SO(5)×U(1) X and introduce a linear field Σ that is an SO(5) fundamental and uncharged under U(1) X . Note that we can ignore the U(1) X charge in our spontaneous symmetry breaking analysis since it's really just 'coming along for the ride' at this point. Σ acquires a vev to break SO(5) → SO(4), Σ = (0, 0, 0, 0, 1) T . (4.38) This is analogous to the qcd chiral condensate. We can now follow the intuition we developed with chiral perturbation theory. The Goldstone bosons of this breaking are given by transforming this vev by the broken generators. A useful parameterization of the four broken generators is Tâ ij = i √ 2 δ k i δ 5 j − δ k j δ 5 i ,(4.39) whereâ ∈ {1, · · · 4}. We refer to the unbroken generators with an undecorated index: T a . The SO(5) group element that acts non-trivially on the vev, exp(ihâTâ/f ), can be written in terms of sines and cosines by separately summing the odd and even terms of the exponential. The linear field Σ can then be decomposed into the Goldstone pieces hâ(x) and a radial component h(x) = hâ(x)hâ(x), Σ = e ihâ(x)Tâ/f Σ = sin(h/f ) h h 1 , h 2 , h 3 , h 4 , h cot(h/f ) . (4.40) With this parameterization, the sm Higgs doublet is H = 1 √ 2 h 1 + ih 2 h 3 + ih 4 . (4.41) Gauge couplings Why ξ is an angle. As a quick exercise, let's see why we said (4.36) should be identified with an angle. Starting from the kinetic term for Σ in (4.40) with electroweak covariant derivative, one finds that the W and Z mass terms are L ⊃ g 2 f 2 4 sin h f \|W \| 2 + 1 2 cos 2 θ W Z 2 . (4.42) Using the relation M W = cos θ W m Z , this tells us that sin h f = v f , (4.43) so that we now see the relation between ξ = v 2 /f 2 and the 'angle' h /f . Note that the vev h = 246 GeV. We would like to write down a Lagrangian for this theory and parameterize the effects of the strong sector on the sm couplings. A useful trick for this is to pretend that the global SO(5) × U(1) X symmetry is gauged and then 'demote' the additional gauge fields to spurions-i.e. turn them off. We can then parameterize the quadratic part of the Lagrangian for the full set of SO(5) [partially spurious] gauge bosons, V µ = A a µ T a + Aâ µ Tâ, and the U(1) X gauge boson, X, by writing down the leading SO(5) × U(1) X -invariant operators: ∆L = 1 2 η µν + q µ q ν q 2 Π X (q 2 )X µ X ν + Π 0 (q 2 )Tr(A µ A ν ) + Π 1 (q 2 )Tr(ΣA µ A ν Σ T ) . (4.44) Where the form factors are completely analogous to (4.25) and (4.26). Contained in this expression are the kinetic and mass terms of the sm electroweak gauge bosons. To extract them, we must expand the form factors Π(q 2 ) in momenta and identify the O(q 0 ) terms as mass terms and the O(q 2 ) terms as kinetic terms. Since the Π X and Π 0 terms include gauge fields in the unbroken directions, they should vanish at q 2 = 0, otherewise masses would be generated for those directions. The Π 1 term, however, selects out the broken direction upon inserting the Σ → Σ 0 and thus contains the Goldstone pole, (4.27). We thus find Π 0 (0) = Π X (0) = 0 Π 1 (0) = f 2 . (4.45) Assuming that the Higgs obtains a vev, one may rotate it into a convenient location (h 1 , · · · , h 4 ) = (0, 0, v/ √ 2, 0) corresponding to the usual sm Higgs vev parameterization. We now assume that H gauge is the sm electroweak group and drop all spurious gauge bosons. Using (4.40), the strong sector contribution to the Lagrangian of these gauge bosons to O(q 2 ) is ∆L q 0 = η µν + q µ q ν q 2 1 2 f 2 4 sin 2 h f B µ B ν + W 3 µ W 3 ν − 2W 3 µ B ν + 2W + µ W − ν (4.46) ∆L q 2 = q 2 2 Π 0 (0)W a µ W a ν + (Π 0 (0) + Π X (0)) B µ B ν , (4.47) where we have used the choice of SO(5) generators in the appendix of [160]. ∆L q 2 gives contributions to the kinetic terms of the gauge bosons. Observe that these are not canonically normalized, but instead can be thought of as shifts in the gauge coupling, ∆ 1 g 2 = −Π 0 (0) ∆ 1 g 2 = − (Π 0 (0) + Π X (0)) . (4.48) Thus if the SU(2) L gauge bosons have a 'pure' gauge coupling g 0 when one turns off the strongly coupled sector, the full observed SU(2) L gauge coupling is (4.49) and similarly for g SM . ∆L q 0 corresponds to contributions the masses of the heavy electroweak gauge bosons. Taking into account the need to canonically normalize with respect to ∆L q 2 , we obtain the usual W ± and Z masses by identifying the sm Higgs vev as v = f sin( h /f ). We see the appearance of the misalignment angle, 1 g 2 SM = 1 g 2 0 − Π 0 (0),ξ = sin 2 h f ≡ v 2 f 2 . (4.50) Finally, by restoring h → h(x) in (4.46) we may determine the composite Higgs couplings to the gauge bosons 19 . The key is the expansion f 2 sin 2 h(x) f = v 2 + 2v 1 − ξh(x) + (1 + 2ξ)h(x) 2 + · · · . (4.51) From this we can make the prediction that the SO(5)/SO(4) composite Higgs couplings to the massive electroweak gauge bosons V = W ± , Z deviate from their sm values, g V V h = 1 − ξg SM V V h g V V hh = (1 − 2ξ)g SM V V hh . (4.52) At this point, these couplings introduce gauge boson loops which are quadratically divergent. These loops go like where we have used the dimensional analysis limit Λ = 4πf . We see that having explained the lightness of the Higgs by appealing to the Goldstone shift symmetry, reintroducing the Higgs couplings to the gauge bosons breaks this shift symmetry and wants to push the Higgs mass back up towards the symmetry breaking scale. In order to avoid this, one additional ingredient called collective breaking (along with light gauge and top partners) is necessary. We present this in Section 4.5. ∼ g 2 16π 2 Λ 2 ∼ g 2 SM (1 − ξ)f 2 , Partial compositeness Having introduced the Higgs couplings to the gauge bosons, we can move on to finding a way to incorporate the Yukawa couplings into composite Higgs models. The way this is done in technicolor is to introduce a four-Fermi interaction that is bilinear in sm fields, e.g. ∆L ∼ (Q L u R )(ψ TC ψ TC ) (4.54) where the (ψ TC ψ TC ) are bilinears of the techni-quarks. The resulting fermion mass is shown in Fig. 11a. This strategy typically runs afoul of constraints on cp violation and flavor-changing neutral currents since one can imagine the composite sector similarly generating a four-fermion operator between sm states unless elaborate flavor symmetry schemes are assumed. Instead of connecting the strong sector to a sm fermion bilinear, we can consider a linear connection. This is known as partial compositeness [106,164] and is shown in Fig. 11b. We assume that instead of (4.54), the elementary fermions mix with a fermionic composite operator, ∆L ∼Q L O Q L , (4.55) where O Q L is a strong sector operator that is interpolated into a composite quark doublet. We assume similar mixing terms for each of the other sm fermions. In order to preserve the sm quantum numbers we must assume that the the sm gauge group is a weakly gauged subgroup of the strongly coupled sector's flavor symmetries. Note that the gauge bosons are also partially composite 20 , as we saw in (4.23). The resulting Yukawa interactions are shown in Fig. 11c. The degree of mixing is now a freedom in our description. Let us parameterize the elementarycomposite mixing by 'angles' , \|observed particle ∼ \|elementary + \|composite . (4.56) We can use this degree of compositeness to control flavor violation. Since the strongest flavor constraints are for the first two generations, we assume that the first two generations have very small mixing with the composite sector. This suppresses dangerous flavor-violating four-fermion operators. On the other hand, we may assume that the third particles are more composite than the first two generations, Since the degree of compositeness also controls the interaction with the Higgs, this means that the third generation particles have a larger Yukawa coupling and, upon electroweak symmetry breaking, have heavier masses. The astute reader will note that this is exactly the same as the flavor structure of the 'realistic' Randall-Sundrum models in Sec. 3.11-3.12. The observation that light fermions can automatically avoid flavor bounds is precisely what we called the 'rs gim mechanism.' This is no surprise since the holographic interpretation of the rs model is indeed one where the Higgs is composite. Let us briefly see how this works with an explicit example. Let us write out (4.55) with a coupling λ Q L and cutoff Λ = 4πf : (4.58) where the power of Λ is chosen so that ∆L has mass dimension four. We assume that at low energies, the operator O Q L dimensionally transmutes into a fermion Ψ Q L with canonical mass dimension. We say that Ψ Q L is an interpolating field for the composite operator O Q L . It is treated as a local field in the same way that one may treat the proton as a local interpolating field for a uud composite operator in qcd. We further assume that Ψ Q L comes with conjugate Ψ c Q L (interpolating a conjugate operator O c Q L ) so that it may form a vectorlike mass, ∆L ⊃ λ Q L Λ dimO Q L −5/2Q L O Q L + h.c.,∆L ⊃ g * Λ 2Ψ Q L Ψ c Q L + h.c., (4.59) which comes from a coupling g * Λ 4−2dimO Q LŌ Q L O c Q L in the uv; the powers of Λ sort themselves out as the operators dimensionally transmute into the interpolating fields. The coupling g * is a characteristic coupling of the strong sector discussed below in (4.92). Together, (4.58) and (4.59) give the mass matrix ∆L = −Λ 2 Q LΨQ LΨ c Q L   0 0 λ Q L 0 0 g * λ Q L g * 0     Q L Ψ Q L Ψ c Q L   . (4.60) 20 In this framework the longitudinal modes of the massive sm gauge bosons pick up this partial compositeness from the Higgs. It is also possible to have a scenario where the transverse modes are partially composite, see [165,166] for explicit realizations. The light eigenstate is: Q physical L = cos θ Q L \|Q L + sin θ Q L \|Ψ Q L , (4.61) with mixing angle sin θ Q L = λ Q L g 2 * + λ 2 Q L . (4.62) We typically expect λ Q L g * so that this simplifies to θ Q L ≈ λ Q L /g * . Breaking electroweak symmetry Having addressed the Higgs couplings to both the sm gauge bosons and fermions, we move on to the Higgs self-couplings. Until now we have simply assumed that the strong sector generates an electroweak-symmetry breaking potential. We now check that this assumption is plausible by arguing that loops involving the third generation quarks generate such a potential; this is similar to the Nambu-Jona-Lasinio model [167,168]. The sm fermions do not form complete SO(5) multiplets. In fact, they cannot even be embedded into SO(5), as we noted in (4.37). The sm fermions can be embedded in the global group G = SO(5) × U (1) X , but certainly do not fill out complete representations. We thus follow the same strategy that we used for the gauge bosons in Sec. 4.4.3. Let us promote the sm fermions to full SO(5) spinor representations, Ψ Q = Q L χ Q Ψ u =   ψ u u R χ u   Ψ d =   ψ d χ d d R   ,r/ p [Π r0 + Π r1 (Γ · Σ)] Ψ r + r=u,dΨ Q/ p [M r0 + M r1 (Γ · Σ)] Ψ r + h.c. (4.64) where, as before, the form factors Π and M are momentum-dependent. We shall focus on only the Q L and t R pieces since they have the largest coupling to the strong sector. Keeping track of conjugate fields. One should be careful with the conjugate fields in the above expression. For the Lorentz group in four and five dimensions, SO(3,1) and SO(4,1), we use the Dirac conjugateΨ ≡ Ψ † γ 0 to form Lorentz invariants. Recall that this is because objects like Ψ † Ψ are not necessarily invariant because representations of the Lorentz group are not unitary-boosts acting on the spinor representation do not satisfy U † U = 1. This is due to the relative sign between the time-like and space-like directions in the Minkowski metric. The Dirac conjugate is a way around this. For the case of the G = SO(5) internal symmetry, however, there is no issue of non-unitarity. Hence no additional Γ 0 (acting on the internal SO(5) space) is necessary in the Lagrangian. To be clear, we can write out the spacetime γ and internal Γ matrices explicitly: Ψ = Ψ † γ 0 = Ψ † γ 0 Γ 0 . (4.65) The matrix Γ · Σ takes the form Γ · Σ = 1 h h cos(h/f ) / h sin(h/f ) / h sin(h/f ) −h cos(h/f ) , (4.66) where / h and / h are appropriate contractions with Pauli matrices. With the above caveat that there is no Γ 0 acting on the SO(5) conjugate, we may write out the Lagrangian for Q L and t R by dropping the spurious components of the Ψ fields, L =Q L/ p Π Q0 + Π Q1 cos h f Q L +t R/ p Π t0 + Π t1 cos h f t R +Q L M u1 h sin h f H c t R , (4.67) where H c = iσ 2 H is the usual conjugate Higgs doublet in the sm 21 . Observe that upon canonical normalization, the top mass can be read off the Yukawa term, m 2 t = v f 2 M 2 t1 (Π Q0 + Π Q1 ) (Π t0 − Π t1 ) , (4.68) where the form factors are evaluated at zero momentum. One may write similar expressions for the other fermions. Observe from (4.67) that the fermions which are more composite (e.g. the top quark) will also experience deviations from their sm couplings depending on ξ, analogously to the deviation of the Higgs-gauge boson couplings, (4.52). In order to determine whether electroweak symmetry is broken, we can now plug this information into the Coleman-Weinberg potential for the Higgs, also known as the [quantum] effective potential. This is the potential term in the effective action after taking into account quantum corrections from integrating out the top quarks. In other words, it is the potential that determines the vacuum expectation value of fields. The result is V CW = −6 d 4 p log Π Q0 + Π Q1 cos h f + log p 2 Π Q0 + Π Q1 cos h f Π t0 − Π t1 cos h f M 2 t1 sin 2 v f . (4.69) Expanding this to first order and keeping the leading order terms in the Higgs gives V CW (h) = α cos h f − β sin 2 h f , (4.70) where α and β are integrals over functions of the form factors where β is typically of the order the top Yukawa. If α ≤ 2β, then the Higgs acquires a vev parameterized by ξ ≡ sin 2 h f = 1 − α 2β 2 . (4.71) This means that a small ξ typically requires a cancellation between α and β. Since these come from different sources, this is generically a tuning in the theory. One can also ask if it was necessary to rely on the top quark. For example, we know that the gauge sector also breaks the Goldstone shift symmetry so that loops of gauge bosons can generate quadratic and quartic terms in the Higgs potential. However, for a vector-like strong sector, gauge loops contribute with the wrong sign to the β term and pushes to align-rather than misalign-the vacuum [169,170]. There are, however, alternate mechanisms to enact electroweak symmetry breaking for a composite Higgs. For example, • mixing the composite Higgs with an elementary state [151], • making use of an explicitly broken global symmetry [171] • enlarging the H gauge so that it cannot be completely embedded into H global [172][173][174]. Collective symmetry breaking The general composite Higgs is a useful framework for working with the Higgs as a psuedo-Goldstone boson. However, we saw in Section 4.4.1 and equation (4.53) that this is not enough to avoid tuning. The source is clear: a pure Goldstone Higgs is protected from quadratic corrections to its mass because of its shift symmetry. This very same shift symmetry prevents the required Higgs couplings to gauge bosons, fermions, and itself. One must break this shift symmetry in order to endow the Higgs with these couplings; this generically reintroduces a dependence on the cutoff, Λ = 4πf . This may make it seem like a no-go theorem for any realistic model of a pseudo-Goldstone Higgs. However, there is a nice way out of this apparent boondoggle called collective symmetry breaking that was originally introduced in 'little Higgs' models [175][176][177] (see [134,135] for reviews) and is now an a key ingredient in composite Higgs models 22 . The idea is that one can separate the scales v and f by introducing new particles which cancel the quadratic divergences at one-loop order. Unlike supersymmetry, these partner particles carry the same spin as the Standard Model particles whose virtual contributions are to be canceled. Further, this cancellation only occurs for one-loop diagrams: higher loop diagrams are expected to contribute quadratically at their naïve dimensional analysis size, but these are suppressed relative to the leading term. The general principle that allows this cancellation is that the shift symmetry is redundantly protected. A process is only sensitive to explicit symmetry breaking-as necessary for sm-like Higgs couplings-if this explicit breaking is communicated by at least two different sectors of the theory. More concretely, the symmetry is only explicitly broken if multiple couplings are non-zero in the theory so that any diagram that encodes this explicit breaking must include insertions from at least two different couplings. This softens the cutoff sensitivity of various operators by requiring additional field insertions that decrease the degree of divergence of loop diagrams. Figure 12: Anatomy of collective symmetry breaking, following the conventions in Fig. 10. G H global H gauge H EW = H global H gauge G G H global H global Collective breaking in action We now demonstrate collective symmetry breaking in a model based on the 'anatomy' in Fig. 12. The reader may find it useful to refer to the explicit example of a simple little Higgs model in Section 4.5.2 below. Instead of a simple global group G, suppose that G = G × G . Each of these factors breaks spontaneously to subgroups H global and H global , respectively. The spontaneous symmetry breaking pattern is thus G = G × G → H global × H global . (4.72) This gives us two linear fields Σ and Σ analogous to (4.40) so that there are two separate sets of Goldstone bosons. We explicitly break G by gauging H gauge ⊂ G. Suppose that both H global and H global are subgroups of H gauge in such a way that both Σ and Σ are charged under H gauge with nonzero charges q and q respectively. A piece of each subgroup is gauged, as shown in Fig. 13a. H global × H global is then explicitly broken to a smaller subgroup, for example a vectorial subgroup identified by the gauging, H. On the other hand, when either q or q is set to zero, only one of the global subgroups is gauged, as shown in Fig. 13b and 13c. In either of these cases, the resulting global symmetry group is still H global × H global . In other words, one requires both q and q to explicitly break H global = H global × H global . When one of the global subgroups is uncharged under the gauged subgroup, say H global , those Goldstone bosons pick up no mass from the gauge sector. For the other global subgroup which is charged under the gauge group, say H global , there are two possibilities: 1. If H gauge ⊆ H global , then loops of the gauge bosons will feed into the mass of the pseudo-Goldstone bosons. In the absence of collective symmetry breaking, this gives a contribution that is quadratic in the cutoff. In the second case, the Higgs mechanism removed the Λ 2 contribution to the pseudo-Goldstone mass, but it also got rid of the pseudo-Goldstones themselves. 23 It is sufficient to consider some subgroup G ⊆ G that contains H global as a proper subgroup (a) q , q = 0 (b) q = 0, q = 0 (c) q = 0, q = 0 Figure 13: Collective symmetry breaking. Upper (blue) and lower (red) blobs represent H and H in Fig. 12. The thick black line represents the gauged symmetry H gauge under which Σ has charge q and Σ has charge q . When either q or q vanishes, the unbroken group is H global × H global . This leads us to consider the case when both G and G (not just their H global subgroups ) are charged under the gauged symmetry. For simplicity, suppose G = G = H gauge so that one gauges the vectorial combination. In this case, both the Σ and Σ fields carrying our Goldstone bosons are charged under the gauge group. The gauge fields become massive by the Higgs mechanism, but there are twice as many Goldstone bosons than it can eat 24 . Indeed, the 'axial' combination of G and G furnishes a set of Goldstone bosons that remain uneaten and are sensitive to explicit breaking effects so that they are formally pseudo-Goldstones. Any contribution to the pseudo-Goldstone mass, however, must be proportional to (gq )(gq ), where g is the gauge coupling. In other words, it requires interactions from both Σ and Σ . The resulting mass term is suppressed since this requires factors of the Σ and Σ vevs to soak up additional boson legs. We now demonstrate this with an explicit example. Why can't you just rotate to a different basis? Based on Fig. 13, one might wonder if we can repartition G = G × G so that the H global and H global subgroups are always both gauged. Alternately, perhaps one can repartition G so that only one subgroup is ever gauged. This cannot be done, even when q = q . The reason is precisely what we pointed out above Sec. 4.3.3: the axial combination of two groups is not itself a group since its algebra doesn't close. Explicit example: (SU(3) → SU(2)) 2 Let us see how this fits together in a simple little Higgs model-though we emphasize that collective symmetry breaking is a generic feature of all realistic composite Higgs models, not just those of little Higgs type. We classify composite Higgs models in Section 4.7 to clarify any ambiguity. Consider the case where G = G = SU(3) and H global = H global = SU(2). We thus have two fields which are linear representations of SU(3) and carry the Goldstone bosons, (4.73) and similarly for Σ . For simplicity let us set f = f ≡ f . The kinetic terms for the Σ fields are Σ = exp   i f   0 2×2 H H † 0       0 0 f   =   0 0 f   + i   H 0   − 1 2f   0 H † H   ,L = \|D µ Σ \| 2 + \|D µ Σ \| 2 = · · · + (gq ) 2 V a µ T a Σ 2 + (gq ) 2 V a µ T a Σ 2 , (4.74) where T a = T a are the generators of the gauged group. To see the contribution to the Higgs mass, one can Wick contract the two gauge bosons in these terms-this is precisely the analog of the 'cat diagram' in Fig. 9. This contraction ties together the gauge boson indices so that the resulting term goes like [loop factor] (gq ) 2 Σ † T a T a Σ = [loop factor] (gq ) 2 2 C 2 Σ † 1 gauge Σ ,(4.75) and similarly for Σ . Here the loop factor contains the quadratic dependence on the cutoff, [loop factor] ∼ Λ 2 /16π 2 , and the factor 1 gauge is the identity matrix in the appropriate gauged subgroup. Here we have used T a T a = C 2 1, where C 2 is the quadratic Casimir operator of the representation 25 . Now let's explicitly demonstrate how collective breaking works. This picks up the Goldstones in the second term on the right-hand side of (4.73) so that there is indeed a Goldstone mass term proportional to Λ 2 = (4πf ) 2 for each set of Goldstones. • On the other hand, in the case where G and G are both gauged with q = q , the matrix 1 gauge becomes a true identity operator, 1 gauge =   1 0 0 0 1 0 0 0 1   . (4.77) Now the global symmetry breaking vevs Σ and Σ break part of the gauge symmetry and the Higgs mechanism tells us that there are gauge bosons that eat would-be Goldstones. Indeed, the first term on the right-hand side of (4.73)-which is no longer projected out by 1 gauge -encodes the mass picked up by the gauge bosons. Observe, however, what has happened to the Λ 2 mass contribution in the previous scenario: it is now canceled by the cross term between the first and third terms on the right hand side of (4.73). In other words, the terms which gave the quadratic sensitivity to the cutoff have vanished. If we were only considering a single SU(3)→SU(2) global symmetry breaking, then we would still be out of luck since the massive gauge bosons would have eaten all of our Goldstone bosons-so even though we got rid of the Λ 2 sensitivity of the pseudo-Goldstone masses, we also would have gotten rid of the pseudo-Goldstones themselves. With foresight, however, we have followed the advice of footnote 24: we have more Goldstones than our gauge bosons can possibly eat. A useful way to parameterize our Goldstones is to follow the convention in (4.16): Σ =exp   i f   0 2×2 V V † 0     exp   i f   0 2×2 H H † 0       0 0 f   (4.78) Σ =exp   i f   0 2×2 V V † 0     exp   −i f   0 2×2 H H † 0       0 0 f   , (4.79) where we have identified the Higgs as the axial combination of global shifts, while the vector combination of Goldstones, V , is eaten by the gauge bosons to become massive. Now the H pseudo-Goldstones only pick up mass from diagrams that involve both the (gq ) and the (gq ) couplings. In other words, it requires a combination of the Σ and the Σ fields. The leading order contribution comes from diagrams of the form Σ Σ Σ Σ ∼ g 4 16π 2 log Λ 2 Σ † Σ 2 . (4.80) Since Σ † Σ = f 2 − 2H † H + · · · , we see that the leading term in the Higgs mass is only logarithmically sensitive to Λ because it required one power each of the Σ and Σ vevs. The Higgs mass sets the electroweak scale to be on the order of f /(4π). This is a factor of (4π) suppressed compared to the global symmetry breaking scale f -generating the hierarchy in ξ that we wanted-and also a further factor of (4π) from the cutoff Λ = 4πf . In this sense, collective symmetry breaking shows us what we can buy for factors of (4π) and why those factors are important in naïve dimensional analysis. Top partners As before, the largest contribution to the Higgs mass comes from the top quark. In the simple scenario above, we have extended our gauge group 26 from SU(2) L to H gauge =SU(3) so we'll need to also extend the usual top doublet to include a partner T L Q = t L b L → Q =   t L b L T L   . (4.81) We must also include a right-handed SU(3) singlet T R as a partner for the T L , in parallel to the usual right-handed t R partner of the sm t L . The prime on the t R -what is normally called t R in 26 For simplicity we ignore the U(1) Y factor, it is straightforward to assign charges appropriately. the sm-is for future convenience. The Yukawa terms for the top quarks are, L top = λ Σ † Qt † R + λ Σ † QT † R + h.c. (4.82) where the fermions are written in terms of Weyl spinors. Other terms, such as Σ † QT R † or Σ † Qt R † , can typically be prohibited by invoking chiral symmetries. Observe that the λ term is invariant under G if Q is a fundamental under G . Similarly, the λ term is invariant under G if Q is a fundamental under G . This is indeed consistent since Q is a fundamental under H gauge which is the diagonal subgroup of G × G . This shows us how collective symmetry breaking is embedded in the Yukawa sector. When only one of the λ terms is nonzero, L top is G × G invariant. However, when both are turned on, the global symmetry is broken down to the diagonal subgroup. This is collective breaking is similar to the breaking of the global U(3) Q × U(3) U × U(3) D flavor symmetry to U(3) by the up-and down-type Yukawas in the Standard Model. If y u = 0 and y d = 0, then the flavor symmetry would be enhanced to U(3) 2 since the right-handed up-type quarks could be rotated independently of the other fields. We can now plug in the expansion (4.78 -4.79) into the Yukawa terms (4.82), ignoring the V terms since we now know those are eaten by the gauge bosons. Expanding the resulting product gives L top =iH † Q(λ T † R − λ t † R ) + f − H † H 2f T L λ t † R + λ T † R . (4.83) From this we can write out the right-handed top eigenstates T R = λ t R + λ T R √ λ 2 + λ 2 t R = i λ T R − λ T R √ λ 2 + λ 2 (4.84) and the resulting top Yukawa, top partner mass, and top partner coupling to H † H, L top = λ t H † Qt † R + λ t f T L T † R − λ t 2f H † HT L T † R ,(4.85) where we see that all of the couplings are simply related to the sm top Yukawa, λ t = √ λ 2 + λ 2 . These relations ensure the cancellation between diagrams that give a Λ 2 contribution to the Higgs mass, h h t λ t λ t + h h λ t f −λ t /f T = O(log Λ). (4.86) Note the symmetry factor of 1/2 in the h 2 T L T † R Feynman rule. For simplicity we also drop an overall √ 2 in the normalization of the h field which is irrelevant for the Λ 2 cancellation. We see that indeed collective symmetry breaking can protect against the reintroduction of quadratic sensitivity to the cutoff by the Yukawa interactions. Just as in the case of natural susy, an important signature of this class of models is to look for the 'partner top' particles which are responsible for the softening of the cutoff dependence of Higgs mass from the top sector. One can search for these objects at the lhc through either pair production, qq/gg → TT , (4.87) or through single production in association with a sm quark, bq → T q qq → T b. (4.88) The top partner decays are fixed by the Goldstone equivalence theorem. The partner top decays approximately 50% of the time to bW , with the remaining decay products split evenly between tZ and th [178]. The lower bound on the top partner mass from vector-like heavy top (also referred to as fourth generation) searches is 700 GeV [179]. One can continue to calculate the Coleman-Weinberg potential in this scenario to check for electroweak symmetry breaking and further study the phenomenology of these models. As discussed below (4.94), in addition to playing an important role generating the Higgs potential, these top partners are of phenomenological significance since they are expected to be lighter than the other strong sector resonances. By virtue of filling out representations of the global group, some of these top partners are expected to have exotic electromagnetic charges, such as Q EM = 5/3; these states are considered to be 'smoking gun' signals of a composite Higgs scenario. We refer the reader to the excellent reviews [134,135] for a pedagogical introduction in the context of the little Higgs. See [180][181][182] for a more general discussion of experimental bounds on top partners. Deconstruction and moose models We now briefly mention some connections with extra dimensional models and introduce a diagrammatical language that is sometimes used to describe the symmetry breaking pattern in composite models. In Section 3.9 we introduced the holographic principle as a connection between strongly coupled 4D theories and weakly coupled theories on a curved spacetime with an extra spatial dimension. This turns out to be a natural tool to get a handle for some of the strong dynamics encoded into the form factors. Indeed, the minimal composite Higgs model described above was developed using these insights [158]. There is, however, another way to connect 5D models to 4D models. 5D models have dimensionful couplings and are manifestly non-renormalizable. One proposal for a uv completion is to discretize ('latticize') the extra dimension [177,183,184]. In this picture, the extra dimension is split into N discrete sites which should no longer be thought of as discrete spacetimes, but rather as nodes in a 'theory space' that describe a gauge symmetry structure on a single 4D spacetime. The bulk gauge symmetry G latticized into a 4D gauged G on each of the N nodes, G G G G At this level the nodes are just N separate gauge groups; after all, this is precisely what we mean by a local symmetry (see [185,186] for a discussion in depth). We next introduce a set of (N − 1) scalar link fields Φ i which are in the bifundamental representation with respect to the N th and (N + 1) th gauge groups: (N i ,N i+1 ). We may draw these link fields as lines between the nodes, G G G G Φ 1 Φ 2 Φ 3 Φ N −1 The arrow on the link field keeps track of the representation with respect to a group: • Arrows leaving a node are fundamental with respect to that group. • Arrows entering a node are anti-fundamental with respect to that group. Now suppose each of these link fields acquires a vev proportional to 1 in their respective G i × G i+1 internal spaces. Each link field would spontaneously break the symmetry G i × G i+1 → G diag . The symmetries are broken down to G. One can diagonalize the mass matrix for the gauge boson-a problem that is mathematically identical to solving the waves in a system of N − 1 springs in series [187]-to find that the spectrum looks like a tower of Kaluza-Klein modes. In fact, the link fields can be identified with the kk modes of the fifth component of the bulk gauge field A 5 . This construction also shows explicitly that the Kaluza-Klein gauge fields in 5D acquire their masses from eating the kk modes of the A 5 , which are here manifestly would-be Goldstone bosons. By coupling matter appropriately, one constructs a uv complete 4D model of a product of gauge groups that gives the same 'low' energy physics as an extra dimension. We refer the reader to the original literature for details [177,183,184] or [74] for a brief summary. Rather than just way to uv complete extra dimensions, deconstructions are also a useful tool for motivating models of chiral symmetry breaking. In fact, they are a manifestation of a more general tool for composite models called moose diagrams 27 [189,190]. One can use this diagrammatic language to construct little Higgs models; indeed, this was the original inspiration for the development of collective symmetry breaking paradigm in Section 4.5. The topology of these diagrams encodes information about spectrum of Goldstone modes [191]. From the dimensional deconstruction of an extra dimension, it's clear that all of the Goldstones are eaten by the kk modes of gauge bosons. More general connections between nodes, however, allow more Goldstones to survive hungry gauge bosons. As an example, we present the 'minimal moose' little Higgs model from [192]. The basic building block is the coset for chiral symmetry breaking, SU(3) L × SU(3) R /SU(3) V . We gauge the electroweak subgroup G EW of SU(3) L and the entire SU(3) R , which we represent schematically with shaded blobs: G EW SU(3) L SU(3) R SU(3) R Σ The minimal moose model actually requires four copies of this basic structure. As before, we only gauge the vectorial G EW of each of the SU(3) L factors and similarly for the SU(3) R factors. In other words, the theory only has two gauge couplings. This is shown schematically in Fig. 14. We note that typically one only draws nodes for the gauge groups so that the usual moose diagram for this model is: Figure 14: Full symmetry structure of the minimal moose little Higgs model. Shaded blobs represent gauged subgroups. We explicitly show that only the 'diagonal' subgroups are gauged. G EW SU(3) RG EW SU(3) R See §4.1 of [135] for a review of this particular model. A full discussion of these moose-based little Higgs models is outside of the scope of these lectures. In addition to the reviews mentioned above [134,135], see [193] for the self-described 'bestest' little Higgs model and [180,194] for a discussion of the status of composite Higgs models after the first run of the lhc. A taxonomy of composite Higgs models Having surveyed the main features of composite Higgs models, let us classify the landscape of such theories. This section is meant to clarify the distinctions between what is colloquially called a 'composite Higgs' versus a 'little Higgs' or a 'holographic composite Higgs' versus a 'dilatonic Higgs. ' We closely follow the discussion in Sections 2 -3 of [132], to which we refer the reader for further details and references. As a warm up and review, recall the Standard Model Higgs potential V (h) = −µ 2 \|H\| 2 + λ\|H\| 4 −→ − 1 2 µ 2 h 2 + λ 4 h 4 .(4.89) Minimizing the potential and matching to experiment yields v 2 = h = µ 2 λ = 246 gev m 2 h = 2µ 2 = (125 gev) 2 ,(4.90) where v has long been known from the masses and couplings of the electroweak gauge bosons, but m 2 h is new data from 2012. This new information tells us that µ = 89 gev and, from the expression for v, that λ = 0.13. Let us now map this onto a convenient parameterization of the Higgs potential in composite Higgs models. V (h) = g 2 SM M 2 16π 2 −ah 2 + b 2f 2 h 4 . (4.91) One can compare this to (4.70). Here g SM is a characteristic Standard Model coupling, such as g 2 SM = N c y 2 t . Implicit in this parameterization is the expectation that the Higgs potential is radiatively generated, giving a g 2 SM /16π 2 prefactor. With this normalization, tree-level contributions appear as coefficients a, b that go like 16π 2 /g 2 SM . The mass scale M is typically that of the new states (e.g. top partners) that cut off the quadratic divergence introduced by the explicit breaking of the Goldstone shift symmetry, as discussed in Section 4.4.3. It is useful to parameterize this in terms of the coupling of these new states to the strong sector g * , M = g * f. (4.92) These states are typically lighter than the cutoff, 4πf , to help with the little hierarchy problem. We expect the lighter mass comes from a weaker coupling to the strong sector, g * , motivating the definition (4.92). This coupling is sometimes written as g * = g ρ in the literature, making the analogy to the coupling of the spin-1 ρ meson in qcd, see Section 4.3.8. (4.92) defines g * as a ratio of mass scales, but when one includes this state in chiral perturbation theory (using the ccwz formalism introduced in Appendix B), this ratio is manifestly the value of the ρππ coupling. In this sense, g ρ is the 'gauge coupling' of the ρ as a massive gauge boson, g ρ = m ρ /f . The experimental information that the sm quartic is λ = 0.13 is strongly suggestive of a loop induced coupling. Using the nda scaling of a strong sector quartic (4.35) and a proportionality factor from an explicit global symmetry breaking sm loop, g 2 SM /16π 2 , we estimate λ loop ≈ 2 1 16π 2 g 2 SM g 2 * ≈ 0.15 × g SM √ N c y t 2 g * 2 2 . (4.93) Here the factor of 2 comes from two top partner polarizations and the scaling with respect to g * = M/f comes from nda [155]. Thus the coupling of the new state is g * 4π and is expected to be weakly coupled. This is a more quantitative version of the statement that the discovery of the 125 gev Higgs signaled the death of technicolor, as we explained qualitatively in Section 4.2. The other implication of this weak coupling is that the new particles that cancel the quadratic sensitivity of the Higgs potential have masses well below the strong coupling scale, M Λ = 4πf ; where we recall the nda cutoff from Section 4.3.7. Comparing (4.91) to (4.89 -4.90) gives v 2 = a b f 2 = (246 gev) 2 m 2 h = 2 g 2 SM 16π 2 M 2 a = 4v 2 g 2 SM g 2 * 16π 2 b = (125 gev) 2 . (4.94) We can restate the discussion below (4.93) in terms of (4.94). Prior to the Higgs discovery, one could have tuned ξ = v 2 /f 2 by, say, increasing the parameter b. With the discovery of a 125 gev Higgs boson, one can no longer do this since increasing b also increases m 2 h . Indeed, this is why prior to the Higgs discovery people said that composite Higgs models predict a heavier Bona-fide composite Higgs 1 1 4π Requires tuning of both a and b. Little Higgs 1 16π 2 g 2 * 4π Tree level quartic, h too heavy. Holographic Higgs 1 1 4π ∼ little Higgs with loop-level quartic. Twin Higgs 1 1 − 16π 2 g 2 * g SM Z 2 rather than collective breaking. Dilatonic Higgs see text Related to rs radion Higgs. [132]. Models must be tuned when phenomenology requires values of the couplings that are very different from the expected magnitudes shown here. Higgs mass of m h ∼ 300 gev. One way to evade making m 2 h > m 2 t is to observe that most of the contributions to b comes from the fermionic top partner resonances. We've been characterizing all of the heavy particle couplings as g * , but in principle the top partners could have a different coupling, g T , in which case g * → g T in (4.94). If this top partner coupling is smaller than the general resonance coupling g T < g * , while also satisfying g T 1 to push the mass up, then one can keep m 2 h < m 2 t while pushing up b to achieve tuning in ξ. This is why one may expect the 'light' top partners described in Section 4.5.3 to have masses lighter than the other strong sector resonances m T ≈ g T f < M . In the remainder of this section we examine five classes of composite Higgs models and classify them according to their natural expectations for a, b, and g * . These are summarized in Table 2. Bona-Fide Composite Higgs The 'bona-fide composite Higgs' models in the first row of Table 2 are the simplest realizations of the Higgs as pseudo-Nambu-Goldstone boson idea: a strongly coupled sector has a global symmetry which is spontaneously broken and yields a Goldstone with the quantum numbers of the Higgs. The Higgs potential is assumed to be radiatively generated by explicit breaking terms so that in the parameterization (4.91), a ∼ b ∼ O(1). From the left-side equation of (4.94), a parametric separation between v and f requires a to be tuned small by an amount ξ in (4.36). Even with this, however, this is a second tuning required on b since the new states are expected to couple to the strong sector with strong couplings, g * ∼ 4π. Thus one finds that the quartic coupling is too large in (4.93) compared to λ = 0.13. In other words, one predicts a Higgs mass that is heavier than observed in (4.94). This is mapped onto a tuning of b. Little Higgs In little Higgs models, collective symmetry breaking naturally gives a hierarchy ξ = v 2 f 2 ∼ g 2 * 16π 2 1. (4.95) The quartic coupling appears at tree-level, λ ∼ g SM . This is shown as b ∼ 16π 2 /g 2 * in Table 2. Prior to the Higgs discovery, this set up was seen to be a feature: one explains the separation between v and f . However, (4.94) shows that this predicts a Higgs mass that is on the order of 500 gev for g SM ∼ 1. Holographic Higgs These models are motivated by ads/cft duals of warped extra dimensional models, as we discussed in Section 3.9. Like the 'bona-fide composite Higgs,' the entire potential for these models are radiatively generated. This thus suffers the same O(ξ = v 2 /f 2 ) to obtain the correct electroweak symmetry breaking scale. Unlike the 'bona-fide composite Higgs,' however, g * is still weak and thus no additional tuning is required to keep the Higgs light. The four-dimensional effective theory (or deconstruction) of this scenario is what is most commonly meant when referring to a [modern] 'composite Higgs' model; see e.g. [158]. The holographic Higgs also has a version of collective symmetry breaking that is a result of locality in 5D [185]. Unlike the little Higgs models above, however, holographic Higgs models have radiative quartics. These models have the minimal amount of tuning: just ξ, which is a tuning of a few percent. Twin Higgs and neutral naturalness Twin Higgs models [195,196] have received a lot of interest after the non-discovery of any toppartners at Run I of the lhc. Rather than protecting the pseudo-Goldstone Higgs from quadratic corrections with collective symmetry breaking, these models impose a Z 2 symmetry that protects the Higgs potential. The key phenomenological feature of this framework is that the partner particles that enact this protection are uncharged under the Standard Model. Since the top partners aren't colored, one no longer expects a large production cross section at the lhc and one avoids the Run 1 bounds. These models are thus often referred to under the banner of 'neutral naturalness' and are considered a last bastion for naturalness against collider bounds. We illustrate the twin mechanism with the toy example presented in [195]; the interested reader is encouraged to read the succinct paper in its entirety. Suppose a theory has a global G =SU(4) symmetry and a field H in the fundamental representation with a symmetry-breaking potential, V (H) = −µ 2 \|H\| 2 + λ\|H\| 4 . (4.96) The field develops a vev \|H\| = m/ √ 2λ ≡ f and breaks SU(4) → SU(3). Now let us gauge a subgroup SU(2) A ×SU(2) B of the global symmetry. We decompose H into a doublet under each gauge group, H A and H B . We may identify A with the Standard Model SU(2) L . As we saw in Section 4.4.3, this gauging generates mass terms for the would-be Goldstone bosons, V ⊃ 9Λ 2 64π 2 g 2 A \|H A \| 2 + g 2 B \|H B \| 2 . (4.97) Next impose a Z 2 'twin' symmetry which swaps A ↔ B. This imposes g A = g B so that the quadratic potential becomes, V ⊃ 9g 2 Λ 2 64π 2 \|H\| 2 + · · · ,(4.98) which respects the original SU(4) symmetry of the theory and thus does not contribute to the mass of the Goldstone bosons. The higher order terms still introduce logarithmically divergent terms that break this SU(4) symmetry. We can see the 'twin' cancellation in the top couplings: L ⊃ −y t H At (A) L t (A) R − y t H Bt (B) L t (B) R . (4.99) The SU(4)→SU(3) breaking imposes h a 2 + h b 2 = f 2 . Expanding the SU(4) fundamental H analogously to (4.40), one may expand to O(h 2 /f 2 ), H A → h, H B → f − h 2 2f . (4.100) Inserting this into (4.99) yields a cancellation that is diagramatically identical to (4.86) with the important difference that the t (B) andt (B) are charged under a 'twin' qcd, but not ordinary qcd. Having demonstrated the basic principle, we refer the reader to the original literature for a demonstration of a complete model. In our phenomenological taxonomy of composite Higgs models, we have written b ∼ O(1−16π 2 /g 2 * ) reflecting that the original twin Higgs models included a tree-level quartic put in by hand to generate the v f hierarchy, though this is not an intrinsic feature of these models. As we have discussed, the observed λ = 0.13 disfavors the inclusion of this tree-level term. Dilatonic Higgs Rather than being a pseudo-Goldstone of an internal global symmetry, this scenario assumes that the Higgs is a dilaton coming from the spontaneous breaking of scale invariance [103,[197][198][199][200][201][202]. We have already explored this scenario in Section 3.10, where we identified the radion in a warped extra dimension as a state which is holographically dual to the dilaton. This is distinct from the 'holographic Higgs' scenario where the Higgs is the Goldstone of an internal global symmetry. In this scenario the vev that breaks scale invariance, f , sets the scale of the potential and is unrelated to the electroweak vev: for example, the order parameter for the breaking of scale invariance talks to all massive particles, whereas the electroweak vev ought to only talk to electroweak doublets. Thus the parameterization in (4.91) is not relevant for comparison with the other composite Higgs models discussed: the dilaton is a completely different type of beast. For a dilaton to play the role of a Higgs, one assumes that the uv theory also has a small explicit breaking of scale invariance to allow a vev f that breaks electroweak symmetry spontaneously. Scale transformations act on spacetime as x → x = e −α x and on mass-dimension-1 objects as φ → φ = e α φ. For a refresher, see [150,203,204]. Suppose that scale invariance is spontaneously broken with some order parameter, f . Applying the rule of thumb from Section 4.3.1, we identify the dilaton, σ, with the parameter of scale transformations: f → χf ≡ e σ/f f. (4.101) Unlike the case for internal symmetries, the dilaton, σ, points in the same direction in field space as the order parameter, f . Thus the potential for the dilaton can be read off the potential for the vev, V (f ). For internal symmetries, V (f ) is trivial since without explicit breaking since these the Goldstones only couple derivatively. For scale symmetries, V (f ) is allowed to have a quartic term, V (f ) ⊃ λf 4 . This term is identified with a cosmological constant. The naïve dimensional analysis size of its coefficient is λ ∼ 16π 2 coming from vacuum bubbles-this is essentially the cosmological constant problem. In order for scale symmetry to have been spontaneously broken, V (f ) must realize its minimum value at f = 0. This is not possible for purely quartic potential, so that one still requires a small explicit breaking of scale invariance that would give a V (f ) with a nontrivial vacuum. From our intuition with the Higgs, one might want to write a breaking term like V (f ) ⊃ −µ 2 f 2 , but this would be a large breaking of scale invariance. Instead, we consider operators that are close to marginal, V (f ) ∼ λf 4 + af 4 log f ∼ λf 4 + af 4− . (4.102) In order for this potential to spontaneously break scale invariance, both terms must be of roughly the same size. Since we expect λ to be large, this means that the 'small explicit breaking' must actually also be large. This is a contradiction to our initial assumption that the theory is approximately scale invariant so that the dilaton is a pseudo-Goldstone boson: with the large explicit breaking of scale invariance, one also expects the dilaton to be a heavy field. In essence this is the source of tuning in dilatonic Higgs models: one needs λ to be small so that the explicit breaking required for f = 0 can also be small which, in turn, allows the dilaton to be light. A detailed discussion of this tuning is presented in Section 5 of [103] and [202]. One way out of this apparent contradiction is supersymmetry, which gives a symmetry reason to protect against the cosmological constant: we saw from the susy algebra that 0\|H\|0 = 0 when supersymmetry is unbroken. Five-dimensional variants have been constructed in [201,202]. As a qualitative example of the above considerations, one may ask whether qcd could furnish a light dilaton since it is an approximately scale-invariant theory for which the chiral condensate is a spontaneous breaking of scale invariance. While α s is small in the uv, we know that it grows in the ir. As the couplings get larger, so does the β function, which is a concrete manifestation of scale invariance being broken by a larger amount. Thus the would-be dilaton in qcd is not a light state in the effective theory. Parameterization of phenomenology We see that the composite Higgs can be probed through its deviations from the sm Higgs couplings. A phenomenological parameterization of the space of light, composite Higgs models is presented in [205] as the 'strongly interacting light Higgs' (silh, pronounced "silch") effective Lagrangian and extended in [206]. We briefly review the main results and refer to [205,206] for a detailed discussion including matching to specific composite Higgs models. We now examine a convenient parameterization of the phenomenological Lagrangian, L silh =L SM H +c H 2v 2 ∂ µ (H † H)∂ µ (H † H) +c T 2v 2 H † ← → D H 2 −c 6 v 2 λ(H † H) 3 + c u 2v 2 y u H † HQ L Hu R + · · · + ic W g 2M 2 W H † σ i ← → H (D ν W νµ ) i + · · · ,(4.103) where H † ← → D µ H = H † D µ H − D µ H † H and the · · · represent similar terms for the other fermions and gauge bosons. The expected sizes of these coefficients arē c H ,c T ,c 6 ,c ψ ∼ v 2 f 2c W,B ∼ M 2 W g 2 * f 2 ,(4.104) where g * is defined in (4.92). Following [206], the operators in (4.103) are normalized with respect to the Higgs vev v rather than the scale f in [205]; this is why the expected values of the barred couplingsc i differ by factors of v/f from the couplings c i ∼ 1 in equation (15) of [205]. The phenomenological Lagrangian (4.103) can be constructed systematically from the nonlinear sigma model including symmetry breaking terms which we assume are parameterized by the sm couplings that break those symmetries: the Higgs quartic coupling λ (breaking the pseudo-Goldstone Higgs shift symmetry) and the Yukawas (breaking shift and flavor symmetries). See Appendix B of [207] for a detailed discussion in terms of naïve dimensional analysis. Following the nda of Section 4.3.8, the general strategy is to write L silh = M 4 g 2 * L U, ∂ M ,(4.105) where U = U (π/f ) is the dimensionless linear field containing the Goldstones (4.11) and the partial derivative carries a factor of M −1 , the scale of new states (4.92), to make it dimensionless. Each Goldstone field π comes with a factor of 1/f . The nda prefactor M 4 g −2 * = M 2 f 2 is derived in (4.33-4.34). Recall that this is precisely the Λ 2 f 2 prefactor in the nda literature except that we replace Λ with a scale which exists in the effective theory, M < Λ [154,155]. This inequality is equivalent to g * < 4π and parameterizes the regime in which the NLΣM is weakly coupled. We then take the dimensionless function L to be a derivative expansion analogous to (4.18); the silh interactions appear at higher order from the term L silh = M 4 g 2 *   · · · + 1 3 π(x) f ← → ∂ M π(x) f 2 + · · ·   (4.106) where π(x) is identified with the Higgs doublet H(x) and the partial derivatives ∂ µ are promoted to sm gauge covariant derivatives. Gauge field strengths are included with factors of m −2 ρ since F µν ∼ [D µ , D ν ]. Thec H andc T terms encode the O(H 4 , ∂ 2 ) interactions after shifting the Higgs by a factor proportional to (H † H)H/f 2 (see [205]). Thec 6 ,c u ,c W (and analogous terms) break the shift symmetries of the NLΣM and carry explicit factors of the sm couplings that break those symmetries: the Higgs quartic interaction, the Yukawas, or sm gauge couplings, respectively. Electroweak precision observables 28 set bounds on composite Higgs models [206,213]; at 2σ: −1.5 × 10 −3 <c T < 2.2 × 10 −3 (4.107) −1.4 × 10 −3 <c W +c B < 1.9 × 10 −3 ,(4.108) coming from theT andŜ parameters respectively. The former condition reflects the requirement of custodial symmetry [214] (see [215] for an introduction) which is assumed in the latter bound. This and other bounds on composite Higgs models coming from Higgs observables are reviewed in [217,218] using a slightly different effective theory parameterization introduced in [160]. In that notation, (4.109) comes from a 2 ≥ 0.84. Further phenomenological bounds and their relations to specific models can be found in [132,205,206,219]. At 2σ, Higgs data constraints the minimal composite Higgs model to satisfy [220] v f 0.5. The bounds on composite Higgs models coming from Higgs observables are reviewed in [206,217,218]. Further phenomenological bounds and their relations to specific models can be found in [132,205,219]. Closing Thoughts We briefly review interconnections between some of the salient ideas in these lectures, acknowledge topics omitted, and point to directions of further study. One of the themes in the latter part of these lectures were weakly coupled descriptions of strong dynamics and we close by highlighting this common thread. Covariant Derivatives Each of the scenarios that we explored carries its own sense of covariant derivative. The most explicit example is in a warped extra dimension, where spacetime is explicitly curved. The holographic principle made use of this geometry: the isometries of ads match the conformal symmetries of the strongly coupled theory near a fixed point. The system is so constrained by these symmetries that the behavior of 5D fields could be identified with the renormalization group flow of 4D operators. Even in supersymmetry-where superspace can be thought of a 'fermionic' extra dimension-we introduced a susy covariant derivative. The practical significance of this covariant derivative is to define chiral superfields, the irreducible representation of N = 1 susy that we use for matter fields. Finally, the nonlinear realizations we used for composite Higgs models also has a geometric structure coming from the coset space. This is seen in the ccwz formalism reviewed in Appendix B, where one identifies covariant derivative and gauge field for the coset space that are necessary to construct invariant Lagrangians. Nonlinear realizations The simplest handle on strong dynamics is to work in an effective theory of pseudo-Goldstone bosons given by the pattern of global symmetry breaking in the strong sector. In composite Higgs models, one addresses the Hierarchy problem by assuming that the Higgs is a pseudo-Goldstone boson associated with the dynamics of a strongly coupled sector that break global symmetries at a scale f . We saw that generically the sm interactions required for a Higgs boson tend to push its mass back up towards the compositeness scale, Λ ∼ 4πf . One way to push the Higgs mass back down is to invoke collective symmetry breaking, which can often be described succinctly using 'moose' diagrams. Holographic and deconstructed extra dimensions We saw that the holographic principle is an alternative way to describe the dynamics of a strongly coupled sector through the use of a higher dimensional theory with a non-trivial geometry. From the extra dimensional perspective, the Higgs mass is natural because it is localized towards the ir brane where the Planck scale is warped down to tev scale. Holographically, this is interpreted as the Higgs being a composite state. Indeed, the minimal composite Higgs model [158] was constructed using holography as a guiding principle. Further, the little Higgs models that first highlighted collective symmetry breaking were motivated by deconstructions of an extra dimension. This picture allowed us to clearly see that the Goldstone modes of the spontaneously broken global symmetries can be identified with the scalar component of a 5D gauge field. In the deconstruction, the gauge bosons from each copy of the gauge group eat these Goldstone modes to become the spectrum of heavy kk modes. In this sense, little Higgs and holographic composite Higgs constructions are similar to gauge-Higgs unification scenarios in 5D where the Higgs is the zero mode of a bulk gauge field, see [221] and references therein. One way to interpret the lightness of the Higgs mass is via locality in the deconstructed extra dimension: the symmetries are only broken on the boundaries and one needs a loop that stretches between the boundaries to generate a Higgs potential. This implies that the loop cannot be shrunk to zero and that the Higgs potential is finite since it can have no short-distance divergences. The natural cutoff is set by the size of the extra dimension. Deconstruction itself, however, is rooted in the idea of a hidden local symmetry in nonlinear models. See [128] for a comprehensive review. A 5D version of the little Higgs in ads was presented in [222]. Shortly after, [185] connected the holographic composite Higgs to a little Higgs theory, relating the ccwz formalism of Appendix B to the hidden local symmetry construction. Natural SUSY and partial compositeness We began these lectures with what appeared to be a completely different subject: supersymmetry. We saw that the natural setting for susy is superspace, which is superficially an 'extra quantum dimension' that is both Grassmannian and spinorial. One way to see how susy solves the Hierarchy problem is to observe that it requires the existence of superpartners (differing by half integer spin) that cancel the loop contributions of particles to superpotential parameters such as the Higgs mass. We saw a similar cancellation when invoking collective symmetry breaking (or the twin Higgs mechanism) in composite Higgs theories with the notable difference that the partner particles had the same spin as their sm counterparts. susy, however, must be broken. These effects feed into the large parameter space of the minimal supersymmetric Standard Model and are required (in the mssm) for electroweak symmetry breaking. The lhc puts tight bounds on the simplest mssm spectra and leads us to consider ways to hide susy. One of these solutions is 'natural susy' where one only maintains the minimal spectrum of superpartners required for the naturalness of the Higgs mass. Among the predictions of natural susy is a light stop and heavy first and second generation quarks. This type of spectrum, however, is automatic when supersymmetrizing the rs model with anarchic flavor 29 . When susy is broken on the uv brane 30 , 5D super fields which are localized near the uv brane are more sensitive to the splitting between the sm and superpartner masses [19,[226][227][228]. Invoking what we know about the anarchic flavor 5D mass spectrum (i.e. localization of the fermion profiles), we come to the conclusions in Table 3. Holographically this is interpreted as supersymmetry being an accidental symmetry in the ir. That is, the strong sector flows to a fixed point that is supersymmetric, even though the theory at the uv is not manifestly supersymmetric. As a particle becomes more composite, it becomes more degenerate in mass with its superpartner. A schematic moose diagram is shown in Fig. 15; note that one of the sacrifices of this realization of natural susy is conventional unification, see e.g. [229]. Naturalness and top partners The three classes of physics beyond the Standard Model that we have explored all generically predict new particles accessible at high energy colliders. For supersymmetry and extra dimensions, these particles were a manifestation of the extended spacetime symmetry under which the sm particles must transform. For a composite Higgs, this reflected a larger global symmetry breaking pattern and included additional fermions that appear necessary to generate an sm-like Higgs potential. At a technical level, we needed new particles to run in Higgs loops to soften the quadratic sensitivity to the cutoff. Since the top quark has the largest coupling to the Higgs, a generic prediction for naturalness are light (i.e. accessible at the lhc) states to cancel the top loop. While these particles may have different spin, the examples we've explored focused on the case where they have the same sm quantum numbers as the top. The color charge of these new particles make them easy to produce at the lhc so that their non-observation is particularly disconcerting. One model building direction out of this puzzle is to consider models where the top partner is not color charged. We saw this in the twin Higgs model in Section 4.7.4. A supersymmetric cousin on these models go under the name of folded susy [230], where the top partners are uncolored but still carry electroweak charges. Non-supersymmetric variants include the quirky little Higgs [231] and the orbifold Higgs [232]. Seiberg duality In the composite Higgs models, same-spin partners cancel the leading sm particle contributions to the quadratically sensitive terms in the Higgs mass. We saw that this is not coincidental, but is in fact imposed by the structure of collective symmetry breaking. In the same way, the protection against quadratic divergences in susy is most clearly understood from the tremendous constraints put on the theory by supersymmetry. Among other things, these constraints impose the holomorphy of the superpotential which, in turn, prevents the perturbative renormalization of any of the superpotential terms. A derivation of this important result is beyond the scope of these lectures, but can be found-along with further implications of susy-in the reviews already mentioned. Supersymmetry turns out to also be a powerful constraint on the behavior of gauge theories. In fact, they allow one to map out the entire phase structure of the supersymmetric generalization of qcd, sqcd. This, in itself, is a topic of depth and elegance which is covered very well in [4,[8][9][10]32]. One key outcome of this exploration in the 1990s was the observation that two distinct supersymmetric non-Abelian gauge theories, shown in Fig. 16, flow to the same ir theory. One theory, sqcd, is a standard SU(N ) supersymmetric gauge theory with F flavors such that F > N + 1. (5.1) In the case where N +1 < F < 3N , this theory is asymptotically free and becomes strongly coupled in the ir. The other theory is an SU(F − N ) gauge theory with F flavors and an additional color singlet 'meson' which is a bifundamental under the SU(F )×SU(F ) flavor symmetry. This theory has a superpotential, W ∼QM Q,(5.2) which can be understood as a loop in the moose diagram since all indices are contracted. When N + 1 < F < 3 2 N , the dual theory is ir free and is perturbative. On the other hand, when 3 2 N ≤ F < 3N , the dual theory is also asymptotically free and flows to the same Banks-Zaks-like non-trivial fixed point that is the same endpoint of the original 'electric' theory's rg flow. The fact that two a priori unrelated theories flow to the same ir theory suggest a compelling interpretation: the initial asymptotically free theory becomes strongly interacting at low energies and can be equivalently described by its dual theory. In the case where N + 1 < F < 3 2 N , the dual theory is ir free and perturbative precisely in the regime where the original theory is not. This is an 'electromagnetic' duality in the sense of exchanging strongly and weakly coupled descriptions of the same physics, similar to the ads/cft correspondence. This Seiberg duality is a powerful handle on strongly coupled physics via a weakly coupled 4D dual description. One popular application was to simplify the construction of models with dynamical susy breaking, see [10,233] for reviews. In some sense this is completely analogous to using chiral perturbation theory to describe low-energy qcd. However, unlike qcd, the low energy ('magnetic') theory is not composed of gauge singlets. In fact, there is an emergent SU(F − N ) gauge symmetry that appears to have nothing to do with the original SU(N ) gauge symmetry of the 'electric' theory. One recent interpretation, however, is that this magnetic gauge group can be identified with the 'hidden local symmetry' in nonlinear models [234], which we previously mentioned in the context of deconstruction and moose models. In this construction, the ρ meson in qcd (the lightest spin-1 meson) is identified as the massive gauge boson of a spontaneously broken gauge symmetry present in the nonlinear Lagrangian. One can also relate Seiberg duality to the ads/cft correspondence through explicit string realizations. Note that (5.1) is typically a different regime from the large N limit invoked in ads/cft. From a purely field theoretical point of view, the ads/cft correspondence can be understood as a duality cascade where a susy gauge theory has a renormalization trajectory that zig-zags between a series of fixed points. This is reviewed pedagogically for a field theory audience in [96]. Multiple guises of strong dynamics In this final section we have touched on multiple ways in which we can address strong dynamics in field theory: nonlinear realizations based on the symmetry breaking structure, holographic extra dimensions, and Seiberg duality in susy. The lesson to take away from this overview is that one should be flexible to think about strong dynamics in different languages. Often the intuition from one understanding of strong dynamics can shed light on constructions based on a different description. One example is the use of Seiberg duality to describe a [partially-]composite electroweak sector based on the 'fat Higgs' model [235]. The idea is to take super-qcd with F = N + 2 flavors so that the magnetic gauge group can be linked with SU(2) L . The realization of this idea in [165] described this in terms of moose diagrams where the magnetic gauge group is 'color-flavor locked' with an externally gauged SU(2) L . This mixes the magnetic gauge bosons with the external gauge bosons so that the observed W and Z are partially composite. Independently, a similar model was presented in [166] where the nature of this mixing was explained in terms of the intuition from a warped extra dimension. In particular, one hope is that one can directly identify the magnetic SU(2) with the electroweak SU(2) L . This, however, is not possible since-as we know from composite model building-at the compositeness scale the näive dimensional analysis expectation is that the composite vector boson couples strongly: g ∼ 4π/ √ N . In other words, if the electroweak gauge bosons are strongly coupled bound states, then one would expect a large residual interaction with other strongly coupled bound states. In the rs language, a composite W and Z would have ir brane localized profiles and this would typically predict very strong couplings. This would require a very large running to squeeze the profile on the ir brane. In the Seiberg dual picture, this requires a very large number of flavors if one maintains that the W and Z are purely composite but have the observed sm couplings, leading one to prefer partial compositeness of these particles. This general framework was later used to construct a model of natural susy in which follows the general deconstruction/moose in Fig. 15 [236]. Omissions We have necessarily been limited in scope. Even among the topics discussed, we have omitted an exploration of susy gauge theories (leading up to Seiberg duality), variants of the 'realistic' rs models (as well as 'universal extra dimension' models), the virtues of different cosets for composite Higgs model building, and an overview of product space (moose-y) little Higgs models. Many explicit calculations were left out and are left to the dilligent reader as exercises, and we only made cursory nods to the phenomenology of these models. In addition to the three major topics covered in these lectures, there are various other extensions to the Standard Model that we have not discussed. Our preference focused on models that address the Higgs hierarchy problem, and as such we have omitted discussions of many important topics such as grand unification, dark matter, flavor, strong cp, cosmology (of which the cosmological constant is the most extreme fine tuning problem), or any of the phenomenology of interpreting possible experimental signals from colliders/telescopes/underground experiments/etc. We have only presented very cursory comparisons to current data; we refer the reader to the appropriate experiments' results pages and conference proceedings for the latest bounds. See also [237] for an overview of the Run I searches for new physics. All of these topics-and perhaps many others-are, in some combination, key parts of a model builder's toolbox in the lhc era. 3 2 A 2 = 1 2M 3 * Λe −A . (A.4) This only has a solution for Λ < 0 so that we're forced to consider ads spaces. This equation is separable with the general solution, e −A(z) = 1 (kz + constant) 2 k = −Λ 12M 3 * . (A.5) To recover (3.28) we identify R = 1/k and impose A = 0 at z = R, setting the constant to zero. The latter choice simply sets the warp factor at the uv brane to be 1. We must remember that the rs space is finite-and has branes at its endpoints-when we solve the M N = µν Einstein equations. These equations depend on the second derivative of A(z) and one should be concerned that this may be sensitive to the energy densities on the branes. This is analogous to the Poisson equation in electrostatics where a second derivative picks up the δ-function of a point charge. In general the branes carry tensions which appear as 4D cosmological constants, Λ ir,uv . Recalling the form of the induced metric √ĝ = g/g 55 , these appear in the action as d 5 x g g 55 Λ ir,uv δ(z − z ir,uv ) ⇒ T µν = 1 √ g δS δg µν = g µν 2 √ g 55 (Λ ir δ(z − R ) + Λ uv δ(z − R)) . (A.6) A.2 RS as an orbifold To better understand the physics of the brane cosmological constants, it is useful to represent the interval with an orbifold S 1 /Z 2 . This is simply the circle y ∈ [−π, π] with the identification y = −y. While this may sound somewhat exotic, such compactifications are common in string theory, and was the original formulation of the rs scenario. Note that y can take any value due to the periodic identification of the circle, while the fixed points at y = 0, π demarcate the physical rs space. The orbifold identification forces us to modify (A.5) by replacing z → \|z\| to preserve the z ↔ −z symmetry.This absolute value, in turn, leads to δ functions in A (z) at the fixed points, A (z) = − 2k 2 (k\|z\| + const) 2 + 4k k\|z\| + const (δ(z − R) − δ(z − R )) . (A.7) The µν Einstein equation then implies − 3 2 η µν − 4k (δ(z − R) − δ(z − R )) k\|z\| + const = η µν 2M 3 * Λ uv δ(z − R) + Λ ir δ(z − R ) k\|z\| + const . (A.8) From this we see that the brane cosmological constants must have opposite values, Λ uv = −Λ ir = 12kM 3 * . (A.9) Recall, further, that k is related to the bulk cosmological constant by (A.5), so that this represents a tuning of the bulk and brane cosmological constants. This is a necessary condition for a static, gravitational solution. Physically, we see that the brane and the bulk cosmological constants are balanced against one another to cause the brane to be flat. A.3 Bulk Fermions in RS The properties of fermions in a curved space can be subtle. In particular, it's not clear how to generalize the usual Dirac operator, iγ µ ∂ µ . In this appendix we review properties of fermions in an extra dimension and then derive the form of the fermion action in rs. A.3.1 The fifth γ matrix Firstly, unlike in 4D where the fundamental fermion representation is a Weyl spinor, 5D Lorentz invariance requires that fermions appear as Dirac spinors. A simple heuristic way of seeing this is to note that in 4D one can construct a γ 5 ∼ γ 0 · · · γ 3 as a linearly independent chirality operator. In 5D, however, γ 5 , is part of the 5D Clifford algebra and is just a normal γ matrix in the z-direction. Note that the normalization of γ 5 is fixed by {γ 5 , γ 5 } = 2η 55 and has a factor of i compared to the usual definition in 4D. One should immediately be concerned: if the 5D fermions are Dirac, then how does one generate the chiral spectrum of the Standard Model matter? As we show below, this follows from a choice of boundary conditions. An excellent reference for the properties of fermions in arbitrary dimension is [240]. A.3.2 Vielbeins In order to write down the fermionic action, we first need to establish some differential geometry so that we may write the appropriate covariant derivative for the spinor representation. We will be necessarily brief here, but refer to [241][242][243] for the interested reader 31 . The familiar γ matrices which obey the Clifford algebra are only defined for flat spaces. That is to say that they live on the tangent space (locally inertial frame) of our spacetime manifold. In order to define curved-space generalizations of objects like the Dirac operator, we need a way to convert spacetime indices M to tangent space indices a. Vielbeins, e a µ (x), are the geometric objects which do this. The completeness relations associated with vielbeins allow them to be interpreted as a sort of "square root" of the metric in the sense that g M N (x) = e a M (x)e b M (x)η ab , (A.10) where η ab = diag(+, −, · · · , −) is the Minkowski metric on the tangent space. For our particular purposes we need the inverse vielbein, e M a (x), defined such that e M a (x)e a N (x) = δ M N e M a (x)e b N (x) = δ b a . (A.11) Spacetime indices are raised and lowered using the spacetime metric g M N (x) while tangent space indices are raised and lowered using the flat (tangent space) metric η ab (x). Physically we may think of the vielbein in terms of reference frames. The equivalence principle states that at any point one can always set up a coordinate system such that the metric is flat (Minkowski) at that point. Thus for each point x in space there exists a family of coordinate systems that are flat at x. For each point we may choose one such coordinate system, which we call a frame. By general covariance one may define a map that transforms to this flat coordinate system at each point. This is the vielbein. One can see that it is a kind of local gauge transformation, and indeed this is the basis for treating gravity as a gauge theory built upon diffeomorphism invariance. A.3.3 Spin covariant derivative The covariant derivative is composed of a partial derivative term plus connection terms which depend on the particular object being differentiated. For example, the covariant derivative on a spacetime vector V µ is D M V N = ∂ M V N + Γ N M L V L . (A.12) The vielbein allows us to work with objects with a tangent space index, a, instead of just spacetime indices, µ. The γ matrices allow us to further convert tangent space indices to spinor indices. We would then define a covariant derivative acting on the tangent space vector V a , D M V a = ∂ M V a + ω a M b V b , (A.13) where the quantity ω a M b is called the spin covariant derivative. Consistency of the two equations implies D M V a = e a N D M V N . (A.14) This is sufficient to determine the spin connection. It is a fact from differential geometry that the spin connection is expressed in terms of the veilbeins via [245] ω ab M = 1 2 g RP e [a R ∂ [M e b] P ] + 1 4 g RP g T S e [a R e b] T ∂ [S e c P ] e d M η cd (A.15) = 1 2 e N a ∂ M e b N − ∂ N e b M − 1 2 e N b (∂ M e a N − ∂ N e a M ) − 1 2 e P a e Rb (∂ P e Rc − ∂ R e Rc ) e c M . (A.16) When acting on spinors one needs the appropriate structure to convert the a, b tangent space indices into spinor indices. This is provided by σ ab = 1 4 [γ a , γ b ] (A.17) so that the appropriate spin covariant derivative is D M = ∂ M + 1 2 ω ab M σ ab . (A.18) A.3.4 Antisymmetrization and Hermiticity The fermionic action on a d-dimensional curved background is 32 S = d d x \|g\| Ψ ie M a γ a ← → D M − m Ψ, (A.19) where the antisymmetrized covariant derivative is defined by a difference of right-and left-acting derivatives ← → D M = 1 2 D M − 1 2 ← − − D M . (A.20) This is somewhat subtle. The canonical form of the fermionic action must be antisymmetric in this derivative in order for the operator to be Hermitian and thus for the action to be real. In flat space we are free to integrate by parts in order to write the action exclusively in terms of a right-acting Dirac operator. Hermiticity is defined with respect to an inner product. The inner product in this case is given by Ψ 1 \|OΨ 2 = d d x \|g\| Ψ 1 OΨ 2 . (A.21) A manifestly Hermitian operator is O H = 1 2 O + O † , where we recall that Ψ 1 \|O † Ψ 2 = OΨ 1 \|Ψ 2 = d d x \|g\| OΨ 1 Ψ 2 . (A.22) The definition of an inner product on the Fock space of a quantum field theory can be nontrivial on curved spacetimes. However, since our spacetime is not warped in the time direction there is no ambiguity in picking a canonical Cauchy surface to quantize our fields and we may follow the usual procedure of Minkowski space quantization with the usual Minkowski spinor inner product. As a sanity-check, consider the case of the partial derivative operator ∂ µ on flat space time. The Hermitian conjugate of the operator is the left-acting derivative, ← − ∂ µ , by which we really mean All of this may seem overly pedantic since integration by parts allows one to go back and forth between the 'canonical' form and the usual 'right-acting only' form of the fermion kinetic operator. Our interest, however, is to apply this to the Randall-Sundrum background where integration by parts introduces boundary terms and so it is crucial to take the canonical form of the Dirac operator as the starting point. The two spin connection terms cancel since γ µ γ 5 γ µ ΨΨ = Ψγ µ γ 5 γ µ Ψ, so that upon including a bulk mass term, S = d 5 x i 2 R z 4 Ψγ M ← → ∂ M Ψ − d 5 x i 2 R z 5 mΨΨ = d 5 x i 2 R z 4 Ψ γ M ← → ∂ M − c z Ψ, (A.36) where c = mR = m/k is a dimensionless parameter that is the ratio of the bulk mass to the curvature. Before we can dimensionally reduce the action straightforwardly, we must write the Dirac operator to be right-acting, i.e. acting on Ψ, so that we can vary with respect to Ψ to get an operator equation for Ψ. Obtaining this is from (A.36) is now a straightforward matter of integration by parts of the left-acting derivative term. Note that it is crucially important that we pick up a derivative acting on the metric/vielbein factor (R/z) 4 . We would have missed this term if we had mistakenly written our original 'canonical action,' (A. 19), as being right-acting only. The integration by parts for the M = µ = 0, · · · , 4 terms proceeds trivially since these directions have no boundary and the metric/vielbein factor is independent of x µ . Performing the M = 5 integration by parts we find S = d 4 x R R dz R z 4 Ψ i / ∂ + iγ 5 ∂ z − i 2 z γ 5 − c z Ψ + (boundary term)\| R R . (A.37) The term in the parenthesis can be identified with the Dirac operator for the Randall-Sundrum model with bulk fermions. The boundary term is (boundary) = (R/z) 4 ψχ − χψ R R , (A.38) where we've written out the Dirac spinor Ψ in terms of two-component Weyl spinors χ and ψ. This term vanishes when we impose chiral boundary conditions, which we review in the next section. In terms of Weyl spinors this gives S = d 4 x R R dz R z 4 ψ χ −∂ z + 2−c z i / ∂ i / ∂ ∂ z − 2+c z χ ψ , (A.39) where we use the two-component slash convention / v = v µ σ µ , / v = v µ σ µ . A.3.6 Chiral boundary conditions The vector-like (Dirac) nature of 5D spinors is an immediate problem for model-building since the Standard Model is manifestly chiral and there appears to be no way to write down a chiral fermion without immediately introducing a partner fermion of opposite chirality and the same couplings. To get around this problem, we can require that only the zero modes of the 5D fermions-those which are identified with Standard Model states-to be chiral. We show that one chirality of zero modes can indeed be projected out, while the heavier Kaluza-Klein excitations are vector-like but massive. We can project out the zero modes of the 'wrong-chirality' components of a bulk Dirac 5D fermion by imposing chiral boundary conditions that these states vanish on the branes. For leftchiral boundary conditions, ψ = 0 on the branes, while for right-chiral boundary conditions, χ = 0 on the branes. These boundary conditions force the 'wrong-chirality' zero mode to be identically zero everywhere. Thus we are guaranteed that both terms in (A.38) vanish at z = R, R for either chirality. Imposing these chiral boundary conditions is equivalent to the statement that the compactified extra dimension is an orbifold. This treatment of boundary conditions for interval compact spaces was first discussed from this viewpoint in [121]. A.4 Gauge fields in RS We now move on to the case of bulk gauge fields. See Section 2 of [77] for a brief, pedagogical discussion of gauge fields in a compact, flat extra dimension. We follow the approach of [246], though we adapt it to follow the same type of derivation espoused above for the fermion propagator. The bulk action is S 5 = d 4 xdz √ g − 1 4 F M N F M N + (brane) + (gauge fixing) (A.47) To derive the propagator, we would like to write the kinetic term in the form A M O M N A N so that we may invert the quadratic differential operator O M N . This require judicious integration by parts including the (R/z) factors from the metric and the measure, √ g. The relevant integration is −R 4z F M N F M N = 1 2 A N ∂ M R z ∂ M A N − A N ∂ M R z ∂ N A N − ∂ M R z A N ∂ [M A N ] , (A.48) where the last term integrates to a boundary term. Observe that this boundary term vanishes for both Dirichlet and Neumann boundary conditions so that it vanishes for µ → ν and 5th component scalar propagators. It does not vanish, however, for the case of vector-scalar mixing. For simplicity, we will drop the term here in anticipation that it will be removed by gauge fixing. With this caveat, the above integration becomes −R 4z F M N F M N = A µ R 2z ∂ 2 η µν − z∂ z 1 z ∂ z η µν − ∂ µ ∂ ν A ν + A 5 R z ∂ z ∂ µ A µ − A 5 R 2z ∂ 2 A 5 . (A. 49) This is now in the desired form: we can read off the quadratic differential operators which encode the propagation of the 5D gauge bosons. Observe that we have a term that connects the 4D vector A µ to the 4D scalar A 5 . We prefer to work with these as separate fields. This term is removed by a judicious choice of gauge fixing. A.4.1 Gauge fixing We must now gauge fix to remove the gauge redundancy which otherwise appears as unphysical states in the propagator. Ideally we would like to pick a gauge where the scalar vanishes A 5 = 0 and the vector has a convenient gauge, say, Lorenz gauge ∂ µ A µ = 0. Unfortunately, these gauges are incompatible. Intuitively this is because we only have a single gauge fixing functional to work with in the path integral so that we are allowed to set at most one expression to vanish. Instead, motivated by the desire to cancel the vector-scalar mixing in (A.49) and to recover the usual R ξ gauge in 4D, we choose a gauge fixing functional L gauge fix = − R z 1 2ξ ∂ µ A µ − ξz∂ z 1 z A 5 2 (A.50) We have introduced a gauge fixing parameter ξ which will play the role of the ordinary R ξ gauge fixing parameter in 4D. We can integrate by parts to convert this to the form A M O M N gauge fix A N , L gauge fix = A µ 1 2ξ R z ∂ µ ∂ ν A ν − A 5 R z ∂ z ∂ µ A µ + A 5 ξ 2 R z ∂ z z∂ z 1 z A 5 . (A.51) Observe that the second term here cancels the unwanted mixing term in (A. 49). Summing this together with the gauge kinetic term gives a clean separation for the kinetic terms for the gauge vector and scalar: L gauge + L gauge fix = A µ R 2z η µν ∂ 2 − 1 − 1 ξ ∂ µ ∂ ν − η µν z∂ z 1 z ∂ z A ν − A 5 R 2z ∂ 2 A 5 + A 5 Rξ 2z ∂ z z∂ z 1 z A 5 (A.52) ≡ A µ 1 2 O µν A ν + A 5 1 2 O 5 A 5 . (A.53) As above, now that we have the action written in terms of right-acting operators on the gauge fields. The 5D equations of motion are simply O µν A ν = 0 O 5 A 5 = 0 . (A.54) We now proceed to do a kk reduction to determine the kk mode properties. A.4.2 KK Decomposition of RS Gauge Bosons Define the kk reductions of the 5D A µ and A 5 fields, A µ (x, z) = 1 √ R n A (n) µ (x)h (n) (z) A 5 (x, z) = 1 √ R n A (n) 5 (x)h (n) 5 (z) . (A.55) For the n th kk mode, we know ∂ 2 A (n) µ (x) = −p 2 A (n) µ (x) = −m 2 n A (n) µ (x), defining the mass of the KK mode, m n . Taking the ξ = 1 gauge for simplicity, the equation of motion for the n th kk mode is a differential equation for the kk masses and profiles, R z −m 2 n − z∂ z 1 z ∂ z h (n) (z) = 0 . (A.56) This is Sturm-Liouville equation of generic form (3.43). where in the case of h (n) we identify p(z) = R z q(z) = 0 w(z) = R z λ = m 2 n . (A.57) The Sturm-Liouville weight, w(z), defines the orthonormality relation (3.44), dz R R z h (n) (z)h (m) (z) = δ mn . (A.58) Observe that the weight differs from that of a scalar field, (3.41), or a fermion (A.45). The general solution for the n th kk mode profile of a bulk gauge field is h (n) (z) = azJ 1 (m n z) + bzY 1 (m n z), (A. 59) where J α and Y α are Bessel functions. The n = 0 modes are constant. A.4.3 The A 5 scalar gauge boson We now turn to the A 5 piece of the 5D gauge field. The equation of motion for the n th kk mode of the A 5 is ∂ z z∂ z 1 z h (n) 5 + m 2 n,5 ξ h (n) 5 . (A.60) One could follow the same analysis as above, but it turns out that there is a convenient shortcut if we compare this to (A.56). Let us make the ansatz that h (n) 5 (z) = 1 m n ∂ z h (n) (z) m 2 n,5 = m 2 n . (A.61) One can confirm that this is a solution to the equation of motion by applying (A.56) and choosing ξ = 1 for consistency with our gauge choice for A µ . The results are general for any ξ, and indeed one recognizes that the equations of motion for the A µ and A 5 kk modes mirror that of massive gauge bosons and the Goldstone modes that they eat in a spontaneously broken gauge theory. This is manifest, for example, in the ξ-dependence in the A 5 kk mass. This behavior is not a coincidence and is, indeed, predicted from the perspective of the extra dimension as a deconstruction with many copies of the gauge group spontaneously breaking into the symmetric combination, see Section 4.6. To be explicit: the n th kk vector, A (n) µ with n > 0, can be understood as the gauge boson of its very own copy of the gauge symmetry. This gauge boson, however, is massive because it picks up the kk mass m n . We know that massive gauge bosons have three polarizations and so expect that the longitudinal polarization came from eating a Goldstone boson. These Goldstone bosons are precisely the A (n) 5 states. The symmetry that was broken spontaneously is the product of gauge symmetries in the deconstruction. In other words: a 5D gauge symmetry can be thought of as a product of 4D gauge symmetries: G 5D ≈ G 4D × G 4D × · · · G 4D . The compactification of the extra dimension breaks G n 4D → G 4D in such a way that the A vanishes. However, one can also realize the Higgs as the fifth component of a 5D gauge field whose 4D vector piece has no zero mode [247][248][249]. The realization of this gauge-Higgs unification scenario in Randall-Sundrum is dual to composite Higgs models [158,250,251] in the sense described in Section 3.9.5. With this application in mind, the zero mode profile of the A 5 is h (0) 5 (z) = az + bz log z . (A.62) The proportionality to z means that such a field is peaked sharply on the ir brane and is indeed phenomenologically suitable to be a Higgs. A.4.4 Gauge boson masses and profiles A sm gauge field must have a zero mode (which is identified with the sm state) so that it must have Neumann boundary conditions (bc). Using the formulae for derivatives of Bessel functions and (A.59), we find Y 0 (m n R)J 0 (m n R ) = J 0 (m n R)Y 0 (m n R ), (A. 63) where m n is the mass of the n th kk mode. We know that m n ∼ n/R and that R R . Thus m n R ≈ 0 for reasonable n. Now invoke two important properties of the J 0 and Y 0 Bessel functions: 1. J 0 (0) = 1 and \|J 0 (x)\| < 1. 2. Y 0 (x → 0) → −∞ and \|Y 0 (x > y 1 )\| < 1 where y 1 is the first zero of Y 0 (x). From this we see that the left-hand side of (A.63) is very large and negative due to the Y 0 (m n R) term while the right-hand side is a product of terms that are O(1) or less. This implies that J 0 (m n R ) ≈ 0. In other words, the kk masses are given by the zeros of J 0 . The first zero is x 1 = 2.405 so that the first kk gauge boson excitation has mass m n ≈ 2.4/R . The solution for the the n th kk mode profile of a sm gauge field is thus h (n) (z) = N z [Y 0 (m n R)J 1 (m n z) − J 0 (m n R)Y 1 (m n z)] . (A. 64) The normalization is fixed by performing the dz integral and requiring canonical normalization of the zero mode 4D kinetic term, d 4 x dz √ gF M N F P Q g M P g N Q = d 4 x dz R z F (0) (x) µν F (0) (x) µν h (0) (z) √ R 2 + · · · . (A. 65) This gives N −2 = log R /R. Finally, for the W and Z bosons, the Higgs vev on the ir brane changes the boundary conditions so that the zero mode profile is not flat. Heuristically it introduces a kink on the profile near the ir brane. Since M Z m 1 , we may treat this as a perturbation to m 0 = 0 so that the Z boson profile is [252] h (0) 66) and similarly for the W . Z (z) = 1 R log R /R 1 − M 2 Z 4 z 2 − 2z 2 log z R , (A. A.5 Caution with finite loops One should be careful when calculating loop diagrams in theories with extra dimensions. When one calculates a finite loop, say a dipole operator, naïve application of effective field theory suggests taking only the lowest kk mode and letting the 4D loop momentum go to k → ∞. This, however, can lead to erroneous results since the loop integral runs over all momenta, including those in the fifth dimension. Only integrating over the 4D directions removes terms that scale like k 2 /M 2 kk which would otherwise make an O(1) finite contribution. This can appear as a dependence on the order in which one does the 4D loop integral versus kk sum; this discrepancy has appeared in the rs gg → h production calculations [253]. One way to avoid this problem is to work in mixed position-momentum space [246]. This was used to calculate rs constraints from f → f γ [254,255] and the muon magnetic moment in [256]. These references include Feynman rules for performing mixed space calculations. For a recent explanation of the subtleties of 5D dipoles and the resolution to puzzles in the previous literature, see [257]. In particular, Section 3 of that paper shows how to quickly estimate the size of 4D couplings from overlap integrals. B The CCWZ Construction The general theory of Goldstone bosons is described in the papers by Callan, Coleman, Wess, and Zumino (ccwz) [122,123]. In this appendix we summarize the ccwz procedure and identify key aspects that are often referred to in the composite Higgs literature. See §19.5 -19.7 of [124], §2.3 of [136], or the introductions of [258,259] for more pedagogical and explicit discussions, the relevant sections of [128], or [260,261] for more depth on how this procedure is applied to the chiral Lagrangian. Since this discussion can be somewhat abstract, we shall include boxes relating each section to the example of chiral perturbation theory (χpt) from Section 4.3. The ccwz construction is a systematic way to write down the interactions of a theory in which a global symmetry G spontaneously broken to H. The G symmetry is thus nonlinearly realized and one can write a theory of Goldstone bosons, as we reviewed for χpt in Section 4.3. ccwz goes beyond this by also providing the ingredients for how to couple non-Goldstone fields to such a theory. B.1 Preliminaries Suppose a Lagrangian is invariant under a global symmetry G, but that G is spontaneously broken to a subgroup H ⊂ G by some order parameter ψ 0 . We assume that ψ 0 is the vev of some field, ψ(x), that transforms as a linear representation of G, ψ(x) → gψ(x) . (B.1) The statement that ψ 0 spontaneously breaks G → G/H means that for any h ∈ H, hψ 0 = ψ 0 . The spontaneous symmetry breaking pattern G → H implies the existence of dim G − dim H Goldstone bosons that take values on a vacuum manifold. This manifold is the coset space G/H ('G mod H'). In particular, the left coset space G/H is an equivalence class of elements g ∈ G modulo elements h ∈ H, g ∼ gh. In other words, any element in g is equivalent to another element g if there exists an h such that g = gh. Note that in general G/H is not a group. for g L,R are matrices in the defining representation of SU(2) L,R . One could also have written this as a matrix acting on a column vector, ψ i (x) → g(g L , g R ) ij ψ j (x), which more closely matches (B.1). However, the bifundamental representation is convenient since the transformation is more intuitive. B.2 Decomposition of the Algebra The generators of G can be divided between two classes: T i which generate the unbroken group H, and X a which do not. This is called the Cartan decomposition. The generators satisfy the following commutation relations for some structure constants f : T i , T j = if ijk T k (B.3) T i , X α = if iαβ X β (B.4) X α , X β = if αβk T k + if αβγ X γ . (B.5) The non-trivial commutator (B.4) is derived by making use of the Cartan inner product for two generators, A\|B ≡ Tr(AB), and the fact that the H and G/H algebras are orthogonal, T \|X = 0. Using the cyclicity of the trace, one finds T i \|[T j , X α ] = T i \|T j X α − T i \|X α T j = [T i , T j ]\|X α = if ijk T k \|X α = 0 . (B.6) This proves that [T i , X α ] is a series of only the X generators. The commutation relation (B.4) can be interpreted as defining the action of the subgroup H on a set of matrices X; it thus implies that the Xs furnish a linear representation of H. If, additionally, there exists a parity transformation P such that P 2 = 1 and P ([g 1 , g 2 ]) = [P (g 1 ), P (g 2 )] and further such that P (X) = −X and P (T ) = +T , then one can further restrict [X α , X β ] = if αβk T k . (B.7) In this case, the coset G/H is a symmetric space. χpt algebra. The generators of the SU(2) algebra are τ i = 1 2 σ i , for i = 1, 2, 3. These satisfy τ i L , τ j L = iε ijk τ k L τ i R , τ j R = iε ijk τ k R τ i L , τ j R = 0 . (B.8) We may re-organize these into generators of the broken symmetry, X = τ i A = τ i L − τ i R , and the unbroken symmetry, T = τ i V = τ i L + τ i R . For convenience we only use lowercase Roman indices. One may check that τ i V , τ j V = iε ijk τ k V τ i A , τ j A = iε ijk τ k V τ i A , τ j V = iε ijk τ k A . (B.9) We see that a special result of chiral symmetry breaking is that [X, X] ∼ T without any component in the broken algebra. This is because SU(2) L ×SU(2) R is a symmetric space with parity transformation P = τ L ↔ τ R . The observation that the algebra of the broken generators does not close is the reason why SU (2) B.3 Decomposition of the Group The distinct G elements gh 1 , gh 2 , gh 3 , . . . ∈ G are all identified with the same element of G/H. For each element of G/H, it is useful to pick a representative element of G, which we denoteĝ. Then any group element g ∈ G may be written in the form g =ĝh, for some h ∈ H. Further, for a compact and connected group, one may further write each ofĝ and h as an exponentiation of elements of the algebra, so that for any g ∈ G, g = e iξ α X α e iu i T i ≡ĝ(ξ)h(u) . (B.11) B.4 Decomposition of the Linear Representation Suppose we have a field ψ(x) which transforms as a non-trivial linear representation of the group G. This field contains the Goldstone degrees of freedom associated with the broken generator directions in field space; that is, the field directions with a flat potential that transform ψ 0 . Let us define an object γ(x) ∈ G that factorizes ψ(x), ψ(x) ≡ γ(x) ψ(x) , (B. 12) in such a way that ψ(x) contains no Goldstone degrees of freedom. Indeed, if one were to ignore all 'radial' (massive) excitations of ψ(x), one may pick this to be the vev, ψ(x) = ψ 0 , so that γ(x) is simply the transformation from ψ 0 → ψ(x). By the invariance of ψ 0 under H transformations, γ(x) is only defined up to right multiplication by any h ∈ H. In other words, we may identify γ(x) with the representative elementγ(x) which is chosen to be the exponentiation of only broken generators, analogously toĝ above. We may now drop the hat onγ for notational clarity. Let us suggestively call the transformation parameter π(x), γ(x) ≡ γ [π] = e iπ α (x)X α . (B.13) The π a (x) are to be identified with the Goldstone bosons. We leave it dimensionless, remembering that the pion field with canonical mass dimension can be restored by taking π a (x) → π a (x) can /f . Suppose the Lagrangian of the theory with respect to the linearly represented field ψ(x) is written in terms of ψ(x) and ∂ψ(x). G-invariants formed out of only ψ(x) don't contain the Goldstone fields, while those made of ∂ψ(x) do. ∂ψ(x) can be written in terms of ψ 0 and the Goldstone fields using (B.12) and (B.13), ∂ µ ψ(x) = γ ∂ µ + γ −1 (∂ µ γ) ψ, (B.14) where we've suppressed the x dependence of γ. Without loss of generality, we can write γ −1 ∂ µ γ = γ † ∂ µ γ in terms of the broken and unbroken generators, γ −1 ∂ µ γ = iD α µ X α + iE i µ T i (B.15) D α µ = D αβ (π)∂ µ π β (B.16) E i µ = E iβ (π)∂ µ π β . (B.17) In the case of a symmetric space, the parity operator, P , gives a short-cut to express the D α µ (x) and E i µ (x). P takes X → −X and T → T . From this we see that it takes, γ[π] → γ[−π] = γ −1 [π] and thus γ −1 ∂ µ γ → γ∂ µ γ −1 . We may thus take the sum and difference of (B.15) with its parity conjugate to derive iD α µ X α = = i∂ µ π α (x) 1 0 ds X α − is[π(x) · X, X α ] − s 2 2! π(x) · X, [π(x) · X, X α ] + · · · . (B.23) 1 2 γ −1 ∂ µ γ − γ∂ µ γ −1 iE i µ T i = 1 2 γ −1 ∂ µ γ + γ∂ µ γ −1 . From this last line we invoke the algebra with respect to the Cartan decomposition, (B.5), which tells us that γ −1 ∂ µ γ is indeed composed of a series of terms in both the broken and unbroken algebras. See §2.8 of [259] for a compact expression. These identities are derived in textbooks on the representations of Lie groups. χpt linear representation. We use the field ψ(x) transforming in the bifundamental of SU(2) L ×SU(2) R , (B.2). The factorization of the linearly realized field ψ(x) into a Goldstone piece and a 'radial' piece, (B.12), is γ(x) ψ(x) = e iπ(x)·τ r(x) 0 0 r(x) e iπ(x)·τ , (B.24) where we represent ψ(x) as the unit 2 × 2 matrix times a scalar function, r(x) 1 2×2 , which represents the radial (massive) degrees of freedom. We may replace ψ(x) with the vev, ψ 0 = r(x) 1 2×2 = f 1 2×2 . Observe that we have explicitly written the left-and right-acting parts of the transformation in the broken (axial) direction so that this is simply (B.10) with a = π(x). Taking ψ(x) → ψ 0 , and peeling off the ψ 0 and corresponding f , one obtains γ(x) = exp (2iπ(x) · τ ). Since this is a symmetric space, (B.18) straightforwardly gives an expression for the D and E. B.5 Transformation of the Goldstones B.5.1 Transformation under a general group element We would like to see how the π a (x) transform under the global group G. We can derive this implicitly from the transformation of the linear field ψ(x) → gψ(x) using (B.12) and (B.13). The decomposition of a general group element g in (B.11) tells us that there exist π and u such that gψ(x) = ge iπ·X ψ(x) ≡ e iπ ·X e iu ·T ψ(x), (B.25) where the primed fields are, in general, nonlinear functions of g and π(x), that is π = π (g, π) and u = u (g, π). We may write this as and an accompanying implicitly defined transformation of the Goldstones, π(x) → π (x). Both the ψ(x) and π(x) transformations have a messy dependence on π and g. We can now appreciate what the non-linear 34 realization has bought us: we now have a transformation rule for the Goldstoneless field ψ(x) which realizes the full symmetry group G as a non-linear representation of the unbroken symmetry group H. If ψ(x) = ψ 0 is the order parameter for this symmetry breaking, then it is H-invariant. However, the decomposition ψ(x) = γ[π] ψ(x) holds for general fields and the transformation law (B.27) is thus way to write the transformation law of a field ψ without reference to its linearization ψ. In low energy qcd, for example, this can be used to describe the interactions of nucleons with pions. 34 The meaning of 'non-linear' is more clear when contrasted with the case of a linear transformation, for which γ[π] → g lin γ[π]g −1 lin = γ[R ab π a ] , (B.28) for some linear representation R of the algebra of a linearly realized group G lin . B.5.2 Transformation under H The story is different for a transformation under the unbroken group, H. In this case, the Goldstones transform linearly with respect to the algebra of H. This serves as a useful counterpoint to the non-linear representation of general G transformations above. To see this, let us transform the field ψ(x), a linear representation of G, by an element h ∈ H. We again make use of the decomposition (B. where each bracketed term can be written as [· · · ] = e iπ ·τ , and the matrix in the center is unchanged. By comparison, for a general transformation g v,a the best we can do is to insert 1 = h −1 v h v for some v(π, g) such that it implicitly defines π by Here v(π, g) is defined such that this relation is true. Thus our general transformation is realized non-linearly, where π → π is now complicated and h v is the H transformation that non-linearly realizes G. 35 To make this more clear, one may expand γ[π] as a power series and insert 1 = h −1 h so that X n = Xh −1 hX · · · h −1 hX, from which point one may use (B.31) to prove the linearity of (B.30). B.6 From Linear to Non-Linear From a linear uv theory, we have identified the Goldstone fields and can integrate out the massive 'radial' modes to obtain a low-energy Lagrangian by taking ψ(x) → ψ 0 . Effective field theory tells us that this uv theory wasn't necessary to construct the low energy theory of Goldstone bosons. So once we have a theory of Goldstone interactions, we may remain agnostic about the specific uv completion of the theory 36 . Prior to the discovery of the Higgs boson-a linear uv completion of the theory of the Goldstone bosons eaten by W ± and Z-the reason why experiments like lep could make precision measurements of the sm without knowing the details of the Higgs is simply that the precision measurements asked precise questions about the non-linear sigma model (NLΣM) of Goldstones that were insensitive to the particular uv completion, linear or otherwise. There are also reasons why one might be interested in keeping around a field like ψ(x) in the effective theory. These radial fields can be used to introduce, say, nucleon excitations in the chiral Lagrangian. The radial modes are identified with excitations along the vev direction ψ 0 . From (B.25), we see that the radial field transforms as ψ → h [u (g, π)] ψ . (B.36) Thus in order to build G-invariants out of the radial fields ψ, it is sufficient to construct H invariants. Said differently, the decomposition ψ(x) = γ[π] ψ(x) converts G-linear representations, ψ, into non-linear realizations of G, π and ψ. B.7 A Low-Energy Lagrangian without the UV Let us now return to (B.14) since we know from the Goldstone shift symmetry that the Goldstones only appear in derivative interactions. The object g −1 ∂g, where g = γ in (B.14), is called the Maurer-Cartan form, it takes an element of the group g ∈ G, differentiates it-pulling out the Lie algebra element based at g-and pulls that generator back to the group identity so that one can compare elements of the algebra on the same tangent space. What about the curious object E µ ? This appears to transform as the gauge field of a local symmetry. The locality of this symmetry is inherited from the x-dependence of the Goldstone fields π(x) and is unsurprising since the coset identification g ∼ gh is local. E µ is thus a 'gauge potential' with respect to the unbroken symmetry H. It encodes the fact that while all symmetries are global, the H symmetry secretly is a subgroup of the larger, spontaneously broken G symmetry. Indeed, differentiating (B.36)-recalling that h(u ) depends on x through its implicit dependence on π(x)-shows that derivatives of the non-Goldstone fields transform inhomogeneously under G. Promoting the partial derivative to a covariant derivative, D µ = ∂ µ → ∂ µ + iE µ , ensures that D µ ψ(x) transforms homogeneously under H. When did H become gauged? The appearance of a covariant derivative and a gauge symmetry may seem surprising in a system where global symmetry G is spontaneously broken to a subgroup H. The appearance of a local symmetry, however, is not surprising since the resulting coset space G/H precisely describes a gauge redundancy. Mathematically, the description of a 'gauged' symmetry is identical to that of a spontaneously broken global symmetry. See, for example, [261] or chapter 7 of [262]. For the mathematically inclined, details of the geometric structure of these theories are presented in [263] and [264]. The punchline is that one can construct a Goldstone boson Lagrangian which is invariant under the full, nonlinearly realized group G, by constructing an H-invariant Lagrangian out of D µ . One can further introduce non-Goldstone fields ψ (not necessarily related to the linear field that gets a vev) so long as one uses the appropriate H covariant derivative, D µ , with the corresponding 'gauge field' E µ . In this way one may include, for example, 'nucleon' excitations to the effective theory. The description above is based on a 'standard realization' of the nonlinearly realized symmetry, (B.25). One of the main results of the ccwz papers was the observation that every non-linear realization can be brought to this standard realization [122,123]. Physically, this means that no matter how one imposes the G/H restriction, the S-matrix elements for the low-energy dynamics will be identical. Explicit examples of this are presented in chapter iv of [260] χpt including nucleons. Explicit examples of how one may invoke the ccwz formalism to extend the effective theory to include heavy particles can be found in chapter iv-7 of [260], §2.3 of [258], or §2.3 of [128]. Figure 1 : 1Heuristic two-loop contributions to the Higgs mass from heavy fermions, Ψ. Even though the Ψ do not directly couple to the Higgs, they reintroduce a quadratic sensitivity to the new scale. Each vertex picks up a factor of the ckm matrix. The gim observation is the fact that the unitarity of the ckm matrix imposes an additional suppression. In the mssm, on the other hand, the squark soft masses introduce an additional source of flavor violation so that the quark and squark mass matrices are misaligned. This manifests itself as flavor-changing mass insertions, ∆m 2 ds , on squark propagators when written in terms of the Standard Model mass eigenstate combinations: rather than W bosons, this diagram is mediated by gluinos which carry much stronger coupling constants α 3 α 2 . Further, Since there are no factors of V CKM , there is no gim suppression. The loop integral goes like d 4 k/k 10 ∼ 1/m 6 susy . Thus we can estimate this contribution to kaon mixing to be Figure 5 : 5Cartoon of the rs scenario with a brane-localized sm. The warp factor, (R/z) 2 , causes energy scales to be scaled down towards the ir brane. Figure 6 : 6Cartoon of the ads/cft correspondence. The isometries of the extra dimensional space enforce the conformal symmetry of the 4D theory. Moving in the z direction corresponds to a renormalization group transformation (rescaling) of the 4D theory.3.9.1 Plausibility check from an experimentalist's perspectiveAs a very rough check of why this would be plausible, consider the types of spectra one expects from an extra dimensional theory versus a strongly coupled 4D theory. In other words, consider the first thing that an experimentalists might want to check about either theory. The theory with an extra dimension predicts a tower of Kaluza-Klein excitations for each particle. The strongly coupled gauge theory predicts a similar tower of bound states such as the various meson resonances in qcd. From the experimentalist's point of view, these two theories are qualitatively very similar. leads to an interesting possibility: could one construct a complete model with no elementary scalar Higgs, but where a condensate breaks electroweak symmetry and scale invariance? Then the dilaton of this theory may have the properties of the sm Higgs. If one can reproduce the observed Higgs mass then it could be very difficult to tell the scenario apart from the sm[103]. Figure 7 : 7A cartoon of the zero mode profiles of various sm particles in the 'realistic' rs scenario. see this since an SU(3) L fundamental contains (u L , d L , s L ), where the first two components form the usual SU(2) L first generation quark doublet. In this way, SU(2) L is simply the upper left 2 × 2 component of the SU(3) L generators. Similarly, hypercharge is a combination of the diagonal generators, Figure 8 : 8Cartoon of the Goldstone excitation for a 'Mexican hat' potential. Image from[148].space associated with the chiral condensate, U 0 ∼ qq . This transforms as a bifundamental with respect to SU(3) L ×SU(3) R , 9) where U L and U R are the transformation matrices under the SU(3) L and SU(3) R respectively. The observation that SU(3) A is broken corresponds to U 0 = 1. Note that this indeed preserves theSU(3) V transformations U L = U R .We now consider the fluctuations U (x) about U 0 -these are what we identify with the Goldstone bosons. Recall the picture of spontaneous symmetry breaking through the 'Mexican hat' potential inFig. 8. The action of an unbroken symmetry does not affect the vev (represented by the ball), while broken symmetries shift the vev along the vacuum manifold. This gives an intuitive picture of how to identify the Goldstone modes:1. Identify a convenient vev, U 0 2. Act on that vev with the broken group elements 3. Promote the transformation parameter to a field, identify these with the Goldstones.For the chiral Lagrangian, our broken symmetries are those for which U L = U † R . Writing U L = exp(i a T a ), we act on U 0 = 1, it is clear that SU(3) V is realized linearly while SU(3) A is realized non-linearly. SU(3) A is not a subgroup of SU(3) L × SU(3) R . While one can divide the algebra of SU(3) L × SU(3) R into axial and vector generators, one should note that there is no such thing as an 'axial subgroup' of SU(3) L × SU(3) R . One can check that the commutation relations of axial generators include vector generators so that the SU(3) A algebra doesn't close by itself. L ×SU(3) R ×U(1) B . The left-and right-chiral quarks are fundamentals under SU(3) L and SU(3) R respectively and have baryon number 1/3. This information, combined with knowing how SU(2) L sits in SU(3) L and (4.8), determines the quantum numbers of the linear field U (x), which transforms as a3 × 3 × 0 under SU(3) L ×SU(3) R ×U(1) B . To 'turn on' the electroweak gauge interactions, we simply promote derivatives to covariant derivatives Figure 10 : 103) L × SU(3) R → SU(3) V . Pattern of symmetry breaking. (left, tree level) Strong dynamics breaks G → H global spontaneously, while H gauge ⊂ G is explicitly broken through gauging. The unbroken group H = H gauge ∩ H global contains the sm electroweak group, SU(2) L × U(1) Y . (right, loop level) Vacuum misalignment from sm interactions shifts the unbroken group H → H and breaks the electroweak group to U(1) EM . The degree of misalignment is parametrized by ξ, the squared ratio of the ewsb vev to the G → H vev. Adapted from[157]. Figure 11 : 11Fermion couplings to the composite sector, represented by shaded blobs. (a): Bilinear coupling of fermions to the composite sector (4.54) lead to fermion masses from the condensate of techniquarks. (b): Partial compositeness scenario. In addition to the Higgs being part of the strong sector, the elementary sm fermions mix linearly with strong sector operators with the same quantum numbers. (c): Yukawa interactions are generated through the strong sector dynamics.Adapted from[160]. dashed line separates the SU(2) L × SU(2) R parts of SO(4) ⊂ SO(5). The ψ and χ fields are spurions. Recall from Sec. A.3.1 that the fundamental spinor representation for SO(5) is a Dirac spinor which decomposes into two Weyl spinors. Do not confuse these Weyl spinors(4.63) with Poincaré representations-these are representations of the global SO(5) internal group. In other words, the entire Ψ multiplet are Weyl spinors with respect to Poincaré symmetry but are Dirac spinors with respct to the internal SO(5) symmetry. The upper half of the Dirac Ψ spinors are charged under SU(2) L while the lower half is charged under SU(2) R . This imposes a U (1) X charge of 1/6 on the Ψ fields to give the correct hypercharge assignments on the sm fields. Now let us parameterize the strong sector dynamics in the couplings of the SO(5) fermions Ψ and the linear field Σ in (4.40) that encodes the composite Higgs. Since the Σ is an SO(5) vector, it can appear in a fermion bilinear as Σ i Γ i , where the Γ are the 5D Euclidean space representation of the Clifford algebra. The effective sm fermion bilinear terms are L = r=Q,u,dΨ 2 . 2If, on the other hand 23 , G ⊂ H gauge , then the would-be Goldstone bosons from G → H global are eaten by the (G/H global ) ∩ H gauge gauge bosons. There is no quadratic sensitivity to the cutoff. • If only the SU(2)= H global = H global parts of G and G were gauged, then there would be two separate sets of pseudo-Goldstone bosons H and H . We plug in the expansion of Σ (4.73) into (4.75) and note that in this case, TheŜ parameter bound comes from the exchange of new spin-1 states of characteristic mass M (analogous to the ρ meson) and goes parametrically likeŜ ∼ M W /M 2 . This pushes M 2.5 tev. The observation of the 125 gev Higgs and the opportunity to measure its couplings offers additional data to fit the phenomenological Lagrangian. For example, thec H and c f (f running over the sm fermions) are related to each other via the couplings of the Higgs to W bosons [216]. The Higgs mass sets (at 3σ)c H ≤ 0.16. (4.109) Figure 15 : 15Schematic moose diagram for natural susy. Figure 16 : 16Moose diagrams for a pair of Seiberg duals. Green nodes are gauged symmetries while white notes are global symmetries. Note that the lines now represent superfields. are the Goldstone bosons. In unitary gauge, ξ → ∞, the A (n) 5 fields decouple, but in general one must include them as internal states in calculations.For the Standard Model gauge fields one selects boundary conditions where the scalar zero mode A χpt coset space. For case of two flavors, G = SU(2) L × SU(2) R and H = SU(2) V , the diagonal subgroup. Let us choose ψ(x) to be a field transforming in the bifundamental of SU(2) L ×SU(2) R . This means that it can be represented as a 2 × 2 matrix where the linear transformation (B.1) is represented by ψ(x) → g L ψ(x)g −1 R , (B.2) A is not a properly subgroup of of SU(2) L ×SU(2) R . A general transformation is parameterized by pairs of 3-vectors, (v, a). The action on the bifundamental field ψ(x) is ψ(x) → e i(v+a)·τ ψ(x)e −i(v−a)·τ . (B.10) General decomposition of γ −1 ∂γ into D and E. The general decomposition (B.15) uses the identity∂ µ e iπ·X = i∂ µ π α 1 0 ds e i(1−s)π·X X α e isπ·X (B.19)and the Baker-Campbell-Hausdorff (bch) relation,e A Be −A = B + [A, B] + 1 2! [A, [A, B]] + · · · , (B.20)to show that you end up with an expansion in the Xs and T s:γ −1 ∂ µ γ = i∂ µ π α (x) e −iπ(−isπ(x)·X X α e isπ(x)·X (B.22) gγ[π] ψ(x) = γ[π ]h[u ] ψ(x) , (B.26)from which we may interpret an induced H-transformation on the Goldstone-less field ψ(x)ψ(x) → ψ (x) = h[u ] ψ(x) , (B.27) the last step we have inserted 1 = h −1 h. Observe that this is of the form (B.26) withψ(x) → ψ (x) = h ψ(x) γ[π] → γ[π ] = hγ[π]h −1 . (B.30)The π transformation, in particular, is a linear representation of H. This can be seen by invoking the algebra of the Cartan decomposition, (B.5), which states that the commutator of a broken and unbroken generator is proportional to the broken generator, [T, X] ∼ T . Using this relation in the bch formula (B.20) shows thathX a h −1 = R ab X b ∈ G/H .(B.31) From this it is clear that (B.30) is a linear representation of the unbroken group H acting on the Goldstone fields that live on the space of broken generators 35 . χpt pion transformations. Let us demonstrate this for SU(2) L ×SU(2) R →SU(2) V . We showed above that the (un-)broken generators are parameterized by 3-vectors a (v) such that the transformation of a bifundamental field ψ(x) by (v, a) is (B.10). Further, the decomposition ψ(x) = γ[π] ψ(x) corresponds to (B.24). The transformation ψ(x) → gψ(x) is thus identified with g v,a γ[π] ψ(x) = e i(v+a)·τ e iπ·τ r(x) 0 0 r(x) e iπ·τ e −i(v−a)·τ . (B.32) When we specialize to the case of a purely vectorlike (unbroken) transformation, a = 0, we may insert factors of 1 = e −iv·τ e iv·τ to see that the π transforms linearly, h v γ[π] ψ(x) = e iv·τ e iπ·τ e −iv·τ e iv·τ r(x) 0 0 r(x) e −iv·τ e iv·τ e iπ·τ e −iv·τ , (B.33) g v,a γ[π] ψ(x) = e i(v+a)·τ e iπ·τ e −i v·τ e i v·τ r(x) 0 0 r(x) e −i v·τ e i v·τ e iπ·τ e −i(v−a)·τ (B.34) ≡ e iπ ·τ r(x) 0 0 r(x) e iπ ·τ . (B.35) Table 1 : 1Matter content of the mssm. Note that we have used 2 = 2 for SU(2) L . the sum of all U(1) charges. We have written a to index only the U(1) factors of the gauge group. Note, however, that usually due to anomaly cancellation. This leads to the very stringent constraint thati q ( a) i = 0 (2.73) STr m 2 = 0. (2.74) These are called A-terms and are the same order as the scalar mass.3 + (susy preserving terms). (2.82) 4. Gaugino masses are generated from corrections to the gauge kinetic term, d 2 θ X M W α W α + h.c. = F M λλ + h.c. + (susy preserving terms). (2.83) Table 2 : 2Taxonomy of composite Higgs models according to the couplings in (4.91) and (4.92); based on Table 3 : 3Holographic picture of natural susy spectra. Superfields localized near the ir brane have a large overlap with the Higgs so that the sm component of the superfield picks up a large mass. Superfields localized near the uv brane have a large overlap with susy breaking so that the 'superpartner' component of the superfield picks up a large mass. Thus light sm fermions have heavy superpartners and vice versa. The expansion of the Maurer-Cartan form into broken and unbroken generators is given in(B.15). Differentiating the transformation rule (B.26),g∂ µ γ[π] ψ + $ $ $ $ $ gγ[π]∂ µ ψ = (∂ µ γ[π ]) h ψ + γ[π ] (∂ µ h) ψ + $ $ $ $ $ $ γ[π ]h∂ µ ψ . (B.37)We have crossed out terms on each side which are identical by (B.26). Now peel off the common factor of γ on each term and multiply each side of this equation from the left byγ −1 [π]g −1 = h −1 [u ]γ −1 [π ] to find, γ −1 [π]∂ µ γ[π] = h −1 γ −1 [π ] ∂ µ γ[π ] h + γ[π ]∂ µ h . (B.38)Comparing this to (B.15), we findiD α µ [π]X α + iE i µ [π]T i = ih −1 D α µ [π ]X α h −1 + ih E i µ [π ]T i + i∂ µ h −1 .(B.39) In other words, the objects D and E defined in (B.15) transform under g ∈ G as where h = h [u (π, g)] as in (B.26). This should look very familiar: D transforms linearly and E transforms like a gauge field. Both transform under G through representations of H rather than the whole group G. This realizes the observation in Section B.6: to write Lagrangians for nonlinear realizations of G/H, we need to construct invariants with respect to only H. The linear object D can indeed be used to construct a simple lowest-order Lagrangian, where we've introduced the symmetry breaking scale f to preserve dimensionality. This should be compared to (4.18), our derivative expansion for in chiral perturbation theory. vs. ccwz. See §2.8 of [259] for an explicit calculation showing that the ccwz and chiral perturbation theory leading-order effective Lagrangians are indeed the same.D α µ → hD α µ h −1 (B.40) E i µ → hE i µ h −1 − ih∂ µ h −1 , (B.41) L = f 2 4 Tr(D µ D µ ), (B.42) χpt See[1] for a recent discussion of naturalness and fine-tuning in the post-Higgs era. One may be used to thinking of covariant derivatives as coming from local symmetries with some gauge field. Here, however, we consider only global susy. Geometrically, the covariant derivative comes from the fact that even rigid superspace carries torsion[15]. This is somewhat unfortunate nomenclature. One would expect the massless mode coming from spontaneously broken susy to be called a Goldstone fermion whereas the 'Goldstino' should refer to the supersymmetric partner of a Goldstone boson coming from the spontaneous breaking of an ordinary symmetry. Non-renormalizable terms in the Kähler potential, for example, modify how the superpotential terms contribute to the scalar potential since one has to rescale fields for them to be canonically normalized. This should have been no surprise given the appearance of Bessel functions. While the Goldberger-Wise mechanism is just one simple option to stabilize the size of the extra dimension, it is close to what actually happens in string compactifications that tacitly uv complete the rs scenario[81]. The trick was inspired by similar calculations in supergravity but otherwise is only related to susy in the sense that the 'superpotential' here also allows one to write first order equations of motion[84]. This itself causes some conceptual issues since the interactions of a purely brane Higgs is incompatible with the boundary conditions required to make the fermion zero modes chiral[121]. A better comparison is Λ = 4πf π ∼ O(gev), where f π is the pion decay constant. 'Typical' qcd states such as the ρ meson have masses of at least this value, m ρ ∼ Λ. We explain the distinction in Section 4.3.7. The name comes from identifying the appearance of this factor in the matrix element for pion decays, e.g. 0\|ūγ µ γ 5 d\|π − ≡ if p µ . This may seem confusing since U (x) transforms as a bifundamental under SU(3) L × SU(3) R . However, components of U (x) are not independent due to the nonlinear constraints of being unitary and having unit determinant. (4.36) parameterizes this separation of scales and quantifies the degree of vacuum misalignment. Note that this is a separation of scales which does not exist in technicolor and is the key to parameterizing how the Higgs remains light relative to the heavier resonances despite not being a 'true' Goldstone boson. The limits ξ → 0 and ξ → 1 correspond to the sm (heavy states completely This is a trivial use of the Higgs low-energy theorem: the low-momentum Higgs couplings are equivalent to promoting the vev to h(x) like[161,162] This theorem can be used, for example, to calculate the Higgs coupling to photons by evaluating the mass dependence of the running of the qed gauge coupling. The application of the theorem to composite Higgs models is explored in[163]. Here we have used the SO(5) basis in[129]. In Section 4.7.4 we present an alternate protection mechanism based on a Z 2 symmetry. This is a manifestation of general outdoors advice: if you (a Goldstone boson) are being chased by a hungry bear (a gauge boson), it is not necessary for you survival that you can outrun it (have zero coupling). It is sufficient that you are with friends whom you can outrun. Collective breaking is, in part, the requirement that you have more slow friends than hungry bears. C 2 (fundamental) = (N 2 − 1)/2N for SU(N ). These diagrams are also called quiver diagrams by string theorists[188]. See[208,209] for pedagogical reviews and[210][211][212] for details. One should note that because 5D spinors are Dirac, N = 1 susy in 5D corresponds to N = 2 susy in 4D. N = 2 was used in[223] to generate Dirac gaugino masses, which can help soften the two-loop quadratic corrections to the Higgs mass. See[224] for a recent analysis of prospects.30 This is one of the ways to interpret anomaly mediation of susy breaking[225]. For a beginner-friendly introduction, see[244] or your favorite general relativity textbook. We write \|g\| to allow for a general sign of g = det g µν . We have implicitly applied to the entire kk tower in (A.41) and then used the orthogonality of solutions. You can also use this in the opposite direction: if you need an aide to write down non-linear interactions of Goldstone bosons, you can always construct a linear uv theory and decouple the radial excitations. Acknowledgements c.c. thanks the organizers of the 2013 European School of High-Energy Physics for the invitation to give these lectures. p.t. is grateful to Brando Bellazzini, Nathaniel Craig, Tony Gherghetta, Roni Harnik, Javi Serra, Yuri Shirman, and and Tim M.P. Tait for enlightening discussions about topics in these lectures. We appreciated feedback from Mohammad Abdullah, Jack Collins, Anthony DiFranzo, Benjamin Lillard, Mario Martone, and Javi Serra who read parts of this manuscript. We thank Gilad Perez and Martijn Mulders for their patience with this manuscript. We thank the Aspen Center for Physics (nsf grant #1066293) for its hospitality during a period where part of this work was completed. c.c. is supported in part by the nsf grant phy-1316222. p.t. is supported in part by the nsf grant phy-1316792 and a uci Chancellor's advance fellowship.A Details of the Randall-Sundrum ScenarioA.1 The RS gravitational backgroundWe have assumed the metric(3.28). In this appendix we derive it from the assumption of a non-factorizable metric of the form ds 2 = e −A(z) η µν dx µ dx ν − dz2(A.1)and check the conditions for which a flat 4D background exists. This generic form of the metric is useful since it is an overall rescaling of the flat metric, that is, it is conformally flat. We can thus use a convenient relation between the Einstein tensors G M N = R M N − 1 2 g M N R of two conformally equivalent metrics g M N = e −A(x) g M N in d dimensions[238],When A = A(z) this is straightforward to calculate by hand for g M N = η M N . Alternately, one may use a computer algebra system to geometric quantities for general metrics, e.g.[239]. We assume a bulk cosmological constant Λ so that the 5D bulk Einstein action isIn the last step we've integrated by parts and dropped the boundary term. We see that the Hermitian conjugate of the partial derivative is negative itself. Thus the partial derivative is not a Hermitian operator. This is why the momentum operator is given byP µ = i∂ µ , since the above analysis then yieldsP † µ =P µ , where we again drop the boundary term and recall that the i flips sign under the bar.where we've used the fact that e M a is a real function with no spinor indices. The second term on the right-hand side can be massaged further,Note that we have used that γ M † = γ 0 γ M γ 0 and, in the last line, that [σ bc , γ a ] = 0. Putting this all together, we can write down our manifestly real fermion action as in (A.19),A.3.5 Application to the RS backgroundWe now apply this machinery to the rs background. The vielbein and inverse vielbein areWe may write out the spin connection term of the covariant derivative as.This can be simplified using the fact that the vielbein only depends on z. The first part is These vanish identically for M = 5. We can now write out the spin-connection part of the covariant derivative,where we've inserted a factor of δ 5 M to cancel the (γ 5 ) 2 when M = 5. Finally, the spin connection part of the covariant derivative isso that the spin covariant derivative isFor all of the geometric heavy lifting we've done, we are led to an anticlimactic result: the spin connection drops out of the action,A.3.7 KK Decomposition of RS FermionsSome care is necessary to dimensionally reduce the fermion equation of motion coming from (A.39): this is a matrix equation in Weyl spinor space with off-diagonal elements. Analogously to what we are used to in flat 4D space, one may 'square' the warped 5D Dirac equation to obtain a scalar equation of motion that is diagonal. For convenience, define the conjugate Dirac operatorsOne recognizes at S = d 4 x dz(R/z) 4Ψ DΨ. Decompose the 5D Dirac fermion Ψ(x, z) in the usual way,Our trick is to applyD to the equation of motion DΨ(x, z) = 0. The combined 'squared' operator DD isDApplying this to a kk mode 33 f (n) (z) gives an expression that is equivalent to the equation of motion. 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Golowich, and B. Holstein, Dynamics of the Standard Model. Cambridge University Press, 1994. Goldstone and Pseudogoldstone Bosons in Nuclear, Particle and Condensed Matter Physics. C Burgess, 10.1016/S0370-1573(99)00111-8arXiv:hep-th/9808176Phys.Rept. 330hep-thC. Burgess, "Goldstone and Pseudogoldstone Bosons in Nuclear, Particle and Condensed Matter Physics," Phys.Rept. 330 (2000) 193-261, arXiv:hep-th/9808176 [hep-th]. T Banks, Modern Quantum Field Theory: A Concise Introduction. Cambridge University PressT. Banks, Modern Quantum Field Theory: A Concise Introduction. Cambridge University Press, 2008. Comments About the Geometry of Nonlinear Sigma Models. R Coquereaux, 10.1142/S0217751X87000910Int.J.Mod.Phys. 21763R. Coquereaux, "Comments About the Geometry of Nonlinear Sigma Models," Int.J.Mod.Phys. A2 (1987) 1763. Nonlinear Realizations and Extended Objects. C Yastremiz, GTCRG-91-7DAMTP-R-91-10C. Yastremiz, "Nonlinear Realizations and Extended Objects,". DAMTP-R-91-10, GTCRG-91-7.	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[ "Radiative Corrections in GSDKP", "Radiative Corrections in GSDKP" ]	[ "R Bufalo \nDepartamento de Física\nUniversidade Federal de Lavras\nCaixa Postal 303737200-000LavrasMGBrazil\n", "T R Cardoso \nDepartamento de Física\nUniversidade Federal de Lavras\nCaixa Postal 303737200-000LavrasMGBrazil\n", "† ", "A A Nogueira \nCentro de Ciências Naturais e Humanas (CCNH) Av. dos Estados 5001\nUniversidade Federal do ABC (UFABC)\nBairro Santa Terezinha CEP\nSanto André09210-580SPBrazil\n", "‡ ", "B M Pimentel \nInstituto de Física Teórica (IFT)\nUniversidade Estadual Paulista (UNESP)\nRua Dr. Bento Teobaldo Ferraz 271\n\nBloco II Barra Funda\n01140-070São PauloCEP, SPBrazil\n" ]	[ "Departamento de Física\nUniversidade Federal de Lavras\nCaixa Postal 303737200-000LavrasMGBrazil", "Departamento de Física\nUniversidade Federal de Lavras\nCaixa Postal 303737200-000LavrasMGBrazil", "Centro de Ciências Naturais e Humanas (CCNH) Av. dos Estados 5001\nUniversidade Federal do ABC (UFABC)\nBairro Santa Terezinha CEP\nSanto André09210-580SPBrazil", "Instituto de Física Teórica (IFT)\nUniversidade Estadual Paulista (UNESP)\nRua Dr. Bento Teobaldo Ferraz 271", "Bloco II Barra Funda\n01140-070São PauloCEP, SPBrazil" ]	[]	We show explicit the first radiative correction for the vertex and photon-photon 4-point function in Generalized Scalar Duffin-Kemmer-Petiau Quantum Electrodynamcis (GSDKP), utilizing the dimensional regularization method, where the gauge symmetry is manifest. As we shall see one of the consequences of the study is that the DKP algebra ensures the functioning of the Ward-Takahashi-Fradkin (WTF) identities in the first radiative corrections prohibiting certain ultraviolet (UV) divergences. This result leads us to ask whether this connection between DKP algebra, UV divergences, and quantum gauge symmetry (WTF) is a general statment.	null	[ "https://arxiv.org/pdf/1803.05728v2.pdf" ]	4,897,982	1803.05728	f47284892678e60e417bcbb7be41cdd23eb0326e	Radiative Corrections in GSDKP September 10, 2018 R Bufalo Departamento de Física Universidade Federal de Lavras Caixa Postal 303737200-000LavrasMGBrazil T R Cardoso Departamento de Física Universidade Federal de Lavras Caixa Postal 303737200-000LavrasMGBrazil † A A Nogueira Centro de Ciências Naturais e Humanas (CCNH) Av. dos Estados 5001 Universidade Federal do ABC (UFABC) Bairro Santa Terezinha CEP Santo André09210-580SPBrazil ‡ B M Pimentel Instituto de Física Teórica (IFT) Universidade Estadual Paulista (UNESP) Rua Dr. Bento Teobaldo Ferraz 271 Bloco II Barra Funda 01140-070São PauloCEP, SPBrazil Radiative Corrections in GSDKP September 10, 2018 We show explicit the first radiative correction for the vertex and photon-photon 4-point function in Generalized Scalar Duffin-Kemmer-Petiau Quantum Electrodynamcis (GSDKP), utilizing the dimensional regularization method, where the gauge symmetry is manifest. As we shall see one of the consequences of the study is that the DKP algebra ensures the functioning of the Ward-Takahashi-Fradkin (WTF) identities in the first radiative corrections prohibiting certain ultraviolet (UV) divergences. This result leads us to ask whether this connection between DKP algebra, UV divergences, and quantum gauge symmetry (WTF) is a general statment. Transition amplitude As we know to study quantum processes we need to write the transition amplitude or, reciprocally, the generating functional (with sources). To construct the transition amplitude we will use the Faddeev-Senjanovic proceeding where we must first do a short study of constraint analysis. The Lagrangian density describing the GSDKP is de-fined by [1] L = i 2ψ β µ ∂ µ ψ − i 2 ∂ µψ β µ ψ − mψψ+ +eA µψ β µ ψ − 1 4 F µν F µν + a 2 2 ∂ µ F µβ ∂ α F αβ ,(1.1) where F µν = ∂ µ A ν − ∂ ν A µ is the usual electromagnetic field-strength tensor and β µ are the DKP matrices that obey the trilinear algebra β µ β ν β θ + β θ β ν β µ = β µ η νθ + β θ η ν µ . The transition amplitude can be written in a covariant form in the non-mixing gauge condition [2] Z = N DA µ DψDψ× exp{i d 4 x[ψ iβ µ ∇ µ − m ψ+ − 1 4 F µν F µν + a 2 2 ∂ µ F µβ ∂ α F αβ + − 1 2ξ ∂ µ A µ 1 + a 2 ∂ µ A µ ]},(1.2) where ∇ µ = ∂ µ − ieA µ . It follows from the above result the infinite chain of equations (Schwinger-Dyson-Fradkin equations) that describes completely our theory (without approximations, due to perturbation theory) and the quantum gauge symmetry of this complet equations manifested in terms of Ward-Takahashi-Fradkin identities [3]. Quantum gauge symmetry Now let us find some identities, arising from gauge symmetry, to analyse before some quantum process of interest . The derivation of (WTF) identities is formally given in terms of the following identity upon the functional generator δ Z η,η, J µ δ α(x) α=0 = 0 (2.1) wherein the infinitesimal gauge transformations are given by Ψ → Ψ + iα(x)Ψ,Ψ →Ψ − iα(x)Ψ and A µ → A µ + 1 e ∂ µ α(x) . Then the expression for the 1PI-generating functional Γ ψ,ψ, A µ is −i eξ 1 + a 2 ∂ µ A µ −ψ δ Γ δψ + δ Γ δ ψ ψ+ + 1 e ∂ µ δ Γ δ A µ = 0. (2.2) This is the equation that will supply all the WFT identities. The first identity we want to point out (in momentum representation) is Λ µ (p, p, k = 0) = − ∂ ∂ p µ Σ −1 (p) . (2.3) where we find out that the vertex is related to the DKP self-energy. The second identity we want to point out (in momentum representation) is p µ Γ µναβ (p, p , k, k ) = 0. (2.4) This identity applies to the study of photonphoton scattering. As we will see, (WTF) identities will play an important role in building and studying the amplitudes that describe the physical processes. Radiative corrections Before starting the study on the quantum radiative corrections [4] it is interesting to gain intuition by doing a quantitative analysis of the types of ultraviolet divergences that could appear in (GSDKP). We will conclude, after a brief power count, that the degree of a divergence in a diagram would be D=4−n− 1 2 n m , which n=(number of vertex) and n m =(number of external mesons lines). With the previous equation, we can classify the types of divergences that have appeared in theory. When D<0, we say that the diagram is superficially convergent. For the first radiative correction to the vertex we have D = 0 (logarithmic divergence) and for the photon-photon process we have D = 0 too. Vertex Now we are going to calculate the first radiative correction associated with the vertex function. The vertex part at the lowest order correction is Λ µ p , p = e 2 µ 4−d d d k (2π) d × β σ S p − k β µ S (p − k) β ν D σ ν (k) ,(3.1) substituting the expressions for their respective propagators Λ µ p , p = −ie 2 µ 4−d m 2 p m 2 d d k (2π) d × β σ [(p −k)(p −k + m) − (p − k) 2 − m 2 ] [(p − k) 2 − m 2 ][(p − k) 2 − m 2 ]k 2 k 2 − m 2 p × β µ [(p −k)(p −k + m) − (p − k) 2 − m 2 ]β σ . (3.2) This expression may be simplified by making use of the Feynman parametrization, and then be cast into a suitable form Λ µ (p , p) = e 2 m 2 p (4π) 2 m 2 1 0 dx 1−x 0 dy× 1−x−y 0 dz[ A µ b 4 − η αν B µ αν 2b 2 − Γ( ε 2 ) 4 4π µ 2 b 2 ε 2 × η αν η λ θ + η νθ η αλ + η θ α η λ ν C µ ανλ θ ]. The term C µ ανλ θ presents a logarithmic divergence. However, by means of using the DKP algebra, one can show that this term actually vanishing because of the identity (η αν η λ θ + η νθ η αλ + η θ α η λ ν )C µ ανλ θ = 0. Showing therefore that there are no divergences on the vertex part. This result confirms the information contained in one of the (WTF) identity eq. (2.3) assuring that the divergence of Σ in the term proportional to the mass m p does not spoil the (WTF) identity. Photon-photon Finally, let us study the light-light scattering. In view of Feynman rules we write the amplitude associated with this scattering as follows (3.5) In this case the structure representing the light-light scattering must not have (UV) divergences because otherwise it would break the gauge symmetry, seen in eq. (2.4). Consequently, as in the first radiative correction of vertex, it is possible to show that DKP algebra prohibits (UV) divergence tr{D µνλ θ ασ γδ πϖρτ }(η ασ η γδ η πϖ η ρτ + perm) = 0. Γ µνλ θ (p , p; p , p) = e 4 µ 4−d d d k (2π) d × tr{β µ S(p + k)β ν S(k)β λ S(p − k)β θ S(k)}.(1 − x − y) bΓ(−1 + ε 2 ) 8[ b 2 4π µ 2 ] ε 2 , Γ(−1 + ε 2 ) [ b 2 4π µ 2 ] ε 2 = [ 2 ε − γ][1 − ε 2 ln( b 2 4π µ 2 )]. Conclusions In this work we show the connection between the DKP algebra, (WTF) identities and the UV divergences for the first radiative correction to the vertex and the photon-photon amplitude. Sometimes, the symmetries of the theory can decrease the (UV) divergence degree of an amplitude. We conclude that gauge symmetry prohibits ultraviolet divergence for the Feynman amplitudes studied but who assures the previous statement is the DKP algebra. This raises the question of what is role of this algebra in the study of the quantum interaction between the DKP fields (mesons) and Podolsky fields (generalized photons). The next step would be formulate the renormalization program and it is possible to study GSDKP at thermodynamical equilibrium with the Matsubara-Fradkin formalism [5,6]. the help of Feynman parameters we are led to write explicitly only the term with a possible (UV) divergence Γ µνλ θ UV (p , p; p , p) = ie 4 (4π) 2 m 4 tr{D µνλ θ ασ γδ πϖρτ }× (η ασ η γδ η πϖ η ρτ + perm) Acknowledgement The generalized scalar Duffin-Kemmer-Petiau electrodynamics, its functional analysis in a covariant quantum dynamics and the thermodynamic equilibrium, P.h.D These, Institute for Theoretical Physics. A A Nogueira, IFT-UnespA. A. Nogueira, The generalized scalar Duffin-Kemmer-Petiau elec- trodynamics, its functional analysis in a covariant quantum dynamics and the thermodynamic equilibrium, P.h.D These, Institute for Theoretical Physics, IFT-Unesp (2016); BRST symmetry and fictitious parameters. A A Nogueira, B M Pimentel, Phys. Rev. D. 9565034A. A. Nogueira and B. M. Pimentel, BRST symmetry and fictitious param- eters, Phys. Rev. D 95, 065034 (2017); . 10.1103/PhysRevD.95.065034https://doi.org/10.1103/PhysRevD. 95.065034. R Bufalo, T R Cardoso, A A Nogueira, B M Pimentel, Generalized Scalar Duffin-Kemmer-Petiau Electrodynamics (GSDKP). 70652002R. Bufalo, T. R. Cardoso, A. A. Nogueira and B. M. Pimentel, Generalized Scalar Duffin-Kemmer-Petiau Elec- trodynamics (GSDKP), Journal of Physics: Conference Series 706, 052002 (2016); . 10.1088/1742-6596/706/5/052002https://doi.org/10.1088/ . R Bufalo, T R Cardoso, A A Nogueira, B M Pimentel, Functional quantization of Generalized Scalar Duffin-Kemmer-Petiau Electrodynamics. R. Bufalo, T. R. Cardoso, A. A. Nogueira and B. M. Pimentel, Functional quantization of Gen- eralized Scalar Duffin-Kemmer- Petiau Electrodynamics, (2015); R Bufalo, T R Cardoso, A A Nogueira, B M Pimentel, Renormalization of Generalized Scalar Duffin-Kemmer-Petiau Electrodynamics. to be publishedR. Bufalo, T. R. Cardoso, A. A. Nogueira and B. M. Pimentel, Renormalization of Generalized Scalar Duffin-Kemmer- Petiau Electrodynamics, to be published. Pimentel, Transition amplitude, partition function and the role of physical degrees of freedom in gauge theories. A A Nogueira, L Rabanal, B , to be publishedA. A. Nogueira, L. Rabanal and B. M. Pi- mentel, Transition amplitude, partition function and the role of physical degrees of freedom in gauge theories, to be pub- lished.	[]
[ "Corner states, hinge states and Majorana modes in SnTe nanowires", "Corner states, hinge states and Majorana modes in SnTe nanowires" ]	[ "Minh Nguyen ", "Nguyen \nInternational Research Centre MagTop\nInstitute of Physics\nPolish Academy of Sciences\nAleja Lotnikow 32/46PL-02668WarsawPoland\n", "Wojciech Brzezicki \nInternational Research Centre MagTop\nInstitute of Physics\nPolish Academy of Sciences\nAleja Lotnikow 32/46PL-02668WarsawPoland\n\nInstitute of Theoretical Physics\nJagiellonian University\nulica S. Łojasiewicza 11PL-30348KrakówPoland\n", "Timo Hyart \nInternational Research Centre MagTop\nInstitute of Physics\nPolish Academy of Sciences\nAleja Lotnikow 32/46PL-02668WarsawPoland\n\nDepartment of Applied Physics\nAalto University\n00076Aalto, EspooFinland\n" ]	[ "International Research Centre MagTop\nInstitute of Physics\nPolish Academy of Sciences\nAleja Lotnikow 32/46PL-02668WarsawPoland", "International Research Centre MagTop\nInstitute of Physics\nPolish Academy of Sciences\nAleja Lotnikow 32/46PL-02668WarsawPoland", "Institute of Theoretical Physics\nJagiellonian University\nulica S. Łojasiewicza 11PL-30348KrakówPoland", "International Research Centre MagTop\nInstitute of Physics\nPolish Academy of Sciences\nAleja Lotnikow 32/46PL-02668WarsawPoland", "Department of Applied Physics\nAalto University\n00076Aalto, EspooFinland" ]	[]	SnTe materials are one of the most flexible material platforms for exploring the interplay of topology and different types of symmetry breaking. We study symmetry-protected topological states in SnTe nanowires in the presence of various combinations of Zeeman field, s−wave superconductivity and inversion-symmetry-breaking field. We uncover the origin of robust corner states and hinge states in the normal state. In the presence of superconductivity, we find inversion-symmetryprotected gapless bulk Majorana modes, which give rise to quantized thermal conductance in ballistic wires. By introducing an inversion-symmetry-breaking field, the bulk Majorana modes become gapped and topologically protected localized Majorana zero modes appear at the ends of the wire. arXiv:2105.11489v2 [cond-mat.mes-hall]	10.1103/physrevb.105.075310	[ "https://arxiv.org/pdf/2105.11489v2.pdf" ]	235,186,991	2105.11489	f784c913c04787a9596b8abdc26b030b17f2206f	Corner states, hinge states and Majorana modes in SnTe nanowires Minh Nguyen Nguyen International Research Centre MagTop Institute of Physics Polish Academy of Sciences Aleja Lotnikow 32/46PL-02668WarsawPoland Wojciech Brzezicki International Research Centre MagTop Institute of Physics Polish Academy of Sciences Aleja Lotnikow 32/46PL-02668WarsawPoland Institute of Theoretical Physics Jagiellonian University ulica S. Łojasiewicza 11PL-30348KrakówPoland Timo Hyart International Research Centre MagTop Institute of Physics Polish Academy of Sciences Aleja Lotnikow 32/46PL-02668WarsawPoland Department of Applied Physics Aalto University 00076Aalto, EspooFinland Corner states, hinge states and Majorana modes in SnTe nanowires SnTe materials are one of the most flexible material platforms for exploring the interplay of topology and different types of symmetry breaking. We study symmetry-protected topological states in SnTe nanowires in the presence of various combinations of Zeeman field, s−wave superconductivity and inversion-symmetry-breaking field. We uncover the origin of robust corner states and hinge states in the normal state. In the presence of superconductivity, we find inversion-symmetryprotected gapless bulk Majorana modes, which give rise to quantized thermal conductance in ballistic wires. By introducing an inversion-symmetry-breaking field, the bulk Majorana modes become gapped and topologically protected localized Majorana zero modes appear at the ends of the wire. arXiv:2105.11489v2 [cond-mat.mes-hall] I. INTRODUCTION SnTe materials (Sn 1−x Pb x Te 1−y Se y ) have already established themselves as paradigmatic systems for studying 3D topological crystalline insulators and topological phase transitions, because the band inversion can be controlled with the Sn content [1][2][3][4], but they may have a much bigger role to play in the future investigations of topological effects. Robust 1D modes were experimentally observed at the surface atomic steps [5] and interpreted as topological flat bands using a model obeying a chiral symmetry [6]. Furthermore, the experiments indicate that these flat bands may lead to correlated states and an appearance of a robust zero-bias peak in the tunneling conductance at low temperatures [7]. Although, there has been a temptation to interpret the zero-bias anomaly as an evidence of topological superconductivity, all the observed phenomenology can be explained without superconductivity or Majorana modes [7][8][9]. On the other hand, the standard picture of competing phases in flat-band systems [10,11] indicates that in thin films of SnTe materials the tunability of the density with gate voltages may allow for the realization of both magnetism and superconductivity at the step defects. The improvement of the fabrication of low-dimensional SnTe systems with a controllable carrier density has become increasingly pressing also because the theoretical calculations indicate that thin SnTe multilayer systems would support a plethora of 2D topological phases, including quantum spin Hall [12,13] and 2D topological crystalline insulator phases [8,14]. The realization of 2D topological crystalline insulator phase would be particularly interesting, because in these systems a tunable breaking of the mirror symmetries would open a path for new device functionalities [14]. Indeed, the recent experiments in thin films of SnTe materials indicate that the transport properties of these systems can be controlled by intentionally breaking the mirror symmetry [15]. Furthermore, SnTe materials are promising can-didate systems for studying the higher order topology [16,17]. Thus, it is an outstanding challenge to develop approaches for probing the hinge and corner states in these systems. In addition to the rich topological properties, SnTe materials are also one of the most flexible platforms for studying the interplay of various types of symmetry breaking fields. The superconductivity can be induced via the proximity effect or by In-doping, and both theory and experiments indicate rich physics emerging as a consequence [18][19][20][21][22][23][24][25]. Interesting topological phases are predicted to arise also in the presence of a Zeeman field [26], which breaks the time-reversal symmetry. In experiments, the Zeeman field can be efficiently applied with the help of external magnetic field by utilizing the huge g factor g ∼ 50 [27,28], or it can be introduced with the help of magnetic dopants [29][30][31]. While the magnetism and superconductivity are part of the standard toolbox for designing topologically nontrivial phases, the SnTe materials offer also unique opportunities for controlling the topological properties by breaking the crystalline symmetries. In particular, it is possible to break the inversion symmetry by utilizing ferroelectricity or a structure inversion asymmetry [32][33][34][35][36][37]. This has already enabled the realization of a giant Rashba effect, and it may be important also for the topology. So far the topological properties of SnTe nanowires have received little attention experimentally, but this may soon change due to the continuous progress in their fabrication [38,39]. In this paper, we systematically study the symmetries and topological invariants in SnTe nanowires and propose to utilize the tunable symmetrybreaking fields for realizing different types of topological states. After describing the system (Sec. II), we consider Zeeman field parallel to the wire and show that depending on the Zeeman field magnitude and the wire thickness there exists four qualitatively different behaviors around the charge neutrality point: trivial insulator regime, one-dimensional Weyl semimetal phase, band-inverted insulator regime and indirect semimetal phase (Sec. III). We show that the band-inverted insulator regime is characterized by a pseudospin texture and an appearance of low-energy states localized at the corners of the wire, whereas the Weyl semimetal phase is protected by a non-symmorphic screw-axis rotation symmetry (4-fold rotational symmetry) in the case of even (odd) thicknesses, and the low-energy states are localized at the hinges of the wire. We uncover how these hinge states are related to the topological corner states appearing in two-dimensional Hamiltonians belonging to the Altland-Zirnbauer class DIII [40] in the presence of a rotoinversion symmetry, and we explicitly construct an analytical formula for a Z 2 topological invariant describing their existence (Sec. IV). In the presence of superconductivity (Sec. V), we find inversionsymmetry-protected gapless topological bulk Majorana modes, which give rise to quantized thermal conductance in ballistic wires. Finally, we show that by introducing an inversion-symmetry-breaking field, the bulk Majorana modes become gapped and topologically protected localized Majorana zero modes appear at the ends of the wire. II. HAMILTONIAN AND ITS SYMMETRIES Our starting is the p-orbital tight-binding Hamiltonian H(k) = m1 2 ⊗1 3 ⊗Σ + t 12 α=x,y,z 1 2 ⊗(1 3 − L 2 α )⊗h α (k α ) +t 11 α =β 1 2 ⊗ 1 3 − 1 2 (L α + αβ L β ) 2 ⊗h α,β (k α , k β )Σ + α=x,y,z λ α σ α ⊗L α ⊗1 8 ,(1) which has been used for describing the bulk topological crystalline insulator phase in the SnTe materials [1] and various topological phases in lower dimensional systems [5,8]. Here we have chosen a cubic unit cell containing eight lattice sites (Fig. 1), Σ is a diagonal 8 × 8 matrix with entries Σ i,i = ∓1 at the two sublattices (Sn and Te atoms), ε αβ is Levi-Civita symbol, L α = −iε αβγ are the 3 × 3 angular momentum L = 1 matrices, σ α are Pauli matrices, and h α (k α ) and h α,β (k α , k β ) are 8 × 8 matrices describing hopping between the nearest-neighbors and next-nearest-neighbor sites, respectively (see Appendix A). In investigations of topological properties it is useful to allow the spin-orbit coupling to be anisotropic, hence λ α , although the reference physical case is λ α ≡ λ. When not otherwise stated we use m = 1.65 eV, t 12 = 0.9 eV, t 11 = 0.5 eV and λ = 0.3 eV. We first consider an infinite nanowire along the zdirection with N x = N y unit cells in x and y directions. The Hamiltonian for the nanowire H 1D (k z ) can be constructed using Hamiltonian (1) and it satisfies a fourfold Figure 1. Schematic view of the system. The blue arrow indicates the screw-axis operation, including a π/2 rotation with respect to z axis and a half-lattice vector translation. The red arrows depict the nearest-neighbor (t12) and the next-nearestneighbor (t11) hopping terms in Hamiltonian (1). The two sublattices, corresponding to Sn and Te atoms, have opposite onsite energies. screw-axis symmetry (see Appendix A) S c (k z ) = P z ⊗ e −i π 4 σz ⊗ e −i π 2 Lz ⊗ s c (k z ) ,(2) where P z and s c (k z ) realize a transformation of the lattice sites, consisting of a translation by a half lattice vector and π/2 rotation with respect to the z axis ( Fig. 1), between the unit cells and inside the unit cell, respectively (see Appendix A). Additionally, there exists also glide plane symmetries M x (k z ) (M y (k z )) consisting of a mirror reflection with respect tox (ŷ) plane and a halflattice translation along z axis and diagonal mirror symmetries M xy (M yx ) with respect to thex +ŷ (x −ŷ) planes. The product of M x (k z ) with M y (k z ) or M xy with M yx yields a twofold rotation symmetry with respect to the z axis. All these symmetry act at any k z can be used to block-diagonalize the 1D Hamiltonian. The mirror symmetry M z (k z ) with respect to the z plane acts on the Hamiltonian as M z (k z )H 1D (k z )M z (k z ) † = H 1D (−k z ) and the inversion symmetry operator can be constructed as I ∝ M x (k z )M y (k z )M z (k z ). If the wire has odd number of atoms in x and y directions, it cannot be constructed from full unit cells, and this influences the symmetries of the system. In particular, for odd thicknesses the screw-axis rotation symmetry is replaced by an ordinary 4-fold rotation symmetry. III. TOPOLOGICAL STATES IN THE PRESENCE OF ZEEMAN FIELD In this section, we study the properties of the system in the presence of Zeeman field H Z = B· σ applied along the wire B = (0, 0, B z ). This field breaks the time-reversal symmetry T and the mirror symmetries M x (k z ), M y (k z ), M xy and M yx , but it preserves the inversion symmetry I, the mirror symmetry M z (k z ) and the screw-axis symmetry S c (k z ). We find that as a function of Zeeman field magnitude and the wire thickness there exists four qualitatively different behaviors around the charge neutrality point: trivial insulator regime, one-dimensional Weyl semimetal phase, band-inverted insulator regime and indirect semimetal phase (Fig. 2). The differences between these phases are summarized in Figs. 3, 4 and 5. In the case of small Zeeman field B z we find a trivial insulating phase or an indirect semimetal phase depending on the wire thickness. Neither of these phases supports in-gap states localized at the ends of the wire. By increasing B z we find that there appears a Weyl semimetal phase for a range of wire thicknesses and Zeeman field magnitudes (Figs. 2 and 4). For even thicknesses of the wire the band crossings (Weyl points) are protected by the nonsymmorphic screw-axis rotation symmetry S c (k z ), which allows us to decompose the Hamiltonian into 4 diagonal blocks, so that the energies of eigenstates belonging to different blocks (indicated with different colors in Fig. 4) can cross. Due to this reason a change in the number of eigenstates belonging to specific eigenvalues of the screwaxis rotation symmetry below the Fermi level as a function of k z can be used as a topological invariant for the Weyl semimetal phase. In the case of odd thicknesses the screw-axis rotation symmetry is replaced by an ordinary 4-fold rotational symmetry, but also this symmetry can be utilized to block diagonalize the Hamiltonian at any k z , and therefore it can protect the Weyl semimetal phase In the regime of large Zeeman field the insulator phase supports a pseudospin texture due to band inversion, resulting in the appearance of localized corner states (see Fig. 5). Dots show the actually computed phase boundaries at discrete values of the wire thickness. in a analogous way. Note that due to non-symmorphic character of S c (k z ) the S c -subblocks of H 1D (k z ) are 4πperiodic and they transform into each other in 2π rotations. Because symmetries guarantee that the spectrum is symmetric around k z = π, this elongated period leads to forced band crossings at k z = π, see Figs. 3-5(a). However, the forced band crossings typically appear away from the zero energy (Figs. 3 and 5) and therefore they do not guarantee the existence of Weyl semimetal phase at the charge neutrality point. In the Weyl semimetal phase shown in Fig. 4 the crossings occur between an electron-like band and a hole-like band carrying different S c eigenvalues so that there necessarily exists states at all energies. As demonstrated by calculating the local density of states (LDOS) in Fig. 4(d) the states connecting the conduction and valence bands are often found to be localized at the hinges of the nanowire. We discuss the origin of these hinge states in Sec. IV. By increasing B z further we find another insulating phase for a wide range of wire thicknesses and Zeeman field magnitudes. In this case, the band dispersions have the camel's back shape [ Fig. 5(a)] which typically appears in topologically nontrivial materials, but we have checked that these band structures can be adiabatically connected to the trivial insulator phase, and therefore the topological nature may only be related to an approximate symmetry of the system. Nevertheless, the bands support a nontrivial pseudospin texture τ p = ψ p (k)\| τ \|ψ p (k) , where the pseudospin operators τ α are the Pauli matrices acting in the sublattice space. In the high-field insulating phase the pseudospin component τ y p is negligible and the pseudospin direction rotates in two-dimensional ( τ x p , τ z p )-space so that its direction is inverted around k z = π [see Fig. 5(b)], whereas in the low-field insulator phase the sublattice pseudospin texture is trivial [ Fig. 3(b)]. Therefore we call the insulating phases as band-inverted and trivial insulators, respectively. The band-inverted insulator phase also supports subgap end states localized at the corners of the wire [ Fig We emphasize that the thin nanowires are used here only for illustration purposes because in these cases the strengths of the Zeeman fields required for realizing the different behaviors of the system are not experimentally feasible. However, with increasing thickness of the nanowires the Weyl semimetal and band-inverted phases occur at smaller values of B z , so that in the case of a realistic thickness they can be accessed with feasible magnitudes of the Zeeman field. The inset of Fig. 2 shows a zoom into the experimentally most relevant regime of the phase diagram. The Weyl points are protected by the screw-axis symmetry (or 4-fold rotation symmetry) which is broken if the Zeeman field is rotated away from the z-axis. However, if we utilize an approximation λ z = 0 and λ x = λ y = λ, there exists also a non-symmorphic chiral symmetry S z (k z ), and it is possible to combine it with M z (k z ) and time-reversal T to construct an antiunitary operator that anticommutes with H 1D (k z ) for any k z . This operator squares to +1 and gives rise to a Pfaffianprotected Weyl semimetal phase also when the Zeeman field is not along the z-axis (Appendix B). We also point out that if the screw-axis symmetry (or 4-fold rotational symmetry) is broken so that the system supports only a 2-fold rotational symmetry (e.g. due to anisotropic spinorbit coupling or rectangular nanowires with N x = N y ), the 2-fold rotational symmetry can still protect the existence of the Weyl points (Appendix C). IV. HINGE STATES We find that in addition to the Weyl semimetal phase at B z = 0 [ Fig. 4(d)], the hinge states appear also in the absence of Zeeman field [ Fig. 6], and they resemble the protected states appearing in higher-order topological phases [16,[41][42][43]. SnTe materials have been acknowledged as promising candidates for higher-order topological insulators but the gapless surface Dirac cones appearing at the mirror-symmetric surfaces make the experimental realization difficult [16]. This problem can be avoided if the system supports a 2D higher-order topological invariant for a specific high-symmetry plane in the k space where the surface states are gapped -this plane is shown in Fig. 7. To explore this possibility, we study the bulk Hamiltonian H(k 1 , k 2 , k 3 ) describing a system with inequivalent atoms at positions (0, 0, ±1/2) and lattice vectors a 1 = (1, 0, 1), a 2 = (0, 1, 1) and a 3 = (0, 0, 2) (see Appendix D 1). We find that the 2D Hamiltonian H(k 1 , k 2 , π) with B = 0 supports edge states [red lines in Fig. 8(a),(b)], but a small energy gap is opened due to λ x = λ y = λ spin-orbit coupling terms. The spectrum is similar both for λ z = 0 . Brillouin zone of the rock-salt crystals. The shaded planes form a k3 = π plane. The blue balls (L points) are the high-symmetry points k1,2 = 0, π within the k3 = π plane. The yellow balls are the fourfold rotation R4 centers at (k1, k2) = (π, π)/2 and (k1, k2) = (3π, 3π)/2. The red and green balls are the rotoinversion centers (W points) at (k1, k2) = (π, 3π)/2 and (k1, k2) = (3π, π)/2, respectively. [ Fig. 8(b)] so that neglecting λ z is a good approximation. Moreover, the numerics indicates that two adjacent edges of the system are topologically distinct leading to appearance of zero-energy corner states at their intersec- tion [Figs. 8(c),(d)]. We find that the presence of the corner states is described by a Z 2 topological invariant. To construct the invariant, we note that H(k 1 , k 2 , π) with λ z = 0 obeys a chiral symmetry S z = σ z ⊗ 1 3 ⊗ τ x , where σ z refers to spin, 1 3 to orbitals and τ x to sublattice degrees of freedom. The Hamiltonian also obeys the time-reversal symmetry T = σ y ⊗ 1 3 ⊗ 1 2 , which anticommutes with S z , so that the Hamiltonian belongs to class DIII. In the eigenbasis of S z the Hamiltonian and time-reversal oper- ator have block-off-diagonal forms H(k 1 , k 2 , π) = 0 u(k 1 , k 2 ) u † (k 1 , k 2 ) 0(3) and T = 0 −i1 6 i1 6 0 .(4) Thus, u(k 1 , k 2 ) T = −u(−k 1 , −k 2 ) and therefore we can define a Pfaffian at the time-reversal invariant points K p = Pf u(K)(5) By utilizing inversion symmery I = 1 2 ⊗ 1 3 ⊗ τ z we get that p is a real number (see Appendix D 2). In our model p can be evaluated explicitly and it takes the form p = (m − 4t 11 )(m + 2t 11 ) 2 − 2mλ 2 .(6) Notice that p is the same for all time-reversal invariant points K in the (k 1 ,k 2 )-plane due to the symmetries of the model (see Appendix D 1). In the usual notation of the 3D Brillouin zone of the rock-salt crystals, the time-reversal invariant points K in the (k 1 ,k 2 )-plane correspond to the L points (see Fig. 7). Interestingly, we find that p does not give a complete description of the presence of the corner states, because we also need to consider the high-symmetry point K = (π/2, 3π/2). This point is a rotoinversion center, so that in the eigenbasis of the rotoinversion operator the m/t 12 Hamiltonian takes a block-diagonal form ��t 12 -4 -2 0 2 4 -3 -2 -1 0 1 2 �=0 �=1 (a) (b)H(K ) =     h 1 0 0 0 0 h 2 0 0 0 0 h 3 0 0 0 0 h 4     .(7) By utilizing the chiral symmetry, inversion symmetry and time-reversal symmetry we find that (see Appendix D 3) det[H(K )] = d 4 ,(8)where d ≡ det h 1 = det h 4 = − det h 2 = − det h 3 , and therefore d changes sign at the zero-energy gap closing occurring at the momentum K . In our model d can be evaluated analytically and it takes the form d = det v 1 = m((m − 2t 11 ) 2 + 4t 2 12 ) − 2(m − 2t 11 )λ 2 . (9) In the usual notation of the 3D Brillouin zone of the rock-salt crystals, the rotoinversion centers K are the W points (see Fig. 7). Figure 10. C + =2 -3 -2 -1 0 1 2 3 0.0 0.5 1.0 1.5 2.0 2.5 3.0 m�t 12 ��t 12 C + =2 C + =0 C + =0 C + =0 C + =0 C + =0 C + =-2 C + =-2 C + =-4 C + =0 C + =0 Topological phase diagram of Hamiltonian H(k1, k2, k3) in the m-λ plane. Colors indicate different mirror Mxy Chern numbers C+ defined in the k1-k3 plane (with k2 = k1). Areas bounded by the dashed line and filled with checkerboard pattern are non-trivial in the sense of ν invariant of Eq. (10), also shown in Fig. 9. The chiral limit is assumed with λz = 0 and λx = λy = λ. The Z 2 invariant ν can be determined using d and p as ν = (1 − sgn(pd))/2.(10) The topological phase diagram in the m-λ plane is given in Fig. 9(a). By comparing to the corner state spectrum shown in Fig. 9(b), we find that ν describes the appearance of the corner states perfectly in our model. In the nontrivial phase ν = 1, there are two localized states at every corner. They are Kramers partners and carry opposite chirality eigenvalues. We emphasized that the topological invariant (10) is not directly related to topological crystalline insulator invariant of the SnTe materials (see Fig. 10). Therefore, we expect that it is possible to find material compositions supporting the higher-order topological phase outside the topological crystalline insulator phase and vice versa. V. MAJORANA MODES IN THE PRESENCE OF SUPERCONDUCTIVITY Majorana zero modes are intensively-searched non-Abelian quasiparticles which hold a promise for topological quantum computing [44][45][46]. The key ingredients for realizing Majorana zero modes are usually thought to be spin-orbit coupling (spin-rotation-symmetry breaking field), magnetic field (time-reversal-symmetry breaking field) and superconductivity [47], and there exists a number of candidate platforms for studying Majorana zero modes including chains of adatoms [48][49][50] and various strong spin-orbit coupling materials in the presence of superconductivity and magnetism [51][52][53][54]. SnTe materials are particularly promising candidates for this purpose because in addition to the strong spin-orbit coupling they offer flexibility for introducing symmetry breaking fields such as superconductivity, magnetism and inversion-symmetry breaking fields. In the presence of induced s−wave superconductivity the Bogoliubov-de Gennes Hamiltonian for the nanowires has the form H sc (k z ) = H 1D (k z ) − µ iσ y ∆ −iσ y ∆ −(H 1D (−k z ) T − µ) ,(11) where σ y acts in the spin space. This Hamiltonian obeys a particle-hole symmetry C sc H sc (−k z ) T C sc = −H sc (k z ),(12) where C sc = 0 1 1 0 .(13) We can utilize C sc to perform a unitary transformation on the Hamiltonian so that in the new basis the Hamiltonian H sc U (k z ) is antisymmetric at k z = 0, π and PfH sc U (k z = 0, π) ∈ R (see Appendix E 1). Since iH sc U (0, π) ∈ R we use real Schur decomposition to evaluate the Pfaffian in a numerically stable way, as suggested in [55]. Therefore, we can define a Z 2 topological invariant as ν sc = (1 − sgn[PfH sc U (0)PfH sc U (π)]) /2.(14) This is the strong topological invariant of 1D superconductors belonging to the class D. In fully gapped 1D superconductors the ν sc = 1 phase supports unpaired Majorana zero modes localized at the end of the wire. The Hamiltonian (11) also satisfies an inversion symmetry I sc (k z ) = I(k z ) 0 0 I(k z ) .(15) The product of C sc and I sc (k z ) is an antiunitary chiral operator, which allows to perform another unitary transformation on the Hamiltonian, so that in the new basis the Hamiltonian H sc V (k z ) is antisymmetric at all values of k z and PfH sc V (k z ) ∈ R (see Appendix E 2). Therefore, consistently with classification of gapless topological phases [56], we can define an inversion-symmetry protected Z 2 topological invariant for all values of k z as ν I (k z ) = (1 − sgn[PfH sc V (k z )]) /2.(16) If this invariant changes as a function of k z there must necessarily be a gap closing. Thus, it enables the possibility of a topological phase supporting inversion-symmetry protected gapless bulk Majorana modes. Here, we use the term Majorana in the same way as it is standardly used in the physics literature, such as Refs. [57], so that it can be used to refer to all Bogoliubov quasiparticles in superconductors. In the presence of inversion symmetry the protection of the gapless Majorana bulk modes is similar to the Weyl points in Weyl semimetals: They can only be destroyed by merging them in a pairwise manner. The experimental signature of the Majorana bulk modes in ballistic wires is quantized thermal conductance G th = (G 0 /2) n T n in units of thermal conductance quantum G 0 = π 2 k 2 B T /(3h), because for ballistic wires with length larger than the decay length of the gapped modes the transmission eigenvalues for the gapless (gapped) modes are T n = 1 (T n = 0) and the number of gapless Majorana bulk modes (apart from phase transitions) is always even. Such kind of quantized thermal conductance is generically expected in ballistic wires in the normal state, but the appearance of quantized thermal conductance in superconducting wires is an exceptional property of this topological phase and it is not accompanied by quantized electric conductance. It turns out that (see Appendix E 3) PfH sc U (k z = 0, π) = PfH sc V (k z = 0, π).(17) This means that in the presence of inversion symmetry the nontrivial topological invariant ν sc = 1 always leads to a change of ν I (k z ) between k z = 0 and k z = π. Thus, in the presence of inversion symmetry there cannot exist fully gapped topologically nontrivial superconducting phase supporting Majorana end modes, but instead ν sc = 1 guarantees the existence of topologically nontrivial phase supporting gapless Majorana bulk modes. To get localized zero-energy Majorana end modes it is necessary to break the inversion symmetry. The inversion symmetry can be broken in SnTe nanowires by utilizing ferroelectricity or a structure inversion asymmetry [32][33][34][35][36][37]. For the results obtained in this paper the explicit mechanism of the inversion symmetry breaking is not important. Therefore, for simplicity we consider a Rashba coupling term H R (k) = λ R · sin k × σ.(18) The magnitude of the Rashba coupling λ R can be considered as the inversion-symmetry breaking field. For λ R = 0 the inversion symmetry is obeyed and only the gapless topological phase can be realized, whereas for λ R = 0 the gapless Majorana bulk modes are not protected and the opening of an energy gap can transform the system into a fully gapped topologically nontrivial superconductor supporting localized Majorana end modes. In Fig. 11 we illustrate the dependence of the topological phase diagrams on the nanowire thickness. For very small thicknesses there exists a large insulating gap at the charge neutrality point in the normal state spectrum (Fig. 3), and therefore the topologically nontrivial phase can be reached only by having either a reasonably large chemical potential µ or Zeeman field B z . However, with increasing thickness of the nanowire the insulating gap decreases and the nontrivial phases are distributed more uniformly in the parameter space. Similarly as in the case of the normal state phase diagram (Fig. 2), we expect that for realistic nanowire thicknesses the topologically nontrivial phase is accessible with experimentally feasible values of the chemical potential and Zeeman field. The structure of the topological phase diagram is quite complicated and it is not easy to extract simple conditions for the existence of the nontrivial phase. It is worth mentioning that the topological region always starts for B z > ∆ and the main effect of increasing (decreasing) ∆ is to shift the nontrivial phases right (left) along B z axis. As discussed above, in the presence of inversion symmetry the ν sc = 1 regions in Fig. 11 correspond to the gapless topological phase. This is indeed the case as demonstrated with explicit calculation in Fig. 12(a). Moreover, if inversion-symmetry breaking field is introduced our numerical calculations confirm the opening of an energy gap in the bulk spectrum and the appearance of the zero-energy Majorana modes localized at the end of the superconducting nanowire [ Fig. 12(b),(c)]. As illustrated in Fig. 11 the topological phase becomes fragmented into smaller and smaller regions in the parameter space upon increasing the wire thickness. Therefore, one might be concerned about the experimental feasibility to observe the topological superconductivity in these systems. The systematic analysis of the dependence of the topological gap on the wire thickness goes beyond the scope of this paper, but in Fig. 13 we have focused on one of the fragmented topological regions in the case of 18 atoms thick nanowires. Our results show that also in this case it is possible to achieve a topological gap on the order of 1K and to realize Majorana zero modes at the end of the wire by breaking the inversion symmetry. Therefore, at least in 18 atoms thick nanowires the observation of the Majorana zero modes is still experimentally feasible. and at the end ρ end (E) (normalized with the maximum value ρmax) of the wire calculated using the Green function method described in Ref. [58]. VI. DISCUSSION AND CONCLUSIONS We have shown that SnTe materials support robust corner states and hinge states in the normal state. The topological nature of these states is related to the approximate symmetries of the SnTe nanowires. Some of the approximations, such as the introduction of anisotropic spin-orbit coupling, are quite abstract technical tricks, but they are extremely useful because they allow us to construct well-defined topological invariants. Moreover, we have checked that our approximations are well-controlled and our results are applicable for realistic multivalley nanowires. We have also shown that the higherorder topological invariant, describing the existence of hinge states, is not directly related to the topological crystalline insulator invariant. Therefore, the nontrivial crystalline insulator and higher-order topologies can appear either separately or together. If they appear together the surface states appearing due to topological crystalline insulator phase can coexist with the hinge states appearing due to higher-order topological phase. The higher-order topological invariant is a 2D invariant related to a high-symmetry plane in the momentum space. This plane corresponds to a fixed value of k z and we have found that both the 2D bulk and the 1D edge are gapped within this plane, so that only the corner states appear. From the practical point of view this means that the surface states arising from the topological crystalline insulator phase and the hinge states arising from the higher-order topological phase are separated in the momentum k z so that they can coexist in ballistic wires where k z is a good quantum number. We have concentrated on relatively thin nanowires. Since the wave functions of the transverse modes transform as a function of the momentum k z and energy from hinge states to surface states and bulk states, the transverse mode energies do not obey simple parametric dependencies as a function of the nanowire thickness and the Zeeman field. This means that the SnTe nanowires cannot be described by using a low-energy effective k.p theory. This is illustrated in the complicated phase diagram of nanowires, which we have discovered. Nevertheless, from the general trends in the thickness dependence we can extrapolate that for realistic nanowire thicknesses the topologically nontrivial phases can be reached with experimentally feasible values of the Zeeman field. Finally, we have found that the superconducting SnTe nanowires support gapless bulk Majorana modes in the presence of inversion symmetry, and by introducing inversion-symmetry-breaking field, the bulk Majorana modes become gapped and topologically protected localized Majorana zero modes appear at the ends of the wire. This finding opens up new possibilities to control and create Majorana zero modes by controlling the inversionsymmetry breaking fields. There exists various possibilities to experimentally probe the corner states, hinge states and Majorana modes. High-quality transport studies are definitely the best way to study these systems. Ideally, the SnTe bulk materials would be insulators where the Fermi level is inside the insulating gap. The interesting physics, including the topological surface states, hinge states and corner states all appear in this range of energies in the nanowires. Unfortunately, in reality the SnTe bulk materials typically have a large residual carrier density due to defects, which poses a significant obstacle for the studies of topological transport effects. Therefore it is of crucial importance to improve the control of the carrier density in SnTe materials. In comparison to the bulk systems the nanowires have the advantage that the carrier density can be more efficiently controlled with gate voltages. Tunneling measurements are possible also in the presence of a large carrier density because one can probe the local density of states as a function of energy by voltage-biasing the tip. One may also try to observe the hinge states and corner states using nano-ARPES but obtaining simultaneously both high-spatial and high-energy resolution is a difficult experimental challenge. The topologically protected gapless Majorana bulk modes could be probed via thermal conductance measurements, and they may also be detectable by measuring electrical shot-noise power or magnetoconductance oscillations in a ring geometry [59]. The Majorana zero modes give rise to various effects, such as a robust zero-bias peak in the conductance [60] and 4π Josephson effect [52,61], but the ultimate goal in the physics of Majorana zero modes is of course to observe effects directly related to the non-Abelian Majorana statistics [45][46][47]. The Majorana zero modes can be realized even if a significant residual carrier density is present as illustrated in our phase diagrams. However, the new experimental challenge in this case is that the topologically nontrivial phase becomes more and more fragmented in thick wires. In this section we give explicit expressions for the different symmetry operators of the nanowire Hamiltonian. Our starting point is the bulk Hamiltonian (1). The nearest-neighbor hopping matrices are h x (k x ) =            0 0 0 0 γ − kx 0 0 0 0 0 0 0 0 γ + kx 0 0 0 0 0 0 0 0 γ − kx 0 0 0 0 0 0 0 0 γ + kx γ + kx 0 0 0 0 0 0 0 0 γ − kx 0 0 0 0 0 0 0 0 γ + kx 0 0 0 0 0 0 0 0 γ − kx 0 0 0 0            ,(A1)h y (k y ) =             0 0 0 0 0 γ − ky 0 0 0 0 0 0 γ + ky 0 0 0 0 0 0 0 0 0 0 γ + ky 0 0 0 0 0 0 γ − ky 0 0 γ − ky 0 0 0 0 0 0 γ + ky 0 0 0 0 0 0 0 0 0 0 γ + ky 0 0 0 0 0 0 γ − ky 0 0 0 0 0             (A2) and h z (k z ) =            0 0 0 0 0 0 0 γ + kz 0 0 0 0 0 0 γ + kz 0 0 0 0 0 0 γ − kz 0 0 0 0 0 0 γ − kz 0 0 0 0 0 0 γ + kz 0 0 0 0 0 0 γ + kz 0 0 0 0 0 0 γ − kz 0 0 0 0 0 0 γ − kz 0 0 0 0 0 0 0            ,(A3) where γ + kα = 1 + e ikα and γ − kα = 1 + e −ikα . The nextnearest-neighbor hopping matrices are h xy (k x , k y ) =           0 φ−x,−y 0 0 0 0 0 0 φx,y 0 0 0 0 0 0 0 0 0 0 θ−x,y 0 0 0 0 0 0 θx,−y 0 0 0 0 0 0 0 0 0 0 θx,−y 0 0 0 0 0 0 θ−x,y 0 0 0 0 0 0 0 0 0 0 φx,y 0 0 0 0 0 0 φ−x,−y 0           ,(A4)h yx (k x , k y ) =           0 θ−x,−y 0 0 0 0 0 0 θx,y 0 0 0 0 0 0 0 0 0 0 φ−x,y 0 0 0 0 0 0 φx,−y 0 0 0 0 0 0 0 0 0 0 φx,−y 0 0 0 0 0 0 φ−x,y 0 0 0 0 0 0 0 0 0 0 θx,y 0 0 0 0 0 0 θ−x,−y 0           ,(A5)h yz (k y , k z ) =           0 0 θ−y,z 0 0 0 0 0 0 0 0 φy,z 0 0 0 0 θy,−z 0 0 0 0 0 0 0 0 φ−y,−z 0 0 0 0 0 0 0 0 0 0 0 0 θ−y,z 0 0 0 0 0 0 0 0 φy,z 0 0 0 0 θy,−z 0 0 0 0 0 0 0 0 φ−y,−z 0 0           ,(A6)h zy (k y , k z ) =           0 0 φ−y,z 0 0 0 0 0 0 0 0 θy,z 0 0 0 0 φy,−z 0 0 0 0 0 0 0 0 θ−y,−z 0 0 0 0 0 0 0 0 0 0 0 0 φ−y,z 0 0 0 0 0 0 0 0 θy,z 0 0 0 0 φy,−z 0 0 0 0 0 0 0 0 θ−y,−z 0 0           ,(A7)h zx (k x , k z ) =           0 0 0 θ−x,z 0 0 0 0 0 0 φx,z 0 0 0 0 0 0 φ−x,−z 0 0 0 0 0 0 θx,−z 0 0 0 0 0 0 0 0 0 0 0 0 0 0 φx,z 0 0 0 0 0 0 θ−x,z 0 0 0 0 0 0 θx,−z 0 0 0 0 0 0 φ−x,−z 0 0 0           (A8) and h xz (k x , k z ) =           0 0 0 φ−x,z 0 0 0 0 0 0 θx,z 0 0 0 0 0 0 θ−x,−z 0 0 0 0 0 0 φx,−z 0 0 0 0 0 0 0 0 0 0 0 0 0 0 θx,z 0 0 0 0 0 0 φ−x,z 0 0 0 0 0 0 φx,−z 0 0 0 0 0 0 θ−x,−z 0 0 0           ,(A9) where θ ±α,±β = e ±ikα + e ±ik β and φ ±α,±β = 1 + e i(±kα±k β ) . To obtain the lower dimensional Hamiltonians, we can first expand the Hamiltonian as H (k) = H 0 (k y , k z ) + e −ikx H 1 (k y , k z ) + e ikx H † 1 (k y , k z ) . Then the 2D Hamiltonian obtained by assuming a finite thickness N x in x-direction is given by 48N x × 48N x ma- trix H 2D (k y , k z ) =            H0 H † 1 0 0 0 · · · 0 H1 H0 H † 1 0 0 · · · 0 0 H1 H0 H † 1 0 · · · 0 . . . . . . . . . . . . . . . . . . 0 · · · · · · 0 0 H0 H † 1 0 · · · · · · 0 0 H1 H0            . (A10) Similarly, we can decompose this 2D Hamiltonian as H 2D (k y , k z ) = H 0 (k z ) + e −iky H 1 (k z ) + e iky H 1 † (k z ) ,(A11) where H 0(1) are 48N x ×48N x matrices. The Hamiltonian for the nanowire with a finite thickness N x (N y ) in x (y) direction is given by 48N x N y × 48N x N y matrix H 1D (k z ) =            H 0 H 1 † 0 0 0 · · · 0 H 1 H 0 H 1 † 0 0 · · · 0 0 H 1 H 0 H 1 † 0 · · · 0 . . . . . . . . . . . . . . . . . . 0 · · · · · · 0 0 H 0 H 1 † 0 · · · · · · 0 0 H 1 H 0            . (A12) Assuming that N x = N y , the nanowire has a screw-axis symmetry, which is described by an operator S c (k z ) = P z ⊗ exp −i π 4 σ z ⊗ exp −i π 2 L z ⊗ s c (k z ) . (A13) Here P z is a N x N y × N x N y matrix that realizes the fourfold rotation of the unit cells. For general N x = N y we have P z such that (P z ) i,j = 1 for i = q + (p − 1)N x and j = p + (N x − q)N x (p, q = 1, . . . , N x ) and (P z ) i,j = 0 otherwise. s c (k z ) is the 8 × 8 matrix acting inside the unit cell s c (k z ) =           0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 e −ikz 0 0 0 0 0 0 e −ikz 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 e −ikz 0 0 0 0 0 0 0 0 e −ikz 0 0           . (A14) The mirror symmetry operators corresponding to the mirror planes perpendicular to x, y and z are Setting λ z = 0 is a good approximation in thin nanowires. In this case, there exists a non-symmorphic chiral symmetry S z (k z ) given by, M x (k z ) = 1 Ny ⊗ m x ⊗ σ x ⊗ (2L 2 x − 1 3 ) ⊗ g(−k z )P z P x , M y (k z ) = m y ⊗ 1 Nx ⊗ σ y ⊗ (2L 2 y − 1 3 ) ⊗ g(−k z )P z P y , M z (k z ) = 1 Ny ⊗ 1 Nx ⊗ σ z ⊗ (2L 2 z − 1 3 ) ⊗ g(k z ), (A15) where g (k z ) = diag e −ikz , e −ikz , 1, 1, e −ikz , e −ikz , 1, 1 , P x =                    (A16) P y =          S z (k z ) = 1 Ny ⊗ 1 Nx ⊗ σ z ⊗ 1 3 ⊗ ΣP z g (k z ) ,(B1) and it is possible to combine it with M z (k z ) and timereversal T to construct an antiunitary operator that anticommutes with H 1D (k z ) for any k z . This operator squares to +1 and therefore it can give rise to a Pfaffianprotected Weyl semimetal phase also when the Zeeman field is not along the z-axis. Fig. 14 shows that there exists a Weyl semimetal phase for a range of Zeeman field magnitudes for all Zeeman field directions in the yz plane. Appendix C: Effects of anisotropic spin-orbit coupling We can study the effects of breaking spatial symmetries by considering anisotropic spin-orbit couplings. In Fig. 15(a) we show the phase diagram of 8 atoms thick nanowire as function of λ z and B z for λ x = λ y = λ. In this case the system still obeys the screw-axis symmetry so that the phase diagram contains the Weyl semimetal phase in addition to the insulator and indirect semimetal phases. The phase boundaries depend on λ z . On the other hand, the screw-axis symmetry is broken if λ x = λ y = λ z . Nevertheless, the system still obeys a two-fold rotational symmetry, which can protect the existence of Weyl points. Indeed, the phase diagram as a function of λ x and B z for λ y = λ z = λ also contains the Weyl semimetal phase as shown in Fig. 15(b). In this section we consider the Hamiltonian for a system with atoms at positions (0, 0, ±1/2) and translation vectors a 1 = (1, 0, 1), a 2 = (0, 1, 1) and (0, 0, 2). For this unit cell the hopping matrices in the bulk Hamiltonian H(k) of Eq. (1) are h x (k) = 0 e ik1 + e i(k3−k1) e −ik1 + e i(k1−k3) 0 , (D1) h y (k) = 0 e ik2 + e i(k3−k2) e −ik2 + e i(k2−k3) 0 , (D2) h z (k) = 0 1 + e ik3 1 + e −ik3 0 ,(D3) and h xy (k) = 2 cos(k 1 + k 2 − k 3 )1 2 h yx (k) = 2 cos(k 1 − k 2 )1 2 h xz (k) = 2 cos(k 1 )1 2 h zx (k) = 2 cos(k 1 − k 3 )1 2 h yz (k) = 2 cos(k 2 )1 2 h zy (k) = 2 cos(k 2 − k 3 )1 2 ,(D4) and the symmetries discussed above are I = 1 2 ⊗ 1 3 ⊗ exp −i k3 2 τ z , M α = σ α ⊗ (2L 2 α − 1 3 ) ⊗ 1 2 , α = x, y M z = σ z ⊗ (2L 2 z − 1 3 ) ⊗ exp −i k3 2 τ z M xy = σ x − σ y √ 2 ⊗ 2 L x − L y √ 2 2 − 1 3 ⊗ 1 2 . Moreover, we also have a four-fold rotation R 4 = exp i π 2 1 2 σ z ⊗ exp i π 2 L z ⊗ 1 2 , and the time-reversal symmetry T = σ y ⊗ 1 3 ⊗ 1 2 . These symmetries act on the Hamiltonian as IH(k 1 , k 2 , k 3 )I † = H(−k 1 , −k 2 , −k 3 ) M x H(k 1 , k 2 , k 3 )M † x = H(k 3 − k 1 , k 2 , k 3 ) M y H(k 1 , k 2 , k 3 )M † y = H(k 1 , k 3 − k 2 , k 3 ) M z H(k 1 , k 2 , k 3 )M † z = H(k 1 − k 3 , k 2 − k 3 , −k 3 ) M xy H(k 1 , k 2 , k 3 )M † xy = H(k 2 , k 1 , k 3 ) R 4 H(k 1 , k 2 , k 3 )R † 4 = H(k 3 − k 2 , k 1 , k 3 ), T H(k)T † = H(−k) T .(D5) We concentrate on the k 3 = π plane. In this case, the standard high-symmetry points are related as M x H(0, 0, π)M † x = H(π, 0, π), M y H(0, 0, π)M † y = H(0, π, π), M y M x H(0, 0, π)M † x M † y = H(π, π, π). Moreover, the k 3 = π plane has four special points (k 1 , k 2 ) = (π/2 + n 1 π, π/2 + n 2 π) (n 1,2 = 0, 1) obeying M x , H( π 2 + n 1 π, π 2 + n 2 π) = 0 M y , H( π 2 + n 1 π, π 2 + n 2 π) = 0. Among these points we find two four-fold rotation centers R 4 , H( π 2 , π 2 , π) = R 4 , H( 3π 2 , 3π 2 , π) = 0,(D8) and two four-fold rotoinversion centers Q 4 , H( π 2 , 3π 2 , π) = Q 4 , H( 3π 2 , π 2 , π) = 0,(D9) where Q 4 = IR 4 .(D10) The rotoinversion centers are mapped onto each other by the four-fold rotation or diagonal mirror R 4 H( π 2 , 3π 2 , π)R † 4 = H( 3π 2 , π 2 , π), M xy H( π 2 , 3π 2 , π)M † xy = H( 3π 2 , π 2 , π),(D11) and the rotation centers are related by the inversion symmetry IH( π 2 , π 2 , π)I † = H( 3π 2 , 3π 2 , π) (D12) Finally, the product of M x and M y yields a two-fold rotation R 2 with respect to the z axis R 2 = iM x M y = σ z ⊗ (2L 2 z − 1 3 ) ⊗ 1 2 ,(D13) and the rotation and rotoinversion centers are also twofold rotation centers R 2 , H( π 2 + n 1 π, π 2 + n 2 π) = 0, n 1,2 = 0, 1. (D14) 2. Pfaffian at the high-symmetry points in the k3 = π plane with λz = 0 approximation By assuming that λ z = 0 and λ x = λ y = λ we find that the Hamiltonian H(k 1 , k 2 , π) satisfies a chiral symmetry S z H(k 1 , k 2 , π)S † z = −H(k 1 , k 2 , π), where S z = σ z ⊗ 1 3 ⊗ τ x .(D15) Therefore, the symmetry class of H(k 1 , k 2 , π) is DIII so that we can find an eigenbasis of S z in which the Hamiltonian takes the form of H(k 1 , k 2 , π) = 0 u(k 1 , k 2 ) u † (k 1 , k 2 ) 0 ,(D16) and the time-reversal symmetry operator is T = 0 −i1 6 i1 6 0 .(D17) Thus, the Hamiltonian satisfies a particle-hole symmetry CH(k 1 , k 2 , π)C † = −H(−k 1 , −k 2 , π) T ,(D18) where C = T S z = 0 1 6 1 6 0 ,(D19) so that u(k 1 , k 2 ) = −u(−k 1 , −k 2 ) T .(D20) Therefore, at the high-symmery points K = (n 1 π, n 2 π) (n 1,2 = 0, 1) u(K) T = −u(K),(D21) and we can define a Pfaffian p = Pf u(K).(D22) Note that all high-symmetry points K are equivalent. The possible values of p are restricted because of the inversion symmetry operator, which can be written in the present basis as I = 0 o o T 0 ,(D23) where o = 0 1 3 −1 3 0 (D24) is an orthogonal matrix. By applying it to the Hamiltonian at the K point we get At the four-fold rotation and rotoinversion points we can use these symmetries to decompose the Hamiltonian into diagonal blocks. We first focus on the rotoinversion center K = (π/2, 3π/2, π) point. In the eigenbasis of Q 4 the Hamiltonian takes a block-diagonal form u(K) = ou(K) † o = ou(K) * o T ,(D25)H(K ) =     h 1 0 0 0 0 h 2 0 0 0 0 h 3 0 0 0 0 h 4     ,(D27) where h 1,...,4 are the 3 × 3 blocks. By ordering the eigenvalues of Q 4 in a suitable way, the chiral symmetry takes a block form S z =     0 s 1 0 0 s 2 0 0 0 0 0 0 s 3 0 0 s 4 0     ,(D28) with s i being 3 × 3 unitary block. This form follows from the fact that Q 4 and S z anticommute. The spectrum of each individual block h i is not symmetric around zero, but the spectrum of h 1 (h 3 ) is opposite to the spectrum of h 2 (h 4 ). Since the blocks also have an odd dimension, the determinants satisfy det h 1 = − det h 2 and det h 3 = − det h 4 . The spectrum of the whole Hamiltonian H(K ) is twice degenerate because of the presence of the symmetry Π = IT with the property ΠΠ = −1 that gives Kramer denegeracy at every k point. This symmetry in the present basis takes a block form of Π = IT =     0 0 0 k 1 0 0 k 2 0 0 k † 2 0 0 k † 1 0 0 0     .(D29) This implies that the blocks h 1 and h 4 (h 2 and h 3 ) have the same spectrum. From this it follows that det[H(K )] = d 4 ,(D30) where d ≡ det h 1 = det h 4 = − det h 2 = − det h 3 , and therefore d changes sign at the zero-energy gap closing occurring at the momentum K . Finally, it is worth noticing that the above construction does not work for the four-fold rotation points. In the eigenbasis of R 4 the Hamiltonian consists of four diagonal blocks h 1,...,4 , where h 1,2 (h 3,4 ) are 2 × 2 (4 × 4) matrices. The rotation R 4 commutes with S z , so that in this basis S z =     s 1 0 0 0 0 s 2 0 0 0 0 s 3 0 0 0 0 s 4     . (D31) From this structure it follows that {h i , s i } = 0 so that spectrum of each block h i is symmetric around zero. Thus, the determinants always satisfy det h 1,2 ≤ 0 and det h 3,4 ≥ 0, and therefore they cannot change sign in a gap closing. metry C sc (H sc (−k z )) T C sc = −H sc (k z ), where C sc can be written in the Nambu space as C sc = 0 1 1 0 . (E1) We can utilize C sc to perform a unitary transformation on the Hamiltonian H sc U (k z ) = U † H sc (k z )U, U = β Λ C −1 ,(E2) where the columns of matrix β are the eigenvectors of C sc and Λ C is a diagonal matrix containing the eigenvalues of C sc , so that C sc β = βΛ C .(E3) Because C sc C sc * = 1, which follows from the D symmetry class, and C sc is unitary we can choose the eigenvectors so that they satisfy β = β * . From this it follows that at k 0 = 0, π (H sc U (k 0 )) T = Λ C −1 β T (H sc (k 0 )) T β Λ C (E4) = Λ C β T C sc (H sc (k 0 )) T C sc β Λ C −1 = − Λ C β T H sc (k 0 )β Λ C −1 = −H sc U (k 0 ) and the particle-hole operator in the new basis becomes identity matrix U T C sc U = Λ C −1 β T C sc β Λ C −1 = 1. (E5) Since the Hamiltonian is antisymmetric at k 0 = 0, π we can define a Pfaffian, which is real because This is the strong topological invariant of 1D superconductors belonging to the class D. Topological invariant for inversion-symmetry protected gapless Majorana bulk modes We can also combine C sc with the inversion symmetry I sc (k z ) to produce an operator A sc (k z ) = C sc I sc (k z ). (E8) whose action on the Hamiltonian is A sc † kz (H sc (k z )) T A sc kz = −H sc (k z ). From the double application of the above equation it follows that A sc A sc * = ±1, where +1 and −1 define different symmetry classes, in analogy to the particle hole symmetry. In our case we get A sc A sc * = +1, following from the D symmetry classs, [C sc , I sc ] = 0 and I sc I sc * = 1. Note that it is enough to know that A sc is unitary and A sc A sc * = +1 to prove that it can be diagonalized by an orthogonal transformation. From unitarity we have A sc = exp(iB) with B being Hermitian. From the latter property we get exp(−iB) = exp(−iB T ), which gives B T = B + 2nπ. By taking trace of this equation we get that n = 0 so B must be real symmetric. Then A sc can be diagonalized by by an orthogonal transformation. Then we find the real eigenbasis γ(k z ) of A sc (k z ), we have: A sc (k z )γ(k z ) = γ(k z )Λ A (k z ).(E10) We define a unitary transformation V (k z ) = γ(k z ) Λ A (k z ) −1 ,(E11) and following the same derivation as in Eq. (E4) we can prove that the transformed Hamiltonian H sc V (k z ) = V (k z ) † H sc (k z )V (k z ),(E12) is antisymmetric for any k z . By utilizing the fact that the Pfaffian of Hamiltonian is real valued, we can now define an inversion-symmetry protected Z 2 topological invariant for all values of k z as ν I (k z ) = (1 − sgn[PfH sc V (k z )]) /2.(E13) If this invariant changes as a function of k z there must necessarily be a gap closing. Therefore, there exists a 1D topological phase supporting inversion-symmetry protected gapless bulk Majorana modes. In the presence of inversion symmetry these gapless Majorana bulk modes can only be destroyed by merging them in a pairwise manner. Relationship between the two Pfaffians We have two antisymmetric forms of Hamiltonian. H sc U (k z ) is antisymmetric at k z = 0, π and H sc V (k z ) is antisymmetric for all values of k z . These antisymmetric Hamiltonians allow us to define topological invariants with the help of their Pfaffians, and thus it is important to know how these Pfaffians are related to each other. Assume we have two antisymmetric Hamiltonians H and H related by a change of basis as H = Q † HQ. Then from antisymmetry of H and H we have that H, QQ T = 0. Then we have two options, either QQ T = 1 and then PfH = det Q PfH or QQ T is equal to a symmetry operator of H and then the Pfaffians of H and H do not have to be proportional. The latter case could lead to two independent invariants. Coming back to our case, we find that V (k z = 0, π)V (k z = 0, π) T = 1, and this operator is indeed related to the inversion symmetry V (k z = 0, π)V (k z = 0, π) T = I sc (k z = 0, π), but it turns out that also a stronger property holds, namely V (k z )V (k z ) T = I sc (k z ). From the last equation we obtain V (k z ) † = V (k z ) T I sc (k z ).(E17) Consequently, H sc V (k z ) = V (k z ) T I sc (k z )H sc U (k z )V (k z ).(E18) Thus the Pfaffians can be related as PfH sc V (k z ) = det V (k z ) Pf [I sc (k z )H sc U (k z )] .(E19) Both sides of the equations are well defined because H sc V (k z ) is antisymmetric for any k z and I sc (k z )H sc U (k z ) is also antisymmetric for any k z despite H sc U (k z ) alone being symmetric only at high-symmetry points. For the right-hand side we get det V (k z ) = exp [24iN x N y k z ] so we always have det V (0) = det V (π) = 1. To calculate the Pfaffian on the right-hand side at k 0 = 0, π we can utilize the operator γ(k 0 ) = γ(k 0 ) * , satisfying det γ(k 0 ) = 1 and γ(k 0 ) T I sc (k 0 )γ(k 0 ) is a diagonal matrix, to obtain PfH sc V (k 0 ) = Pf γ(k 0 ) T I sc (k 0 )γ(k 0 )γ(k 0 ) T H sc U (k 0 )γ(k 0 ) = Pf −γ − (k 0 ) T H sc U (k 0 )γ − (k 0 ) Pf γ + (k 0 ) T H sc U (k 0 )γ + (k 0 ) = (−1) d−/2 Pf γ − (k 0 ) T H sc U (k 0 )γ − (k 0 ) Pf γ + (k 0 ) T H sc U (k 0 )γ + (k 0 ) = PfH sc U (k 0 ). Here, γ(k 0 ) = [γ + (k 0 ), γ − (k 0 )], the columns of γ ± (k 0 ) are the eigenvectors of I sc (k 0 ) corresponding to eigenvalues ±1 and we have utilized the fact that the dimension d − of the eigenvalue −1 subspace is always a multiple of 4. Therefore, the two Pfaffians are always equal at the highsymmetry momenta k 0 = 0, π. Figure 2 . 2Phase diagram as function of the nanowire thickness (dimensions of the square cross-section) and Zeeman field Bz. The different phases are: insulator phase (blue), Weyl semimetal phase (red) and indirect semimetal phase (grey). Figure 3 .Figure 4 . 34(a) Low-energy band structure in the trivial insulator phase. The different colors indicate the various Sc eigenvalue subspaces. (b) The sublattice pseudospin texture for a pair of bands with the same Sc eigenvalue. The texture is in-plane because τy is negligible. (c) Energy spectrum of 200 atoms long wire showing no end-states. (d) LDOS for the bulk state highlighted in plot (c). The thickness of the wire is 4 atoms and Bz = 0.1t12. (a) Low-energy band structure in the Weyl semimetal phase for 4 atoms thick nanowire and Bz = 0.3t12. The different colors indicate various Sc eigenvalue subspaces. (b) Band crossings around kz = π. (c) Energy spectrum of 200 atoms long wire. (d) Band structure of 26 atoms thick nanowire at Bz = 0.012t12. At the band-crossing point (red dot) the states are localized in the hinges of the wire. Inset: LDOS (normalized with the maximum value) projected to the square cross section of the wire. Figure 5 . 5(a) Low-energy band structure in the band-inverted insulator phase for the 4 atoms thick nanowire and Bz = 0.4t12. (b) The sublattice pseudospin texture for a pair of bands with the same Sc eigenvalue. (c) Energy spectrum for 200 atoms long wire. The red points indicate eight corner states and black points are bulk states. (d) LDOS as a function of z for the corner states (red line) and a bulk state (black line). Inset: LDOS (normalized with the maximum value) as a function of x and y for the corner states in a 6 atoms thick nanowire. Figure 6 . 6[Fig. 8(a)] and λ z = λ Low-energy band structure and the LDOS (normalized with the maximum value) demonstrating the existence of hinge states in the absence of Zeeman field B = 0 for 50 atoms thick nanowire. Figure 7 7Figure 7. Brillouin zone of the rock-salt crystals. The shaded planes form a k3 = π plane. The blue balls (L points) are the high-symmetry points k1,2 = 0, π within the k3 = π plane. The yellow balls are the fourfold rotation R4 centers at (k1, k2) = (π, π)/2 and (k1, k2) = (3π, 3π)/2. The red and green balls are the rotoinversion centers (W points) at (k1, k2) = (π, 3π)/2 and (k1, k2) = (3π, π)/2, respectively. Figure 8 . 8(a) Low-energy spectrum of the Hamiltonian H(k1, k2, k3 = π) for λz = 0 in the case of open boundary conditions in the a2 direction. The width of the system is 50 unit cells. (b) The low-energy spectrum in the case of uniform spin-orbit coupling λα = λ (α = x, y, z). (c) Spectrum for λz = 0 in the case of open boundary conditions in both a1 and a2 directions. The red dots show the 8 zero-energy corner states. The system size is 50 × 50 unit cells. (d) LDOS of the corner states. Here, n1 and n2 label the unit cells in the directions of a1 and a2, respectively. Figure 9 . 9(a) Topological invariant ν [Eq. (10)] of the 2D Hamiltonian H(k1, k2, π) in the chiral limit λz = 0 and λx = λy = λ as function of m and λ. The shaded region is the nontrivial phase ν = 1 supporting corner states. (b) Energy of a state being closest to the zero energy in the system with all edges open, as function of m and λ. The red lines are phase boundaries from (a). The system size is 50 × 50 unit cells Figure 11 .Figure 12 . 1112Topological phase diagrams for SnTe nanowires in presence of induced superconductivity. The thicknesses of the nanowires are (a) 8, (b) 10 and (c) 18 atoms. The blue regions indicate parameter regimes where νsc = 1. In the presence of inversion symmetry they correspond to a topological phase supporting gapless Majorana bulk modes. If the inversion symmetry is broken they correspond to fully gapped topological superconducting phase supporting Majorana end modes. In the numerical calculations we have used ∆ = 0.01 eV. The topological region always starts for Bz > ∆ and the main effect of increasing (decreasing) ∆ is to shift the nontrivial phases right (left) along Bz axis. Band structures for 8 atoms thick superconducting nanowires (a) in the presence of inversion symmetry λR = 0 and (b) in the absence of inversion symmetry λR = (0, 0.05, 0) eV. (c)The spectrum for 400 atoms long superconducting nanowire, for λR = (0, 0.05, 0) eV, demonstrating the existence of zero-energy Majorana modes localized at the end of the wire (red dots). For illustration purposes, we have computed the spectra for very thin nanowires with Bz = 0.16 eV, µ = 0.91 eV and ∆ = 0.1 eV. However, due to the general arguments presented in the text, qualitatively similar results are expected also for experimentally feasible values of Bz, µ and ∆ in thicker nanowires. Figure 13 . 13Low-energy band structures for 18 atoms thick superconducting nanowires (a) in the presence of inversion symmetry λR = 0 and (b) in the absence of inversion symmetry λR = (0.02, 0.03, 0) eV. (c) Zoom into the relevant regime of the phase diagram. We have chosen Bz = 0.02 eV, µ = 0.5913 eV and ∆ = 0.01 eV (indicated by the red dot in the phase diagram). (d) Local density of states in the bulk ρ bulk (E) Appendix B: Weyl semimetal phase for λz = 0 Figure 14 .Figure 15 . 1415The direct band gap ∆ as function of By and Bz for 4 atoms thick nanowire and λz = 0. In the black region ∆ = 0 and the system is in a 1D Weyl semimetal phase. Phase diagram as function of (a) λz and Bz for λx = λy = λ and (b) λx and Bz for λy = λz = λ. The different phases are: insulator phase (blue), Weyl semimetal phase (red) and indirect semimetal phase (grey). The thickness of the nanowire is 8 atoms.Appendix D: Higher-order topological invariant 1. Hamiltonian and symmetries in the k3 = π plane where we have used Eq. (D20) and o T = −o. Using the general properties of the Pfaffian we getp = Pf u(K) = Pf ou(K) * o T = det o Pf u(K) * = p .(D26) This means that p is a real number. 3 . 3Determinant at the rotation points in the k3 = π plane with λz = 0 approximation [ PfH sc U (k 0 )] * = PfH sc U (k 0 ) T = Pf[H sc U (k 0 )].(E6)Therefore we can define a Z 2 topological invariant as ν sc = (1 − sgn[PfH sc U (0)PfH sc U (π)]) /2. ACKNOWLEDGEMENTSThe work is supported by the Foundation for Polish Science through the IRA Programme co-financed by EU within SG OP. W.B. also acknowledges support by Narodowe Centrum Nauki (NCN, National Science Centre, Poland) Project No. 2019/34/E/ST3/00404.In the presence of induced superconductivity the BdG Hamiltonian H sc (k z ) always satisfies a particle-hole sym- Topological crystalline insulators in the SnTe material class. 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[]	[ "G Allemandi [email protected]†e-mail:[email protected]‡e-mail:[email protected] \nDipartimento di Matematica\nUniversità degli Studi di Torino\nVia Carlo Alberto 1010123TorinoItaly\n", "M Francaviglia \nDipartimento di Matematica\nUniversità degli Studi di Torino\nVia Carlo Alberto 1010123TorinoItaly\n", "M Raiteri \nDipartimento di Matematica\nUniversità degli Studi di Torino\nVia Carlo Alberto 1010123TorinoItaly\n" ]	[ "Dipartimento di Matematica\nUniversità degli Studi di Torino\nVia Carlo Alberto 1010123TorinoItaly", "Dipartimento di Matematica\nUniversità degli Studi di Torino\nVia Carlo Alberto 1010123TorinoItaly", "Dipartimento di Matematica\nUniversità degli Studi di Torino\nVia Carlo Alberto 1010123TorinoItaly" ]	[]	We try to give hereafter an answer to some open questions about the definition of conserved quantities in Chern-Simons theory, with particular reference to Chern-Simons AdS3 Gravity. Our attention is focused on the problem of global covariance and gauge invariance of the variation of Noether charges. A theory which satisfies the principle of covariance on each step of its construction is developed, starting from a gauge invariant Chern-Simons Lagrangian and using a recipe developed in [2] and [27] to calculate the variation of conserved quantities. The problem to give a mathematical well-defined expression for the infinitesimal generators of symmetries is pointed out and it is shown that the generalized Kosmann lift of spacetime vector fields leads to the expected numerical values for the conserved quantities when the solution corresponds to the BTZ black hole. The fist law of black holes mechanics for the BTZ solution is then proved and the transition between the variation of conserved quantities in Chern-Simons AdS3 Gravity theory and the variation of conserved quantities in General Relativity is analysed in detail.	10.1088/0264-9381/20/3/307	[ "https://arxiv.org/pdf/gr-qc/0211098v2.pdf" ]	15,932,306	gr-qc/0211098	9670d2c9b7f11de9f98aea7aff1641ef2d9a504f	4 Dec 2002 G Allemandi [email protected]†e-mail:[email protected]‡e-mail:[email protected] Dipartimento di Matematica Università degli Studi di Torino Via Carlo Alberto 1010123TorinoItaly M Francaviglia Dipartimento di Matematica Università degli Studi di Torino Via Carlo Alberto 1010123TorinoItaly M Raiteri Dipartimento di Matematica Università degli Studi di Torino Via Carlo Alberto 1010123TorinoItaly 4 Dec 2002Dedicated to Matteo Raiteri, born in the Fifth of October 2002 *Covariant Charges in Chern-Simons AdS 3 Gravity We try to give hereafter an answer to some open questions about the definition of conserved quantities in Chern-Simons theory, with particular reference to Chern-Simons AdS3 Gravity. Our attention is focused on the problem of global covariance and gauge invariance of the variation of Noether charges. A theory which satisfies the principle of covariance on each step of its construction is developed, starting from a gauge invariant Chern-Simons Lagrangian and using a recipe developed in [2] and [27] to calculate the variation of conserved quantities. The problem to give a mathematical well-defined expression for the infinitesimal generators of symmetries is pointed out and it is shown that the generalized Kosmann lift of spacetime vector fields leads to the expected numerical values for the conserved quantities when the solution corresponds to the BTZ black hole. The fist law of black holes mechanics for the BTZ solution is then proved and the transition between the variation of conserved quantities in Chern-Simons AdS3 Gravity theory and the variation of conserved quantities in General Relativity is analysed in detail. Introduction Chern-Simons field theories have been widely studied in the past decades as a possible model to analyse the classical and quantum behaviour of the gravitational field. Efforts were focused towards rewriting gravity as a gauge theory with gauge group the Poincaré group or the (anti) de Sitter group. To this purpose, in place of the Hilbert-Einstein Lagrangian, a Chern-Simons Lagrangian was considered in which the gauge potential is a linear combination of the vielbein and the spin connection. This is possible in all odd dimensions and particularly in dimension three, where field equations reproduce exactly the Einstein field equations; see [1,3,4,15,42] where it is shown that 2 + 1 gravity with a negative cosmological constant can be formulated as a Chern-Simons theory (see also [16] for higher dimensional Chern-Simons gravity). In particular it was found in [14] that a particular solution of Chern-Simons theory corresponds to the well-known BTZ black hole [5]. The interest in 3-dimensional Chern-Simons theory as a possible and simpler model to analyse (2 + 1)-dimensional gravity was also strengthened by the observation that the thermodynamics of higher dimensional black holes can be understood in terms of the BTZ solution. The BTZ solution provides indeed a model for the geometry of a great amount of black hole solutions relevant to string theory, the geometry of which can be factorized in the product of BT Z × M , where M is a simple manifold; see [30]. According to the renewed interest in Chern-Simons theories a lot of papers dealing with the problem of gauge symmetries and gauge charges for Chern-Simons theories have appeared in the recent years, all addressed to analyse the origin of the gravitational boundary degrees of freedom and, eventually, to understand the statistical mechanical origin of Bekenstein-Hawking entropy via a micro-and grand-canonical calculation (see [3,4,10,15] and references therein). Motivated by this state of affairs, in the present article we deal with the problem of developing a totally covariant formulation for Chern-Simons conserved quantities, with particular attention to SL(2, IR) × SL(2, IR) ≃ SO(2, 2) Chern-Simons theory in dimension three (with this gauge choice, Chern-Simons theory is well suited to describe AdS gravity; see e.g [1,3,4,14,15,42]). This issue is tackled, first of all, by framing Chern-Simons theory into the mathematical domain of gauge natural theories, which provide a unifying mathematical arena to describe all classical Lagrangian field theories and, in particular, are fundamental to mathematically analyse the interaction of gravity with fermionic and bosonic matter; see e.g. [22,23]. The gauge natural approach is essentially based on the Lagrangian formulation of field theories in terms of fiber bundles and the Calculus of Variations on jet bundles. Hence, in order to have a gauge natural formulation of Chern-Simons theory, the first problem we meet is to construct a covariant Chern-Simons Lagrangian, where covariance is referred both to spacetime diffeomorphisms and gauge transformations. Indeed, despite the field equations are covariant for both spacetime diffeomorphisms and gauge transformations, the Chern-Simons Lagrangian is not covariant for gauge transformations. To solve this problem we define a covariant Lagrangian for the theory which differs from the usual Chern-Simons Lagrangian just for the addition of a divergence term. In such a way a covariantized Lagrangian is obtained and, in the meantime, the field equations remain unchanged. The Lagrangian we calculate hereafter is the same "covariantized" Lagrangian obtained in [8] through the use of the "transgression" formula [18]. In the same paper [8] a calculation of conserved quantities based on Noether theorems and the covariant formula for the superpotential of the theory were obtained. Nevertheless, we shall show in the present paper that the superpotential, even though it is a welldefined mathematical object, in practice it is not suited to calculate conserved quantities. Indeed when conserved quantities are explicitly calculated for the Chern-Simons solution corresponding to the BTZ black hole via the integration of the superpotential on a circle enclosing the horizon, the expected values for mass and angular momentum are not recovered even if the circle tends to spatial infinity (see equation (33) below). This is a rather undesiderable result. In fact, even if the same physical solution of a field theory can be obtained from two, or more, different field theories (whose field content is nevertheless equivalent on-shell) it would be reasonable that physical observables depend solely on the given solution, e.g. the black hole, and not on the theory the solution comes from. Roughly speaking, mass, angular momentum and charge of a black hole are physical properties of the black hole itself and they should not change if the same black hole solution is obtained either from the Hilbert Lagrangian of General Relativity or from the Chern-Simons Lagrangian. Hence, the Noether charges which generate conserved quantities have to be somehow tied to the solution under examination rather than to the Lagrangian which generates the equations of motion. Since the Chern-Simons Lagrangian can be mapped into the Hilbert Lagrangian only modulo divergence terms, this basically means that the general formula for the conserved currents we are interested in has to be linked only to the homology class of the Lagrangian (i.e., it has not to depend on additional divergence terms). In this way conserved quantities are related to the equations of motion and do not depend on the representative chosen inside the homology class of Lagrangians. A step towards the solution of this problem can be found in the papers [32], whereby conserved quantities are obtained from the equations of motion via a cascade equation. Similarly, in the present paper we shall mainly be concerned with the issue of developing a fully covariant approach to conserved quantities in Chern-Simons gravity which could lead to a direct correspondence with charges in General Relativity, thus giving the expected values for mass, angular momentum and entropy for the BTZ solution. The main problems to be solved are two. The first one deals with the very basic object which has to enter in the definition of conserved quantities. Indeed, we will check that the superpotential does not reproduce the correct numerical values, even if in a first approach to conserved quantities it seems to be the best candidate to describe global charges. The reason of these discrepancies is mainly due to the fact that the naive definition of charges via the superpotential alone does not take into a proper account the role played by boundary terms. Boundary conditions have to be imposed on the dynamical fields, namely Dirichlet or Neumann conditions, specifying which are the intensive or extensive variables or, equivalently, which are the control-response parameters. The choice of boundary conditions basically reflects onto the choice of a background solution the starting solution has to be matched with. Physically speaking, the background fixes the zero level for all conserved quantities ( [6,11,17,20,25]). A way to overcome the problems which come from the background fixing procedure is to define the variation of conserved quantities, as suggested in [17,25,27,32]. In this way a covariant analysis of boundary terms,á la Regge-Teitelboim [38], can be implemented, leading to the covariant ADM formalism for conserved quantities. This is exactly the approach we shall develop also for Chern-Simons theory. The second problem we are faced with is related to the choice of the symmetry generators. This mathematical problem is by no means trivial and it deserves a careful investigation. Indeed, in a natural theory such as General Relativity the action of spacetime vector fields on the dynamical fields is unambiguously defined. This means that we know how the fields are Lie dragged along spacetime directions and this is enough to build a mathematical consistent theory of conserved quantities. On the contrary, in a theory with gauge invariance (such as Chern-Simons theory as well as any gauge natural theory; see [34]) there is no canonical way to construct conserved quantities starting from a given spacetime vector field. Indeed, in those theories there is no canonical way to lift the spacetime diffeomorphisms on the configuration space of the theory, i.e. the target space where fields take their values. On the contrary, there may exist different non canonical (but nevertheless global and hence well-defined) ways to perform lifts of spacetime diffeomorphisms. This in turn implies that the transformation rules of the dynamical fields under spacetime diffeomorphisms are not defined until a preferred lift procedure has been somehow selected. For example we shall see that in the Chern-Simons theory there are exactly three distinct dynamical connections. All of them can be equally well used, from a mathematical viewpoint, to define lifted vector fields on the configuration bundle starting from spacetime diffeomorphisms. These vector fields, in their turn, enter into the definition of Lie derivatives of the dynamical fields and they eventually lead to different definitions of conserved quantities. All these definitions provide well-defined and mathemathical meaningful expressions. But which of them is also physically meaningful? In this paper we have tried to answer this question by testing all the admissible definitions of conserved quantities with the BTZ black hole solution. The numerical results obtained suggest that the only viable definition of (the variation of) conserved quantities inside the Chern-Simons gravity framework is the one based on the generalized Kosmann lift of spacetime diffeomorphisms [22,28]. Conserved quantities computed with the generalized Kosmann lift reproduce, in fact, exactly the expected values for the BTZ mass, angular momentum and entropy, while other choices do not lead to meaningful results. Moreover, when we perform the transition from Chern-Simons AdS 3 gravity to General Relativity, the formula expressing the (variation of the) conserved quantities in Chern-Simons gravity is mapped exactly into the formula for (the variation of) the conserved quantities in General Relativity found in [6,11,27,33], as one should expect. The present paper is organized as follows. In Section 2 we define the covariant Lagrangian for SL(2, IR)×SL(2, IR) Chern-Simons theory. In Section 3 we illustrate the geometric framework for the Hamiltonian and symplectic formulation of the theory and we derive a general formula to calculate the variation of conserved quantities. In Section 4 we analyse the problem of defining the lift to the configuration bundle for an infinitesimal generator of symmetries over spacetime and we define the generalized Kosmann lift. In Section 5 we explicitly calculate the transition of the variation of conserved quantities from Chern-Simons theory to (2 + 1)-dimensional gravity. In the Appendix A are summarized the notations and the formulae entering the calculations of Section 5. In Appendix B the formalism developed throughout the paper is applied to the general anti-de Sitter solution and to the one-particle solution of Chern-Simons theory. The Covariant Chern-Simons Lagrangian The 3-dimensional Chern-Simons Lagrangian can be written as: L CS (A) = κ 4π ǫ µνρ Tr A µ d ν A ρ + 2 3 A µ A ν A ρ d 3 x(1) where κ is a constant which will be fixed later, while A µ = A i µ J i are the coefficients of the connection 1-form A = A µ dx µ taking their values in any Lie algebra g with generators J i . By fixing g = sl(2, IR) and choosing the generators J 0 = 1 2 0 −1 1 0 , J 1 = 1 2 1 0 0 −1 , J 2 = 1 2 0 1 1 0 (2) we have [J i , J j ] = η lk ǫ kij J l and Tr(J i J j ) = 1/2 η ij , with η = diag(−1, 1, 1) and ǫ 012 = 1. Hence, the Lagrangian (1) can be explicitly written as: L CS (A) = κ 8π ǫ µνρ η ij A i µ d ν A j ρ + 1 3 ǫ ijk A i µ A j ν A k ρ d 3 x = κ 16π ǫ µνρ η ij F i µν A j ρ − 1 3 ǫ ijk A i µ A j ν A k ρ d 3 x (3) where F i µν = d µ A i ν − d ν A i µ + ǫ i jk A j µ A k ν is the field strength. We then consider two independent sl(2, IR) dynamical connections A andĀ, the evolution of which is dictated by the Lagrangian L CS (A,Ā) = L CS (A) − L CS (Ā)(4) which is nothing but the difference of two Chern-Simons Lagrangians (3), one for each dynamical connection. Field equations ensuing from (4) are of course:    η ij ǫ µνρ F i µν = 0 η ij ǫ µνρF i µν = 0(5) Starting from the fields A,Ā it is then possible (see [1,3,14,42]) to define two new dynamical fields, e i and ω i , through the rule: A i = ω i + 1 l e iĀi = ω i − 1 l e i (l = constant)(6) In terms of the new (e, w) variables field equations (5) become: R ij = − 1 l 2 e i ∧ e j , T i = de i + ω i j ∧ e j = 0(7) where ω i = 1/2 η ij ǫ jkl ω kl and R i j = dω i j + ω i k ∧ ω k j . Equations (7) are nothing but Einstein's equations with cosmological constant Λ = −1/l 2 written in terms of the triad field e i and the torsion-free spin connection ω i j . Moreover the Lagrangian (4), in the new variables, reads as: L CS (A(w, e),Ā(w, e)) = κ 4π l √ g(g µν R µν − 2Λ) + d µ k 8π η ij ǫ µνρ A i νĀ j ρ = κ 4π l √ g(g µν R µν − 2Λ) + d µ k 4πl η ij ǫ µνρ e i ν ω j ρ (8) with g µν = η ij e i µ e j ν and R µν = R i jρν e j µ e ρ i being the Ricci tensor of the metric g. Notice, however, that the transition from Chern-Simons theory to General Relativity displays some theoretical undesiderable features. Indeed, while Chern-Simons equations of motion are manifestly covariant with respect to spacetime diffeomorphism as well as with respect to gauge transformations, the Chern-Simons Lagrangian (3) is not gauge invariant. If we consider a pure gauge transformation with generator ξ i acting on the gauge potential, i.e. δ ξ A i µ = D µ ξ i(9) from (3) we obtain: δ ξ L CS (A) = d µ κ 8π ǫ µνρ η ij A i ν d ρ ξ j(10) The fact that (4) is not gauge invariant becomes even more explicit in expression (8) where the non invariant term is ruled out into the divergence part. The simplest way to overcome this drawback is to push the divergence term appearing in the right hand side of (8) into the left hand side, defining in this way a global covariant Chern-Simons Lagrangian L CScov : L CScov (A,Ā) = κ 8π ǫ µνρ η ij A i µ d ν A j ρ + 1 3 ǫ ijk A i µ A j ν A k ρ d 3 x − κ 8π ǫ µνρ η ijĀ i µ d νĀ j ρ + 1 3 ǫ ijkĀ i µĀ j νĀ k ρ d 3 x (11) − d µ k 8π η ij ǫ µνρ A i νĀ j ρ d 3 x or, equivalently: L CScov (A,Ā) = k 8π ǫ µνρ η ijF i µν B j ρ + η ijDµ B i ν B j ρ + 1 3 ǫ ijk B i µ B j ν B k ρ d 3 x(12) whereD µ is the covariant derivative with respect to the connectionĀ and we set B i µ = A i µ −Ā i µ . Being B i µ tensorial, expression (12) tranforms as a scalar function under gauge transformations (and as a scalar density under diffeomorphisms). Taking into account definition (6) we now obtain: L CScov (A(ω, e),Ā(ω, e)) = κ 4π l √ g (g µν R µν − 2Λ)(13) meaning that the Chern-Simons Lagrangian (11) is mapped exactly (i.e. without undesiderable non invariant boundary terms) into the Hilbert Lagrangian for General Relativity with a negative cosmological constant, provided we set κ = l 4G(14) being G the Newton's constant (and setting c = 1). We remark that the Lagrangian (11), or equivalently (12), is the same "covariantized" Lagrangian already obtained in [8] through the use of the "transgression" formula established by Chern and Simons (see [18]). It was shown in [8] that a "covariantization" procedure can be applied to each Chern-Simons Lagrangian in dimension three, independently on the relevant gauge group of the theory. Variation of Noether Charges The approach here proposed to calculate the fundamental parameters of the theory, such as mass, angular momentum and gauge charges, is based on a geometrical Lagrangian formalism for classical field theories. In this framework conserved currents and conserved quantities can be calculated by means of the first and the second Noether theorem as shown in [7], [24], [27]. The Hamiltonian formalism for the theory can also be derived identifying the variation of the Hamiltonian with the variation of the Noether current with respect to a vector field transversal to a Cauchy hypersurface in spacetime [27]. The variation of energy is then naturally defined as the on-shell value of the variation of the Hamiltonian. The advantages which derive in using this approach are related to the fact that all quantities we are going to introduce (Noether currents, Noether charges and symplectic forms) are both covariant and gauge invariant, thereby having a global geometrical interpretation. Physically speaking this means that all formulae retain their validity independently on the observer, i.e. independently on the coordinate system in which formulae can be expressed and independently on the spacetime splitting into space + time. The whole theory is also independent on the addition of divergence terms to the Lagrangian. Hence we have not to care about choosing a representative inside the cohomology class of the Lagrangians. We only have to care that the representative Lagrangian be covariant in order to frame the whole theory of conserved quantities in a wellposed geometric background (this is the ultimate reason why we have chosen the Lagrangian (11) in place of (4): the latter Lagrangian is not gauge invariant!). Moreover we shall see that in presence of Killing vector fields the variation of the Noether charges, which are naively defined through integration on a (n − 2)dimensional surface B in spacetime, does not change inside the homology class of 2-dimensional surfaces to which B belongs (see [27]). Roughly speaking this is the mathematical property which will allow to formulate the first law of black holes mechanics. In addition, in the framework we shall develop we do not have to impose a priori boundary conditions to make the variational principle well defined. Boundary conditions just assume a fundamental role, a posteriori, in the formal integration of the variational equation which defines the Noether charges, e.g. the variation of energy. Different boundary conditions on the fields (corresponding to different ways in which the physical system can interact with the outside) can be imposed on the same variational equation leading to different physical interpretations of the results, e.g. internal energy for Dirichlet boundary conditions, free energy for Neumann boundary conditions and so on; see [2,11,17,27,33] We shall assume that the reader is already familiar with the geometrical language of fiber bundles and with the calculus of variations on jet bundles (see, e.g. [24,34,39]). We just recall few notions in order to fix the notation. As it is common use in a geometric approach to field theories the Lagrangian L(j k ϕ) = L(j k ϕ) ds is considered as a global horizontal m-form on the k-order prolongation of the configuration bundle Y −→ M (which is a fiber bundle over the base manifold M , with dim M = m), L is the Lagrangian density, ds is the volume form on M (in a coordinate chart ds = dx 1 ∧ . . . ∧ dx m ) while ϕ are the fields of the theory, considered as sections of the configuration bundle, i.e. ϕ : M −→ Y . In the sequel we shall be mainly concerned with first order theories, i.e. theories in which the Lagrangian depends only on the fields together with their first derivatives (for higher order theories we refer the interested reader to [27]). As it is well known, the variation of the Lagrangian, after integration by parts, splits into the sum of two terms, called, respectively, the Euler-Lagrange and the Poincaré-Cartan morphisms: δ X L(j 1 ϕ) =< E(L, j 2 ϕ), X > +d < F(L, j 1 ϕ), X >(15) where X is any vertical vector field on the configuration bundle (namely, X = δϕ ∂ ∂ϕ describes the infinitesimal deformation of the dynamical fields), while < , > denotes the canonical pairing between differential forms and vector fields. Locally: < E(L, j 2 ϕ), X > = ∂ L ∂ϕ A − d µ ∂ L ∂(dµϕ A ) X A ds < F(L, j 1 ϕ), X > = ∂ L ∂(dµϕ A ) X A ds µ (ds µ = ∂ µ ⌋ds)(16) where we have collectively labelled the fields with the index A = 1, . . . , n = dim Y − dim M . The Euler-Lagrangian morphism selects the critical sections ϕ (i.e. the physical field solutions) through the field equations E(L, j 2 ϕ) = 0. According to [24,40], for any given projectable vector field Ξ on the manifold Y projecting onto the vector field ξ on M and locally described by: Ξ = ξ µ ∂ µ + X A ∂/∂ϕ A , ξ = ξ µ ∂ µ(17) we say that the Lagrangian L admits a 1-parameter group of symmetries generated by the vector field Ξ if it satisfies the following property: δ Ξ L(j 1 ϕ) :=< δL, j 1 £ Ξ ϕ >= £ ξ L(18) where £ Ξ ϕ = T ϕ • ξ − Ξ • ϕ is the geometrically defined Lie derivative of the section ϕ with respect to Ξ and £ ξ denotes the usual Lie derivative of differential forms; see [34,39]. From equations (15) and (18) it follows that the Noether currents generated by the infinitesimal symmetry Ξ can be written as: Ȩ (L, ϕ, Ξ) =< F(L, j 1 ϕ), £ Ξ ϕ > −i ξ L(j 1 ϕ)(19) and satisfy the conservation law dȨ (L, ϕ, (19) is then a (m − 1)-form closed on-shell which can be integrated on any hypersurface Σ of spacetime M . The field theories we shall be concerned with from now on are the gauge natural theories (see [20,22,28,34]). In gauge natural theories the configuration bundle Y is associated to a given principal bundle P G −→ M , with Lie group G. Moreover each projectable vector field on P , which can be locally written as Ξ) = − < E(L, j 2 ϕ), £ Ξ ϕ >. The Noether currentΞ P = ξ µ (x)∂ µ + ξ i (x, g) ρ i (having denoted with ρ = (g ∂/∂g), g ∈ G, a basis for right invariant vector field on P in a trivialization (x, g) of P ) canonically induces a vector field (17) on the configuration bundle for which the property (18) holds true. Roughly speaking gauge natural theories are the ones which admit the group Aut(P ) of the automorphisms of a principal bundle P as group of symmetries. For instance, Yang-Mills theories on a dynamical background as well as the covariant Chern-Simons Lagrangian (11) are examples of gauge natural theories. Each vector field Ξ P = ξ µ ∂ µ + ξ i ρ i on the relevant principal bundle P induces on the bundle of connections (the sections of which are the connection 1-forms A i µ ) the vector field Ξ = ξ µ ∂ µ + Ξ i µ ∂ ∂A i µ Ξ i µ = −d µ ξ ν A i ν − C i jk A j µ ξ k − d µ ξ i(20) Notice that vertical vector fields, i. e. the generators of "pure" gauge transformations, denoted from now on as , are the ones for which ξ µ = 0; in this case Ξ i µ = −D µ ξ i . Each vector field (20) satisfies the property (18), thereby inducing a Noether current (19). Specifically, in the Chern-Simons theory (11), the relevant principal bundle P of the theory is a SL(2, IR) principal bundle and the connections A andĀ are two different sections of the associated configuration bundle Y = J 1 P/SL(2, IR) which is called the bundle of connections; see [34,39]. The generalization to gauge natural theories of the second Noether theorem (see [20,24,25,26]) states that in each gauge natural theory the Noether current can be canonically split, through an integration by parts, into two terms: Ȩ (L, ϕ, Ξ) =Ȩ (L, ϕ, Ξ) + d[U (L, ϕ, Ξ)](21) called, respectively, the reduced currentȨ , which identically vanishes on shell since it is proportional to field equations, and the superpotential U of the theory. The reduced currentȨ is unique while the superpotential U is unique modulo cohomology. It can be uniquely fixed by choosing a specific connection and integrating covariantly (see [26]). Example 3.1 The superpotential U (L CScov , A,Ā, Ξ) = 1/2 U µν ds µν written in local coordinates (with ds µν = ∂ ν ⌋∂ µ ⌋ds), associated to the Lagrangian (12) and relative to the vector field (20) has been calculated in [8]: U µν = κ 8π ǫ µνρ η ij B i ρ ξ j (V ) +ξ j (V )(22) where Ξ P = ξ µ ∂ µ + ξ i ρ i denotes now a generic vector field on the SL(2, IR) principal bundle P of the theory and ξ i (V ) = ξ i + A i µ ξ µ ,ξ i (V ) = ξ i +Ā i µ ξ µ(23) are the vertical parts of Ξ P with respect to A i µ andĀ i µ , respectively. We recall in fact that a vector field Ξ P on a principal bundle P can be split, once a given connection A i µ has been chosen, into the sum of its horizontal and vertical parts: Ξ P = ξ µ ∂ (h) µ + ξ i (V ) ρ i(24) where we have set: ∂ (h) µ = ∂ µ − A i µ ρ i ξ i (V ) = ξ i + A i µ ξ µ(25) Notice that both the components ξ i (V ) andξ i (V ) in (23) trasform as vectors under gauge transformations since the non-tensorial character of ξ i is cured by the non-tensorial character of A i µ . Assuming that the topology of spacetime is diffeomorphic to Σ × IR (see Appendix A), the Noether charge relative to the vector field Ξ can be calculated from the formula: Q B (L CScov , A,Ā, Ξ) = B U (L CScov , , A,Ā, Ξ)(26) where B ≃ S 1 is a 1-dimensional surface embedded into Σ. Now, it would seem physically reasonable to define the energy as the Noether charge relative to a spacetime vector field ξ transverse to Σ, i.e. by simply setting Ξ = ξ µ ∂ µ in formula (22), i.e. ξ i = 0. Nevertheless, as it was already pointed out in [8], this prescription is not admissible from a mathematical viewpoint since the group Diff(M ) of spacetime diffeomorphisms is not a global invariance group for the theory. Indeed, in gauge natural theories the symmetry group is the group Aut(P ) and Diff(M ) is not a canonical and natural subgroup of it. Namely, the splitting of Aut(P ) into Diff(M ) and is by no means canonical and it is meaningful only locally (we shall enter into the details of the matter below). Roughly speaking, this means that, given a spacetime vector field ξ, there is no canonical way to define a vector field Ξ P on the principal bundle and consequently to define the associated vector field Ξ on the configuration bundle Y which enters (26). However, we can define, in a non canonical but nevertheless global way, a lift of spacetime vector fields through a dynamical connection 1 . In the theory under examination there are three different dynamical connections: the original connections A i µ andĀ i µ and their combination ω i µ = (A i µ +Ā i µ )/2; accordingly, different lifts can be defined. In [8] the horizontal lift with respect to the connectionĀ was considered. These choices correspond to set in (22): ξ j (V ) = 0, ξ j = −Ā j µ ξ µ(27) thus leading to the Noether charge: Q B (L CScov , Ξ) = B κ 8π η ij B i ρ B j µ ξ µ dx ρ(28) (other choices will be considered in detail hereafter). We can now check the viability of formula (28) by specifying it for the BTZ solution. In the sl(2, IR) basis (2) and in the coordinate system (x µ ) = (t, ρ, ϕ) on spacetime, the connections A = A i µ dx µ J i andĀ =Ā i µ dx µ J i corresponding to the (exterior of) the BTZ black hole solution are given by [3,14,15] (±) A i µ =   ± r+±r− l 2 sinh ρ 0 r+±r− l sinh ρ 0 ±1 0 r+±r− l 2 cosh ρ 0 ± r+±r− l cosh ρ  (29) where the (+) is referred to A while (−) is referred toĀ. For the triad field e i µ we have: e i µ = l 2 (+) A i µ − (−) A i µ =   r+ l sinh ρ 0 r − sinh ρ 0 l 0 r− l cosh ρ 0 r + cosh ρ  (30) where M = r 2 + + r 2 − 8Gl 2 , J = r + r − 4Gl(31) are, respectively, the mass and the angular momentum of the black hole. The BTZ metric g of components g µν = η ij e i µ e j ν then becomes: g = − sinh 2 ρ(r + /l dt − r − dφ) 2 + l 2 dρ 2 + cosh 2 ρ(r − /l dt − r + dφ) 2(32) Defining the surface B as the surface of constant t and constant ρ, from formula (28) and the solution (29) we obtain: Q B (L CScov , (±) A,∂ t ) = J l Q B (L CScov , (±) A,∂ φ ) = 1 4Gl cosh 2 ρ r 2 + − sinh 2 r 2 −(33) 1 The best known of these ways is the horizontal lift which consists simply in setting ξ i (V ) = 0 in formula (24). having denoted with∂ t and∂ φ , respectively, the lift of the spacetime vector fields ∂ t and ∂ φ through the prescription (27). These results do not agree with the expected physical quantities M and J, respectively, even in the limit ρ −→ ∞; see [5,9,10,14]. This can be seen as an hint that the superpotential or the lift chosen in [8], or both, are not suited in calculating physical observables, at least in the Chern-Simons formulation of (2 + 1) gravity. Let us then consider again the expression (19) and let us perform the variation of the Noether current Ȩ (L, ϕ, Ξ) with a (vertical) vector field X = δϕ ∂ ∂ϕ . We obtain: δ X Ȩ (L, ϕ, Ξ) = δ X < F(L, j 1 ϕ), £ Ξ ϕ > −i ξ [δ X L(j 1 ϕ)] = δ X < F(L, j 1 ϕ), £ Ξ ϕ > −£ ξ < F(L, j 1 ϕ), X > −i ξ < E(L, j 2 ϕ), X > +d[i ξ < F(L, j 1 ϕ), X >] = ω(ϕ, X, £ Ξ ϕ) + d(i ξ < F(L, j 1 ϕ), X >) −i ξ < E(L, j 2 ϕ), X >(34) where, in passing from the first to the second equality, we made use of (15) and of the rule £ ξ = d i ξ + i ξ d. In (34) we have denoted with ω(ϕ, X, £ Ξ ϕ) the naive symplectic current [13,31]: ω(ϕ, X, £ Ξ ϕ) = δ X < F(L, j 1 ϕ), £ Ξ ϕ > −£ ξ < F(L, j 1 ϕ), X >(35) For field theories described through a first order Lagrangian the Poincaré-Cartan morphism < F(L, j 1 ϕ), X > depends on the fields ϕ together with their first derivatives and depends linearly on the components of the vector field X (see (16)). Hence formula (35), by using Leibniz rule, can be rewritten: ω(ϕ, X, £ Ξ ϕ) = < δ X F(L, j 1 ϕ), £ Ξ ϕ > − < £ Ξ F(L, j 1 ϕ), X > + < F(L, j 1 ϕ), δ X (£ Ξ ϕ) − £ Ξ X >(36) Moreover, for first order theories, the Poincaré-Cartan morphism, which in practice is given by the derivatives of the Lagrangian with respect to the first derivatives of the fields, is nothing but the mathematical object describing the generalized momenta conjugated to the dynamical fields (see [25]). In trying to establish a correspondence with Classical Mechanics we could say that the first two terms in (36) correspond to the expression δpq −ṗ δq which generates the Hamilton equations of motion through the rule δH =qδp −ṗδq (in this analogy the time derivative of Classical Mechanics is replaced, in field theories, by the Lie derivative £ Ξ ). What about the further contribution to (36)? Carrying on the analogy with Classical Mechanics this term would correspond to the expression p [δ(d t q) − d t (δq)] which is clearly equal to zero in Classical Mechanics since the time derivative commute with the variation of the configuration variables. The situation is quite different in field theories. To realize this property let us consider gauge theories (i.e. theories where the dynamical field is the gauge potential A i µ ). In this case we have: δ X (£ Ξ A i µ ) = δ X ξ ρ d ρ A i µ + d µ ξ ρ A i ρ + d µ ξ i + C i jh A j µ ξ h = ξ ρ d ρ (δ X A i µ ) + d µ ξ ρ (δ X A i ρ ) + C i jh (δ X A j µ )ξ h +d µ (δ X ξ i ) + C i jh A j µ (δ X ξ h ) = £ Ξ (δ X A i µ ) + D µ (δ X ξ i )(37) so that: δ X (£ Ξ A i µ ) − £ Ξ (δ X A i µ ) = D µ (δ X ξ i )(38) If we consider the case in which the components ξ i are built out of the dynamical fields (see, e.g. formula (27)) the variations δ X ξ i are different from zero and the term (38) does not vanish. However, through an integration by parts procedure, the naive symplectic current (36) for gauge theories splits as: ω(ϕ, X, £ Ξ ϕ) =ω(ϕ, X, £ Ξ ϕ) + d[τ (ϕ, X, Ξ)] + f (E(L, ϕ, X) (39) whereω (ϕ, X, £ Ξ ϕ) =< δ X F(L, j 1 ϕ), £ Ξ ϕ > − < £ Ξ F(L, j 1 ϕ), X >(40) is the reduced symplectic (m − 1)-form and f (E(L, ϕ, X)) denotes a term proportional to the equations of motion. 2 Collecting together formulae (34) and (39) we have: δ X Ȩ (L, ϕ, Ξ) =ω(ϕ, X, £ Ξ ϕ) + d τ (ϕ, X, Ξ) + i ξ < F(L, j 1 ϕ), X > + E.M.(41) where E.M means terms proportional to the Euler-Lagrange morphism E(L, j 2 ϕ, X) and hence vanishing on-shell. On the other hand, from formula (21) we have δ X Ȩ (L, ϕ, Ξ) = δ XȨ (L, ϕ, Ξ) + d[δ X U (L, ϕ, Ξ)](42) Comparing (41) with (42) we finally obtain: 2 We stress that a splitting similar to (39) occurs also for higher order natural theories, e.g. General Relativity, even though its origin is quite different (see [27]). Indeed, the divergence term τ does not arise from the third term of (36), which is identically vanishing for natural theories, but from an integration by parts applied to the first two contributions. In both cases, i.e. in the first order gauge natural theories considered herein as well as in the higher order natural theories treated in [2,27], it is the analogy with Classic Mechanics which suggests how to perform the splitting (39). All the terms in (39) which are not in the form δpq −ṗ δq have to be decomposed, through integrations by parts, under the form d[τ (ϕ, X, Ξ)] + f (E(L, ϕ, X)). δ XȨ (L, ϕ, Ξ) + d[δ X U (L, ϕ, Ξ) − i ξ < F(L, ϕ), X > −τ (ϕ, X, Ξ)] = =ω(ϕ, X, £ Ξ ϕ) + E.M.(43) This formula can be seen as the counterpart in field theories of the variational equation δH =q δp −ṗδq + [d t (∂L/∂q) − ∂L/∂q]δq of Classical Mechanics. This analogy suggests to define the variation δ X H of the Hamiltonian density conjugate to the vector field Ξ as follows: δ X [H(L, ϕ, Ξ)] = (44) = δ XȨ (L, ϕ, Ξ) + d[δ X U (L, ϕ, Ξ) − i ξ < F(L, ϕ), £ Ξ ϕ > −τ (ϕ, X, Ξ)] so that δ X H(L, ϕ, Ξ) =ω(ϕ, X, £ Ξ ϕ) + E.M.(45) Given a Cauchy hypersurface Σ the variation δ X H of the Hamiltonian is simply defined as δ X H = Σ δ X H. We remark that the right hand side of the equation (43) does not contain divergence terms at all. This means that the divergence terms d[δ X U − i ξ F − τ ] in (43) exactly cancel out the divergence terms arising in the variation δ XȨ , i.e. d ∂Ȩ ∂(dϕ) δ X ϕ = −d(δ X U − i ξ F − τ )(46) hence leading to a formula for δH which is divergence-free and which gives rise to the proper Hamilton equations of motion (45). The definition (44) of the variation of the Hamiltonian density is close in spirit with (and it can be seen as a covariant generalization of) the original idea of Regge and Teitelboim to handle boundary terms: all boundary terms arising in the variation δH of the Hamiltonian are added (with a minus sign) into the definition of the naive Hamiltonian in order to define (the variation of) a new Hamiltonian function endowed with a well defined variational principle and hence suited to be used as the generator of the allowed surface deformations; see [27,32,38]. The terms of (44) under the exterior differential are the only ones surviving on shell, i.e. when we consider a field ϕ which is a solution of field equations (we remind thatȨ is proportional to field equations) and a variation δ X performed along the space of solutions (i.e. δ XȨ = 0). The variation δ X Q of the Noether charges relative to a particular solution ϕ, relative to the surface Σ and relative to the vector field Ξ, are hence defined as the integral on the boundary ∂Σ: δ X Q ∂Σ (L, ϕ, Ξ) = ∂Σ δ X U (L, ϕ, Ξ) − i ξ < F(L, ϕ), X > −τ (ϕ, X, Ξ) (47) From equation (45) it then follows that δ X Q satisfies, on-shell, the master equation: δ X Q(L, ϕ, Ξ) = Σω (ϕ, X, £ Ξ ϕ)(48) Remark 3.2 Definition (47) and equation (48) deserve now some further comment. 1-First of all we stress that the (variation of) conserved quantities is not defined only through the (variation of) of the superpotential since two more corrective terms have to be added in the definition. This is one of the reasons why formula (33) leads to a wrong result. 2- The definition (47) do not depend on the representative L chosen inside the homology class [L] of Lagrangians (two Lagrangians L and L ′ belong to the same class [L] if they differ only for divergence terms, which entails that they give rise to the same equations of motion). This property descends from (46): all the terms (δ X U − i ξ F − τ ) which constitute the density of the the variation of the charges can be obtained directly and altogether from the reduced current Ȩ . SinceȨ is basically a linear combination of field equations with coefficients given by the vector field Ξ (see e.g. [20], [24] for a rigorous proof of this statement) the only mathematical data we need for defining δ X Q are the equations of motion and the generator of symmetries Ξ. Nevertheless, even though (46) can be used, in practice, as an operative schema to calculate explicitly δ X Q (we point out that this is exactly the approach followed by Julia and Silva in [32]), from a theoretical point of view, formula (46) hides the symplectic informations which are instead manifest in (48). (40) we see that, if Ξ is a Killing vector for the solution, i. e. £ Ξ ϕ = 0, the reduced symplectic formω does vanish. From (48) we obtain then: 3-Let us then consider formula (48). Recalling definition δ X Q(L, ϕ, Ξ) = ∂Σ δ X Q(L, ϕ, Ξ) = 0 (δ X Q = δ X U − i ξ F − τ )(49) This latter equation is nothing but the conservation law of conserved quantities. Indeed, let us assume that a metric on spacetime can be built out of the dynamical fields, let us consider a (local) foliation of the m-dimensional spacetime M into space + time and let us consider a timelike hypersurface B, namely a world tube in M ; referring to the example (32) B would correspond to a surface with constant ρ. Let us then denote by B t1 and B t2 , respectively, the (m − 2)surfaces generated by the intersection of B with two spacelike hypersurfaces Σ t1 and Σ t2 at constant times t 1 and t 2 . Since B t1 ∪ B t2 defines a boundary in B, from (49) we infer that: Bt 1 δ X Q(L, ϕ, Ξ) − Bt 2 δ X Q(L, ϕ, Ξ) = 0 (50) so that, if £ Ξ ϕ\| B = 0,then δ X Q is conserved in time. On the other hand let us consider a portion D of a spacelike hypersurface Σ t0 at a given time t 0 (for example a generic surface of constant t for the BTZ spacetime (32)) and let us suppose that the (oriented) boundary ∂Σ is formed by the disjoint union of two (m − 2) surfaces S and S ′ , e.g. two circles of constant ρ in the spacetime the metric of which is defined by (32). Assuming that Ξ is a Killing vector on Σ, we have S δ X Q(L, ϕ, Ξ) − S ′ δ X Q(L, ϕ, Ξ) = 0 (51) Referring now to the black hole solution and denoting by S and S ′ , respectively, the spatial infinity and the horizon, formula (51) explains why conserved quantities such as mass and angular momentum, which are naively calculated at spatial infinity S, are related to properties of the horizon S ′ : this comes from the homological properties of δ X Q. 4-Formula (47) has a drawback: in fact it provides only the variation δ X Q of conserved quantities and the conserved quantities Q are obtained only after a formal integration. Nevertheless the integrability of (47) is not a priori assured. It depends on the boundary conditions δ X ϕ\| ∂Σ we impose. Starting from the same expression, different boundary conditions may lead to different results corresponding to different physical interpretations of conserved quantities. For example the recipe (47) has been proven to give the expected values for the quasilocal energy for Einstein and Einstein-Maxwell theories once the variational equation is solved with Dirichlet boundary conditions: see [2] and [27]. We want hereafter to generalize the above formalism to the case of Chern-Simons theories with applications to Chern-Simons AdS 3 gravity and to BTZ black holes. We assume as a Lagrangian for the theory the covariant Lagrangian (12). The superpotential is given by (22) and (23) while the Poincaré-Cartan morphism has been calculated in [8] to be the following: < F(L CScov , A,Ā), X >= κ 8π {ǫ µνρ η ij B j ρ δ X (A i ν +Ā i ν )}ds µ(52) Hence, from definition (36) and the property (38) we obtain (compare with (39)): ω µ (A,Ā, X, Ξ) =ω µ (A,Ā, X, Ξ) + d ν τ µν (A,Ā, X, Ξ) − κ 8π ǫ µνρ η ij δ X ξ i {F j νρ +F j ρν }(53) with: ω µ (A,Ā, X, Ξ) = κ 8π ǫ µνρ η ij [δ X B j ρ £ Ξ (A i ν +Ā i ν ) − £ Ξ B j ρ δ X (A i ν +Ā i ν )] = κ 2πl ǫ µνρ η ij [δ X e j ρ £ Ξ ω i ν − £ Ξ e j ρ δ X ω i ν ](54) and τ µν (A,Ā, X, Ξ)) = κ 4π ǫ µνρ η ij B j ρ δ X (ξ i )(55) We stress that the third term in the right hand side of (53) identically vanishes on shell, so that (53) reproduces exactly the structure (39). We also stress that the reduced symplectic formω as given by (54) provides us the correct symplectic structure for General Relativity once we have identified e j ρ with the vielbein and ω i ν with the spin connection, according to (6). Then e j ρ and ω i ν can be recognized as dynamical variables conjugated to each other, i.e. they form a pair of (q, p) variables in the appropriate phase space. Inserting (22), (52) and (55) into the definition (47) we finally obtain: δ X Q(L CScov , A,Ā, Ξ) = κ 4π η ij B [ξ i (V ) δ X A j µ −ξ i (V ) δ XĀ j µ ]dx µ (56) with ξ i (V ) = ξ i + A i µ ξ µ ,ξ i (V ) = ξ i +Ā i µ ξ µ . Notice that the above definition for the variation of the charges in Chern-Simons theory is clearly covariant as well as gauge invariant (indeed δ X A, δ XĀ , ξ (V ) andξ (V ) are all gauge vectors). Generalized Kosmann Lift Once the explicit formula (56) for the variation of charges has been established we are faced with another problem. Our goal is to make use of the same formula (56) in order to calculate different physical quantities, such as energy and angular momentum, for the BTZ black hole. To do that we have to appropriately choose a vector field Ξ on the configuration bundle Y the projection ξ of which on M is the generator of time traslations and angular rotations. As already remarked this is not an obvious choice. Indeed the vector fields Ξ projecting onto the same spacetime vector field are far from being unique! Given a vector Ξ all the vectors obtained from it through the addition of a generic vertical vector give in fact rise to the same projected vector. Hence the fundamental problem is to find a physically reasonable mathematical rule to lift up to the configuration bundle a given spacetime vector field. While in natural theories there exists a preferred rule, namely the natural lift (see, e.g. [34]), this is not at all the case in gauge natural theories. We now try to reformulate the problem in terms of local coordinates. In the gauge natural theory we are analysing, i.e. the theory described by (11), an infinitesimal generator of Lagrangian symmetries is any vector field Ξ of the kind (20), which is functorially associated to a given vector field: Ξ P = ξ µ ∂ µ + ξ i ρ i(57) on the relevant SL(2, IR) principal bundle of the theory. Given a spacetime vector field ξ = ξ µ ∂ µ the problem to define its lift up to the configuration bundle corresponds to the problem of defining a rule for constructing the components ξ i in (57) starting from the components ξ µ , the dynamical fields A i µ ,Ā i µ together with their derivatives. This rule must be mathematically well-defined in the sense that ξ i (V ) andξ i (V ) must transform as vectors under gauge transformations. For example, this requirement forbids us to simply set ξ i = 0 since this choice in general is not globally defined. In the sequel we shall consider some of the admissible lifts we are allowed to construct. One of them is known in geometrically oriented literature as the generalized Kosmann lift of vector fields. The Kosmann lift was defined for the first time in [22] in order to establish a connection between the ad hoc definition of Lie derivative of spinor fields given in [35] and the general theory of Lie derivatives on fiber bundles. We shall not enter here into the mathematical details the definition of the Kosmann lift is based on. For this issue we refer the interested reader to [28] where an exhaustive bibliography can also be found. We shall only specialize to the present case the formalism there developed. To this end we just outline that the generalized Kosmann lift we are going to define takes its values in the Lie algebra so(1, 2) ≃ sl(2, IR). The (generalized) Kosmann lift is then available in the theory we are analysing since SO(1, 2) is a reductive Lie subgroup of GL(3, IR); see [28]. Given a spacetime vector field ξ = ξ µ ∂ µ the Kosmann lift K(ξ) of ξ on the principal SL(2, IR) bundle locally reads as: Ξ P = K(ξ) = ξ µ ∂ µ + ξ i (K) ρ i (58) ξ i (K) = 1 2 η ij ǫ jkl ξ [k h η l]h (59) ξ k h = e k ρ (e µ h d µ ξ ρ − ξ µ d µ e ρ h )(60) We again point out that the above expression for ξ i (K) is obtained via skewsymmetrization of the components ξ k h and hence it takes values in the Lie algebra so(1, 2). Moreover the components ξ k h , in their turn, are built out only from the components ξ µ and their derivatives. They are independent on any dynamical connection, as it is clearly shown by (60). However using the spin connection induced by the frame e a µ : ω a bµ = e a α (Σ α βµ − Σ β α µ + Σ µβ α )e β b , Σ α βµ = e α c ∂ [β e c µ](61) the expression for the Kosmann lift can be rewritten in the following way: ξ k h = e k ρ (e µ h d µ ξ ρ − ξ µ d µ e ρ h ) = e µ h (ω) ∇ µ (e k ν ξ ν ) − ω k hµ ξ µ(62) The latter formula shows explicitly that the components ξ a (V ) = ξ a (K) + A a µ ξ µ andξ a (V ) = ξ a (K) +Ā a µ ξ µ correctly transform as gauge vectors. Indeed we have: ξ a (V ) = 1 2 η ab ǫ bcd e µd (ω) ∇ µ (e c ν ξ ν ) + (A cd µ − ω cd µ )ξ µ = 1 2 η ab ǫ bcd e µd (ω) ∇ µ (e c ν ξ ν ) + 1 l e a µ ξ μ ξ a (V ) = 1 2 η ab ǫ bcd e µd (ω) ∇ µ (e c ν ξ ν ) + (Ā cd µ − ω cd µ )ξ µ = 1 2 η ab ǫ bcd e µd (ω) ∇ µ (e c ν ξ ν ) − 1 l e a µ ξ µ(63) The relations (59) and (62) between the infinitesimal symmetry generators ξ µ and ξ a (K) suggest how different global lifts can be defined. Indeed, in Chern-Simons theory with g = sl(2, IR) and three dynamical connections A,Ā and ω = A+Ā 2 , we have the possibility to introduce different lifts which are mathematically well defined global lifts. They can be obtained formally from (62) by arbitrarily replacing one of the dynamical connections A,Ā and ω into the expression for the covariant derivative and another one into the last term of (62). For example we can choose the lifts defined by means of: ξ k h = e µ h (A) ∇ µ (e k ν ξ ν ) − ω k hµ ξ µ(64) where we have used the dynamical connections A and ω. Interchanging in this way the dynamical connections A,Ā and ω we can define nine different lifts; notice, however, that two of them, namely, the one with A,Ā and the one with A, A, are both identical to the Kosmann lift (62) owing to the splitting (6). These lifts are global as much as the Kosmann lift and there is no mathematical prescription to select one among them. However it can be easily shown that each different lift we can define gives different values for the Lie derivatives of the dynamical fields and also different Noether conserved quantities. The choice we make among them in evaluating the variation of global charges (following the recipe of formula (56)) has therefore to be dictated by pure physical considerations. We then proceed as follows. Evaluating the different possible lifts for the two spacetime vector fields which generate, respectively, time translations and angular rotations for BTZ solution (29), we construct the corresponding variations of conserved quantities according to (56) and we look at the ones reproducing the expected values of mass and angular momentum. That seems to be a physically acceptable criterium to select among the different lifts. For instance, performing the calculations using the Kosmann lift we obtain the correct values for mass and angular momentum: Q(L CScov , K(∂ t ), B) = r 2 + +r 2 − 8Gl 2 = M (65) Q(L CScov , K(∂ φ ), B) = r+r− 4Gl = J(66) independently on the radius of the circle B (to be rigorous, the results (65) and (66) are correct modulo a constant of integration which can be viewed as the charge of a background solution and can therefore be set equal to zero fixing, in this way, the zero level for the measurement of the charges). For each one of the other six possible lifts we obtain instead non integrable expressions which, obviously, have no interest here. Just to show one let us consider the lift of formula (64); we obtain then for the conserved quantity associated with the lift of ξ = ∂ t the following: δQ = κ l 3 r + δr + + r − δr − − r − cosh 2 (ρ) δr + + r + sinh 2 (ρ) δr −(67) which is manifestly non integrable since δ δr+ δQ δr− = δ δr− δQ δr+ . We stress that the only lift providing the expected results is then the generalized Kosmann lift. This justifies a posteriori the choice of the Kosmann lift to construct the infinitesimal generator of symmetries for our theory. We also stress again that, among all the possible lifts, the Kosmann lift is the only one which does not involve a connection whatsoever (see (60)). In a practical language we could say that, among the different lifts, the generalized Kosmann lift is the "most natural". 3 Let us now consider the results (65) and (66). As we already pointed out these numerical values are independent on the radius of the circle B on which integration is performed. This fact is not surprising. Indeed we know, see equation (48) and expression (54), that, on shell, the master formula (56) obeys the equation: κ 4π η ij ∂Σ [ξ i (V ) δ X A j µ −ξ i (V ) δ XĀ j µ ]dx µ = Σ κ 2πl ǫ µνρ η ij [δ X e j ρ £ Ξ ω i ν − £ Ξ e j ρ δ X ω i ν ](69) so that the left hand side is a homological invariant iff the right hand side is vanishing. Let us then calculate the Lie derivatives £ Ξ e j ρ and £ Ξ ω i ν . From the general formula £ Ξ A i µ = ξ ρ F i ρµ + D µ ξ i (V ) and the splitting (6) we have: £ Ξ e a µ = ξ ρ d ρ e a µ + d µ ξ ρ e a ρ + ǫ a bc e b µ ξ c £ Ξ ω a µ = ξ ρ d ρ ω a µ + d µ ξ ρ ω a ρ + ω Dµ ξ a(70) When we take Ξ equal to the (generalized) Kosmann lift (60) we obtain: £ K(ξ) e a µ = 1 2 e νa £ ξ g µν £ K(ξ) ω i kµ = − 1 2 e ν k e ρi D µ (£ ξ g ρν ) + £ ξ Γ ρ νµ e i ρ e ν k(71) where £ ξ denotes the usual Lie derivative with respect to the spacetime vector field ξ. Notice that £ K(ξ) g µν = η ij (£ K(ξ) e i µ ) e j ν + e i µ (£ K(ξ) e j ν ) = £ ξ g µν = ∇ µ ξ ν + ∇ ν ξ µ (72) so that the generalized Kosmann lift reproduces the usual Lie derivative of the metric with respect to diffeomorphisms. Both derivatives (71) are vanishing for the BTZ metric when ξ is either ∂ t , ∂ ϕ or any linear combinations of them with constant coefficients. In all these cases the right hand side of (69) is vanishing so that: δ X Q(L CScov , K(∂ t + Ω∂ ϕ ), B 1 ) = δ X Q(L CScov , K(∂ t + Ω∂ ϕ ), B 2 )(73) where Ω is a constant and B 1 and B 2 are two homologic surfaces, i.e. they form the boundary ∂Σ of a two dimensional surface Σ. The above formula allows us to formulate the first law of black hole dynamics. Indeed if we set Ω equal to the (constant) angular velocity of the BTZ black hole (see [3]): Ω = r − 8Glr +(74) and B 1 equal to any circle of constant radius enclosing the black hole horizon the left hand side of (73) turns out to be: δQ(K(∂ t + Ω∂ ϕ ), B 1 ) = δQ(K(∂ t ), B 1 ) + ΩδQ(K(∂ ϕ ), B 1 ) = δM + ΩδJ(75) where in the last equality we made use of the results (65) and (66). On the other hand if in (73) we take B 2 equal to the outer horizon H of the black hole (i.e. ρ = 0) we have, after some algebraic calculations: δQ(K(∂ t + Ω∂ ϕ ), H) = T δS (76) where T = (r 2 + −r 2 − ) 2πl 2 r+ is the temperature for the BTZ black hole and S = (2πr+) 4G is one quarter of the horizon area. Equating (75) and (76) we obtain that the first law of black holes mechanics T δS = δM + ΩδJ(77) holds true also in the domain of a Chern-Simons formulation of (2 + 1) gravity. Transition to General Relativity Variables In this section we shall show that formula (56), once we specialize it for the Kosmann lift and we make use of the splitting (6), reproduces exactly the formula for the variation of conserved quantities in General Relativity found elsewhere, see [27]. The notations and the basic formulae relative to the ADM foliation of spacetime entering the calculations of this section are summarized in the Appendix A. For the sake of clarity we recall here the formula (56): δ X Q B (L CScov , A,Ā, Ξ) = κ 4π B η ij [ξ i (V ) δA j µ −ξ i (V ) δĀ j µ ]dx µ , κ = l 4G(78) We now insert the splitting (6) for the connections A andĀ in the above expression , and we make use of the expression (63) for ξ i (V ) andξ i (V ) obtained via the Kosmann lift. This yields δ X Q B (L CScov , A,Ā, K(ξ)) = κ 4πl B ǫ aij [e νj e i ρ ∇ ν ξ ρ δe a µ + e a ν ξ ν δω ij µ ]dx µ (79) where ω a µ = 1 2 ǫ a ij ω ij µ . In the following we shall calculate the conserved quantities (79) with respect to the generalized Kosmann lift of a spacetime vector field ξ µ = N u µ + N µ (see Appendix A for the notation) and our goal will be to express the above formula in terms of metric quantities adapted to an orthogonal foliation of spacetime. First of all we remind that, in the notations summarized in the Appendix A, we have: B √ gf µ dx µ = B f µ ǫ µνρ n ν u ρ √ σ(80) for any 1-form f = √ gf µ dx µ , being σ = det(σ µν ). By making use of the properties ǫ aij e i µ e j ν = det(e) ǫ ρµν e ρ a , ǫ µνρ ǫ αβγ = δ µνρ αβγ , taking into account (90), (94) and the relations e λ a δe a λ = δ √ g/ √ g, √ g = N V √ σ (see Appendix A) the first term under integration in (79) can be expressed as: B ǫ aij e νj e i ρ ∇ ν ξ ρ δe a µ dx µ = B ∇ µ ξ ν (u µ n ν − u ν n µ ) δ √ σ(81) If now we apply Leibniz rule: u ν ∇ µ ξ ν = ∇ µ (u ν ξ ν ) − ξ ν ∇ µ u ν = −∇ µ N − ξ ν ∇ µ u ν (82) n ν ∇ µ ξ ν = ∇ µ (n ν ξ ν ) − ξ ν ∇ µ n ν = −ξ ν ∇ ν n ν(83) and we recall formulae (99) and (98) we get: B ǫ aij e νj e i ρ ∇ ν ξ ρ δe a µ dx µ = B δ √ σ[u µ ξ ν Θ µν + n µ ∇ µ N − n µ ξ ν K µν ] = = B δ √ σ[2N (a µ n µ ) − 2N µ n ν K µν ](84) where we made use of h µ α a µ = h µ α ∇ µ N/N . Calculations are more involved for the second term of (79). We start from the compatibility condition: D γ e i µ = ∇ γ e i µ + ω i jγ e j µ = 0 (∇ γ e i µ = d γ e i µ − Γ ρ µγ e i ρ )(85) to express the spin connection as ω i jµ = −e ν j ∇ µ e i ν . After some algebraic calculations we then obtain: B ǫ aij e a ν ξ ν δω ij µ dx µ = B ξ ν δ γαβ νµσ u α n β e µ i e σ j δe j ρ ∇ γ e iρ + δ(∇ γ e iσ ) √ σ = 2 B √ σ[ξ γ δ(n β ∇ γ u β ) − N δ(∇ γ n γ ) + N e iβ δe γ i ∇ γ n β ] Since e i(β δe i γ) = 2δg βγ , using formulae (93) together with (99) and (98), we can rewrite: B ǫ aij e a ν ξ ν δω ij µ dx µ = 2 B √ σ{N δĶ − N α δ(n β K β α ) + N 2 K αβ δσ αβ } (86) = B 2N δ( √ σĶ ) − 2 √ σN α δ(n β K β α ) − N √ σ(Ķ σ αβ − Ķ αβ )δσ αβ Summing up the two terms (84) and (86) and multiplying the result with κ/4πl = 1/16πG, we finally obtain the explicit formula for the variation of the conserved quantity relative to the Kosmann lift of the vector field ξ. It reads: δ X Q B (L CScov , A,Ā, K(ξ)) = B N δ X Ȩ − N α δ X J α + N √ σ 2 s αβ δ X σ αβ (87) where:            Ȩ = 1 8πG √ σĶ J α = 1 8πG √ σσ µ α K ν µ n ν s µν = 1 8πG [(n α a α )σ µν − Ķ σ µν + Ķ µν ](88) are, respectively, the quasilocal energy density, the quasilocal angular momentum density and the surface pressure; see [9,11]. Formula (87) reproduces exactly the formula for the variation of conserved quantities found in [6,9,11,27,33] for General Relativity. We stress again that the full correspondence between the variation of conserved quantities in the SL(2, IR) Chern-Simons (covariant) theory and (2 + 1) General Relativity essentially depends on the choice of the generalized Kosmann lift, thereby selecting the Kosmann lift, whenever it can be defined (see [28]), as the preferred lift to be considered in the domain of gauge natural theories. This is also in accordance with the results of [29,37] where the Kosmann lift was used to calculate the superpotential in the tetrad-affine formulation of General Relativity, as well as with the results of [21] where the same lift was considered in the domain of BCEA theories. Nevertheless we again point out that the generalized Kosmann lift is just one among the various possibilities and there does not exist a mathematical reason to select it, while there exists only, a posteriori, a physical justification for its choice; see [37]. Acknowledgments We are grateful to A. Borowiec of the University of Warsaw and to L. Fatibene, M. Ferraris and M. Godina of the University of Torino for useful discussions and suggestions on the subject. We mention that this research has been performed under no support from the Italian Ministry of Research. A The Orthogonal Foliation of Spacetime We consider a three dimensional region D ⊆ M of a Lorentzian three dimensional manifold (M, g) and a foliation of it into spacelike hypersurfaces {Σ t }, being t ∈ IR the parameter of the foliation. Denoting by B t the boundary B t = ∂Σ t of each leaf of the foliation the union B = ∪ t∈I R B t defines a timelike hypersurface, the unit normal of which we denote by n µ . We shall restrict the attention solely to the case of orthogonal foliations, i.e. foliations for which it holds u µ n µ \| B = 0, having denoted by u µ the components of the unit vector field which is everywhere orthogonal to each Σ t (see, e.g. [11,12]). On the hypersurface B the spacetime metric g can be decomposed as: g µν = σ µν + n µ n ν − u µ u ν(89) where σ µν is the metric induced by g µν on the surface B t . The metric can be also expressed in terms of the triad fields e i µ as g µν = η ij e i µ e j ν where η ij = diag(−1, 1, 1). In the sequel, on each surface B t we shall use the following notation: u µ = e 0 µ n µ = e 1 µ(90) so that g results to be: g µν = σ µν + e 1 µ e 1 ν − e 0 µ e 0 ν . Obviously we have: u µ u µ = e 0 µ e 0µ = −1, n µ n µ = e 1 µ e 1µ = 1, u λ e λ a = δ 0 a and n λ e λ a = δ 1 a . In a system of coordinates (t, ρ, ϕ) adapted to the foliation and for which B is a constant ρ hypersurface, the components of the vectors u µ , n µ can be expressed as: u µ = (−N, 0, 0) n µ = (0, V, 0)(91) where N is the ordinary lapse in a foliation-adapted ADM decomposition of the metric while V is called the radial lapse. The vector field ξ = ∂ t is defined, in terms of the ADM lapse and shift, as: ξ µ = N u µ + N µ(92) and the orthogonal condition u µ n µ \| B = 0 implies ξ µ n µ \| B = N µ n µ \| B = 0. The variations δ X g of the metric with respect to a vertical vector field X = δg µν ∂ ∂gµν can be written as: δ X g µν = − 2 N u µ u ν δN − 2 N h α(µ u ν) δN α + h α (µ h β ν) δh αβ(93) where h µν = g µν +u µ u ν are the components of the metric on each Σ t . Moreover, from (91) we have: δu µ = δN N u µ (94) δn µ = δV V n µ The extrinsic curvature of the generic hypersurface Σ t embedded in M is defined by: K µν = −h α µ ∇ α u ν(95) while the extrinsic curvature of the hypersurface B embedded in M results to be: Θ µν = −γ α µ ∇ α n ν(96) where γ µν = g µν −n µ n ν is the metric on B induced by g. The extrinsic curvature of B t as a surface embedded in Σ t can be expressed, for each t ∈ IR, as: Ķ µν = −σ α µ D α n ν(97) where D is the metric covariant derivative with respect to the metric h of Σ. The extrinsic curvatures of B, Σ and B can be expressed each one in terms of the others via the relation [11,12]: Θ µν = Ķ µν + (n α a α )u µ u ν + 2σ α(µ u ν) K αβ n β where a is the covariant acceleration of the normal u, i.e. a µ = u ρ ∇ ρ u µ . Denoting by b the covariant acceleration of the normal n, i.e. b µ = n ρ ∇ ρ n µ , we obtain the two formulae: ∇ ρ u µ = −K ρµ − u ρ a µ ∇ ρ n µ = −Θ ρµ + n ρ b µ (99) B More Examples Apart from the BTZ solution, other solutions have been found for Chern-Simons field equations in the framework of Chern-Simons Gravity. We are going to analyse the Chern-Simons anti-de Sitter solution [4] and the single particle solution [19,36]. B.1 The anti-de Sitter solution Let us consider a three dimensional manifold M with a boundary ∂M which has the topology of a torus. We assume (ω,ω, ρ) 4 as coordinates over M such that the boundary is located at ρ = ∞ and the torus is labelled by complex coordinates (ω,ω). A solution of the SL(2, I C) Chern-Simons field equations depending on the coordinates on the torus can be found in [4] and it corresponds to the Euclidean anti-de Sitter solution. In the coordinates chosen a general SL(2, C) connection reads as A = A i J i with A i = A i ω dω + A ī ω dω + A i ρ dρ. 5 Imposing the gauge condition: A ρ = iJ 3(102) B.2 The particle solution In the same sl(2, IR) basis (2) and in the coordinates system (t, ρ, ϕ) the exact solution corresponding to a particle source in (2 + 1) dimensional gravity turns out to be: A i µ =    ρ 2 + γ 2 0 ρ 2 + γ 2 0 1 √ ρ 2 +γ 2 0 ρ 0 ρ    (110) A i µ =    − ρ 2 +γ 2 0 ρ 2 +γ 2 0 − 1 √ ρ 2 +γ 2 0 ρ 0 −ρ   (111) where γ = 1 − α,γ = 1 −α and π(α +α) is the deficit angle of the conical singularity introduced by the particle in the geometry of spacetime (see e.g. [19,41]). Using formula (56) it is possible to calculate the conserved quantities relative to the infinitesimal generators of symmetries in spacetime, suitably lifted by means of the generalized Kosmann lift. The variations of mass and angular momentum turn out to be: δ X Q B (L CScov , K(∂ t ), A,Ā) = − 1 l (γδ X γ +γ δ Xγ ) δ X Q B (L CScov , K(∂ φ ), A,Ā) = −(γδ X γ −γ δ Xγ )(112) Integrating the above expressions around the conical singularity we obtain for the mass and the angular momentum the following values: Q B (L CScov , K(∂ t ), A,Ā) = − 1 2l (γ 2 +γ 2 ) + const = 1 l L + 0 + L − 0 + const Q B (L CScov , K(∂ φ ), A,Ā) = − 1 2 (γ 2 −γ 2 ) + const = L + 0 − L − 0 + const (113) where we have set L + 0 = − 1 2 (γ 2 ) and L − 0 = − 1 2 (γ 2 ). These results are in accordance with the values found by Martinec in [36] for the ADM mass and spin of the particle. We point out that it is not natural, i.e. functorial, in a rigorous mathematical language, since it does not preserve commutators:[K(ξ), K(η)] ab = K([ξ, η]) ab + 1 2 e [a µ (£η g µσ £ ξ gσν )e b]ν(68)Only if at least one of the vector fields ξ or η is a conformal Killing vector for the metric then [K(ξ), K(η)] = K([ξ, η]). This is the reason why the generalized Kosmann lift is sometimes referred to as "the quasi-natural" lift in gauge theories; see[28]. These complex coordinates on spacetime are related to the usual spherical coordinates by means of the following expressions: and it is an exact solution of Einstein equations (with negative cosmological constant Λ = −1/l 2 ) depending on two arbitrary functions L(ω),L(ω).We perform the calculations to obtain, from (56) with η = δ and κ = −l/4G, the variation of the conserved quantity relative to the generalized SO(3) Kosmann lift of the vector field ξ = ∂ t = i ∂ ∂ω − ∂ ∂ω . 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[ "X-ray emission from the galaxies in Abell 2634", "X-ray emission from the galaxies in Abell 2634" ]	[ "I Sakelliou \nDepartment of Physics and Astronomy\nUniversity of Southampton\nSO17 1BJSouthampton\n", "M R Merrifield \nDepartment of Physics and Astronomy\nUniversity of Southampton\nSO17 1BJSouthampton\n" ]	[ "Department of Physics and Astronomy\nUniversity of Southampton\nSO17 1BJSouthampton", "Department of Physics and Astronomy\nUniversity of Southampton\nSO17 1BJSouthampton" ]	[ "Mon. Not. R. Astron. Soc" ]	It is difficult to detect X-ray emission associated with galaxies in rich clusters because the X-ray images of the clusters are dominated by the emission from their hot intracluster media (ICM). Only the nearby Virgo cluster provides us with information about the X-ray properties of galaxies in clusters. Here we report on the analysis of a deep ROSAT HRI image of the moderately-rich cluster, Abell 2634, by which we have been able to detect the X-ray emission from the galaxies in the cluster. The ICM of Abell 2634 is an order of magnitude denser than that of the Virgo cluster, and so this analysis allows us to explore the X-ray properties of individual galaxies in the richest environment yet explored.By stacking the X-ray images of the galaxies together, we have shown that their emission appears to be marginally resolved by the HRI. This extent is smaller than for galaxies in poorer environments, and is comparable to the size of the galaxies in optical light. These facts suggest that the detected X-ray emission originates from the stellar populations of the galaxies, rather than from extended hot interstellar media. Support for this hypothesis comes from placing the optical and X-ray luminosities of these galaxies in the L B -L X plane: the galaxies of Abell 2634 lie in the region of this plane where models indicate that all the X-ray emission can be explained by the usual population of X-ray binaries. It is therefore probable that ram pressure stripping has removed the hot gas component from these galaxies.	10.1046/j.1365-8711.1998.01212.x	[ "https://arxiv.org/pdf/astro-ph/9710092v1.pdf" ]	15,934,106	astro-ph/9710092	8597b47320323c9992c6a467fe51418ee72904ed	X-ray emission from the galaxies in Abell 2634 7 January 2018 I Sakelliou Department of Physics and Astronomy University of Southampton SO17 1BJSouthampton M R Merrifield Department of Physics and Astronomy University of Southampton SO17 1BJSouthampton X-ray emission from the galaxies in Abell 2634 Mon. Not. R. Astron. Soc 00000007 January 2018Printed 7 January 2018arXiv:astro-ph/9710092v1 9 Oct 1997 (MN L A T E X style file v1.4)galaxies: clusters: individual: A2634 -galaxies: interactions -inter- galactic medium -X-rays: galaxies It is difficult to detect X-ray emission associated with galaxies in rich clusters because the X-ray images of the clusters are dominated by the emission from their hot intracluster media (ICM). Only the nearby Virgo cluster provides us with information about the X-ray properties of galaxies in clusters. Here we report on the analysis of a deep ROSAT HRI image of the moderately-rich cluster, Abell 2634, by which we have been able to detect the X-ray emission from the galaxies in the cluster. The ICM of Abell 2634 is an order of magnitude denser than that of the Virgo cluster, and so this analysis allows us to explore the X-ray properties of individual galaxies in the richest environment yet explored.By stacking the X-ray images of the galaxies together, we have shown that their emission appears to be marginally resolved by the HRI. This extent is smaller than for galaxies in poorer environments, and is comparable to the size of the galaxies in optical light. These facts suggest that the detected X-ray emission originates from the stellar populations of the galaxies, rather than from extended hot interstellar media. Support for this hypothesis comes from placing the optical and X-ray luminosities of these galaxies in the L B -L X plane: the galaxies of Abell 2634 lie in the region of this plane where models indicate that all the X-ray emission can be explained by the usual population of X-ray binaries. It is therefore probable that ram pressure stripping has removed the hot gas component from these galaxies. INTRODUCTION The advent of the Einstein observatory changed the belief that early-type galaxies contain little interstellar gas by revealing hot X-ray emitting halos associated with many of them (e.g. Forman et al. 1979). Subsequent X-ray observations led to the conclusion that these galaxies can retain large amounts (up to ∼ 10 11 M⊙) of hot (T ∼ 10 7 K) interstellar medium (Forman, Jones & Tucker 1985;Trinchieri & Fabbiano 1985;Canizares, Fabbiano & Trinchieri 1987). However, this picture might be expected to be different for galaxies that reside near the centres of rich clusters of galaxies, since their properties must be affected by their dense environment. For example, we might expect interstellar medium (ISM) gas to be stripped from the galaxy by the ram pressure resulting from the passage of the galaxy through the intracluster medium (ICM) (Gunn & Gott 1972). Stripping of the ISM can also result from tidal interactions with other nearby galaxies (Richstone 1975;Merritt 1983Merritt , 1984. The most dramatic and well-studied example of a galaxy which appears to be in the process of being stripped of its ISM is the elliptical galaxy M86 in the Virgo cluster, which shows a 'plume' of X-ray emission emanating from it (Forman et al. 1979;White et al. 1991;Rangarajan et al. 1995). In addition to the mechanisms which remove the ISM of a galaxy, gas can also be replenished. The gravitational pull of a galaxy attracts the surrounding ICM. This gas ends up being concentrated in or behind the galaxy, depending on the velocity of the galaxy relative to the ICM (see, for example, Sakelliou, Merrifield & McHardy 1996). Stellar winds can also replenish the hot gas in a galaxy's ISM. All the processes mentioned above take place simultaneously. The relative importance of each process depends on: the galaxies' velocities; the local density of the ICM; the number density of galaxies; their orbits in the cluster; and the gravitational potential of each galaxy. It is therefore a priori difficult to say which mechanism dominates in the cores of rich clusters of galaxies, and hence whether cluster galaxies are surrounded by the extensive X-ray emitting halos that we see associated with galaxies in the field. Unfortunately, X-ray observations of rich clusters have generally not been of high enough quality to answer this c 0000 RAS question, since any emission from the galaxies is hard to detect against the high X-ray background produced by the cluster's ICM (Canizares & Blizzard 1991;Vikhlinin, Forman & Jones 1994;Bechtold et al. 1983;Grebenev et al. 1995;Soltan & Fabricant 1990;Mahdavi et al. 1996). In the cases where galaxy X-ray emission has been reported, the studies have been restricted to a few bright cluster galaxies, and it has not proved possible to investigate the general galaxy population in a statistically complete manner. In order to search for X-ray emission from galaxies in a moderately rich environment, we obtained a deep ROSAT HRI observation of the core of the rich cluster Abell 2634, which is a nearby (z=0.0312) centrally-concentrated cluster of richness class I. In §2.1 we describe the analysis by which the X-ray emission from the galaxies in this cluster was detected. In §2.2 we explore the properties of this X-ray emission, and show that the galaxies in this cluster lack the extensive gaseous halos of similar galaxies in poorer environments. In §3, we show how this difference can be attributed to ram pressure stripping. X-RAY OBSERVATIONS AND ANALYSIS The core of Abell 2634 was observed with the ROSAT HRI in two pointings, in January and June 1995, for a total of 62.5 ksec. The analysis of these data was performed with the IRAF/PROS software. Inspection of the emission from the cD galaxy and other bright X-ray sources in the images from the two separate observations indicates that the two sets of observations do not register exactly and that a correction to the nominal ROSAT pointing position is required. Therefore, the second set of observations was shifted by ∼ 2.0 arcsec to the east and ∼ 0.8 arcsec to the south; such a displacement is consistent with typical ROSAT pointing uncertainties (Briel et al. 1996). Both images were then registered with the optical reference frame to better than an arcsecond. A grey-scale image of the total exposure is shown in Fig. 1. The image has been smoothed with a Gaussian kernel of 8 arcseconds dispersion. At the distance of Abell 2634, 1 arcsec corresponds to 900 pc. ⋆ This deep image of Abell 2634 reveals the largescale X-ray emission from the hot ICM of the cluster and a few bright sources, which are numbered on Fig. 1. Source 1 is the cD galaxy NGC 7720, located near the centre of Abell 2634. It hosts the prototype wide-angle tailed radio source 3C 465 (e.g. Eilek et al. 1984). Source 2 is a background cluster at a redshift of cz ≃ 37, 000 km s −1 (Pinkney et al. 1993;Scodeggio et al. 1995). For the rest of the X-ray bright sources, the Automatic Plate Measuring machine, run by the Royal Greenwich Observatory in Cambridge, was used to obtain optical identifications. Table 1. gives the positions of these sources as determined from the X-ray image, and the class of their optical counterparts. The position of source 7 coincides with a faint object in the Palomar sky survey, but there is also a nearby star, and source 11 does not seem to have a discernible optical counterpart. All these sources were masked out in the subsequent analysis. The positions of galaxies that are members of Abell 2634 are also indicated on Fig. 1. Pinkney et al. (1993) collected the redshifts of ∼150 galaxies that are probable members Abell 2634 (on the basis that their redshifts lie in the range 6, 000 < cz < 14, 000 km s −1 ), and Scodeggio et al. (1995) have increased the number of galaxies whose redshifts confirm that they are cluster members up to ∼ 200. The sample of redshifts is complete to a magnitude limit of 16.5, and from this magnitude-limited sample we have selected those galaxies that appear projected within a circle of 15 arcmin radius, centered on the cD galaxy. This selection yields 62 galaxies, of which the vast majority are of type E and S0 -only 10 are classified as spirals or irregular. The positions of these galaxies are taken from the CCD photometry of Pinkney (1995) and Scodeggio et al. (1995), and are accurate to ∼ 1 arcsec. They are marked as crosses on Fig. 1. Inspection of Fig. 1 reveals several cases where the location of a galaxy seems to coincide with an enhancement in the cluster's X-ray emission, and it is tempting to interpret such enhancements as the emission from the galaxy's ISM. However, it is also clear from Fig. 1 that the X-ray emission in this cluster contains significant small-scale fluctuations and non-uniformities. We must therefore consider the possibility that the apparent associations between galaxy locations and local excesses in the X-ray emission may be chance superpositions. We therefore now present a more objective approach to searching for the X-ray emission from cluster galaxies. Detection of the cluster galaxies Before adopting an approach to detecting the emission from cluster galaxies, we must first have some notion as to how bright we might expect the emission to appear in this deep HRI image. Previous X-ray observations have shown that the X-ray luminosities of E and S0 galaxies in the 0.2-3.5 keV energy band range from ∼ 10 39 to ∼ 10 42 erg s −1 (Kim, Fabbiano & Trinchieri 1992a, b;Forman et al. 1985). These limits at the distance of Abell 2634 correspond to fluxes of 5 × 10 −16 to 5 × 10 −13 erg s −1 cm −2 . We have used the PIMMS software to convert these limits to count rates for the ROSAT HRI detector. The emission from the galax-ies was modeled by a Raymond-Smith plasma (Raymond & Smith 1977) with a temperature kT = 0.862 keV and a metal abundance of 25% solar; these quantities are consistent with the values previously found from observations of early-type galaxies (Kim et al. 1992a;Matsushita et al. 1994;Awaki et al. 1994). The absorption by the galactic hydrogen was also taken into account by using the column density given by Stark et al. (1992) for the direction of Abell 2634 (NH = 4.94×10 20 cm −2 ). These calculations predict that the 62.5 ksec HRI observation of this cluster should yield somewhere between ∼ 1 and ∼ 1200 counts from each galaxy. Motivated by this prediction of a respectable, but not huge, number of counts per galaxy, we set out to detect emission associated with cluster galaxies. We are trying to detect this fairly modest amount of emission against the bright background of the ICM emission. We therefore seek to improve the statistics by stacking together the X-ray images in the vicinity of the 40 E and S0 galaxies marked in Fig. 1. Fig. 2. presents a contour plot of the combined image, which covers a region of 1 arcmin radius around the stacked galaxies. The centre of the plot coincides with the optical centres of the individual galaxies. Clearly, there appears to be X-ray emission associated with the cluster galaxies, and it is centered at their optical positions. This coincidence provides us with some confidence that the X-ray and optical frames are correctly registered. We have also constructed a composite brightness profile for the 40 galaxies by adding the unsmoothed counts detected in concentric annuli centered on each galaxy. The width of each annulus in this profile was set at 6 arcsec and the local background, as measured in an annulus between 1.0 and 2.0 arcmin around each galaxy, was subtracted. The resulting profile is presented in Fig. 3. Once again, the excess of emission in the vicinity of the cluster galaxies is apparent. In order to assess the significance of this detection, we generated 100 sets of simulated data from randomly selected points on the image. The diffuse emission from the ICM varies systematically with radius, and so we might expect the probability that a galaxy is coincidentally aligned with a clump in the ICM emission to vary systematically with radius. Further, the sensitivity of the HRI varies with radius, and so the detectability of the emission from a single galaxy will vary with radius as well. We therefore constructed the simulated data sets by extracting counts from the HRI image at the same radii as the true galaxy locations, but at randomized azimuthal angles. The mean profile and the RMS fluctuations amongst the simulated data sets are shown in Fig. 3. As might be expected, the average number of counts in these random data sets is zero; the larger RMS error bars at small radii reflect the smaller sizes of these annuli. From a χ 2 comparison between the observed galaxy profile and the simulated profile, we can conclude that there is less than 0.1% probability that the apparent peak in the galaxy emission is produced by chance. Thus, the detection of emission from the galaxies in Abell 2634 is highly statistically significant. Origin of the X-ray emission As mentioned in the introduction, early-type galaxies have been found to retain large amounts of hot gas, which extends far beyond the optical limits of the galaxies. X-ray binaries also contribute to the total emission, and they become more dominant in X-ray faint galaxies. We might also expect some of the emission to originate from faint active galactic nuclei (AGNs) in the cores of these galaxies. Although none of the galaxies in our sample has been reported as an active galaxy, there is increasing dynamical evidence that the vast majority of elliptical galaxies contain central massive black holes (van der Marel et al. 1997; Kormendy et al. 1996aKormendy et al. , 1996b; for a review Kormendy & Richstone 1995), and so we might expect some contribution from low-level activity in such systems. We therefore now see what constraints the observed X-ray properties of the galaxies in Abell 2634 can place on the origins of the emission. The extent of the X-ray emission One diagnostic of the origins of the X-ray emission is the measurement of its spatial extent. AGN emission should be unresolved by the HRI, while emission from X-ray binaries should be spread over a similar spatial scale as the optical emission, and halos of hot gas should be still more extended. In order to assess the spatial extent of the X-ray emission, we need to characterize the PSF in this HRI observation. The point sources detected in these data are more extended than the model PSF for the HRI detector given by Briel et al. (1996) as is seen in Fig. 4, where four of the sources are fitted by this model PSF (dash line). This discrepancy can be attributed to residual errors in the reconstruction of ROSAT's attitude, which broaden the PSF in long integrations. We have therefore empirically determined the PSF that is appropriate for this observation by fitting the profiles of the point sources with a Gaussian PSF model (Fig. 4, solid line). Only sources 1, 3, 4, 8 are used for the determination of the width of the Gaussian. Source 6 is very elongated and can not be represented by a symmetrical function. The mean dispersion of the best-fit model was found to be (4.1 ± 0.1) arcsec. All of the point sources detected in this image have widths consistent with this value, and so there is no evidence that the PSF varies with radius. We therefore adopt this PSF for the emission from all the galaxies in the observation. Figure 5 shows the comparison between the adopted PSF and the emission from the cluster galaxies. The emission appears to be more extended than the PSF; fitting the data to the PSF yields a χ 2 value of 14.2 with 9 degrees of freedom, which is marginally consistent with the emission being unresolved. We can obtain a better fit by modeling the radial profile of the emission using a Gaussian, which we convolve with the PSF to model the observed profile. Fitting this model to the observations, we find that the intrinsic width of the X-ray emission is 4.3 +2.2 −2.8 arcsec. The best-fit model is also shown in Fig. 5. The radius of the X-ray halos of early-type galaxies with optical luminosities comparable to those in this cluster has been shown to be ∼ 20 -60 kpc (e.g. Fabbiano et al. 1992), with the lower values characterizing optically fainter galaxies. At the distance of Abell 2634 these values correspond to ∼ 20 -60 arcsec, much larger than the upper limit of ∼ 7 arcsec we found for the extent of the galactic X-ray emission. Thus, the X-ray emission from the cluster galaxies, although apparently extended, clearly does not originate from the large halos of hot gas found around comparable galaxies in poorer environments. One possible explanation for the spatial extent of the emission from these galaxies is that it could arise from errors in the adopted positions for the galaxies. Such errors would broaden the distribution of X-rays when the data from different galaxies are co-added even if the individual sources are unresolved. However, the zero-point of the X-ray reference frame is well tied-down by the detected point sources in the field. Further, the optical locations of the galaxies come from CCD photometry with positional errors of less than an arcsecond. It therefore cannot explain the ∼ 4 arcsec extent of the observed X-ray emission. We therefore now turn to the extent of the X-ray emission that we might expect from X-ray binaries. Nearly half of the early-type galaxies that we use for our analysis have been imaged in the I-band by Scodeggio, Giovanelli & Haynes (1997). They have fitted the optical galaxy profile with a de Vaucouleur law, and found a mean value for their effective radii of ∼8 arcsec, with only 4 galaxies smaller than 3 arcsec and another 3 larger than 13 arcsec. These values are directly comparable to the spatial extent of the X-ray emission derived above. Thus, it would appear that the ob-servations are consistent with what we would expect if the X-ray emission from the galaxies in Abell 2634 originates from X-ray binaries in these systems, although we have not ruled out the possibility that some fraction of the emission comes from AGN. The luminosity of the X-ray emission A further test of the origins of the X-ray emission in the cluster galaxies comes from its luminosity. It has been found that the blue luminosities of galaxies correlates with their X-ray luminosities, with the optically brighter galaxies being more luminous in X-rays (e.g. Forman et al. 1985;Fabbiano et al. 1992). This correlation for the early-type galaxies in the Virgo cluster is presented in Fig. 6. The optical and X-ray luminosities of these galaxies are taken from Fabbiano et al. (1992). The line in this plot divides the LB − LX plane into two distinct galaxy types (Fabbiano & Schweizer 1995). In addition to the differences in the ratio of X-ray-to-optical luminosities, galaxies in these two regions have been shown to possess different spectral properties. The spectra of the X-ray bright galaxies [group (I)] are well fitted by Raymond-Smith models of 1 keV temperature, and it is believed that these galaxies retain large amounts of hot ISM. In the spectra of the X-ray faint galaxies of group (II), on the other hand, a hard component is present; X-ray binaries are believed to be the major source of the X-rays in these galaxies. In order to see where the galaxies of Abell 2634 lie in this plot, we must calculate their optical and X-ray luminosities. Butcher & Oemler (1985) have measured J and F optical magnitudes for a large number of galaxies in Abell 2634. We have converted these magnitudes to the blue band by applying the colour relations provided by Oemler (1974) and Butcher & Oemler (1985), correcting for galactic extinction, and using the appropriate K-correction. We find that the absolute blue magnitude of the galaxies in the HRI image lie in the range from −18.5 to −21.4. We have divided these galaxies into three groups according to their optical luminosities: group A (0.4 − 2.0) × 10 10 L ⊙ with 17 galaxies; group B (2.1 − 3.7) × 10 10 L ⊙ with 8 galaxies; and group C (3.8 − 5.5) × 10 10 L ⊙ with only 2 galaxies. The X-ray luminosity of each group was obtained by repeating the analysis of §2.1 using just the galaxies in each sub-sample. Using the PIMMS software we converted the observed count rate from the HRI image to X-ray luminosity in the energy range 0.2-3.5 keV. The thermal model used for the conversion is the same that was used by Fabbiano et al. (1992) to derive the plot shown in Fig. 6, and is discussed in §2.1. Figure 6. X-ray luminosity versus blue luminosity for the earlytype galaxies. The line delineates the boundary between the locations of galaxies where the hot ISM makes a significant contribution to the total emission [region (I)], and the locations of galaxies where the entire emission can be ascribed to X-ray binaries [region (II)]. The locations in this plane of galaxies that belong to the Virgo cluster are marked by filled circles. The average properties of galaxies in Abell 2634 lying in different optical luminosity ranges are indicated by crosses. The resulting values for optical and X-ray luminosities in each sub-sample are shown in Fig. 6. The horizontal error bars represent the width of each optical luminosity bin and the vertical ones show the errors in the measured X-ray luminosities. This plot shows that the galaxies in our sample follow the established correlation: the optically-brighter galaxies are also more luminous in the X-rays. The existence of this correlation also implies that the detected X-ray flux from the galaxies in Abell 2634 is not dominated by a few bright galaxies, but that the optically-fainter galaxies also contribute to the detected X-ray emission. The galaxies in Abell 2634 probe the fainter end of the LB -LX relation as covered by Virgo galaxies. It should be borne in mind that there is a bias in the Virgo data which means that the two data sets in Fig. 6 are not strictly comparable. At the lower flux levels, a large number of Virgo galaxies have not been detected in X-rays, and so this plot preferentially picks out any X-ray-bright Virgo galaxies. For the Abell 2634 data, on the other hand, the co-addition of data from all the galaxies in a complete sample means that the data points represent a true average flux. However, it is clear that the X-ray fluxes from galaxies in these two clusters are comparable. The similarity between the X-ray properties of galaxies in these two clusters is of particular interest because their environments differ significantly. The galaxies from the Virgo cluster shown in Fig. 6 lie in a region between 360 kpc and 2 Mpc from the centre of the cluster. Recent ROSAT PSPC observations have shown that the number density of the hot ICM of this cluster drops from 3 × 10 −4 to 3 × 10 −5 cm −3 in this region (Nulsen & Böhringer 1995). The galaxies from Abell 2634 that have gone into this plot lie in the inner 0.8 Mpc of Abell 2634, and in this region the number density of the ICM varies between 1 × 10 −3 and 2 × 10 −4 cm −3 (Sakelliou & Merrifield 1997). Thus, the galaxies in the current analysis come from a region in which the intracluster gas density is, on average, an order of magnitude higher that surrounding the Virgo cluster galaxies. The location of the galaxies in region (II) of Fig. 6 adds weight to the tentative conclusion of the previous section that the X-ray emission from these galaxies can be explained by their X-ray binary populations, since any significant ISM contribution would place them in region (I). Similarly, the low X-ray fluxes of these galaxies leaves little room for a significant contribution from weak AGN. If the X-ray binary populations are comparable to those assumed by Fabbiano & Schweizer (1995) in calculating the dividing line in Fig. 6, then essentially all the X-ray emission from these galaxies can be attributed to the X-ray binaries. Thus, any average AGN emission brighter than a few times 10 40 erg s −1 can be excluded, as such emission would also move the galaxies into region (I) of the LB -LX plane. Spiral galaxies Having discussed the X-ray properties of the early-type galaxies in Abell 2634 at some length, we now turn briefly to the properties of the spiral galaxies in the cluster. Abell 2634 is a reasonably rich system, and we therefore do not expect to find many spiral galaxies within it. Indeed, only 7 of the 62 galaxies whose redshifts place them at the distance of Abell 2634, and which lie within the field of the HRI, have been classified as spirals. The statistics are correspondingly poor when the X-ray emission around these galaxies is coadded: the combined profile is shown in Fig. 7, together with the results from the control simulations (see §2.1 for details). It is clear from this figure that the spirals have not been detected in this observation, and a χ 2 fit confirms this impression. The failure to detect these galaxies is not surprising. Not only are there relatively few of them, but their X-ray luminosities are lower than those of early-type galaxies. In the Einstein energy band (0.2 -3.5 keV), their luminosities have been found to lie in the range ∼ 10 38 to ∼ 10 41 erg s −1 (Fabbiano 1989). Modeling this emission using a Raymond-Smith model with a higher temperature than for the earlytype galaxies, as appropriate for spiral galaxies (Kim et al. 1992a), we find that the expected count rate for these galaxies is a factor of ∼ 40 lower than for the ellipticals in the cluster. It is therefore unsurprising that we fail to detect the small number of spiral galaxies present in the cluster. DISCUSSION In this paper, we have detected the X-ray emission from the normal elliptical galaxies in Abell 2634. The limited spatial extent of this emission coupled with its low luminosity is consistent with it originating from normal X-ray binaries in the galaxies' stellar populations. These galaxies do not seem to have the extended hot ISM found around galaxies that reside in poorer cluster environments. We therefore now discuss whether this difference can be understood in terms of the physical processes outlined in the introduction. Intuitively, the simplest explanation for the absence of an extensive halo around a cluster galaxy is that it has been removed by ram pressure stripping as the galaxy travels through the ICM. A simple criterion for the efficiency of this process can be obtained by comparing the gravitational force that holds the gas within the galaxy to the force due to the ram pressure, which tries to remove it (Gunn & Gott 1972). The gravitational force is given by: FGR ∼ G M gal Mgas R 2 gal(1) where M gal is the total mass of the galaxy, Mgas is the mass of the X-ray emitting gas, and R gal is the radius of the galaxy's X-ray halo. For typical values for the masses of the galaxy and the gas of 10 12 M ⊙ (Forman et al. 1985) and 5 × 10 9 M ⊙ (e.g. Canizares et al. 1987) respectively, and a mean value for R gal of 40 kpc (Canizares et al. 1986), which is a representative value for galaxies of the same optical luminosity as the galaxies in Abell 2634, equation (1) implies that FGR ∼ 1 × 10 30 N. The force due to ram pressure is described by: FRP = ρICM v 2 gal πR 2 gal = µ mp n v 2 gal πR 2 gal where ρICM is the density of the ICM, µ is the mean molecular weight, mp is the proton mass, n is the number density of the ICM, and v gal is the galaxy velocity. From the velocity dispersion profile of Abell 2634 presented by den Hartog & Katgert (1996) we find that the velocity dispersion, σv, in the inner 15 arcmin of this system is ≈ 710 km s −1 . Assuming an isotropic velocity field, the characteristic three-dimensional velocity of each galaxy is hence v gal = √ 3 σv ≈ 1230 km s −1 . The number density of the ICM in the same inner region has been derived from recent ROSAT PSPC data (Schindler & Prieto 1997) and this HRI observation (Sakelliou & Merrifield 1997), and is found to vary from 1 × 10 −3 cm −3 down to 2 × 10 −4 cm −3 , consistent with previous Einstein observations (Jones & Forman 1984;Eilek et al. 1984). Inserting these values into equation (2), we find FRP ∼ (1 − 10) × 10 30 N. Thus, the ram pressure force exerted on the galaxies in Abell 2634 is found to be larger than the force of gravity, and so we might expect ram pressure stripping to be an effective mechanism for removing the ISM from these galaxies. In poorer environments, the density of the ICM is likely to be at least a factor of ten lower, and the velocities of galaxies will be a factor of ∼ 3 smaller. We might therefore expect FRP to be a factor of ∼ 100 lower in poor environments. Since such a change would make FRP < FGR, it is not surprising that galaxies in poor environments manage to retain their extensive halos. The absence of extensive X-ray halos around the galaxies in Abell 2634 implies that ram pressure stripping dominates the processes of accretion and stellar mass loss which can replenish the ISM. By carrying out similar deep X-ray observations of clusters spanning a wide range of ICM properties, it will be interesting to discover more precisely what sets of physical conditions can lead to the efficient ISM stripping that we have witnessed in Abell 2634. Figure 1 . 1Grey-scale ROSAT HRI image of the core of the cluster Abell 2634. The image has been smoothed using a Gaussian kernel with a dispersion of 8 arcsec. The positions of the cluster galaxies with measured redshift are marked with crosses. Figure 2 . 2Contour plot of the combined image of all the earlytype galaxies that belong to Abell 2634. The pixel size of the image is 2 arcsec and it has been smoothed with a Gaussian kernel of 2 pixels. The center of the plot coincides with the optical centres of the galaxies. The contour lines are from 20 to 100 per cent the peak value and are spaced linearly in intervals of 5 per cent. Figure 3 . 3The combined surface brightness profile of all the earlytype galaxies (filled squares) normalized to one galaxy. Open squares represent the average profile from the simulations. Figure 4 . 4Surface brightness distribution of the bright sources in the HRI image. (s1, s3, s4, and s8). Source 1 is the central cD galaxy. The profile is fitted by the appropriate HRI PSF for the distance of the point source from the centre of the image (dash line) and a Gaussian (solid line). The calculated width (σ) of the best fit Gaussian is also given. Figure 5 . 5The combined surface brightness distribution of the 40 early-type galaxies, normalized to one galaxy. The profile is fitted by the measured HRI PSF (dashed line) and the spatiallyextended model (solid line). Figure 7 . 7The combined surface brightness profile of the spiral galaxies that lie in the field of view of the HRI (filled squares) normalized to one galaxy. Open squares represent the average profile from the simulations. Table 1 . 1Bright SourcesSource α(J2000) δ(J2000) ID/notes h m s • ′ ′′ 1 23 38 29.1 27 01 53.5 cD galaxy 2 23 37 56.1 27 11 31.3 cluster 3 23 39 01.6 27 05 35.9 star 4 23 39 00.5 27 00 27.9 star 5 23 38 31.7 27 00 30.5 nothing 6 23 38 41.5 26 48 04.1 star 7 23 38 19.8 26 56 41.5 ? 8 23 38 07.4 26 55 52.8 star 9 23 37 57.5 26 57 30.1 galaxy ? 10 23 37 45.3 26 57 53.1 two objects 11 23 37 26.2 27 08 14.6 ? ⋆ Here, as throughout this paper, we have adopted a Hubble constant of H 0 = 50 km s −1 Mpc −1 . c 0000 RAS, MNRAS 000, 000-000 c 0000 RAS, MNRAS 000, 000-000 ACKNOWLEDGEMENTSWe are indebted to the referee, Alastair Edge, for a very insightful report on the first incarnation of this paper. We thank Jason Pinkney for providing us with the positions and redshifts of the galaxies and Rob Olling for helpful discussions. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. Much of the analysis was performed using iraf, which is distributed by NOAO, using computing resources provided by STARLINK. MRM is supported by a PPARC Advanced Fellowship (B/94/AF/1840). . H Awaki, R Mushotzky, T Tsuru, A C Fabian, Y Fukazawa, M Loewenstein, K Makishima, H Matsumoto, K Matsushita, T Mihara, T Ohashi, G R Ricker, R J Serlemitsos, Y Tsusaka, T Yamazaki, PASJ. 4665Awaki H., Mushotzky R., Tsuru T., Fabian A. C., Fukazawa Y., Loewenstein M., Makishima K., Matsumoto H., Matsushita K., Mihara T., Ohashi T., Ricker G. R., Serlemitsos R. J., Tsusaka Y., Yamazaki T., 1994, PASJ, 46, L65 . 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[ "Stability of thin liquid films and sessile droplets under confinement", "Stability of thin liquid films and sessile droplets under confinement" ]	[ "Fabian Dörfler \nMax-Planck-Institut für Intelligente Systeme\nHeisenbergstr. 370569StuttgartGermany\n\nIV. Institut für Theoretische Physik\nUniversität Stuttgart\nPfaffenwaldring 5770569StuttgartGermany\n", "Markus Rauscher \nMax-Planck-Institut für Intelligente Systeme\nHeisenbergstr. 370569StuttgartGermany\n\nIV. Institut für Theoretische Physik\nUniversität Stuttgart\nPfaffenwaldring 5770569StuttgartGermany\n", "S Dietrich \nMax-Planck-Institut für Intelligente Systeme\nHeisenbergstr. 370569StuttgartGermany\n\nIV. Institut für Theoretische Physik\nUniversität Stuttgart\nPfaffenwaldring 5770569StuttgartGermany\n" ]	[ "Max-Planck-Institut für Intelligente Systeme\nHeisenbergstr. 370569StuttgartGermany", "IV. Institut für Theoretische Physik\nUniversität Stuttgart\nPfaffenwaldring 5770569StuttgartGermany", "Max-Planck-Institut für Intelligente Systeme\nHeisenbergstr. 370569StuttgartGermany", "IV. Institut für Theoretische Physik\nUniversität Stuttgart\nPfaffenwaldring 5770569StuttgartGermany", "Max-Planck-Institut für Intelligente Systeme\nHeisenbergstr. 370569StuttgartGermany", "IV. Institut für Theoretische Physik\nUniversität Stuttgart\nPfaffenwaldring 5770569StuttgartGermany" ]	[]	The stability of nonvolatile thin liquid films and of sessile droplets is strongly affected by finite size effects. We analyze their stability within the framework of density functional theory using the sharp kink approximation, i.e., on the basis of an effective interface Hamiltonian. We show that finite size effects suppress spinodal dewetting of films because it is driven by a long-wavelength instability. Therefore nonvolatile films are stable if the substrate area is too small. Similarly, nonvolatile droplets connected to a wetting film become unstable if the substrate area is too large. This instability of a nonvolatile sessile droplet turns out to be equivalent to the instability of a volatile drop which can atttain chemical equilibrium with its vapor.	10.1103/physreve.88.012402	[ "https://arxiv.org/pdf/1302.2029v1.pdf" ]	32,660,955	1302.2029	14e378e9002436111716ee2a083881f4331a2433	Stability of thin liquid films and sessile droplets under confinement 8 Feb 2013 (Dated: February 11, 2013) Fabian Dörfler Max-Planck-Institut für Intelligente Systeme Heisenbergstr. 370569StuttgartGermany IV. Institut für Theoretische Physik Universität Stuttgart Pfaffenwaldring 5770569StuttgartGermany Markus Rauscher Max-Planck-Institut für Intelligente Systeme Heisenbergstr. 370569StuttgartGermany IV. Institut für Theoretische Physik Universität Stuttgart Pfaffenwaldring 5770569StuttgartGermany S Dietrich Max-Planck-Institut für Intelligente Systeme Heisenbergstr. 370569StuttgartGermany IV. Institut für Theoretische Physik Universität Stuttgart Pfaffenwaldring 5770569StuttgartGermany Stability of thin liquid films and sessile droplets under confinement 8 Feb 2013 (Dated: February 11, 2013)numbers: 6808Bc wetting6843-h chemisorption/physisorption: adsorbates on surfaces6803Cd surface tension and related phenomena8260Nh thermodynamics of nucleation Keywords: wettingnanofluidics The stability of nonvolatile thin liquid films and of sessile droplets is strongly affected by finite size effects. We analyze their stability within the framework of density functional theory using the sharp kink approximation, i.e., on the basis of an effective interface Hamiltonian. We show that finite size effects suppress spinodal dewetting of films because it is driven by a long-wavelength instability. Therefore nonvolatile films are stable if the substrate area is too small. Similarly, nonvolatile droplets connected to a wetting film become unstable if the substrate area is too large. This instability of a nonvolatile sessile droplet turns out to be equivalent to the instability of a volatile drop which can atttain chemical equilibrium with its vapor. I. INTRODUCTION Dewetting of fluid films and the ensuing formation of sessile droplets are both part of everyday experience. Moreover these mechanisms are important for the functioning of biological systems as well as for numerous technological processes. Dewetting on homogeneous substrates and the subsequent formation of droplets have been studied both experimentally [1][2][3][4][5][6][7][8][9] and theoretically [10][11][12][13][14][15][16][17] in great detail. Both mechanisms can be understood quantitatively within the well established theory of wetting phenomena [18][19][20][21]. More recently, wetting and dewetting on structured surfaces receives increasing attention, in particular with a view on controlling the dewetting process on patterned surfaces [22][23][24][25][26][27][28][29] as well as in the context of microfluidics [30][31][32][33][34][35]. Chemical patterns consisting of lyophilic and lyophobic patches as well as topographic patterns such as pits and grooves effectively lead to a lateral confinement of wetting films and droplets. It is well known that confinement modifies the structural and thermodynamic properties of condensed matter. In small scale systems these finite size effects can either stabilize or destabilize certain structures. For example, systems exhibiting a long-wavelength instability are characterized by a critical wavelength such that fluctuations with larger wavelengths grow exponentially in time. This type of instability is suppressed in systems smaller than this critical wavelength. On the other hand, certain structures can only exist if they are larger than a certain critical size, such as droplets which, at least within classical nucleation theory, have to be larger than the critical nucleus. This means that certain structures * [email protected] † [email protected] are suppressed by finite size effects, or, to put it differently, the availability of large space can stabilize them. Spinodally unstable flat films show a long-wavelength instability such that the dependence of their stability on the substrate size is obvious. Droplets of nonvolatile fluids, however, are usually considered to be stable. But they are in chemical equilibrium with an adsorbate or a wetting film connected to them [18][19][20][21] which, on very large substrates, acts like a reservoir: a spherical droplet of 100 nm radius has the same volume as an adsorbate layer with an effective thickness of 1Å on a substrate of 6.5 × 6.5 µm 2 . Therefore, an isolated droplet of a nonvolatile fluid placed on a macroscopically large substrate is expected to be unstable with respect to the formation of a film. If a droplet is volatile, i.e., in chemical equilibrium with its vapor, it is expected to be unstable, too, but with respect to evaporation or condensation and the formation of an equilibrium wetting layer. The Laplace pressure in droplets decreases upon increasing their diameter while the pressure in wetting films is determined by the disjoining pressure. In the case of stable films it increases with their thickness. In a stationary situation the pressure in the droplet is balanced by the pressure in the connected film. Moving a small amount of fluid from a droplet into its attached film increases the pressure in the drop and, as the thickness of the film increases, also the pressure in the film. However, due to volume conservation, the larger the substrate the smaller is the increase of the ensuing film thickness and therefore the smaller is the increase of pressure in the film. This implies, that beyond a certain substrate size the pressure increase in the drop is larger than the pressure increase in the film and the drop will dissipate into the large film [35]. On the other hand, a substrate of limited size can only support droplets with a base radius smaller than half the substrate diameter. This means, that one can expect that there is a window of droplet sizes for stable droplets as shown for two-dimensional droplets with small slopes (i.e., liquid ridges with a small contact angle) in Ref. [36]. Since droplet volumes scale with the third power of the droplet radius while the volume of the wetting or adsorbate film scales with the second power of the substrate diameter, the influence of the wetting or adsorbate film on droplet stability is most important on the nanoscale because in this case the volumes of the liquid in the droplet and in the film are comparable. In addition, due to the non-vanishing width of the three-phase-contact line there is a minimal size for well defined droplets [10,37], which gives rise to an additional contribution to the finite size effects. In this spirit, here we study the influence of substrate size on the stability of flat films and of droplets using the framework of density functional theory within the sharp kink approximation, i.e., by minimizing the corresponding effective interface Hamiltonian [38] in the presence of an effective interface potential [39,40]. II. EFFECTIVE INTERFACE HAMILTONIAN Within the capillary model for nonvolatile fluids [41][42][43] interfaces and contact lines are geometrical objects of zero volume and area, respectively, and the free energy of a fluid in contact with a substrate is given by bulk, interface, and line contributions which are proportional to the volume, interface areas, and contact line lengths, respectively. Within this macroscopic model, finite size effects occur only if the three-phase-contact line of a droplet reaches the lateral boundary of the substrate. Wetting transitions and the dependence on temperature and pressure of the thickness of wetting layers cannot be described within this macroscopic model. For this reason, in order to access mesoscopic scales, we resort to the effective interface model as the simplest non-trivial model to describe a fluid in contact with a substrate. It can be derived from a classical density functional theory using the so-called sharp-kink approximation [38,44]. As in the capillary model, also in this approach interfaces are only two-dimensional manifolds but contact lines, such as the three-phase-contact line between fluid, vapor, and substrate have a nonzero width as a result of explicitly taking into account the finite range of intermolecular interactions (for reviews see Refs. [20,21]). Accordingly, within this model line tensions emerge and are not input parameters [45][46][47][48][49][50][51]. The effective, local interface Hamiltonian H for a liquid film in Monge parameterization z = h(x, y) on a homogeneous substrate with the substrate-liquid interface A located in the xy-plane reads H[h] = A dx dy σ 1 + (∂ x h) 2 + (∂ y h) 2 + φ(h) + δµ h ,(1) with the liquid-gas interface tension σ. φ(z) is the effective interface potential [19,39,40] and it describes the effective interaction between the liquid-vapor interface and the liquid-substrate interface. The last term δµ = ∆ρ ∆µ is the product of the undersaturation ∆µ = µ coexistence (T ) − µ at temperature T and the number density difference ∆ρ = ρ liquid − ρ vapor between the coexisting phases, and thus it measures the thermodynamic distance from the bulk two-phase coexistence line. Within mean field theory the equilibrium configuration of the liquid-vapor interface minimizes H[h]. The Monge parameterization is restricted to single valued interface configurations z = h(x, y) so that droplets with contact angles larger than 90 • cannot be described this way. Therefore we rewrite Eq. (1) in a parameter free form also used in the finite element code employed below. For arbitrary parameterizations of the liquid-gas interface r(u, v) = (x(u, v), y(u, v), z(u, v)) the area of the surface element is dA = ∂r ∂u × ∂r ∂v du dv = Gn du dv with the interface normal vectorn pointing into the gas phase and G = ∂r ∂u × ∂r ∂v . In the Monge parameterization this reduces to G = 1 + \|∇h(r)\| 2 , i.e., the first term in the square brackets in Eq. (1). The effective interface Hamiltonian can be written in terms of an integral over the liquid-vapor interface S H[r] = S dA · {σn(u, v) + [φ(z(u, v)) + δµ z(u, v)]ê z } ,(2) withê z as the normal vector of the substrate-liquid interface pointing into the liquid phase, i.e., in z-direction. The existence of a classical density functional has been proven for grand canonical ensembles [52]. Nonetheless the functional in Eq. (1) has been used successfully to describe also equilibrium shapes of nonvolatile fluids (i.e., in the canonical ensemble) by fixing the liquid volume V via a Lagrange multiplier p. In this case, δµ is not an independent parameter. It turns out, that upon adding the constant term δµ V (which is independent of the droplet shape) to the functional in Eq. (1), δµ and p multiply the same terms such that δµ can be absorbed into p. It will turn out later (see Eq. (7)) that p is the pressure difference between the liquid and the vapor, and for droplets one has p > 0, given the choice of sign for the Lagrange multiplier contribution as in Eq. (3). Since in a nonvolatile system the liquid and the vapor are not in thermodynamic equilibrium, the pressures do not have to be equal. This leads to a variation principle for the equilibrium shape of the liquid-vapor interface of nonvolatile fluids. The equilibrium shape r (eq) (u, v) minimizes the functional ( S dA ·ê z = A) F [r(u, v)] = S dA · σn(u, v)+ φ(z(u, v)) − p z(u, v) − V A ê z .(3) In the case of the laterally homogeneous substrates −1 −0.5 0 0 1 h i /h 0 2 3 z / h 0 φ(z) / φ 0 Π(z) / (φ 0 /h 0 ) φ″(z) / (φ 0 /h 0 2 ) FIG. 1. The effective interface potential φ(z) (full red line) according to Eq. (4) and the corresponding disjoining pressure Π(z) = −φ ′ (z) (dashed blue line) in units of φ0 and φ0/h0, respectively. The positions of the minimum of φ(z) at z = h0 and of its i nflection point at z = hi ≡ 6 √ 3 h0 ≈ 1.2 h0 are indicated by vertical dotted lines. Also shown is φ ′′ (z) (dash-dotted green line) which appears in the second variation operatorÔ h in Eq. (10) and which determines the stability of flat film solutions (see Eq. (17)). considered in this paper, the effective interface potential φ(z) does not explicitly depend on the lateral coordinates (x, y). However, due to the formation of droplets one can still find non-trivial solutions to the minimization problem in Eq. (3). The structure of φ(z) depends on the types of intermolecular interactions involved. The simplest effective interface potential for long-ranged dispersion forces (described by Lennard-Jones type interactions) and at temperatures below the wetting temperature has the form φ(z) = φ 0 h 8 0 3 z 8 − 4 h 2 0 3 z 2 .(4) The potential has a minimum of depth −φ 0 at z = h 0 and an inflection point at z = h i ≡ 6 √ 3 h 0 ≈ 1.2 h 0 (see Fig. 1). The potential is negative for z > 3 1/2 h 0 and approaches zero from below for z → ∞. The shape of φ(z) corresponds to that of a continuous wetting transition [19]. A. Minimizing the free energy functional Within mean-field theory the minimum of the effective interface functional F containing the volume constraint (Eq. (3)) renders the interfacial free energy for the corresponding stable equilibrium configuration. The functional in Eq. (3) can be minimized numerically by means of an adaptive finite element algorithm implemented by the software Surface Evolver [53]. Therein, the liquid-vapor interface is represented by a mesh of oriented triangles and, by means of a gradient projection method, iteratively evolves towards the configuration of minimal F (for an example see Fig. 2) below. B. Variations of the effective interface Hamiltonian Within the framework of variational calculus, a stable equilibrium profile corresponds to a vanishing first variation and a negative second variation of the functional F . In order to calculate them we return to the Monge parameterization and introduce the perturbed interface configuration z =h(x, y) withh(x, y) = h(x, y) + ǫ Ψ(x, y) and p = p + ǫ ψ, where 0 < ǫ ≪ 1 is a small dimensionless parameter. It is straightforward to show that the first vari- ation δ (1) F of F ([h],p) = F ([h], p) + ǫ δ (1) F + ǫ 2 δ (2) F + O(ǫ 3 ) with respect to the interface configuration is given by (h = h(x, y)) δ (1) F = A dx dy Ψ [−2 σ H h + φ ′ (h) − p] + ψ A dx dy h − V A ,(5) with the mean curvature H h = (∂ 2 x h) [1 + (∂ y ) 2 ] − 2 (∂ x h) (∂ y h) (∂ x ∂ y h) + (∂ 2 y h) [1 + (∂ x h) 2 ] 2 1 + (∂ x h) 2 + (∂ y h) 2 3(6) of the unperturbed surface and φ ′ (h) denoting the derivative of the effective interface potential with respect to the local film thickness. The Euler-Lagrange equation corresponding to the vanishing of δ (1) F is 2 σ H h + Π(h) + p = 0(7) together with V = A dx dy h,(8) where Π(h) = −φ ′ (h) is the disjoining pressure, which describes the effective interaction between the substrate surface and the film surface, and 2 σ H h is the Laplace pressure, which follows from the interface tension of the fluid surface. For equilibrium interface configurations the sum of the disjoining pressure and of the Laplace pressure is constant. The variation with respect to the Lagrange multiplier p leads to the volume constraint (see Eq. (8)). The second variation δ (2) F of F with respect to the film height can be written as a form quadratic in the perturbation Ψ: δ (2) F = A dx dy ΨÔ h Ψ + 2 ψ Ψ ,(9) with the self-adjoined operator O h = −σ ∂ x ∂ y ·   1+(∂yh) 2 [1+(∂xh) 2 +(∂yh) 2 ] 3 2 (∂xh) (∂yh) [1+(∂xh) 2 +(∂yh) 2 ] 3 2 (∂xh) (∂y h) [1+(∂xh) 2 +(∂yh) 2 ] 3 2 1+(∂xh) 2 [1+(∂xh) 2 +(∂yh) 2 ] 3 2   · ∂ x ∂ y + φ ′′ (h),(10) and with the second derivative φ ′′ (h) of the effective interface potential. For the model potential given in Eq. (4) φ ′′ (h) is shown in Fig. 1. It is positive for small h and negative for large h. The second variation with respect to the Lagrange multiplier is identical to zero. The mixed variation with respect to p and h leads to the second term in Eq. (9) which due to ψ = const. vanishes for perturbations Ψ(x, y) which conserve the volume. The stability of a solution of the Euler-Lagrange equation (7) is determined by the spectrum of eigenvalues ofÔ h . A solution is linearly stable if all eigenvalues are positive. III. THIN FILMS AND NANO-DROPLETS On a chemically homogeneous substrate with an area A there exist two distinct classes of solutions of the Euler-Lagrange equation (7). One consists of f lat films with h (eq) (x, y) = h f = V /A.(11) The other class consists of nontrivial droplet solutions with one or many droplets smoothly connected to a wetting film. Here we focus on solutions with a single droplet because in general two or more droplets connected via a wetting film are unstable with respect to coarsening. In the following we discuss the stability of flat films and such droplets as a function of the substrate area A, of the excess liquid volume V ex = V − A h 0 = (h f − h 0 ) A,(12) and of material properties encoded in φ(h)/σ. A. Flat films For flat films with homogeneous thickness h f the Euler-Lagrange equation (7) reduces to p + Π(h f ) = 0.(13) This means that for any size of the substrate area a homogeneous flat film obeying Eq. (13) is obviously a solution of the Euler-Lagrange equation. It represents either a local maximum, a local minimum, or a saddle point of the free energy functional in Eq. (3). The curvature of the interface is zero and thus the liquid gas interface tension drops out. If h f minimizes the effective interface potential one has p = 0 (assuming that φ(h) is differentiable). For a flat interface the operatorÔ h in Eq. (10), which determines the linear stability of the flat film solution, reduces toÔ h = −σ (∂ 2 x + ∂ 2 y ) + φ ′′ (h f ).(14) The corresponding eigenvalue problem has the form of a stationary single particle Schrödinger equation with a potential which is constant across the domain of the substrate. In Fig. 1 φ ′′ (z) is shown for the model potential from Eq. (4). The inverse surface tension plays the role of the mass. The eigenvalue spectrum of this operator depends on the shape of the domain and on the boundary conditions at its borders. Boundary conditions corresponding to actual substrates of finite size are rarely compatible with a flat film solution because usually there is a bending of the interface at the edge of the domain. For example, at the edge of a lyophilic patch on a lyophobic substrate the film thickness will go to zero (or at least to a microscopically small value) and at the brim of a flat piece of substrate the fluid film either continues onto the side walls or ends with thickness zero. The two simplest types of mathematical boundary conditions, which allow for flat film solutions, are either periodic boundary conditions or a Neumann type boundary condition which corresponds to zero slope of the film surface at the domain boundaries. The latter would correspond to upright side walls with an equilibrium wetting angle of 90 • at a pit-shaped substrate. However, even for such a setup, the interplay of the long-ranged forces from the substrate and from the side wall would lead to a bending of the film surface [54,55]. For a square substrate with edge length L = √ A and with Neumann boundary conditions the eigenvalue problem corresponding toÔ h can be factorized by separating the variables and the eigenfunctions are given by plane waves. The degeneracy of the eigenfunctions character-ized by wave vectors of equal modulus is alleviated by the boundary condition. Assuming the two edges of the substrate to be aligned with the x-axis and with the y-axis, respectively, the eigenfunctions are given by Ψ nm (x, y) ∝ cos 2 π n L x cos 2 π m L y ,(15) with n, m ∈ N 0 . Since Ψ (−n)m = Ψ n(−m) = Ψ (−n)(−m) = Ψ nm we only consider non-negative indices. Since we consider a nonvolatile system there is volume conservation, i.e., A dx dy Ψ nm = 0 and therefore either n or m have to be positive. The corresponding eigenvalues are given by λ nm = σ 2 π L 2 n 2 + m 2 + φ ′′ (h f ).(16) Therefore the film is linearly stable, i.e., min n,m λ nm > 0, for (2 π) 2 A > − φ ′′ (h f ) σ . (17) For substrates of infinite, i.e., macroscopic, size A this is the case only if φ ′′ (h f ) > 0. For the model effective interface potential in Eq. (4) the latter inequality holds for h f < h i ≡ 6 √ 3 h 0 ≈ 1.2 h 0 ,(18) Since h i > h 0 (see Fig. 1) films with negative excess volumes (i.e., h f < h 0 (see Eq. (12))) exhibit φ ′′ (h f ) > 0 so that, according to Eq. (17), they are linearly stable for any substrate size A = L 2 . However, even for φ ′′ (h f ) < 0 flat films are linearly stable if the substrate size L is below the critical value L c = 2 π σ/\|φ ′′ (h f )\|. This perturbation analysis does not yield any information about the nonlinear stability of film solutions, i.e., whether a flat film has a lower free energy than a droplet. B. The nano-droplet configuration For a given area A and a certain ratio V ex /(A h 0 ) (see Eq. (12)), nano-droplets with a nonzero pressure p > 0 minimize the free energy F in Eq. (3). This is due to the interplay of the surface free energy densities and the effective interface potential, in combination with the nonvolatility of the liquid and the finite area A of the solidliquid interface. Since the difference between the liquidsubstrate and the gas-substrate surface tensions is given by σ + φ(h 0 ) Young's law [56] reads [19]: cos θ eq = 1 − φ 0 σ .(19) θ eq denotes the equilibrium contact angle of a macroscopic drop. The influence of the ratio φ 0 /σ on the shape of a nano-droplet is shown in Fig. 2. A suitable definition of the contact angle of a nano-droplet is to determine the curvature of its surface at the apex, to inscribe the corresponding cap of a sphere which intersects the asymptote . During the iterative minimization process, the mesh size of the triangulation has been coupled to the evolution of the interface shape in an adaptive way in order to optimize the spatial resolution locally. The lateral boundary conditions are implemented by a constraint on the boundary vertices, such that their lateral coordinates are fixed during the minimization process while the perpendicular height coordinate can evolve freely, effectively corresponding to neutral wetting (contact angle 90 • ) at vertical side walls (not shown) or Neumann boundary conditions. of the attached wetting film thus forming a contact angle [51]. For the systems studied here, this contact angle is smaller than θ eq . The wetting film surrounding the nano-droplet is almost flat, i.e., 2 σ \|H h \| ≪ \|Π(h)\| (see Eq. (7)). According to this Euler-Lagrange equation (7), the spatially constant pressure p is approximately given by p ≈ −Π(h (eq) w ),(20) and thus h (eq) w > h 0 implies p > 0 (see Fig. 1). The height h (eq) w of the wetting film, the pressure p, the disjoining pressure Π(h (eq) w ) of the wetting film, and the ratio between the drop free energy F drop and the free energy F film of a flat film with the same excess volume are shown in Table I for several values of V ex . For decreasing values of V ex /(A h 0 ) with constant A, p increases. This is mainly due to the increasing curvature of the liquid-vapor interface. For the same reason the pressure in macroscopic drops also increases with decreasing volume. While the free energy F drop of large drops turns out to be smaller than the free energy F film of a flat film with the same excess volume, the situation is reversed for V ex /(A h 0 ) < 0.06 (the critical excess volume lies between 0.05 A h 0 and 0.06 A h 0 ). This means that nano-droplets below a certain size become metastable or unstable. is the height of the film at the edge of the numerical domain. The pressure p as the value of the Lagrange multiplier for fixing the volume and the disjoining pressure Π(h (eq) w ) of the wetting film surrounding the droplet as calculated from the numerically determined h (eq) w are balanced according to Eq. (20). Accordingly, the differences between the third and fourth column indicate the level of numerical accuracy. F drop /F film is the ratio of the (mean-field) surface free energy of a nano-droplet and the free energy of a flat film with a height h f = Vex/A + h0. Table I. Although the domain for the numerical calculation is rectangular, the droplet shape is to a good approximation radially symmetric (r = x 2 + y 2 ). The wetting film height for both profiles is h (eq) w = 1.01 h0 and the contact angle is αeq = 55 • compared with θeq = 60 • for the corresponding macroscopic drop. The free energies for these profiles agree up to the third digit. Vex/(A h0) h (eq) w /h0 − 1 p h0/σ −Π(h (eq) w ) h0/σ F drop /F film 0.h / h w (eq) r / h 0 α eq h c (eq) h (eq) C. Morphological transition In order to analyze the morphological phase transition between nano-droplets and flat films as indicated by the numerical data discussed above, we minimize the effective interface Hamiltonian F in Eq. (3) in the subspace of interface shapes h c (x, y) describing a spherical cap sitting on top of a flat wetting film (see Fig. 3). For a given total volume of liquid, these trial profiles are parameterized by the contact angle α and the wetting film height h w . The latter determines the fluid volume available for the drop connected to the film and the contact angle determines the drop shape. This ansatz reduces the problem of min-imizing F in Eq. (3) to a minimization problem of the function F (α, h w ) = F [h c (x, y)](21) depending on the two variables α and h w with the minimum at α eq and h (eq) w . The corresponding minimizing profile is denoted by h (eq) c . In contrast to the direct, full numerical minimization of the free energy functional in Eq. (3), the function F (α, h w ) provides also a free energy landscape in the parameter space (α, h w ). Since for these two-parameter trial functions the wetting film is perfectly flat, the Laplace pressure vanishes and instead of Eq. (20) one has p = −Π(h (eq) w ).(22) In the macroscopic limit, i.e., upon increasing both A and V ex such that A h 2 0 → ∞ with V ex A h 0 = const(23) one finds h (eq) w → h 0 for the droplet solution because in this limit the Laplace pressure 2 σ H h as well as the disjoining pressure at the cap apex vanish. The reason for this is that the curvature of the droplet surface goes to zero if the drop size diverges and that the disjoining pressure vanishes for large distances from the substrate surface. Therefore the sum of the disjoining pressure and of the Laplace pressure, i.e., −p, also vanishes (see Eq. (7)). The Lagrange multiplier p does not depend on the position along the droplet surface and, according to Eq. (22), the disjoining pressure on the wetting film is also zero. Therefore a macroscopic liquid cap with volume V ex is formed above the level h (eq) w = h 0 where Π(h 0 ) = 0. The numerical minimization of F (α, h w ) also yields, in this limit, α eq → θ eq with θ eq given by Eq. (19). Figure 4 shows the free energy landscape F (α, h w ) for a large drop. All points in the parameter space {α, (h w − h 0 )/(h f − h 0 ) < 1} correspond to droplet solutions (see Eq. (11)), i.e., α = 0. The line (h w − h 0 )/(h f − h 0 ) = 1 corresponds to a flat film solution for which F (α, h w ) is independent of α because the volume of the droplet is zero. The global minimum of the free energy is located at α eq ≈ θ eq = 60 • and h (eq) w ≈ h 0 . The contour lines of the free energy landscape close to the minimum in Fig. 4 are almost parallel to the α axis and hence shape fluctuations of the liquid cap with a constant cap volume are more likely than volume fluctuations, i.e., fluctuations of the wetting film height h w . As shown in the inset of Fig. 4 the equilibrium angle α eq approaches the macroscopic equilibrium contact angle θ eq from below. In Fig. 4 the excess volume is chosen such that h f < h i , i.e., according to Eq. (18) the film configuration is linearly stable. Nonetheless, the droplet solution is the global minimum of F (α, h w ). However, as shown in Fig. 5(a) there is a minimal droplet size V ex below which droplets cannot exist: reducing the droplet size the Laplace pressure in the droplet increases until it cannot be counterbalanced by the negative disjoining pressure −p = Π(h eq w ) (see Eq. (22)) in the film (Π(z) has a minimum of finite depth (see Fig. 1)) and the droplet drains into the film. In Fig. 5(a) there is also a second branch of droplet solutions which are unstable and which have a pressure p intermediate between the pressure of the metastable or stable droplets and of the flat film. For a given value of V ex such a droplet solution corresponds to the saddle point in the two-dimensional parameter space between the two (local) minima given by the droplet solution and the flat film solution. Upon reaching the macroscopic limit, the unstable droplet branch asymptotically approaches the flat film pressure from below ( Fig. 5(a)). This means, that the thickness h (eq) w of the wetting film surrounding the unstable droplets approaches the thickness h f = V ex /A + h 0 of the flat film solution. Therefore the volume inside the unstable droplets (i.e., above h w ) decreases monotonically as the macroscopic limit is approached. Figure 5 corresponds to Fig. 12 in Ref. [36] where, however, the volume rather than the pressure is plotted as a function of the substrate size without discussing the stability of the solutions. We conclude that in this respect in essence there is no qualitative difference between the quasi two-dimensional ridges studied in Ref. [36] and the three-dimensional systems studied here. For large excess volumes with h f > h i , according to Eq. (18), the flat film solution is linearly unstable. Therefore it should represent a saddle point or a maximum in the free energy landscape. The droplet solution should represent the global minimum. However, as shown in Fig. 5(b) the flat film solution for h f = 1.25 h 0 > h i is either stable or metastable, but not unstable within this only two-dimensonal parameter space considered here. In addition there is an unphysical branch of unstable droplet solutions with pressures above the pressure of the flat film solution. The reason for this artefact is, that a slightly undulated film cannot be represented in this two-dimensional parameter space; but spinodal dewetting occurs via the growth of such small perturbations. According to Eq. (17), the critical substrate size, below which the instability is suppressed by the finite size effects, is A/(π h 2 0 ) ≈ 6, i.e., much smaller than A/(π h 2 0 ) ≈ 11.2, the smallest substrate size for which the present two-dimensional parameter space analysis predicts the existence of droplet solutions (see Fig. 5(b)). In view of this inconsistency we conclude that the results obtained within this approximate scheme for very small substrate sizes are unreliable. However, the actual stability of droplets in the macroscopic limit is correctly covered within this model. At the morphological transition a flat film and a droplet of equal volume have the same free energy but different pressure. In the theory of thermodynamic phase transitions, it is common to consider transitions between shows the equilibrium angle αeq upon approaching the macroscopic limit as described by Eq. (23) for the same excess volume as used in the main figure; αeq approaches θeq from below. The error bars are due to numerical inaccuracies. With h f /h0 = Vex/(A h0) + 1 = 1.06 < hi/h0 = 1.2 (see Eq. (18)) the flat film solution with (hw − h0)/(h f − h0) = 1 is expected to be linearly stable (i.e., metastable). But the number of data points calculated here is to small in order to be able to detect the corresponding free energy barrier (cf. Fig. 9 for a smaller substrate). states of different volume (or density) but equal pressure (or more general, between states with equal intensive state variables but distinct extensive ones). These states can spatially coexist with each other. However, the morphological transition between a flat film and a droplet is of a different nature in the sense that the droplet solution and the flat film solution do not coexist with each other in space: the system as a whole switches from one solution to the other. This is not to be confused with the coexistence between a droplet and the wetting film to which it is connected. While the pressure in the wetting film and the pressure in the droplet are equal, this droplet configuration does not represent a bona fide thermodynamic phase: its pressure changes with size whereas from a proper thermodynamic phase one would expect to be able to produce systems of different size but with the same pressure. In fact, Eq. (1) has the structure of a Ginzburg-Landau Hamiltonian but the potential Φ(h) has its second minimum at h → ∞. In this sense the droplet as a whole amounts to an interfacial region. The free energy landscape for finite systems with various excess volume ratios V ex /(A h 0 ) are shown in Figs. 6- Fig. 6, the droplet configuration with α eq ≈ 57 • and h For the large excess volume in (eq) w − h 0 ≈ 0.2 (h f − h 0 ) is the global minimum. The flat film solution with h f = 1.1 h 0 < h i is linearly stable as expected for the chosen effective interface potential (see Eq. (18)). Upon decreasing the excess volume the free energy of the droplet solution increases and the minimum becomes shallower (see Fig. 7). At a certain excess volume, the flat film solution becomes the stable solution and the droplet solution becomes metastable. Reducing the excess volume even further, the free energy minimum corresponding to a droplet solution becomes more and more shallow until it finally merges with the corresponding saddle point (see Fig. 8), and vanishes completely. This leaves the film solution as the only stable solution. This morphological transition is visualized even better by forming vertical cuts of the free energy landscape at fixed h w , i.e., parallel to the α-axis and by seeking the minimum of the free energy F within each cut as a function of α. This renders F min (h w ) = min α F (α, h w ). In Fig. 9 the corresponding the minimal free energy F min (h w ) is shown as a function of the wetting film thickness h w . The energy scale is normalized by the free energy F f (V ex ) of the corresponding flat film solution (compare Figs. 4 to 8). For very small excess volumes the free energy as a function of the wetting film thickness is mono- flat film (at V ex /(A h 0 ) ≈ 0.057). This marks the point of the morphological transition between a flat film and a droplet solution. Increasing the excess volume even further the droplet solution becomes more stable. According to the inset of Fig. 9 it seems that the flat film solution (i.e., (h w − h 0 )/(h f − h 0 ) = 1) becomes unstable for V ex /(A h 0 ) ≥ 0.2 as expected from Eq. (18). However, for this latter value a tiny free energy barrier cannot be ruled out on the basis of the available numerical data. Figure 10 shows the pressure p as a function of the excess volume for a homogeneous film of thickness h f (upper curve) and for the droplet solution (lower curve). The upper branch is exact while the lower branch is calculated by minimizing the approximate expression for the free energy F (α, h w ) defined in Eq. (21). According to Eq. (22), for both branches one has p = −Π(h eq w ). Figure 10 also shows pressure values obtained by numerical minimization of the full functional F (for which, according to Eq. (20), p ≈ −Π(h eq w )). The pressure in the flat films (upper curve) is given by p = −Π(h f ) with h f /h 0 = V ex /A + 1 (see Eq. (22)) and it has a maximum at V ex /(A h 0 ) = 0.2, corresponding to h f = h i . For excess volumes smaller than V ex /(A h 0 ) = 0.2 the flat film solution is metastable or stable. For larger excess volumes, the flat film solution is linearly unstable. However, the spinodal wavelength is extremely large close to the pressure maximum such that, according to Eq. (17) and for the given substrate size, the instability actually sets in only for V ex /(A h 0 ) > 0.2011. In Fig. 10 for 0.05 < V ex /(A h 0 ) < 0.2 there are two curves below the curve corresponding to the flat film solution; the upper one (red circles with crosses) corresponds to a saddle point in the free energy landscape and the lower one corresponds to a (potentially local) minimum. Both branches represent droplet solutions. The unstable branch ends at V ex /(A h 0 ) = 0.2, i.e., at the maximum of the pressure in the flat film solution. The three curves in Fig. 10 form a hysteresis loop. The value of V ex /(A h 0 ), at which the transition (thin vertical line in Fig. 10) between a flat film and a droplet occurs, can be obtained either by comparing free energies directly or via a Maxwell construction (see Fig. 11). The latter can be shown by integrating p = ∂F /∂V = ∂F /∂V ex (due to Eq. (3) and since dV /dV ex = 1 due to Eq. (12)) along p(V ): F (V ) − F (V 0 ) = V V0 p(V ) dV . Starting the integration at the volume V eq at which the free energy of the film (upper branch) and the stable droplet (lowest branch) are equal (see Fig. 11) one integrates up to V = V i , i.e., the volume of a film of thickness h i at which the unstable droplet branch merges with the film branch. The result, i.e., the sum of area (1) and (2) in Fig. 11, is the difference of the free energies of a film with volume V i and a film with volume V eq . At V i one switches to the unstable droplet branch and integrates down to its end at V m . The result is the difference between area (1) and the sum of area (3) and area (4). From there one continues on the metastable droplet branch up to V eq , which adds area (4). As a result, the difference of the free energy of a flat film of volume V eq and a stable droplet of the same volume is the difference between area (1) and area (3). For the chosen model interface potential in Eq. (4) the flat film solution becomes linearly unstable at the value of V ex /(A h 0 ) (i.e., 0.2 in Fig. 10), where the unstable droplet curve merges with the flat film curve. In Fig. 10 the excess volume is expressed in terms of the substrate area. In order to discuss whether the minimal droplet size is determined by the interface potential or by the substrate size, one could fix the excess volume V ex (as a measure for the droplet size) and the substrate potential and vary the substrate size A. But the excess volume is defined as the fluid volume above the height h 0 (see Eq. (12)) and increasing the substrate area A for fixed V ex means effectively reducing the droplet size. The droplet volume V d = V − A h w above the height of the wetting film h w is a more suitable measure for the droplet size. For this reason in Fig. 12 we plot the droplet volume V d as a function of the excess volume V ex for two substrate sizes. The data are obtained in the following way: for each fixed value of A and of V ex (i.e., for fixed total volume V = V ex + A h 0 ) the interfacial free energies as shown in Figs. 6-8 are calculated. The position of local and global minima and of saddle points (corresponding to stable, metastable, and unstable droplet or flat film solutions) are determined numerically, in particular the Fig. 10. The color code corresponds to the one in Fig. 10: green, blue, and red indicate stable, metastable, and unstable states, respectively. V0 denotes the volume of a film of thickness h0, Vm is the minimal volume required to form a droplet, Veq is the volume at which the free energies of the flat film and of the stable droplet are equal. For V ր Vi the branch of metastable flat films turns into a branch of unstable flat films. There also the branch of unstable droplets merges into the flat film branch. V V V V V p 0 m i eq droplet h i −Π( ) wetting film thickness h w from which one can determine the droplet volume V d = V ex − A (h w + h 0 ). As in Fig. 10 The nonexistence of droplet solutions for too small values of V ex can be rationalized by considering a further simplified reduced expression for the free energy. Neglecting the influence of the disjoining pressure on the spherical cap the minimization problem for F (α, h w ) yields (see Eq. (21) and up to the constant substrateliquid surface tension) F = (A − r 2 π) [σ + φ(h w )] + σ S d ,(24) with r = 2 h d R − h 2 d denoting the base radius of the d rop (taken at z = h w ) and S d = 2 π R h d denoting the surface area of a spherical cap of height h d and radius R. The volume of the spherical cap is given by V d = free energy V d / h 0 3 V ex / (A h 0 ) (F (h d , h w ) = A 1 + 2 h w h d − 2 V h d + π 3 h 2 d φ(h w ) + σ A + π h 2 d(25) as a function of the droplet height h d rather than the droplet contact angle α. The minimum of F (h d , h w ) follows from the zeroes of its first derivatives with respect to h d and h w . Using the above expressions for V and V d one obtains from ∂F (h d , h w )/∂h d = 0 R φ(h w ) + σ h d = 0.(26) Using this expression together with the above expressions for V and V d one obtains from ∂F (h d , h w )/∂h w = 0, after reintroducing α via the geometric condition r = R sin α, Π(h w ) = − 2 σ R 1 − π R 2 sin 2 α A .(27) Apart from a small correction (which is small if A is large compared with the base area π R 2 sin 2 α of the droplet) Eq. (27) tells that the disjoining pressure in the film and the Laplace pressure 2 σ H h = −2 σ/R (see Eq. (6)) in the droplet are equal (according to Eq. (7) both are equal to p). Using the geometric relation cos α = 1 − h d /R in Eq. (26) we also get 0 as obtained graphically from (a) (black squares and magenta circles, respectively) and from the analytic approximation described in the main text (full black and dotted magenta line, respectively). cos α = 1 + φ(h w ) σ .(28) In the macroscopic limit R → ∞ in Eq. (27) implies Π(h w ) → 0, i.e., h w → h 0 so that φ(h w ) → φ(h 0 ) = −φ 0 (see Fig. 1), and therefore α → θ eq (see Eq. (19)). As a function of α, R, and h w the total conserved fluid volume is V = A h w + π R 3 3 (2 + cos α) (1 − cos α) 2 .(29) For a given value of α Eqs. (27) and (29) R(h w , h f , A) = 3 3 A (h f − h w ) π 3 + φ(hw) σ φ(hw) σ 2(30) due to V = A h f . Accordingly, one can consider both sides of Eq. (27) as a function of h w as shown in Fig. 13(a) where α is approximated by θ eq = arccos(1 − φ 0 /σ). The right hand side of Eq. (27) increases (decreases in absolute value) upon increasing h f /h 0 = V /(A h 0 ). For large A/R 2 and h w ≈ h 0 , the right hand side of Eq. (27) is approximately given by − φ 0 h 0 2 3 √ 3 3 σ φ0 3 + φ0 σ A π h 2 0 h f −hw h0 .(31) The two curves only intersect if the fluid volume (or h f = V /A) is sufficiently large (see the three blue dashed curves in Fig. 13(a)). For sufficiently large excess volumes, i.e., for sufficiently large h f there are two intersections in Fig. 13(a). Because for fixed total volume V increasing h w (i.e., increasing the amount of liquid in the film) means decreasing the droplet volume, the intersection at the larger values of h w corresponds to the unstable solution while the intersection at the smaller value of h w corresponds to the stable droplet solution. (The unstable droplet is always smaller than the stable one.) In the macroscopic limit A/h 2 0 → ∞ with fixed h f = h 0 + V ex /A (see Eq. (23)) the stable solution moves to h w → h 0 . This means that the volume of the stable droplet gets very large because due to h w → h 0 the whole excess volume goes into the droplet. In the macroscopic limit, the unstable solution moves to h w → h f . We can obtain the corresponding leading behavior by the following procedure. First we insert R(h w , V, A) as obtained from Eq. (29) into Eq. (27) and we replace cos α by the expression in Eq. (28). After substituting V = A h f we expand both sides in powers of h w − h f and we obtain in leading order h f − h w ∼ 1/A [57]. As a consequence, in the macroscopic limit the volume V d = A (h f − h w ) of the unstable droplet should converge to a finite value. However, this primitive model only applies to large droplet volumes and therefore this result for unstable drops might turn out to be an artefact of the approximations used. As shown in Fig. 13 Fig. 13(a)). If one makes the additional approximations of using α ≈ θ eq and of reducing the right hand side of Eq. (27) to the Laplace pressure by neglecting the term ∼ A −1 in the denominator of the right hand side of Eq. (27), one can determine h c f and h c w analytically with the result h c f /w − h 0 ∝ A −1/4 (see Fig. 13(b)). The drop volume is V d = V − A h w = A (h f − h w ) . Thus the volume of the smallest possible droplet diverges for A → ∞ as V c d = (h c f − h c w ) A ∝ A 3/4 . IV. SUMMARY AND CONCLUSIONS We have studied the stability of nonvolatile flat films and droplets on smooth and chemically homogeneous substrates with finite surface area A. The analysis is based on density functional theory within the so-called sharp kink approximation, i.e., by minimizing the effective local interface Hamiltonian with the effective interface potential shown in Fig. 1. The stability of flat films and of nano-droplets is strongly affected by finite size effects. We have shown that in these systems (i) spinodal dewetting can occur only if the substrate area A is large enough to support the shortest unstable wavelength, (ii) there is a minimal size for droplets connected to a surrounding wetting layer, (iii) droplets are unstable with respect to drainage into a connected wetting films if the substrate area is too large, and (iv) that fluctuations of the droplet shape under the constraint of a fixed volume are more likely than volume fluctuations. Our findings are manifestations of the general rule that long-wavelength instabilities are suppressed by finite size effects. The shortest instability wavelength L c = 2 π σ/\|φ ′′ (h f )\| of spinodal dewetting depends on the material properties, i.e., on the surface tension σ and on the effective interface potential φ(z), as well as on the average film thickness h f = V /A, whereas V is the conserved total liquid volume. In particular for film thicknesses close to inflection points of φ(z) and for thick films this wavelength becomes very large. For differentiable effective interface potentials the second derivative has a maximum (typically at a thickness of a few h 0 where φ ′ (h 0 ) = 0). Therefore the spinodal wavelength L c of films with the corresponding thickness has a minimum. Experimentally spinodal wavelengths of the order of microns have been reported [2,3]. This means that spinodal dewetting can be suppressed by structuring the surface, e.g., by a periodic pattern of hydrophilic and hydrophobic stripes, the latter ones with a width smaller than L c . The width of the hydrophilic stripes which is necessary to stabilize the film has to be determined separately. We have calculated the shape of nano-droplets numerically as shown in Fig. 2 and we have determined the thickness h w of the wetting film on which the nanodroplet resides (see Table I). Using a subset of trial function for the droplet shape which are parameterized by the contact angle of the droplet and by the wetting film thickness h w (see Fig. 3) we have mapped the free energy landscape of the system (see . In contrast to macroscopic drops (see Fig. 4), for nanodroplets the influence of the wetting film to which they are connected cannot be neglected. If the excess volume V ex = V − A h 0 = (h f − h 0 ) A is fixed, there is a minimal substrate size below which no droplet solutions exist (see Fig. 5). Conversely, for a fixed substrate size A one can find droplet solutions only above a critical (excess) volume (see Figs. 7 and 10). This is reminiscent of classical nucleation theory which also leads to the notion of a critical nucleus size. However, in the latter case one usually considers unbounded systems such that one cannot obtain stable droplet solutions at all. In the present case, the conserved total volume of fluid is distributed between a finite sized wetting film and a droplet; this allows for stable droplet solutions. As illustrated in Fig. 11 the volume V (or excess volume V ex = V − A h 0 ) at which the free energy of the flat film solution (a film of homogeneous thickness h f = V /A) equals the free energy of the stable droplet (indicated by a thin vertical line in Fig. 10) can be determined by a Maxwell construction. This construction is based on the observation that the Lagrange multiplier p (i.e., the pressure difference between the liquid and the vapor phase) is given by p = ∂F /∂V , i.e., by the partial derivative with respect to the chosen total volume V (see Eq. (3)). The size of the smallest possible droplet increases (see Fig. 12) and the thickness of the wetting film surrounding the droplet decreases upon increasing the substrate area (see Fig. 13). Within a suitable approximation of the free energy we have found that the volume V c d of the smallest possible droplet diverges upon increasing the substrate size A as V c d /h 3 0 ∝ (A/h 2 0 ) 3/4 . The proportionality factor depends on the equilibrium contact angle θ eq and for nonzero contact angles it is of the order of unity with 0.077 as a lower bound (realized at θ eq = 180 • ). For h 0 ≈ 1Å this means that the minimal droplet volume on substrates of size A = 1 mm 2 , 1 µm 2 , and (100 nm) 2 equals that of a cube of edge length 300 nm, 10 nm, and 1.4 nm, respectively. On the same substrate the volumes of the connected wetting films of thickness 1Å fit into cubes of an edge length of 4.6 µm, 46 nm, and 10 nm, respectively, i.e., they are much larger. Our results show that nonetheless the finite extent of the substrate surface plays a significant role for the droplet formation and the associated morphological phase transition. configurations h (eq) of nano-droplets as obtained by numerical minimization of the functional F in Eq. (3) based on the effective interface potential in Eq. (4) for A/(π h 2 0 ) = 100 2 and under the constraint Vex/(A h0) = 0.5, with φ0/σ = 0.5 (θeq = 60 • , left panel (a)), and φ0/σ = 0.1 (θeq = 26 • , right panel (b)). In the projected side view (bottom row), the underlying substrate of area A is indicated in blue. Lengths are given in units of h0. Drop heights are measured from above the wetting film thickness h FIG. 3 . 3Vertical cut through the apex of a fully numerically obtained interface profile h (eq) (red squares) and the corresponding approximate profile h (eq) c (full blue line) consisting of a spherical cap resting on a flat film. The profiles correspond to A/(π h 2 0 ) = 100 2 , Vex/(A h0) = 0.5, and φ0/σ = 0.5, which are the parameters corresponding toFig. 2(a)and to the bottom line in FIG. 4 . 4The approximate interfacial free energy F (α, hw) close to the macroscopic limit as described by Eq.(23): A/(π h 2 0 ) = 10 8 , Vex/(A h0) = 0.06, and for φ0/σ = 0.5 corresponding to θeq = 60 • . F f = A (σ + φ(h f )) is the free energy of the flat film solution for these parameters; h f −h0 = Vex/A. The global minimum is located at α ≈ θeq = 60 • and hw ≈ h0. The contour lines are almost parallel to the α-axis. (The contour lines shown range from 0.9792 to 0.98 in steps of 0.0001 and from 0.98 to 1.0 in steps of 0.001.) The inset FIG. 5 . 5The pressure p = −Π(h (eq) w ) (see Eq. (22)) (in units of p f = −Π(h f )) calculated from the approximate free energy in Eq. (21) as a function of the substrate size A and of the excess volume Vex for (a) Vex/(A h0) = 0.04 and (b) Vex/(A h0) = 0.25, i.e., for a fixed homogeneous film thickness h f = 1.04 h0 < hi and h f = 1.25 h0 > hi, respectively, and φ0/σ = 0.5. Note that for a fixed ratio Vex/(A h0) = vex one has Vex/h 3 0 = π vex x 2 with x = [A/(π h 2 0 )] 1/2 .Open blue and full green symbols indicate metastable and stable states, respectively, and circles with crosses indicate unstable droplet solutions. Stable and metastable flat film solutions are indicated by boxes and stable and metastable droplets by circles. The vertical line indicates the morphological transition between stable films and stable droplets as obtained via numerical comparison of the corresponding two free energies. (A Maxwell construction for determining this transition point is discussed in, cf.,Fig. 11.) A → ∞ corresponds to the macroscopic limit. (a) For h f < hi, as in the present case corresponding to Vex/(A h0) = 0.04, the flat film solution is stable or metastable for all A. Droplets (lowest branch) occur for A/(π h 2 0 ) 140 and they are stable for A/(π h 2 0 ) 180. (b) Within the present free energy approximation, the flat film solution is stable or metastable (although it should be unstable according to Eq. (18)) and there is an unphysical unstable branch of droplet solutions (top branch). Droplets are stable or metastable for for all substrate sizes A. FIG. 6 .−FIG. 7 . 67The approximate interfacial free energy F (α, hw) for A/(πh 2 0 ) = 100 2 , Vex/(A h0) = 0.10, and φ0/σ = 0.5.F f = A (σ + φ(h f ))is the free energy of the flat film solution for these parameters; h f − h0 = Vex/A. The global minimum representing a stable nano-droplet is located at (h (eq) w − h0)/(h f − h0) ≈ 0.2 and αeq ≈ 57 • , i.e., close to but smaller than the macroscopic equilibrium contact angle θeq = 60 • for this system. The flat film solution h h0)/(h f − h0) = 1) is metastable; for this solution there is no dependence on α. Contour lines are shown in the range 0.9802 to 0.9809 in steps of 0.0001 and from 0The approximate free energy F (α, hw) as defined in Eq. (21) for A/(πh 2 0 ) = 100 2 , Vex/(A h0) = 0.06, and φ0/σ = 0.5 (i.e., for the same parameters as inFig. 6but for a smaller value of Vex). F f = A (σ + φ(h f )) is the free energy of the flat film solution for these parameters; h f − h0 = Vex/A. The contact angle corresponding to the global minimum is αeq ≈ 55 • at (h(eq) w − h0)/(h f − h0) ≈ 0.04, i.e., smaller than in Fig. 6. The flat film solution h (eq) w = h f (so that (h (eq) w − h0)/(h f − h0) = 1) is metastable and exhibits no dependence on α. Contour lines are shown in the range 0.99902 to 0.9991 in steps of 0.00002 and from 0.9992 to 1.003 in steps of 0.0002. FIG. 8 .FIG. 9 .FIG. 10 . 8910The approximate interfacial free energy F (α, hw) for A/(πh 2 0 ) = 100 2 , Vex/(A h0) = 0.048, and φ0/σ = 0.5 (i.e., for the same parameters as in Figs. 6 and 7 but for a smaller value of Vex). F f = A (σ + φ(h f )) is the free energy of the flat film solution for these parameters; h f − h0 = Vex/A. For this excess volume the droplet solution has disappeared and the flat film solution h(eq) w = h f (so that (h (eq) w −h0)/(h f −h0) = 1)is the global minimum. Contour lines are shown in the range 0.999 to 1.006 in steps of 0.00025. w − h 0 ) / (h f − h 0 ) V ex / (A h 0 ) The minimal free energy Fmin(hw) = minα F (α, hw) as a function of the wetting film thickness hw for A/(πh 2 0 ) = 100 2 , φ0/σ = 0.5, and several values of Vex/(A h0) ranging from 0.048 to 0.3. F f = A (σ + φ(h f )) is the free energy of the corresponding flat film solution for these parameters. The morphological transition between flat films and nanodroplets occurs between Vex/(A h0) = 0.058 and 0.056. For Vex/(A h0) > 0.2 the flat film solution appears to becomes unstable as expected from Eq. (18) (in the inset see the enlarged view of the region near hw = h f ). The symbols indicate the points calculated numerically. For Vex/(A h0) = 0.3 a small barrier cannot be ruled out on the basis of the available numerical data. tonically decreasing and the only minimum which occurs is the one corresponding to a flat film of thickness h f so that (h w − h 0 )/(h f − h 0 ) = 1. For intermediate excess volumes (0.05 V ex /(A h 0 ) 0.058 in Fig. 9) there is a second minimum corresponding to a metastable droplet. With increasing V ex this droplet minimum deepens until it is as deep as the minimum corresponding to the The pressure p in units of σ/h0 as a function of Vex/(A h0) for a fixed substrate size, A/(π h 2 0 ) = 100 2 , and φ0/σ = 0.5 as obtained from the approximate free energy expression F (α, hw). Global minima (full green), local minima (open blue), and saddle points or local maxima (symbols with red crosses) are shown. The upper branch corresponds to flat films (boxes) and the lower one to nano-droplets (circles). Within the reduced model we have p = −Π(h eq w ) (see Eq. (22)). The pressure values obtained from a numerical minimization of Eq. (3) (black diamonds; p ≈ −Π(h eq w ), see Eq. (20)) agree well with the results obtained from the approximate free energy. The dashed vertical line indicates the volume at which h f = hi. At this volume the unstable droplet branch merges with the flat film branch. The full vertical line indicates the morphological transition between the film and the droplet configurations. 11. Sketch of the Maxwell construction leading to the position of the thin full vertical line in , for large V ex there are three branches of solutions (flat film solutions with V d = 0, unstable droplet solutions, and metastable or stable droplet solutions). For small V ex there are only flat film solutions. The size V c d = V d (V ex = V c ex ) of the smallest metastable droplet (which is identical to the size of the largest unstable droplet) increases with the substrate area, as well as the value V c ex of the corresponding excess volume. However, V c ex /(A h 0 ) decreases upon increasing A (compare Figs. 12(a) and (b)). This means, that the thickness h c f = V c ex /A of the flat film solution corresponding to the minimal droplet also decreases upon an increase of the substrate area. d (3 R − h d ) and the total fluid volume by V = V d + A h w . It is convenient to write the volume constrained FIG. 12 . 12The droplet volume V d = V − A hw as function of Vex/(A h0) (i.e., as a function of V = Vex + A h0) for φ0/σ = 0.5 and (a) A/(h 2 0 π) = 100 2 and (b) A/(h 2 0 π) = 1000 2 as obtained from the approximate expression F (α, hw) for the free energy: for fixed V = Vex + A h0 the free energy landscapes (see, e.g., Figs. 6-8) have been calculated and the wetting film thicknesses hw of the droplet solutions-if they existhave been determined. The upper branch corresponds to the stable (green) or metastable (blue) droplet solution. For (a) this is the lower branch inFig. 10. The points on the abscissa correspond to stable (green) or metastable (blue) flat film solutions. The comparison between (a) and (b) shows that the minimal droplet size V c d (for which the unstable and the metastable droplet branches meet) increases upon increasing the substrate area while the corresponding excess volume V c ex in units of the substrate area decreases. 13. (a) Disjoining pressure Π(hw) with h0 < hw < h f (calculated for the model potential in Eq. (4)) in the wetting film (solid red line) and the right hand side of Eq. (27) with R(hw, h f , A) from Eq. (29) (dashed blue lines) for A/(π h 2 0 ) = 10 4 and φ0/σ = 0.5 as a function of the wetting film thickness hw for h f = 1.04 h0, h f = h c f = 1.05005 h0, and h f = 1.06 h0, which fixes V = A h f for a given A (see Eq. (12)). We also show the right hand side of Eq. (27) for A/(π h 2 0 ) = 10 5 and h f = h c f = 1.02587 h0 (dash-double-dotted cyan line), as well as for A/(π h 2 0 ) = 10 6 and h f = h c f = 1.01394 h0 (dash-dotted green line). The thin dotted red line shows the linear fit to Π(hw) at hw = h0. 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[ "Prepared for submission to JHEP Black string first order flow in N = 2, d = 5 abelian gauged supergravity", "Prepared for submission to JHEP Black string first order flow in N = 2, d = 5 abelian gauged supergravity" ]	[ "Dietmar Klemm [email protected] \nDipartimento di Fisica\nUniversità di Milano\nINFN\nSezione di Milano\nVia Celoria 16I-20133MilanoItaly\n", "Nicolò Petri [email protected] \nDipartimento di Fisica\nUniversità di Milano\nINFN\nSezione di Milano\nVia Celoria 16I-20133MilanoItaly\n", "Marco Rabbiosi [email protected] \nDipartimento di Fisica\nUniversità di Milano\nINFN\nSezione di Milano\nVia Celoria 16I-20133MilanoItaly\n" ]	[ "Dipartimento di Fisica\nUniversità di Milano\nINFN\nSezione di Milano\nVia Celoria 16I-20133MilanoItaly", "Dipartimento di Fisica\nUniversità di Milano\nINFN\nSezione di Milano\nVia Celoria 16I-20133MilanoItaly", "Dipartimento di Fisica\nUniversità di Milano\nINFN\nSezione di Milano\nVia Celoria 16I-20133MilanoItaly" ]	[]	We derive both BPS and non-BPS first-order flow equations for magnetically charged black strings in five-dimensional N = 2 abelian gauged supergravity, using the Hamilton-Jacobi formalism. This is first done for the coupling to vector multiplets only and U(1) Fayet-Iliopoulos (FI) gauging, and then generalized to the case where also hypermultiplets are present, and abelian symmetries of the quaternionic hyperscalar target space are gauged. We then use these results to derive the attractor equations for near-horizon geometries of extremal black strings, and solve them explicitely for the case where the constants appearing in the Chern-Simons term of the supergravity action satisfy an adjoint identity. This allows to compute in generality the central charge of the two-dimensional conformal field theory that describes the black strings in the infrared, in terms of the magnetic charges, the CY intersection numbers and the FI constants. Finally, we extend the r-map to gauged supergravity and use it to relate our flow equations to those in four dimensions.	10.1007/jhep01(2017)106	[ "https://arxiv.org/pdf/1610.07367v3.pdf" ]	54,194,962	1610.07367	d9034300d3aaf324e89849f222bacce2eab8c870	Prepared for submission to JHEP Black string first order flow in N = 2, d = 5 abelian gauged supergravity 16 Jan 2017 Dietmar Klemm [email protected] Dipartimento di Fisica Università di Milano INFN Sezione di Milano Via Celoria 16I-20133MilanoItaly Nicolò Petri [email protected] Dipartimento di Fisica Università di Milano INFN Sezione di Milano Via Celoria 16I-20133MilanoItaly Marco Rabbiosi [email protected] Dipartimento di Fisica Università di Milano INFN Sezione di Milano Via Celoria 16I-20133MilanoItaly Prepared for submission to JHEP Black string first order flow in N = 2, d = 5 abelian gauged supergravity 16 Jan 2017Black HolesSupergravity ModelsBlack Holes in String TheoryAdS- CFT Correspondence arXiv:161007367v3 [hep-th] We derive both BPS and non-BPS first-order flow equations for magnetically charged black strings in five-dimensional N = 2 abelian gauged supergravity, using the Hamilton-Jacobi formalism. This is first done for the coupling to vector multiplets only and U(1) Fayet-Iliopoulos (FI) gauging, and then generalized to the case where also hypermultiplets are present, and abelian symmetries of the quaternionic hyperscalar target space are gauged. We then use these results to derive the attractor equations for near-horizon geometries of extremal black strings, and solve them explicitely for the case where the constants appearing in the Chern-Simons term of the supergravity action satisfy an adjoint identity. This allows to compute in generality the central charge of the two-dimensional conformal field theory that describes the black strings in the infrared, in terms of the magnetic charges, the CY intersection numbers and the FI constants. Finally, we extend the r-map to gauged supergravity and use it to relate our flow equations to those in four dimensions. Introduction Exact solutions to supergravity theories, like black holes, domain walls, plane-fronted waves etc. have been instrumental in various developments of string theory, for instance in holography or black hole microstate counting. Generically, one is interested both in supersymmetric backgrounds and in solutions that break supersymmetry, like nonextremal black holes. The latter play an important role for example in holographic descriptions of condensed matter systems at finite temperature, or of the quark-gluon plasma. The supergravity equations of motion are coupled, nonlinear, second-order partial differential equations, and as such quite difficult to solve analytically, even in presence of a high degree of symmetry. A possible way out is to consider instead the Killing spinor equations, which are of first order in derivatives, and imply (at least when the Killing vector constucted as a bilinear from the Killing spinor is timelike) the second-order equations of motion. In this way, however, one obtains only supersymmetric solutions, and therefore interesting objects like nonextremal or extremal non-BPS black holes are excluded a priori. A more general possibility that still involves solving first-order equations, is the Hamilton-Jacobi approach. This includes the (symmetry-reduced) Killing spinor equations as a special subcase, but is quite easily generalizable to extremal non-BPS-or even nonextremal black holes. Using Hamilton-Jacobi theory is essentially 1 equivalent to writing the action as a sum of squares, which can be seen as follows: Suppose that, owing to various symmetries (like e.g. staticity and spherical symmetry), the supergravity action can be dimensionally reduced to just one dimension 2 , such that I = dr 1 2 G ΛΣq ΛqΣ − U (q) ,(1.1) where r is a radial variable (the 'flow' direction), the q Λ (r) denote collectively the dynamical variables, U (q) is the potential and G ΛΣ (q) the metric on the target space parametrized by the q Λ , with inverse G ΛΣ . Now suppose that U can be expressed in terms of a (fake) superpotential W as U = E − 1 2 G ΛΣ ∂W ∂q Λ ∂W ∂q Σ ,(1.2) where E is a constant. Then, the action (1.1) becomes I = dr 1 2 G ΛΣ q Λ − G ΛΩ ∂W ∂q Ω q Σ − G Σ∆ ∂W ∂q ∆ + d dr (W − Er) ,(1.3) which is up to a total derivative equal to I = dr 1 2 G ΛΣ q Λ − G ΛΩ ∂W ∂q Ω q Σ − G Σ∆ ∂W ∂q ∆ . (1.4) The latter is obviously stationary if the first-order flow equationṡ q Λ = G ΛΩ ∂W ∂q Ω (1.5) 1 ' Essentially' means that in many flow equations obtained in the literature by squaring an action, the rhs of (1.5) is not a gradient, or, in other words, the flow is not driven by a (fake) superpotential (no gradient flow). 2 When there is less symmetry, e.g. for rotating black holes, one obtains a field theory living in two or more dimensions, instead of a mechanical system [1,2]. In this case, the Hamilton-Jacobi formalism has to be generalized to the so-called De Donder-Weyl-Hamilton-Jacobi theory [1]. hold. But (1.2) is nothing else than the reduced Hamilton-Jacobi equation, with W Hamilton's characteristic function, while (1.5) represents the expression for the conjugate momenta p Λ = ∂L /∂q Λ = G ΛΣq Σ in Hamilton-Jacobi theory 3 . First-order flow equations, derived either by writing the dimensionally reduced action as a sum of squares or from the Hamilton-Jacobi formalism, appear for many different settings in the literature, both in ungauged and gauged supergravity, and for BPS-, extremal non-BPS-and even nonextremal black holes, cf. e.g. [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] for an (incomplete) list of references. In particular, ref. [18] establishes general properties of supersymmetric flow equations for domain walls in five-dimensional N = 2 gauged supergravity coupled to vector-and hypermultiplets. Here we shall consider magnetically charged black strings in five-dimensional N = 2 gauged supergravity, and obtain first-order flow equations for them. This is done first for the Fayet-Iliopoulos (FI)-gauged case, and then generalized to include also hypermultiplets, where abelian symmetries of the quaternionic hyperscalar target manifold are gauged. Extremal magnetic black strings interpolate between (so-called 'magnetic') AdS 5 at infinity and AdS 3 × Σ (with Σ a two-dimensional space of constant curvature) at the horizon. Holographically, this corresponds to an RG flow across dimensions from a 4d field theory to a two-dimensional CFT in the infrared [19][20][21][22][23]. By plugging the near-horizon data into the flow equations, one gets the attractor equations. We solve the latter in full generality under the additional assumption that the 'adjoint identity' (A.3) holds. This enables us to compute the central charge of the 2d CFT that describes the black strings in the IR, in terms of the string charges, the FI parameters (or, more generally, the moment maps, if also hypermultiplets are present), and the constants C IJK that appear in the Chern-Simons term of the supergravity action. The remainder of this paper is organized as follows: In the next section, we briefly review N = 2, d = 5 Fayet-Iliopoulos gauged supergravity. In section 3, we derive the BPS flow equations for static black strings, and generalize them in 4 to the nonsupersymmetric case, by using a simple deformation of the BPS superpotential. After that, in section 5, the presence of (abelian) gauged hypermultiplets is taken into account as well. In 6, the flow equations are solved in the near-horizon limit, where the geometry contains an AdS 3 factor. This leads to the attractor equations for black strings, that we subsequently solve in full generality, and compute the central charge of the 2d CFT that describes the black strings in the IR. We conclude in section 7 with some proposals for extensions of our work. An appendix contains some useful relations in very special geometry and the construction of the r-map in the gauged case. N = 2, d = 5 Fayet-Iliopoulos gauged supergravity The bosonic Lagrangian of N = 2, d = 5 FI-gauged supergravity coupled to n v vector multiplets is given by [24,25] 4 e −1 L = 1 2 R − 1 2 G ij ∂ µ φ i ∂ µ φ j − 1 4 G IJ F I µν F Jµν + e −1 48 C IJK µνρσλ F I µν F J ρσ A K λ − g 2 V , (2.1) where the scalar potential reads V = V I V J 9 2 G ij ∂ i h I ∂ j h J − 6h I h J . (2.2) Here, V I are FI constants, ∂ i denotes a partial derivative with respect to the real scalar field φ i , and h I = h I (φ i ) satisfy the condition V ≡ 1 6 C IJK h I h J h K = 1 . (2.3) Moreover, G IJ and G ij can be expressed in terms of the homogeneous cubic polynomial V which defines a 'very special geometry' [26], G IJ = − 1 2 ∂ ∂h I ∂ ∂h J log V\| V=1 , G ij = ∂ i h I ∂ j h J G IJ \| V=1 . (2.4) Further useful relations can be found in appendix A. We note that if the five-dimensional theory is obtained by gauging a supergravity theory coming from a Calabi-Yau compactification of M-theory, then V is the intersection form, h I and h I ≡ 1 6 C IJK h J h K correspond to the size of the two-and four-cycles and the constants C IJK are the intersection numbers of the Calabi-Yau threefold [27]. BPS flow for a static black string Very special real Kähler manifolds can be viewed as the pre-image of the supergravity r-map [28,29]. This suggests to consider the five-dimensional spacetime as a Kaluza-Klein uplift of the usual static black holes in four dimensions. Moreover, a pure string solution in d = 5 supports only magnetic charges, thus the field configuration reads ds 2 = e 2T (r) dz 2 + e −T (r) −e 2U (r) dt 2 + e −2U (r) dr 2 + e 2ψ(r)−2U (r) dσ 2 κ , F I = p I f κ (θ)dθ ∧ dφ , φ i = φ i (r) , (3.1) where dσ 2 κ = dθ 2 +f 2 κ (θ)dϕ 2 is the metric on the two-dimensional surfaces Σ = {S 2 , H 2 } of constant scalar curvature R = 2κ, with κ ∈ {1, −1}, and f κ (θ) = 1 √ κ sin( √ κθ) = sin θ κ = 1 , sinh θ κ = −1 . (3.2) Plugging the ansatz (3.1) into the equations of motion following from (2.1) yields a set of ordinary differential equations that can be derived from the one-dimensional effective action S eff = dr e 2ψ U 2 + 3 4 T 2 − ψ 2 + 1 2 G ij φ i φ j − V eff , V eff = κ − e 2ψ−2U −T g 2 V − 1 2 e 2U +T −2ψ G IJ p I p J , (3.3) imposing in addition the zero energy condition H eff = 0. In the Hamilton-Jacobi formulation the latter becomes the partial differential equation e −2ψ (∂ U W ) 2 − (∂ ψ W ) 2 + 4 3 (∂ T W ) 2 + 2G ij ∂ i W ∂ j W + V eff = 0 , (3.4) where W is Hamilton's characteristic function that we will sometimes refer to also as (fake) superpotential. A solution of (3.4) permits to write the action as a sum of squares and to derive a set of first-order flow equations by setting the squares to zero 5 . Guided by the fourdimensional case [9,13,17], the ansatz for the simplest non-trivial solution is W = ae U + T 2 p I h I + be 2ψ−U − T 2 V I h I , (3.5) where a, b are constants. Using (A.1), one can show that (3.5) solves (3.4) if one imposes a = − 3 4 , b = 3 2 g , V I p I = − κ 3g . (3.6) The last of (3.6) is a sort of Dirac quantization condition for the linear combination V I p I of the magnetic charges in terms of the inverse gauge coupling constant g −1 . This solution for W leads to the first-order flow U = − 3 4 e U + T 2 −2ψ p I h I − 3 2 g e −U − T 2 V I h I , ψ = −3g e −U − T 2 V I h I , T = 2 3 U , φ i = 3G ij − 1 2 e U + T 2 −2ψ p I ∂ j h I + g e −U − T 2 V I ∂ j h I . (3.7) One can check that (3.7) coincides with the system obtained in [30] from the Killing spinor equations. In particular, it is easy to verify that the supersymmetric magnetic black string solution of [31] satisfies (3.7). Moreover, introducing a new radial coordinate R and the warp factors f and ρ such that U = 3 2 f , ψ = 2f + ρ , T = f , dR dr = e −3f ,(3.8) and specifying to the stu model, one shows that (3.7) is precisely the system of equations derived in appendix 7.1 of [19] from the Killing spinor equations. Non-BPS flow One of the main advantages of the Hamilton-Jacobi formalism is to allow for a simple generalization of the first-order flow driven by (3.5) to a non-BPS one. Similar to the case of N = 2, d = 4 abelian gauged supergravity [14,17], one introduces a 'field rotation matrix' S I J such that G LK S L I S K J = G IJ . (4.1) A nontrivial S (different from ±Id) allows to generate new solutions from known ones by 'rotating charges'. This technique was first introduced in [6,7], and generalizes the sign-flipping procedure of [32]. Using (4.1), one easily verifies that W = − 4 3 e U + T 2 h I S I J p J + 2 3 ge 2ψ−U − T 2 V I h I (4.2) satisfies again the Hamilton-Jacobi equation (3.4), provided the modified quantization condition V I S I J p J = − κ 3g (4.3) holds. This leads to the first-order flow driven byW , U = − 3 4 e U + T 2 −2ψ h I S I J p J − 3 2 g e −U − T 2 V I h I , ψ = −3g e −U − T 2 V I h I , T = 2 3 U , φ i = 3G ij − 1 2 e U + T 2 −2ψ ∂ j h I S I J p J + g e −U − T 2 V I ∂ j h I . (4.4) An interesting example for which (4.1) admits nontrivial solutions, is the model V = h 1 h 2 h 3 = 1 (cf. e.g. [30]), for which G IJ = δ IJ 2(h I ) 2 . (4.5) In this case a particular solution of (4.1) is given by S I J =   1 0 0 0 2 0 0 0 3   , (4.6) with I = ±1. These matrices form a discrete subgroup D = (Z 2 ) 3 ⊂ GL(3, R). Since there are two equivalent BPS branches, the independent solutions correspond to elements of the quotient group D/Z 2 . Inclusion of hypermultiplets We now generalize our analysis to include also the coupling to n H hypermultiplets. The charged hyperscalars q u (u = 1, · · · , 4n H ) parametrize a quaternionic Kähler manifold with metric h uv (q), i.e., a 4n H -dimensional Riemannian manifold admitting a locally defined triplet K v u of almost complex structures satisfying the quaternion relation h st K x us K y tw = −δ xy h uw + ε xyz K z uw ,(5.1) and whose Levi-Civita connection preserves K up to a rotation, ∇ w K v u + ω w × K v u = 0 ,(5.2) where ω ≡ ω u (q) dq u is the connection of the SU(2)-bundle for which the quaternionic manifold is the base. The SU(2) curvature is proportional to the complex structures, Ω x ≡ dω x + 1 2 ε xyz ω y ∧ ω z = −K x . (5.3) Here we shall consider only gaugings of abelian isometries of the quaternionic Kähler metric h uv . These are generated by commuting Killing vectors k u I (q). For each Killing vector one can introduce a triplet of moment maps, P x I , such that D u P x I ≡ ∂ u P x I + ε xyz ω y u P z I = −2Ω x uv k v I . (5.4) One of the most important relations satisfied by the moment maps is the so-called equivariance relation. For abelian gaugings it has the form 1 2 xyz P y I P z J − Ω x uv k u I k v J = 0 . (5.5) The bosonic Lagrangian is now given by 6 e −1 L = 1 2 R − 1 2 G ij ∂ µ φ i ∂ µ φ j − h uv∂µ q u∂µ q v − 1 4 G IJ F I µν F Jµν + e −1 48 C IJK µνρσλ F I µν F J ρσ A K λ − g 2 V ,(5.6) with the covariant derivative∂ µ q u = ∂ µ q u + 3gA I µ k u I ,(5.7) and the scalar potential V = P x I P x J 9 2 G ij ∂ i h I ∂ j h J − 6h I h J + 9h uv k u I k v J h I h J .(5.∂ µ e G IJ F Jµν + 1 4 µλρσν C IJK ∂ µ A J λ ∂ ρ A K σ = 6geh uv k u I∂ ν q v . (5.9) Imposing the ansatz (3.1), the t-, θ-and z-components of (5.9) are automatically satisfied, while the r-and ϕ-components become respectively h uv k u I q v = 0 , k u I p I = 0 . (5.10) The remaining equations of motion can be derived from the effective action S eff = dr e 2ψ U 2 + 3 4 T 2 − ψ 2 + 1 2 G ij φ i φ j + h uv q u q v − V eff , V eff = κ − e 2ψ−2U −T g 2 V − 1 2 e 2U +T −2ψ G IJ p I p J ,(5.11) supplemented by the Hamiltonian constraint H eff = 0. The latter leads to the Hamilton-Jacobi equation e −2ψ (∂ U W ) 2 − (∂ ψ W ) 2 + 4 3 (∂ T W ) 2 + 2G ij ∂ i W ∂ j W + h uv ∂ u W ∂ v W + V eff = 0 . (5.12) Guided by the FI-gauged case and by previous work in four dimensions [17], we use the ansatz W = ce U + T 2 Z + de 2ψ−U − T 2 L ,(5.G IJ , C IJK → 1 6 C IJK , k I → 2k I , A I → 3 2 A I , g → 3 2 g. where Z = p I h I , L = Q x W x , Q x = p I P x I , W x = h J P x J . (5.14) Using some relations of very special geometry as well as (5.1), (5.3) and (5.5), one can show that (5.13) solves indeed (5.12) provided that c = − 3 4 , d = − 9 2 κg 2 , Q x Q x = 1 9g 2 . (5.15) The solution (5.13) leads then to the first-order flow equations U = − 3 4 e U + T 2 −2ψ Z + 9 2 κg 2 e −U − T 2 L , T = 2 3 U , ψ = 9κg 2 e −U − T 2 L , φ i = G ij − 3 2 e U + T 2 −2ψ ∂ j Z − 9κg 2 e −U − T 2 ∂ j L , q u = − 9 2 κg 2 e −U − T 2 h uv ∂ v L . (5.16) One can recast (5.16) into a form very similar to that of the first-order flow in four dimensions, cf. eqns. (3.43) in [17]. Integrating T = 2 3 U and plugging this into the remaining equations of (5.16), one gets T = − 1 2 e 2T −2ψ Z + 3κg 2 e −2T L , ψ = 9κg 2 e −2T L , φ i = G ij − 3 2 e 2T −2ψ ∂ j Z − 9κg 2 e −2T ∂ j L , q u = − 9 2 κg 2 e −2T h uv ∂ v L .e 2ψ e −2T h I + 9g 2 κe 2ψ−4T Q x P x J G IJ − p I = 0 . (5.18) Note that the FI case can be recovered imposing P 1 I = P 2 I = 0 and P 3 I = V I . Then the charge quantization condition Q x Q x = 1/(9g 2 ) boils down to Q 3 = p I V I = ±κ/(3g) (use κ 2 = 1), while L in (5.14) becomes L = ± κ 3g h J V J , which is the expression appearing in (3.5). The two signs correspond to the two equivalent BPS branches; in section 3 the lower sign was chosen. Attractors and central charge of the dual CFT In this section we want to investigate the near-horizon configurations of the black string. To keep things simple, we shall first concentrate on the hyperless FI-gauged case considered in section 3, and set g = 1. The geometry is of the type AdS 3 × Σ with Σ = {S 2 , H 2 }, and we assume that the scalars stabilize regularly at the horizon, i.e., φ i = 0. Note that a similar problem was solved in four dimensions in [33] for the case of symmetric special Kähler manifolds with cubic prepotential 7 . In the coordinates (t, R, z, θ, φ), where R was introduced in (3.8), the metric (3.1) takes the form ds 2 = e 2f (−dt 2 + dR 2 + dz 2 ) + e 2ρ dσ 2 κ , (6.1) and the first-order flow equations (3.7) become f = −e f (h I V I + 1 2 e −2ρ Z) , ρ = −e f (h I V I − e −2ρ Z) , φ i = 3G ij e f (∂ j h I V I − 1 2 e −2ρ ∂ j Z) , (6.2) where the primes now denote derivatives w.r.t. R. For a product space AdS 3 × Σ we have e 2f = R 2 AdS 3 R 2 , e 2ρ = R 2 H . (6.3) Plugging this together with φ i = 0 into (6.2), one obtains a system of algebraic equations whose solution fixes the near-horizon values of the scalars in terms of the charges and the FI parameters, h I V I = 2 3R AdS 3 , Z = R 2 H h I V I , ∂ i Z = 2R 2 H ∂ i h I V I . (6.4) For the ansatz (6.3), the FI-version of (5.18) (obtained by taking Q x P x J = Q 3 P 3 J = −κV J /(3g)) reduces to e f +2ρ (e −2f h I ) − 3e 2ρ G IJ V J − p I = 0 . (6.5) Using (6.3) and (6.4), this can be rewritten as p I + 3R 2 H G IJ V J = 3Zh I . (6.6) We want to solve the attractor equations (6.4) (or equivalently (6.6)) in order to express R AdS 3 , R H and h I in terms of p I and V I . To this end, contract the third relation of (A.2) with V I to get G ij ∂ i h I V I ∂ j h J = − 2 3 V J + 2 3 h I V I h J . (6.7) With (6.4), this becomes R 2 H V J = − 3 4 G ij ∂ i Z∂ j h J + Zh J . (6.8) Using h I = 1 6 C IJK h J h K and (A.2), one obtains R 2 H V J = 1 6 C JKL p K h L . (6.9) Let us introduce the charge-dependent matrix C p IJ ≡ C IJK p K . (6.10) Using the adjoint identity (A.3), one easily shows that C p IJ is invertible, with inverse C IJ p = 3 C IJK C KM N p M p N − p I p J C p , (6.11) where C p = C IJK p I p J p K . (6.9) implies then h I = 6R 2 H C IJ p V J . (6.12) Plugging (6.12) into (2.3), one can derive a general expression for R H in terms of the intersection numbers, the charges and the FI parameters, R 2 H = (36C IJK C IM p C JN p C KP p V M V N V P ) − 1 3 . (6.13) Using this in (6.12) gives the values of the scalars at the horizon, h I = 6C IJ p V J (36C KLM C KN p C LP p C M R p V N V P V R ) 1 3 . (6.14) Contracting (6.12) with V I and using the first equation of (6.4) as well as (6.13), we obtain an expression for the AdS 3 curvature radius R AdS 3 , R AdS 3 = (36C IJK C IM p C JN p C KP p V M V N V P ) 1 3 9C RS p V R V S . (6.15) Finally, one can plug (6.11) into (6.13), (6.14) and (6.15), and use (A.3) to write the solutions of (6.4) and (6.6) as R 2 H = (C IJK (p)V I V J V K ) − 1 3 , h I = 6κ C p p I + 3κC IJK C KLM p L p M V J (C N P R (p)V N V P V R ) 1 3 , R AdS 3 = C p 27 (C IJK (p)V I V J V K ) 1 3 C LM N C N RS p R p S V L V M − 1 9 , (6.16) where C IJK (p) = − 108 C p 2C IJK − 9 C p p (I C JK)M C M N P p N p P + 9 C p p I p J p K . (6.17) The central charge of the two-dimensional conformal field theory that describes the black strings in the infrared [19,35,36], is given by [37] c = 3R AdS 3 2G 3 ,(6.18) where G 3 denotes the effective Newton constant in three dimensions, related to G 5 by 1 G 3 = R 2 H vol(Σ) G 5 . (6.19) In what follows, we assume Σ to be compactified to a Riemann surface of genus g, with g = 0, 2, 3, . . .. The unit Σ has Gaussian curvature K = κ, and thus the Gauss-Bonnet theorem gives vol(Σ) = 4π(1 − g) κ . (6.20) Using (6.19) and (6.20) in (6.18) yields for the central charge c = 6π(1 − g)R AdS 3 R 2 H κG 5 . (6.21) The curvature radii R AdS 3 and R H can be expressed in terms of the constants C IJK , the magnetic charges p I and the FI parameters V I by means of (6.16). This leads to c = 2π(1 − g)C p κG 5 (9C IJK C KM N p M p N V I V J − 1) . (6.22) If the hyperscalars are running, one has to consider also the near-horizon limit of the last equation of (5.17). Assuming q u = 0 at the horizon and using (5.4), one easily derives the algebraic condition k u I h I = 0 . (6.23) As far as the remaining equations of (5.17) are concerned, one can follow the same steps as in this section, with the only difference that V I has to be replaced everywhere by −3κQ x P x I . Final remarks Let us conclude our paper with the following suggestions for possible extensions and questions for future work: • Try to solve the flow equations in presence of hypermultiplets obtained in section 5 for some specific models, e.g. like those considered in [18], to explicitely construct black strings with running hyperscalars, similar in spirit to the black holes found in [38]. To the best of our knowledge, no such solutions are known up to now. • Derive first-order equations for electrically charged black holes (rather than magnetically charged black strings) in five-dimensional matter-coupled gauged supergravity. • Extend our work to the nonextremal case, similar to what was done in [5,8,10,11,15] in different contexts. Up to now, the only known nonextremal black string solutions in AdS 5 were constructed in [39] for minimal gauged supergravity. • It would be interesting to see how the BPS flow equations derived in sec. 5 arise precisely in the general classification scheme of supersymmetric solutions obtained in [40]. • We have checked that our central charge (6.22) agrees with the results of [20,35], where black string solutions corresponding to D3-branes at a Calabi-Yau singularity have been studied in detail. It may be of some interest to use the flow equations obtained in section 5 to study more complicated type IIB configurations, as was initiated in [23]. Work along these directions is in progress. G ij ∂ i h I ∂ j h J = G IJ − 2 3 h I h J , G ij ∂ i h I ∂ j h J = 4 9 G IJ − 2 3 h I h J , G ij ∂ i h I ∂ j h J = − 2 3 δ I J + 2 3 h I h J . (A.2) In the special case where the tensor T ijk that determines the Riemann tensor of the vector multiplet scalar manifold M (cf. [24] for details) is covariantly constant 8 , one has also C IJK C J (LM C P Q)K δ JJ δ KK = 4 3 δ I(L C M P Q) , (A.3) which is the adjoint identity of the associated Jordan algebra [24]. Using (A.3) and defining C IJK ≡ δ II δ JJ δ KK C I J K , one can show that G IJ = −6C IJK h K + 2h I h J . (A.4) B Down to d = 4 via r-map A natural question arising in the discussion of (5.17) is the relation with the flow equations of [17,41], coming from the abelian gauged supergravity theory in d = 4. An interesting way to answer this question is to extend the general r-map construction in ungauged supergravity [29] to the gauged case. B.1 Construction of the r-map The first step is a Kaluza-Klein reduction along the z-direction (i.e., along the string), by using the ansatz 9 ds 2 5 = e φ √ 3 ds 2 4 + e − 2 √ 3 φ (dz + K µ dx µ ) 2 , A I = B I dz + C I µ dx µ + B I K µ dx µ . (B.1) Defining K µν = ∂ µ K ν − ∂ ν K µ and C I µν = ∂ µ C I ν − ∂ ν C I µ , the five-dimensional Lagrangian (5.6) reduces to 10 e −1 4 L (4) = R (4) 2 − 1 8 e − √ 3φ K µν K µν − 1 4 G IJ e − φ √ 3 (C Iµν + B I K µν )(C J µν + B J K µν ) − 1 2 e 2φ √ 3 G IJ ∂ µ B I ∂ µ B J − 1 2 G IJ ∂ µ h I ∂ µ h J − 1 4 ∂ µ φ∂ µ φ − h uv∂µ q u∂µ q v − e −1 4 16 µνρσ C IJK C I µν C J ρσ B K + 1 3 K µν K ρσ B I B J B K + C I µν K ρσ B J B K − e √ 3φ g 2 B I k u I B J k v J h uv − g 2 e φ √ 3 V 5 . (B.2) 8 This implies that M is a locally symmetric space. 9 In this subsection µ, ν, . . . are curved indices for the four-dimensional theory, and the dilaton is related to the function T in (3.1) by T = −φ/ √ 3. Further details on the notation and the theory in d = 4 can be found in [17]. 10 We choose µνρσz 5 = − µνρσ 4 . Now we want to rewrite L (4) in the language of N = 2, d = 4 supergravity, by using the identifications of the ungauged case [42]. The coordinates of the special Kähler manifold, Kähler potential, Kähler metric and electromagnetic field strengths are given in terms of five-dimensional data respectively by z I = −B I − ie − φ √ 3 h I , e K = 1 8 e √ 3φ , g IJ = 1 2 e 2φ √ 3 G IJ , F Λ µν = 1 √ 2 (K µν , C I µν ) , (B.3) where capital greek indices Λ, Σ, . . . range from 0 to n v +1. If we introduce the matrices R ΛΣ = − 1 3 B 1 2 B J 1 2 B I B IJ , I ΛΣ = −e − √ 3φ 1 + 4g 4gJ 4g I 4g IJ , (B.4) where we defined B IJ =C IJK B K , B I = C IJK B J B K , B = C IJK B I B J B K , g = g IJ B I B J , g IJ B J = g I = gĪ = gĪ J B J , (B.5) the Lagrangian (B.2) can be cast into the form e −1 4 L (4) = R 2 − g IJ ∂ µ z I ∂ µzJ − h uv∂µ q u∂µ q v + 1 4 I ΛΣ F Λµν F Σ µν + 1 8 e −1 4 µνρσ R ΛΣ F Λ µν F Σ ρσ −Ṽ , (B.6) with the four-dimensional potential given bỹ V = g 2 e φ √ 3 V 5 + e √ 3φ g 2 h uv k u I k v J B I B J . (B.7) The underlying prepotential of the special Kähler manifold turns out to be F = 1 6 C IJK X I X J X K X 0 , (B.8) chosen the parametrization X I /X 0 = z I = −B I − ie −φ/ √ 3 h I [42] . The actual novelties with respect to the ungauged case are the potential and the covariant derivative acting on the hyperscalars. The former reads Now the first two terms in the second line of (B.9) combine to give − 1 2 I ΛΣ (the inverse of I ΛΣ defined above), while the last two terms yield −4X IX J . Fixing furthermore g 4 = 3 √ 2g, one has thus V g 2 = −9e φ √ 3 P x I P x J h I h J − 1 2 G IJ + 9e φ √ 3 h uv k u I k v J h I h J + 9e √ 3φ h uv k u I k v J B I B J = 18P x I P x J 1 4 e φ √ 3 G IJ + 1 2 e √ 3φ B I B J − 4 e √ 3φ 8 (e − φ √ 3 h I )(e − φ √ 3 h J ) − 1 2 e √ 3φ B I B J +72 e √ 3φ 8 h uv k u I k v J (e − 2φV =g 2 4 P x Λ P x Σ − 1 2 I ΛΣ − 4X ΛX Σ + 4h uv k u Λ k v Σ X ΛX Σ P x 0 =0,k u 0 =0 =g 2 4 V 4 P x 0 =0,k u 0 =0 , (B.10) which is precisely the truncated potential of the four-dimensional theory. The final point to take care of is the covariant derivative of the hyperscalars, ∂ µ q u = ∂ µ q u + 3gC I µ k u I = ∂ µ q u + g 4 A I 4µ k u I . (B.11) We have therefore shown that the r-map can be extended to the case of gauged supergravity, where the scalar fields have a potential. B.2 Comparison with the flow in N = 2, d = 4 gauged supergravity The result of the preceeding subsection is completely general and interesting by itself, however our aim is to use this mapping to compare the flow (5.16) for a black string in d = 5 with the flow equations for black holes in four dimensions obtained in [17,41]. Specifying to a purely magnetic charge configurationp Λ = (0, p I / √ 2), purely electric couplings with P x 0 = 0, k u 0 = 0, and restricting to imaginary scalars, z I = −ie −φ/ √ 3 h I , the quantities defining (B.12) become Z 4 = 3 2 √ 2 e T /2pI h I = 3 4 e T /2 Z , Q x 4 = P x Ip I = 1 √ 2 Q x , W x 4 = − i 2 √ 2 e −T /2 P x I h I = − i 2 √ 2 e −T /2 W x , L 4 = Q x 4 W x 4 , (B.14) where the quantities Z, Q x and W x were defined in section 5. Note that axions are absent, since for magnetically charged black strings the z-components B I of the fivedimensional gauge potentials vanish. For this choice, (B.13) becomes e 2iα = 1. Moreover, taking in account that g 4 = 3 √ 2g and choosing e iα = −1, the function (B.12) boils down to (5.13). On the other hand, the Hamilton-Jacobi equation satisfied by (B.12), namely (3.40) of [17], becomes (5.12) once the dictionary is imposed. This proves the expected equivalence between the flows in five and four dimensions. equation for φ i together with (h I ) = φ i ∂ i h I and (A.2), the equations for T and φ i can be rewritten as √ 3 h 3I h J + B I B J ) . (B.9) The latter are driven by the Hamilton-Jacobi functionW 4 = e U Re(e −iα Z 4 ) − κg 4 e 2ψ−U Im(e −iα L 4 ) , (B.12)where the phase α is defined bye 2iα = Z 4 + iκg 4 e 2(ψ−U ) L 4 Z 4 − iκg 4 e 2(ψ−U )L 4 . (B.13) For further discussions of the relationship between the Hamilton-Jacobi formalism and the firstorder equations derived from a (fake) superpotential cf.[3,4]. The indices I, J, . . . range from 1 to n v + 1, while i, j, . . . = 1, ..., n v . 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[ "How to deal with PCR composition problem at E 0 10 17 eV", "How to deal with PCR composition problem at E 0 10 17 eV" ]	[ "Galkin V I [email protected] \nFaculty of Physics of Lomonosov\nMoscow State University Leninskie Gory\n119991MoscowRussian Federation\n", "Anokhina A M [email protected] \nFaculty of Physics of Lomonosov\nMoscow State University Leninskie Gory\n119991MoscowRussian Federation\n", "Bakhromzod R \nFaculty of Physics of Lomonosov\nMoscow State University Leninskie Gory\n119991MoscowRussian Federation\n", "Mukumov A \nFaculty of Physics of Lomonosov\nMoscow State University Leninskie Gory\n119991MoscowRussian Federation\n" ]	[ "Faculty of Physics of Lomonosov\nMoscow State University Leninskie Gory\n119991MoscowRussian Federation", "Faculty of Physics of Lomonosov\nMoscow State University Leninskie Gory\n119991MoscowRussian Federation", "Faculty of Physics of Lomonosov\nMoscow State University Leninskie Gory\n119991MoscowRussian Federation", "Faculty of Physics of Lomonosov\nMoscow State University Leninskie Gory\n119991MoscowRussian Federation" ]	[]	Basic ideas of muon tracker technique for the solution of primary cosmic ray (PCR) composition problem in the energy range 10 17 − 10 18 eV are presented. The approach uses MC simulation data made with CORSIKA6.990 for "Pamir-XXI" site conditions. Similar technology can certainly be developed for other observation levels and interaction models. One can probably extend it to much higher primary energies.	null	[ "https://arxiv.org/pdf/1507.03150v1.pdf" ]	117,569,841	1507.03150	5976bfae153f100b2a51046615cfc0ada388599f	How to deal with PCR composition problem at E 0 10 17 eV 11 Jul 2015 Galkin V I [email protected] Faculty of Physics of Lomonosov Moscow State University Leninskie Gory 119991MoscowRussian Federation Anokhina A M [email protected] Faculty of Physics of Lomonosov Moscow State University Leninskie Gory 119991MoscowRussian Federation Bakhromzod R Faculty of Physics of Lomonosov Moscow State University Leninskie Gory 119991MoscowRussian Federation Mukumov A Faculty of Physics of Lomonosov Moscow State University Leninskie Gory 119991MoscowRussian Federation How to deal with PCR composition problem at E 0 10 17 eV 11 Jul 2015arXiv:1507.03150v1 [astro-ph.HE] Basic ideas of muon tracker technique for the solution of primary cosmic ray (PCR) composition problem in the energy range 10 17 − 10 18 eV are presented. The approach uses MC simulation data made with CORSIKA6.990 for "Pamir-XXI" site conditions. Similar technology can certainly be developed for other observation levels and interaction models. One can probably extend it to much higher primary energies. Introduction It is useless to repeat all the words that show the importance of the study of primary mass composition at super high energies covered by the extensive air shower (EAS) method. Many different groups using different detector set-ups have dealt with the problem but the answer is still vague. The reason for such a poor result, from our viewpoint, lies in the inadequacy of experimental methods used. To be more precise, EAS characteristics that are experimentally measured (X max , N µ /N charged ) do not contain enough information on the mass of the primary particle. A natural way out of such a situation is to look for more informative measures of the primary mass. In 1-100 PeV energy range the EAS Cherenkov light angular distribution characteristics prove to be appropriate for the solution of mass composition problem [1,2]. Generally speaking, this technique can be used at higher energies as well but the detector array should be made much larger while still being rather dense and, thus, must be very expensive. Here we present another approach for E 0 100 PeV which seems to be more cost-effective and, probably, will work even for the extremely high energy showers. Main idea of the new method EAS is a particle cascade originated by a primary cosmic ray, probably proton or other nucleus of super high energy. The primary particle undergoes a few strong interactions with air nuclei (N, O, Ar) giving life to quite a number of secondary particles. The most numerous EAS component is the electromagnetic one (γ, e + , e − ) which is characterized by a radiation length (37 g/cm 2 ) that is about 3 times smaller than hadron interaction length. On the contrary, the range of the next to most numerous component, which is the muon one, can be much larger and for E µ ∼ 10 GeV is comparable with the shower length (10-30 km). One can conclude that the EAS muons with E µ ≥ 1 GeV can yield important information on the longitudinal shower development. In this aspect muons resemble the EAS Cherenkov light which proved to be the most informative component of all. Crucial question is how to collect the data on the PCR mass composition borne by muons, i.e. what characteristic of the muon component is sensitive to the primary mass and, thus, to be measured in EAS experiment. Usually, the muon content N µ of the shower is used as a measure of the primary particle mass. Unfortunately, any procedure for N µ estimation involves the measurement of muon densities all over the area swept by EAS. The measurement is made by local detectors of limited area ∼1 m 2 spread widely which results in rather large uncertainties of statistical (due to small detector area) and physical (due to cascade fluctuations) origin. Finally, it turns out that N µ uncertainty kills its sensitivity to the primary mass. As the ultimate results of KASKADE-Grande [3] show, N µ parameter is only capable of dividing all the events into two groups in primary mass which is definitely not enough to solve the composition problem. Certainly, one should think of finding more sensitive parameters, preferably directly measurable ones. One of the possibilities is to use the similarity between the muons and Cherenkov light. An optical detector observing an EAS from large ( 300 m) core distance reveals a Cherenkov light pulse presenting roughly an electron cascade curve. The latter can also be seen by an imaging telescope which field of view is divided into sensitive areas of small size < 1 • in diameter (pixels). Because relativistic muons can reach the observation level even from as high as the first interaction of the primary particle, one can think of some muon analogue of the imaging Cherenkov telescope, namely, of a muon tracker as a detector of muon angular distribution which should be sensitive to the primary mass. A priory it is not clear a) whether such a sensor can distinguish even between primary proton and iron nucleus events, cascade fluctuations taken into account; b) what the design of such a tracker should be; c) how must the experimental data on muon angular distribution be processed in order the primary mass information to be extracted. Full MC simulation presumably can answer the questions. Artificial shower generation The present study is made in the framework of the "Pamir-XXI" project which determines the observation level (4250 m above sea level) and the primary energy range of interest (10 16 − 10 18 eV for secondary charged particle methods). "Pamir-XXI" will include an optical part [2] incorporating a net of fast detectors and imaging telescopes that, working together, are capable of dealing with PCR mass composition problem in 100 TeV -100 PeV energy range. At higher energies the event rate with the basic (minimal) optical set-up will be low and one should think of a charged particle detector set-up to take over the task. In view of all written above, one should first check 10 17 − 10 18 eV energy range for a muon angular distribution sensitivity to the primary mass. The following EAS samples were generated: -100 PeV proton (p), nitrogen (N) and iron nucleus (Fe) initiated vertical showers, sample volumes 50; -1 EeV p, N and Fe initiated vertical showers, sample volumes 30. For the generation of all samples CORSIKA6.990/QGSJET01 [4] was used without thinning with energy threshold 1 GeV for all secondaries. Also 100 PeV p, N and Fe initiated vertical shower samples of volume 30 were generated by CORSIKA6.990/QGSJET-II to analyze the interaction model effect on muon angular distribution. Muon angular distribution processing: how to distinguish between different primary nuclei The following considerations defined the choice of the main parameters of the muon tracker and data handling procedure (area, angular resolution, typical core distance, threshold energy, azimuthal angle limitations): -tracker area should be large enough to detect a number of muons in each shower such that a histogram in muon polar angle α (with respect to the shower axis, see -typical core distance R for the primary energies of interest should ensure, on one hand, large enough number of detected muons (i.e. R being not too large), on the other hand, a sufficient base to scan the muon cascade curve from the side (i.e. R being not too small); -tracker angular resolution should be high enough for the polar angle bins to be made rather narrow to scan at least a few kilometers above the observation level, still the tracker must not be too expensive due to the angular resolution requirements; -energy threshold must ensure that the detected muons can come from high above the observation level being abundant at the same time; -azimuthal angle φ (see Fig.1) limitations appear due to the fact that the muons of interest should come mainly from the shower core and not from its periphery; -muon energy threshold and azimuthal angle limitations should not reduce the number of detected muons substantially. Preliminary tests with E 0 =10 PeV showers showed that the tracker must be of area S ∼ 100 m 2 and work at core distance R 60 m but the number of detected muons was rather low under such conditions (E 0 , S, R) to form angular histograms. With E 0 = 10 17 , 10 18 eV we use S = 100 m 2 , R = 100 m within this work but our approach can definitely be applied to other S and R, the above presented considerations taken into account. Naive muon angular distribution processing includes histogramming all muons of a sample in α above 1 GeV within azimuthal angle φ band [-0.2,0.2], bins in α being of equal width. An example of p and F e histogram comparison is shown in Fig.2. Marker abscissa correspond to the centers of histogram bins, marker ordinates represent bin contents averaged over a sample while error bars show r.m.s. deviatons of the contents. One can see no definite separation of the two distributions. Next step of the naive procedure is the presentation of the two samples in 2-dimensional feature space. A few different angular histogram shape measures r ij = w i /w j were tried , here w i is the i-th bin content of α−histogram of a shower. The reason for the use of the distribution shape parameters and not absolute muon bin contents is simple: relative features are less interaction model dependent and are more robust (i.e. less sample dependent). Fig.3 presents some examples of p and F e samples in 2-dimensional feature spaces: one can see substantial sample overlapping in all spaces so that no obvious criterion for sample separation can be found. Discouraging result of the use of the naive procedure forces one to look for some radical improvements of the approach. The first improvement comes from a consideration that equal width bins do not reflect the real scale of the muon cascade and one should optimize the bin grid. The second one takes into account the steep shape of the α-distribution and rather large ρ ± ijk = w i ±w j w k : w 2 +w 4 w 3 = ρ + 243 , w 2 −w 4 w 3 = ρ − 243 , etc. Interaction model: CORSIKA/QGSJET01 . fluctuations of the bin contents: generally speaking, one should reduce the number of histogram bins to 4 or 5 leaving the α-range about the same, which will suppress the fluctuations and make the separability of the samples better. To carry out the two improvements one is to introduce some quantitative measure of sample separability. We use Bhattacharyya distance D B under an assumption that the angular bin contents behave as multivariate normal distribution. Each sample is characterized by its own mean vector µ k ≡ w k and covariance matrix Σ k , k = p, F e. Then Bhattacharyya distance for p-Fe pair takes the form: D p−F e B = 1 8 (µ p − µ F e ) T Σ −1 (µ p − µ F e ) + 1 2 ln detΣ detΣ p · detΣ F e , Σ = Σ p + Σ F e 2 . For simplicity one can assume that Σ k matrices are diagonal which means the independence of different bins of the histogram. The latter assumption is not really true but does not change the result substantially. After the measure of the class (sample) separability is set one can design a procedure to tune the bin width vector h so that D B = D B h approaches its maximum for a concrete pair of samples. Fig.4 shows the result of such optimization for p − F e pair of E 0 = 10 17 eV. One can see definite differences between the two histograms as the bin contents (designated by markers with error bars) do not overlap for bins 2 to 4, unlike the histograms in Fig.2. Thus, we form appropriate histogram shape measures ρ ± ijk = w i ±w j w k , i, j, k = 2, 3, 4, i = j, i = k, j = k and present p and F e samples in different 2-dimensional feature spaces in Fig.5. Note that the two samples are fully separated for all combinations of ρ ± ijk considered. Fig.6 shows the same p and F e histograms as in Fig.4 but with N-histogram added. The corresponding feature space presentations are given in Fig.7. Though the bin widths were optimized for better p − F e separation, N-sample overlaps p-and F e-samples only partially for all ρ ± ijk combinations in Fig.7. One can also optimize the α bin grid for the best separation of p − N or N − F e sample pairs. The corresponding feature space plots are presented in Fig.8 and Fig.9 for E 0 = 10 18 eV. Sample separation is good for all ρ ± ijk combinations presented. We also carry out one more improvement of the bin width optimization algorithm, namely, a simultaneous maximization of three pair separation while optimizing the widths. Such improvement can be made in many different ways. We do this as follows: Fig.10 and Fig.11 made with CORSIKA6.990/QGSJET-II in order to check whether the change of the interaction model can spoil the muon angular distribution sensitivity to the primary particle mass. Fig.12 shows the corresponding sample plots in different 2-dimensional feature spaces. One can see that after the change of the interaction model the sensitivity still persists though the set of the most suitable feature spaces may differ for different models. Conclusion 1. A new muon tracker approach to PCR mass composition problem in EAS detection is based on high energy ( 1 GeV) muon angular distribution measurements by a large area (∼100 m 2 ) tracking detector of ∼ 0.3 • angular resolution placed at core distance R∼100 m to view the shower sideways. 2. In order to separate showers initiated by different primary nuclei, a special processing of the muon angular distributions is required which maximizes the differences between event histograms and, thus, makes it possible to distinguish between different groups of primary nuclei. 3. Concrete realizations of event selection algorithm made for "Pamir-XXI" conditions show that "complete" separation of at least three groups of nuclei is possible which makes the new method second to none among the charged particle methods in the primary energy range considered. 4. The new approach presumably must work under different observation conditions: observation levels and/or primary energy ranges. Certainly, the optimal α-histogram bin widths will differ in different cases. 5. The dependence of the method on the hadron/nucleus interaction model should be studied next but it is clear now that the sensitivity of muon angular distribution to the primary mass will hold because the general properties of the air shower remain the same no matter which interaction model is used. All in all, large area muon tracker can become an important instrument for the solution of the PCR mass composition problem at E 0 10 17 eV. Figure 1 : 1Detection geometry for vertical shower: polar α and azimuthal φ of a muon track. Fig. 1 Figure 2 : 12vert.EAS at 4250m, muon ADF, R=100m, Ethr=1GeV, phi=-+0.2rad, 50+50 events Comparison of muon angular distributions for 100 PeV p and Fe initiated shower in case of uniform grid in α (all histogram bins have equal widths). Interaction model: COR-SIKA/QGSJET01 . Figure 3 :Figure 4 : 34Comparison of 100 PeV p and Fe shower samples in different 2-dimensional feature spaces in case of uniform grid in α. Features used: r ij and r ij − r kj . Interaction model: CORSIKA/QGSJET01 . Comparison of muon angular distributions for 100 PeV p and Fe initiated shower in case of optimum grid in α. Interaction model: CORSIKA/QGSJET01 . Figure 5 : 5Comparison of 100 PeV p and Fe shower samples in different 2-dimensional feature spaces in case of p-Fe-optimum (non-uniform) grid in α. Features used: Figure 6 : 6Same asFig.4but with N-histogram added. Note that the polar angle grid was optimized with respect to p-N separation only. Figure 7 : 71. calculate D B,max for each of three pairs p-Fe, p-N, N-Fe (D p−F e B,max , D p−N B,max , D N −F e B,max ) by varying the vector h; Same as Fig.5 but with N-sample added. Figure 8 : 8Comparison of 1 EeV p and N shower samples in different 2-dimensional feature spaces in case of p-N-optimum grid in α. Interaction model: CORSIKA/QGSJET01 . 2. minimize a function R p−N −F e h R p−N −F e h = D p− Figure 9 :Figure 10 : 910Comparison of 1 EeV N and Fe shower samples in different 2-dimensional feature spaces in case of N-Fe-optimum grid in α. Interaction model: CORSIKA/QGSJET01 . Comparison of 100 PeV p, N and Fe shower samples in different 2-dimensional feature spaces in case of R p−N −F e -optimum grid in α. Interaction model: CORSIKA/QGSJET01 . keeping the contributions of different pairs in balance. Resulting sample plots for 100 PeV and 1 EeV showers are presented in Figure 12 : 12Comparison of 100 PeV p, N and Fe shower samples in different 2-dimensional feature spaces in case of R p−N −F e -optimum grid in α. Interaction model: CORSIKA/QGSJET-II . Figure 11: Comparison of 1 EeV p, N and Fe shower samples in different 2-dimensional feature spaces in case of R p−N −F e -optimum grid in α. Interaction model: CORSIKA/QGSJET01 .100PeV p-N-Fe separation by (w2+w4/w3,w2-w3/w4), 30+30+30100PeV p-N-Fe separation by (w2-w4/w3,w2+w3/w4), 30+30+30 100PeV p-N-Fe separation by (w2-w3/w1,w3-w4/w2), 30+30+30 100PeV p-N-Fe separation by (w2+w3/w4,w3-w4/w2), 30+30+30, respectively. Sample separation is rather good for all 2-dimensional feature spaces considered. Finally, we apply the R p−N −F e -minimization algorithm to 100 PeV p, N and Fe samples w2+w4/w3 12 13 14 15 16 17 18 w2-w3/w4 12 13 14 15 16 17 18 19 20 Fe N p 1EeV p-N-Fe separation by (w2+w4/w3,w2-w3/w4), 30+30+30 w2-w4/w3 10 11 12 13 14 15 16 w2+w3/w4 14 16 18 20 22 Fe N p 1EeV p-N-Fe separation by (w2-w4/w3,w2+w3/w4), 30+30+30 w2+w4/w3 12 13 14 15 16 17 18 w2-w4/w3 10 11 12 13 14 15 16 Fe N p 1EeV p-N-Fe separation by (w2+w4/w3,w2-w4/w3), 30+30+30 w2+w4/w3 12 13 14 15 16 17 18 w2+w3/w4 14 16 18 20 22 Fe N p 1EeV p-N-Fe separation by (w2+w4/w3,w2+w3/w4), 30+30+30 w2+w4/w3 12 13 14 15 16 17 18 w3-w4/w2 -0.01 -0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 Fe N p 1EeV p-N-Fe separation by (w2+w4/w3,w3-w4/w2), 30+30+30 w2+w3/w4 14 16 18 20 22 w3-w4/w2 -0.01 -0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 Fe N p 1EeV p-N-Fe separation by (w2+w3/w4,w3-w4/w2), 30+30+30 w2+w4/w3 8 10 12 14 16 18 20 22 w2-w3/w4 10 15 20 25 30 Fe N p w2-w4/w3 6 8 10 12 14 16 18 w2+w3/w4 10 15 20 25 30 35 40 Fe N p w2-w3/w1 4 6 8 10 12 14 16 18 w3-w4/w2 -0.04 -0.02 0 0.02 0.04 0.06 0.08 Fe N p w2+w4/w3 8 10 12 14 16 18 20 22 w2+w3/w4 10 15 20 25 30 35 40 Fe N p 100PeV p-N-Fe separation by (w2+w4/w3,w2+w3/w4), 30+30+30 w2-w3/w1 4 6 8 10 12 14 16 18 w3-w4/w1 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Fe N p 100PeV p-N-Fe separation by (w2-w3/w1,w3-w4/w1), 30+30+30 w2+w3/w4 10 15 20 25 30 35 40 w3-w4/w2 -0.04 -0.02 0 0.02 0.04 0.06 0.08 Fe N p . V I Galkin, T A Dzhatdoev, Moscow Univ. Phys. Bull. 65V.I. Galkin, T.A. Dzhatdoev, Moscow Univ. Phys. Bull., Vol. 65, Issue 3, 195-202 (2010). . V I Galkin, T A Dzhatdoev, Bull. Russian Acad. Sci.: Phys. 75V. I. Galkin, T. A. Dzhatdoev, Bull. Russian Acad. Sci.: Phys., Vol. 75, Issue 3, 309-312 (2011). . A S Borisov, V I Galkin, J. Phys.: Conf. Ser. 40912089A.S. Borisov, V.I. Galkin, J. Phys.: Conf. Ser. 409 012089. . W D , KASKADE-Grande CollaborationPhys. Rev. Lett. 107171104W.D. Apel et al. (KASKADE-Grande Collaboration), Phys. Rev. Lett. 107, 171104 (2011). . W D , KASKADE-Grande CollaborationPhys. Rev. D. 8781101W.D. Apel et al. (KASKADE-Grande Collaboration), Phys. Rev. D 87, 081101(R), (2013). . W D , KASKADE-Grande CollaborationAstropart. Phys. 47W.D. Apel et al. (KASKADE-Grande Collaboration), Astropart. Phys. 47 (2013) 54-66. CORSIKA: A Monte Carlo code to simulate extensive air showers. D Heck, Report FZKA. 6019D.Heck, CORSIKA: A Monte Carlo code to simulate extensive air showers, Report FZKA 6019, Forschungszentrum Karlsruhe, Germany, 1998.	[]
[ "arXiv:0807.0185v2 [hep-lat] Series expansions of the density of states in SU(2) lattice gauge theory", "arXiv:0807.0185v2 [hep-lat] Series expansions of the density of states in SU(2) lattice gauge theory" ]	[ "A Denbleyker ", "Daping Du ", "Yuzhi Liu ", "Y Meurice ", "A Velytsky ", "\nDepartment of Physics and Astronomy\nEnrico Fermi Institute\nThe University of Iowa Iowa City\n52242IowaUSA\n", "\nand HEP Division and Physics Division, Argonne National Laboratory\nUniversity of Chicago\n5640 S. Ellis Ave., 9700 Cass Ave60637, 60439Chicago, ArgonneIL, ILUSA, USA\n" ]	[ "Department of Physics and Astronomy\nEnrico Fermi Institute\nThe University of Iowa Iowa City\n52242IowaUSA", "and HEP Division and Physics Division, Argonne National Laboratory\nUniversity of Chicago\n5640 S. Ellis Ave., 9700 Cass Ave60637, 60439Chicago, ArgonneIL, ILUSA, USA" ]	[]	We calculate numerically the density of states n(S) for SU (2) lattice gauge theory on L 4 lattices. Small volume dependence are resolved for small values of S. We compare ln(n(S)) with weak and strong coupling expansions. Intermediate order expansions show a good overlap for values of S corresponding to the crossover. We relate the convergence of these expansions to those of the average plaquette. We show that when known logarithmic singularities are subtracted from ln(n(S)), expansions in Legendre polynomials appear to converge and could be suitable to determine the Fisher's zeros of the partition function.	10.1103/physrevd.78.054503	[ "https://arxiv.org/pdf/0807.0185v2.pdf" ]	119,275,998	0807.0185	a9ead19e765beabb446df25e03a0783c887e2013	arXiv:0807.0185v2 [hep-lat] Series expansions of the density of states in SU(2) lattice gauge theory 22 Jul 2008 A Denbleyker Daping Du Yuzhi Liu Y Meurice A Velytsky Department of Physics and Astronomy Enrico Fermi Institute The University of Iowa Iowa City 52242IowaUSA and HEP Division and Physics Division, Argonne National Laboratory University of Chicago 5640 S. Ellis Ave., 9700 Cass Ave60637, 60439Chicago, ArgonneIL, ILUSA, USA arXiv:0807.0185v2 [hep-lat] Series expansions of the density of states in SU(2) lattice gauge theory 22 Jul 2008(Dated: July 22, 2008)numbers: 1115-q1115Ha1115Me1238Cy We calculate numerically the density of states n(S) for SU (2) lattice gauge theory on L 4 lattices. Small volume dependence are resolved for small values of S. We compare ln(n(S)) with weak and strong coupling expansions. Intermediate order expansions show a good overlap for values of S corresponding to the crossover. We relate the convergence of these expansions to those of the average plaquette. We show that when known logarithmic singularities are subtracted from ln(n(S)), expansions in Legendre polynomials appear to converge and could be suitable to determine the Fisher's zeros of the partition function. I. INTRODUCTION Quantum Chromodynamics is a widely accepted theory of strong interactions. From a theoretical point of view, understanding the large distance behavior in terms of the weakly coupled short distance theory has been an important challenge. The connection between the two regimes can be addressed meaningfully using the lattice formulation. In the pure gauge theory (no quarks) described with the standard Wilson's action, no phase transition between the weak and strong coupling regime has been found numerically for SU (2) or SU (3) and the theory should be in the confining phase for all values of the coupling. Recently, convincing arguments have been given [1,2] in favor of the smoothness of the renormalization group flows between the two fixed points of interest, putting the confining picture on more solid mathematical ground. The absence of phase transition discussed above suggests that it is possible to match the weak coupling and the strong coupling expansions of the lattice formulation. However, if we consider these two expansions, for instance for the average SU (2) plaquette as a function of β = 4/g 2 , we see in Fig. 1 that there is a crossover region (approximately 1.5 < β < 2.5) where none of the two expansions seem to work. This behavior is probably related to singularities in the complex β plane [3,4] that are not completely understood. In the case of the one plaquette model [5], taking the inverse Laplace transform with respect to β (Borel transform) of the partition function * [email protected] † [email protected] ‡ [email protected] § [email protected] ¶ [email protected] yields a function that has better convergence properties. It would be interesting to know if this feature persists on V = L 4 lattices. In this article, we study expansions of the inverse Laplace transform of the partition function (the density of states) of SU (2) lattice gauge theory on symmetric 4 dimensional lattices. The density of states is denoted n(S) and defined precisely in Sec. II. It gives a relative measure of the number of ways to get a value S of the action. Knowing n(S), we can calculate the partition function and its derivatives for any real or complex value of β. In particular, it could be used to determine the Fisher's zeros of the partition function [6,7,8]. The choice of SU (2) is motivated by the existence of a particular symmetry [9] which allows to determine the behavior of n(S) near its maximal argument without extra calculation. In Sec. III, we explain why ln(n(S)) is expected to scale like the volume and can be interpreted as a "color entropy". Numerical calculations of n(S) obtained by patching plaquette distributions multiplied by the inverse Boltzmann weight at values of β increasing by a small increment are presented in Sec. IV. The article is focused on comparisons with numerical data on a 6 4 lattice where finite volume effects are not too large and plaquette distributions broad enough to allow a smooth patching. The values of n(S) on such lattice are compared with those on a 4 4 and 8 4 lattice. It is interesting to note that the volume dependence is resolvable only for small values of S where a behavior ln(S)/V is observed for ln(n(S)) The numerical results are compared with expansions that can be obtained from the strong (Sec. V) and weak (Sec. VI) coupling expansions of the average plaquette. Intermediate orders in these expansions show a good overlap for values of S that correspond to the crossover. We then show that the convergence of the new series can be related empirically to those of the series for the average plaquette. The weak coupling expansion determines the logarithmic singularities of ln(n(S)) at both boundaries. When these singularities are subtracted we obtain a bell-shaped function that can be approximated very well by Legendre polynomials (Sec. VII). We conclude with possible applications for the calculations of the Fisher's zeros and open problems. II. THE DENSITY OF STATES We consider the standard pure gauge partition function Z = l dU l e −βS ,(1) with the Wilson action S = p (1 − (1/N )ReT r(U p )) .(2) and β ≡ 2N/g 2 . We use a D dimensional cubic lattice with periodic boundary conditions. For a symmetric lattice with L D sites, the number of plaquettes is N p ≡ L D D(D − 1)/2 .(3) In the following, we restrict the discussion to the group SU (2) and D = 4. For SU (2), one can show [9] that the maximal value of S is 2N p . We define the average plaquette: P ≡ S/N p = −d(lnZ/N p )/dβ .(4) Inserting 1 as the integral of delta function over the numerical values S of S in Z, we can write Z = 2Np 0 dSn(S)e −βS ,(5) with n(S) = l dU l δ(S − p (1 − (1/N )ReT r(U p ))) (6) We call n(S) the density of states . A more general discussion for spin models [10] or gauge theories [11] can be found in the literature where the density of states is sometimes called the spectral density. From its definition, it is clear that n(S) is positive. Assuming that the Haar measure for the links is normalized to 1, the partition function at β = 0 is 1 and consequently we can normalize n(S) as a probability density. A first idea regarding the convergence properties of various expansions can be obtained from the single plaquette model [5]. In that case, we have n 1pl. (S) = 2 π S(2 − S) .(7) The large β behavior of the partition function is determined by the behavior of n(S) near S = 0. In this example, n(S) ∝ √ S for small S, implies that Z ∝ β −3/2 at leading order. Successive subleading corrections can be calculated by expanding the remaining factor √ 2 − S in powers of S and integrating over S from 0 to ∞. If we factor out the leading behavior, we obtain a power series in 1/β. The large order behavior of this power series is determined by the large order behavior of the expansion of √ 2 − S, itself dictated by the branch cut at S = 2. One can see [5] that the S-integration over the whole positive real axis converts an expansion with a finite radius of convergence into one with a zero radius of convergence. On the other hand, if the S-integration is carried over the interval [0, 2], the resulting series converges but the coefficients need to be expressed in terms of the incomplete gamma function. From this example, one may believe that it is easier to approximate n(S) than the corresponding partition function. However, it is not clear that these considerations will survive the infinite volume limit. Note also that the behavior of n(S) near S = 2 can be probed by taking β → −∞ in agreement with the common wisdom that the large order behavior of weak coupling series can be understood in terms of the behavior at small negative coupling. It was showed [9] that if the lattice has even number of sites in each direction and if the gauge group contains −1 1, that it is possible to change βReT rU p into −βReT rU p by a change of variables U l → −U l on a set of links such that for any plaquette, exactly one link of the set belongs to that plaquette. This implies Z(−β) = e 2βNp Z(β)(8) This symmetry implies that n(2N p − S) = n(S) .(9) In the following, we will be working exclusively with SU (2) which contains −1 1 and lattices with even numbers of sites in every direction. We will thus assume that Eq. (9) is satisfied and we only need to know n(S) for 0 ≤ S ≤ N p . III. VOLUME DEPENDENCE In this section, we discuss the volume dependence of the density of state. We make this dependence explicit by writing n(S, N p ). Given the density of states, we can always write f (x, N p ) ≡ ln(n(xN p , N p ))/N p .(10) The function is nonzero only if 0 ≤ x ≤ 2. The symmetry (9) implies that f (x, N p ) = f (2 − x, N p )(11) In the statistical mechanics interpretation of the partition function (where β is an inverse temperature), f (x, N p ) can be interpreted as a density of entropy. The existence of the infinite volume limit requires that lim Np→∞ f (x, N p ) = f (x) ,(12) with f (x) volume independent. In the same limit, the integral ( 5) can be evaluated by the saddle point method. The maximization of the integrand requires f ′ (x) = β .(13) We believe that f (x) is strictly increasing for 0 < x < 1 with an absolute maximum at x = 1. By symmetry, this would imply that f (x) is strictly decreasing for 1 < x < 2. We also believe that f ′ (x) is strictly decreasing and that Eq. (13) has a unique solution (with positive β if 0 < x < 1 and negative β if 1 < x < 2). The numerical study of Sec. IV is in agreement with these statements, but we are not aware of mathematical proofs. Assuming that Eq. (13) has a unique solution, the infinite volume solution should be x = P the average plaquette defined above. We can then convert an expansion for P into an expansion of f . If we want to include the volume dependence, the distribution has a finite width, and we should expand about the saddle point and perform the integration. In the following, we will work at large but finite volume and residual volume dependence in f will be kept implicit in equations. The behavior of f (x) for small x, can be probed by studying the model at large positive β (weak coupling expansion discussed in Sec. VI). On the other hand, at small values of β (strong coupling expansion discussed in Sec. V), the partition function is dominated by the behavior of f (x) near its peak value x = 1. For convenience, we introduce notations suitable for the study of the density of state near x = 1 g(y) ≡ f (1 + y) .(14) g(y) is then an even function defined for −1 < y < 1. IV. NUMERICAL CALCULATION OF n(S) To find n(S) numerically we will use a Monte Carlo simulation to create configurations of SU(2) for different values of β. In the following example we will follow the steps we will use to find n(S) for a volume of 6 4 . We will start with 550 different sets of data ranging from β = 0.02 to β = 11.00 in steps of 0.02 and with sizes of 10 5 configurations. To join the data from different values of β we will first create histograms of each set of data, each of these histograms is roughly Gaussian in shape. We then filter out the data that has statistics that are lower than half of the maximum bin. We can then remove the beta dependence by multiplying the height of each bin by e βS . We will be left with a series of arches which when overlayed on each other form the curve n(s). To create this overlay we will start with the lowest β, which will correspond to the peak of n(s), and then take the logarithm of this. We will then look at the neighboring β and do the same thing but then shifting it up or down so that the average distance in the bins overlapping with the first is zero. We will then continue in this manner until the supply of datasets has been exhausted. A portion of this process can be seen in Fig. 2. We then average the points for each bin together and divide both the bin width and height by N p and shift the top of the curve to zero to make the final output, which can be seen in Fig. 3 for both 4 4 and 6 4 . We see that they overlap well. We now consider the difference between two different volumes, as shown in Fig. 4. We can see in Fig. 5 that as we get closer to S/N p = 1 this difference turns into noise, and as we get closer to S/N p = 0 we see a volume dependence growing. The results reported here correspond to the difference between 6 4 and 4 4 . We have also studied the difference between 8 4 and 6 4 and found consistent results. Calculation at larger volumes are much more computationally expensive and require many more sets of data because of the narrow width of the distributions. V. STRONG COUPLING EXPANSION In this section, we discuss the strong coupling expansion of the logarithm of the density of state. We will work with the shifted function g(y) defined in Eq. (14). The strong coupling expansion of P can be extracted from the expansion of lnZ given in Ref. [12,13] using appropriate rescalings (for instance the β used there is one half of the β used here). The expansion is of the form P (β) ≃ 1 + m=1 a 2m−1 β 2m−1 .(15) The values of the coefficients are given in Table I. With periodic boundary conditions, the low order coefficients are volume independent. This can be understood from the exact translation invariance for the low order strong coupling graphs that provides a multiplicity that cancels exactly the 1/N p in Eq. (4). Volume dependence may appear for graphs wrapping around the torus. The simplest such graph is a straight line that closes into itself due to the periodic boundary condition. It appears at order β 2L and has a reduced translation multiplicity since translation along the graph does not generate a new graph. This type of graphs produce 1/L corrections that to the best of our knowledge have not been studied quantitatively. In the following, we will ignore such effects, but a study of the contribution of graphs with a nontrivial topology would certainly be interesting. We will plug the expansion of P in the expansion g(y) ≃ m=0 g 2m y 2m .(16) At lowest order we have y ≃ a 1 β and the saddle point Eq. (13) yields 2g 2 y ≃ 2g 2 a 1 β ≃ β which implies g 2 = 1/(2a 1 ). This procedure can be followed order by order in β. The results are shown in Table I. Since n(S) is zero for S = 0 and 2N p , we expect logarithmic singularities at x = 0 and 2 for f (x) and y = ±1 for g(y). This singularities will cause the strong coupling series to diverge when \|y\| ≥ 1. Consequently, we define the subtracted function h(y) ≡ g(y) − A(ln(1 − y 2 )) . The coefficient A will be calculated using the weak coupling expansion in Sec. VI. In the infinite volume limit, we have A = 3/4. Expanding h(y) ≃ m=0 h 2m y 2m ,(18) we obtained coefficients that are showed in Table I for A = 3/4. The coefficients g 2m and h 2m are also shown on a logarithmic scale in Fig. 8. This graph shows that the two types of coefficients become rapidly of the same order, which indicates singularities in the complex y plane for smaller values of \|y\| than the ones at ±1. In Fig. 9, we show the error made at successive order of the strong coupling expansion of the plaquette. We then show successive approximation of f (x) (Fig. 10) and the corresponding errors (Fig. 11). We can now compare the apparent convergence of P and f . From Fig. 9 we see that the larger order errors cross between β = 1.5 and 2. For values of β larger, incresing the order increases the error. This is the sign of a finite radius of convergence [14]. Similarly, the larger order errors for f cross for x between 0.5 and 0.6 which are approximately the values of P in the β interval of crossing. Consequently, it seems like the convergence properties of the two expansions are the same (finite radius of convergence). VI. WEAK COUPLING EXPANSION In this section, we discuss the weak coupling expansion of f (x). The starting point is the expansion of P in inverse powers of β P (β) ≃ m=1 b m β −m .(19) We then assume the behavior f (x) ≃ A ln(x) + m=0 f m x m .(20) Using the saddle point Eq. (13), and using the leading large β and small x terms we find β ≃ A/x ≃ A/(b 1 /β) ,(21) which implies A = b 1 at infinite volume. The procedure can be pursued order by order without difficulty. The result for the two lowest orders is f 1 = b 2 /b 1 f 2 = (b 3 b 1 − b 2 2 )/(2b 2 1 ) Numerical experiments indicate that the two series have the same type of growth (power or factorial). Note that f 0 cannot be fixed by the saddle point equation. The overall height of f depends on the behavior near x = 1 (if we insist on normalizing n(S) as probability density) and it seems unlikely that it can be found by a weak coupling expansion. At finite volume, the saddle point calculation of P should be corrected in order to include 1/V effects (V the number of sites , L D for a symmetric lattice). If we perform the Gaussian integration of the quadratic fluctuations, and use the V dependent value of b 1 given in Eq. (23) below, we find after a short calculation that the coefficient A of ln(x) is A = (3/4) − (5/12)(1/V ) .(22) This leading coefficient correction, predicts a difference of −0.0013ln(x) for the difference between f (x) for a 4 4 and 6 4 and is roughly consistent with Fig. 7. Our next task is to find the values of b m . A closed form expression can be found [15,16] for b 1 . For the case N c = 2 and D = 4, we obtain b 1 = (3/4)(1 − 1/(3V )) .(23) The (−1/(3V )) comes from the absence of zero mode (−1/V ) in a sum calculated in [15] plus the contribution of the zero mode with periodic boundary conditions (+2/(3V )) calculated in [16]. Numerical values for b 2 can be found in Ref. [15] and for b 3 in Ref. [17] . In these Refs., several sums are calculated numerically at particular volumes that do not include 6 4 . Rough extrapolations from the existng data indicate that for V = 6 4 uncertainties are less than 0.0002 for b 2 and 0.0008 for b 3 . For β ≥ 3, these effects are close to the numerical errors for P . In the following, we use the approximate values b 2 = 0.1511 and b 3 = 0.1427 for V = 6 4 . We are not aware of any calculation of b m for m ≥ 4 for SU (2). In the case of SU (3), calculations up to order 10 [18] and 16 [19] are available and show remarkable regularities. Using the assumption [4] that ∂P/∂β has a logarithmic singularity in the complex β plane and integrating, we obtained [20] the approximate form m=1 b m β −k ≈ C(Li 2 (β −1 /(β −1 m + iΓ)) + h.c ,(24) with Li 2 (x) = k=1 x k /k 2 .(25) We believe that at zero temperature, the new parameter Γ which measures the (small) distance from the singularity to the real axis in the 1/β plane stabilizes at a nonzero value in the infinite volume limit. For reasons not fully understood, this parametrization of the series turns out to work very well for SU (3). For instance, by fixing the value of Γ in the middle of the allowed range and using the values of b 9 and b 10 , we obtain values of the lower order coefficients with a relative error of 0.2 percent for b 8 and that increases up to 5 percent for b 3 . In the limit Γ = 0, the parametrization provides simple predictions for instance b 3 /b 2 ≃ (4/9)β m . The location of the Fisher's zeros for SU (2) [8] suggests β m = 2.18. This implies b 3 /b 2 ≃ 0.969 in good agreement with our numerical estimate b 3 /b 2 ≃ 0.944. In the following we use the values β m = 2.18, Γ = 0.18/β 2 m ≃ 0.038 (see [8]) and we fixed C = 0.0062 in order to reproduce b 3 . The numerical values of b m and the corresponding values of f m are displayed in Table II. We have compared the weak coupling expansion of P with numerical values in the case V = 6 4 . The results are shown in Fig. 12. In the region where the curves are smooth, the error decrease with the order and appears to accumulate. This is very similar to the case of SU (3) [20]. However, it is clear that more reliable estimates for m ≥ 4 would be desirable for SU (2). It should be noted that for large β, the noise in the error is at the same level as the numerical error on P . This would not be the case if we had not included the contribution of the zero mode to b 1 as shown in the second part of Fig. 12. We have compared the weak coupling expansion of f (x) with numerical values in the case V = 6 4 . The results are shown in Fig. 13. The differences are resolved in Fig Logarithm of the absolute value of the difference between the numerical data and the weak coupling expansion of P at successive orders (above). For reference, we give the estimated numerical error on P. The graph below is the same except that we have not included the zero mode in b1. VII. EXPANSION IN LEGENDRE POLYNOMIALS We now consider the function h(y), which is g(y) with the logarithmic singularity subtracted as defined in Eq. (17). This is a bell shaped even function defined on the interval [−1, 1] and shown on Fig. 16. We can expand this function in terms of the even Legendre polynomials. h(y) = m=0 q 2m P 2m (y)(26) The q 2m can be determined from the orthogonality relations with interpolated values of h(y) to perform the integral. A minor technical difficulty is that we do not have numerical data all the way down to y = −1. This is because as y → −1 + , or in other words x → 0 + , β → +∞ where the plaquette distribution becomes infinitely narrow. Consequently there is a small gap in the numerical data that needs to be filled. Fortunately, this is precisely where the weak coupling expansion works well. Using the weak coupling expansion (including the overall constant), subtracting Aln(x(2 − x)) and shifting to the y coordinate, we obtained the approximate behavior near y = −1 for the 6 4 data: h(y) ≃ 0.2145 + 1.2961y + 0.5261y 2 + 0.1109y 3 (27) In order to estimate the error associated with this approximation we have compared with an extrapolation of a quadratic fit of the leftmost part of the data. In order to give an idea of the volume effects, we have also used the second method on a 4 4 lattice. The results are shown in Table III. This indicates that the variations are small, increase with the order in relative magnitude and that the volume effects are stronger than the dependence on the extrapolation procedure. The logarithm of the coeffi- cients is shown in Fig. 15 which illustrate the exponential decay of the coefficients. The expansion provides excellent approximation of h(y) shown in Fig. 16. The errors are resolved in Fig. 17. It is also possible to calculate P (β) by solving the saddle point Eq. (13) using successive approximations for h. This is shown in Figs. 18 and 19. The spikes in the error graphs correspond to change of sign of the errors. It is important to notice that the quality of the approximations improves with the order in all region of the interval. VIII. CONCLUSIONS We have calculated the density of states for SU (2) lattice gauge theory. The intermediate orders in weak and strong coupling agree well in an overlapping region of action values as shown in Fig. 20. However, the large or- der behaviors of these expansions appear to be similar to the corresponding ones for the plaquette. Volume effects can be resolved well for small actions values. Corrections to the saddle point estimate need to be developed systematically. Aprroximation of a subtracted quantity by Legendre polynomials looks very promising and works well uniformly. We plan to use this approximate form to look for Fisher's zeros. The density of states can be calculated in more general situations. For instance, Z(β, {β i }) = and χ i a complete set of SU (2) characters. This is a type of action which naturally arises in the RG studies of SU (N ) lattice gauge theories. It is possible to apply exact renormalization group transformation [1,2] or the MCRG procedure [21] to the partition function in order to define the couplings. Following the analogy between f ′ = β and V ′ = J for the effective potential V in presence of a source J in scalar models, it would be interesting to study finite size effects from this point of view. FIG. 1 : 1Weak and strong coupling expansions of the average plaquette P for SU (2) at various orders in the weak and strong coupling expansion compared to the numerical values. FIG. 2 : 2Close-up of the patching process for 6 4 . FIG. 3 : 3Results of patching for 4 4 and 6 4 . FIG. 4 : 4The difference between ln(n(S))/Np for 4 4 and 6 4 . FIG. 5 : 5The difference between ln(n(S))/Np for 4 4 and 6 4 with β > 0.5 (close up). FIG. 6 :FIG. 7 : 67The difference between ln(n(S))/Np for 4 4 and 6 4 with β < 0.5 (close-up). The difference between ln(n(S))/Np for 4 4 and 6 4 divided by ln(S/Np) FIG. 8 :FIG. 9 : 89Logarithm of the absolute value of g2m and h2m Logarithm of the absolute value of the difference between the numerical data and the strong coupling expansion of P at successive orders. For reference, we give the estimated numerical error on P. FIG. 10 : 10Numerical value of f (x) compared to the strong coupling expansion at successive orders. FIG. 11 : 11Logarithm of the absolute value of the difference between the numerical data and the strong coupling expansion of f at successive orders. For reference, we give the estimated numerical error on f . . 14. In these graphs we have taken f 0 = −0.14663 which maximizes the length of the accumulation line on the left of Fig. 14. .690 5.3207 15 63.702 9.6945TABLE II: Weak coupling coefficients defined in Sec. VI. The choice of b1 corresponds to V = 6 4 . FIG. 12: Logarithm of the absolute value of the difference between the numerical data and the weak coupling expansion of P at successive orders (above). For reference, we give the estimated numerical error on P. The graph below is the same except that we have not included the zero mode in b1. FIG.13: Numerical value of f (x) compared to the weak coupling expansion at successive orders. FIG. 14 : 14Logarithm of the absolute value of the difference between the numerical data and the weak coupling expansion of f at successive orders. For reference, we give the estimated numerical error on f . FIG. 15 : 15Legendre polynomial coefficients q2m with the three methods described in the text. 00008 0.00026 0.00028 10 -0.00094 -0.00069 -0.00067 FIG.16: h(y) together with the expansion in Legendre polynomials up to order 20. FIG. 17 :FIG. 18 : 1718Logarithm of the absolute value of the difference between the numerical data for h(y) and expansions in Legendre polynomials at successive P together with the expansion in Legendre polynomials up to order 20. FIG. 19 :FIG. 20 : 1920Logarithm of the absolute value of the difference between the numerical data for P and expansions in Legendre polynomials at successive orders Weak and strong coupling expansion of f at a few intermediate orders. 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[ "STABILITY OF THE LOGARITHMIC SOBOLEV INEQUALITY VIA THE FÖLLMER PROCESS", "STABILITY OF THE LOGARITHMIC SOBOLEV INEQUALITY VIA THE FÖLLMER PROCESS" ]	[ "Ronen Eldan ", "Joseph Lehec ", "Yair Shenfeld " ]	[]	[]	We study the stability and instability of the Gaussian logarithmic Sobolev inequality, in terms of covariance, Wasserstein distance and Fisher information, addressing several open questions in the literature. We first establish an improved logarithmic Sobolev inequality which is at the same time scale invariant and dimension free. As a corollary, we show that if the covariance of the measure is bounded by the identity, one may obtain a sharp and dimension-free stability bound in terms of the Fisher information matrix. We then investigate under what conditions stability estimates control the covariance, and when such control is impossible. For the class of measures whose covariance matrix is dominated by the identity, we obtain optimal dimension-free stability bounds which show that the deficit in the logarithmic Sobolev inequality is minimized by Gaussian measures, under a fixed covariance constraint. On the other hand, we construct examples showing that without the boundedness of the covariance, the inequality is not stable. Finally, we study stability in terms of the Wasserstein distance, and show that even for the class of measures with a bounded covariance matrix, it is hopeless to obtain a dimension-free stability result. The counterexamples provided motivate us to put forth a new notion of stability, in terms of proximity to mixtures of the Gaussian distribution. We prove new estimates (some dimension-free) based on this notion. These estimates are strictly stronger than some of the existing stability results in terms of the Wasserstein metric. Our proof techniques rely heavily on stochastic methods.	10.1214/19-aihp1038	[ "https://arxiv.org/pdf/1903.04522v1.pdf" ]	119,132,464	1903.04522	8db8db06ae9330e4bbd9f9c40a2e4f9aa48274b1	STABILITY OF THE LOGARITHMIC SOBOLEV INEQUALITY VIA THE FÖLLMER PROCESS 11 Mar 2019 Ronen Eldan Joseph Lehec Yair Shenfeld STABILITY OF THE LOGARITHMIC SOBOLEV INEQUALITY VIA THE FÖLLMER PROCESS 11 Mar 2019arXiv:1903.04522v1 [math.PR] We study the stability and instability of the Gaussian logarithmic Sobolev inequality, in terms of covariance, Wasserstein distance and Fisher information, addressing several open questions in the literature. We first establish an improved logarithmic Sobolev inequality which is at the same time scale invariant and dimension free. As a corollary, we show that if the covariance of the measure is bounded by the identity, one may obtain a sharp and dimension-free stability bound in terms of the Fisher information matrix. We then investigate under what conditions stability estimates control the covariance, and when such control is impossible. For the class of measures whose covariance matrix is dominated by the identity, we obtain optimal dimension-free stability bounds which show that the deficit in the logarithmic Sobolev inequality is minimized by Gaussian measures, under a fixed covariance constraint. On the other hand, we construct examples showing that without the boundedness of the covariance, the inequality is not stable. Finally, we study stability in terms of the Wasserstein distance, and show that even for the class of measures with a bounded covariance matrix, it is hopeless to obtain a dimension-free stability result. The counterexamples provided motivate us to put forth a new notion of stability, in terms of proximity to mixtures of the Gaussian distribution. We prove new estimates (some dimension-free) based on this notion. These estimates are strictly stronger than some of the existing stability results in terms of the Wasserstein metric. Our proof techniques rely heavily on stochastic methods. 1. Introduction 1.1. Overview. The logarithmic Sobolev inequality is one of the fundamental Gaussian functional inequalities [21]. The inequality was proven independently in the information-theoretic community by Stam [27] and in the mathematical-physics community by Gross [18]. The form of the inequality which we consider in this paper states that for any nice enough probability measure µ on R n , H(µ \| γ) ≤ 1 2 I(µ \| γ).(1) Here γ is the standard Gaussian measure on R n with density γ(dx) = (2π) − n 2 e − \|x\| 2 2 dx, and H(µ \| γ), I(µ \| γ) are the relative entropy and relative Fisher information respectively: H(µ \| γ) = R n log dµ dγ dµ and I(µ \| γ) = R n ∇ log dµ dγ 2 dµ. The inequality (1) is sharp as can be seen by taking µ to be any translation of γ, and in fact these are the only equality cases as was proved in [7]. This characterization naturally leads to the question of stability. That is, supposing that the deficit δ(µ) := 1 2 I(µ \| γ) − H(µ \| γ) is small, in what sense is µ close to a translate of γ? The study of stability questions for Gaussian inequalities is an ongoing active area of research with many applications [16], [24]. The precise notion of stability is context-dependent, but a common thread is the desire to make the stability estimates dimension-free. This is because the Gaussian measure itself is inherently infinite-dimensional, so we expect functional inequalities about Gaussian measures in R n to extend to infinite dimensions. Indeed, the infinite-dimensional nature of the logarithmic Sobolev inequality is crucial to its applications to quantum field theory, which was the original motivation of Gross. For example, it was proven in a series of works [8], [24,25], [13], [3] that the Gaussian isoperimetric inequality (which implies the log-Sobolev inequality) enjoys such dimension-free estimates. The logarithmic Sobolev inequality however, turns out to be much more delicate. Up until now, the state of affairs was that if only minimal assumptions are imposed on µ, then natural dimension-free stability estimates were almost completely absent (but see [20] for such estimates in terms of Stein deficit). 1.2. Fisher information matrix and deficit. Our first observation is that the log-Sobolev inequality can be self-improved in a dimension-free way. This observation then leads to natural stability results, provided that cov(µ) Id n . Let us formulate first the log-Sobolev inequality in an alternative way. Define the entropy and Fisher information of µ with respect to the Lebesgue measure by H(µ \| L) = R n log dµ dx dµ, and I(µ \| L) = R n ∇ log dµ dx 2 dµ. The log-Sobolev inequality (1) then reads H(µ \| L) − H(γ \| L) ≤ 1 2 (I(µ \| L) − n) . It is well known (see [4] and [11] for example), that the above inequality can be improved via scaling. Let X ∼ µ and let σ > 0. Computing the entropy and Fisher information of the law of σX, and optimizing over σ, shows that H(µ \| L) − H(γ \| L) ≤ n 2 log I(µ \| L) n . ( Inequality (2) is known as the dimensional logarithmic Sobolev inequality. Our first result shows that this bound is sub-optimal, and that one should consider the individual eigenvalues of the Fisher information matrix: I(µ \| L) := R n ∇ log dµ dx ⊗2 dµ. This matrix is of course related to the Fisher information via Tr[I(µ \| L)] = I(µ \| L). Theorem 1. Let µ be a probability measure on R n . Then H(µ \| L) − H(γ \| L) ≤ 1 2 log det [I(µ \| L)] .(3) Theorem 1 improves upon (2) by the AM/GM inequality. Note also that (3) is at the same time scale invariant and dimension-free: both sides of the inequality behave additively when taking tensor products. Remark 1. A certain type of a reverse form of Theorem 1 is known when the measure is logconcave. Observe first the integration by parts identity I(µ \| L) = − R n ∇ 2 log dµ dx dµ. The reverse form of Theorem 1 then asserts that if µ is log-concave and if log det is moved inside the integral in the right-hand side of (3), then the inequality is reversed, see [1]. See also [6] for a simpler proof based on the functional Santaló inequality. Self-improvements of the form of (2) and (3) lead to stability results for the log-Sobolev inequality, provided that the covariance of µ is bounded by the identity. Define the function ∆(t) := t − log(1 + t) for t > −1. It was observed in [4] that if E µ [\|x\| 2 ] ≤ n, then (2) implies that δ(µ) ≥ n 2 ∆ I(µ \| γ) n .(4) From (4) one can deduce weaker but more amenable stability statements. For example, δ(µ) ≥ c n W 4 2 (µ, γ)(5) for some universal constant c, see [4] for the details. Here, W 2 (µ, γ) is the Wasserstein twodistance between µ and γ. In general, the p-Wasserstein distance (p ≥ 1) for probability measures µ, ν is defined as W p (µ, ν) := inf X,Y E[ \|X − Y \| p ] 1/p ,(6) where the infimum is taken over all couplings (X, Y ) of (µ, ν). A problematic feature of both bounds, (4) and (5), is that they are dimension-dependent. On the other hand, we can deduce from Theorem 1 the following dimension-free estimate. Corollary 2. Let µ be a probability measure on R n such that E µ [x ⊗2 ] Id n , and let {β i } n i=1 be the eigenvalues of its Gaussian Fisher information matrix I(µ \| γ). Then δ(µ) ≥ 1 2 n i=1 ∆(β i ).(7) Again, by concavity of the logarithm, (7) is a strict improvement on (4). To see how Corollary 2 follows from Theorem 1, note that (3) can be rewritten as δ(µ) ≥ 1 2 n i=1 ∆(α i − 1),(8) where α 1 , . . . , α n are the eigenvalues of the Fisher information matrix of µ with respect to the Lebesgue measure. Using the integration by parts identity I(µ \| L) − Id n = I(µ \| γ) + Id n − E µ [x ⊗2 ] we see that if E µ [x ⊗2 ] Id n , then I(µ \| L) − Id n I(µ \| γ) 0. Since ∆ is increasing on [0, +∞), the inequality (7) thus follows from (8). Remark 2. Corollary 2 bears an interesting formal resemblance to the following result. Let T be the Brenier map from µ to γ and let {κ i (x)} n i=1 be the eigenvalues of the map DT (x) − Id n . Then it can be shown [10] that δ(µ) ≥ n i=1 E µ [∆(κ i )]. For further appearances of the map ∆ as a cost function in transportation distance, see [4]. Let us note that although Theorem 1 (and thus Corollary 2) follow from a simple scaling argument (see section 2), it is arguably the first natural dimension-free stability result, under minimal assumptions on µ. Until now, the only known dimension-free estimates of the form of Corollary 2, are the results of [16] where strong assumptions were imposed on µ, namely that it satisfies a Poincaré inequality. Our emphasis on the eigenvalues of the Fisher information matrix rather than their average (Fisher information) also seems to be new. Covariance and Gaussian mixtures. As we saw, in order to get stability estimates for the deficit from the self-improvements of the log-Sobolev inequality, we need to assume that E µ [\|x\| 2 ] ≤ n. The phenomenon that the size of cov(µ) serves as a watershed for stability estimates has already been observed in the literature, but the precise connection has remained unclear. Indeed, [20] raises the question regarding the relation between the distance of the covariance of µ from the identity, and the possible lower bounds on the deficit. Our next result completely settles this question. Theorem 3. Let µ be a probability measure on R n and let λ := {λ i } n i=1 be the eigenvalues of cov(µ). Then δ(µ) ≥ 1 2 n i=1 1 {λ i <1} (λ −1 i − 1 + log λ i ).(9) In particular, if cov(µ) Id n , then δ(µ) ≥ 1 2 n i=1 (λ −1 i − 1 + log λ i ) = δ(γ λ ) where γ λ is a Gaussian measure on R n having the same covariance matrix as µ. On the other hand, this becomes completely wrong if we remove the hypothesis on the covariance matrix, even in dimension 1: there exists a sequence (µ k ) of mixtures of Gaussian measures on R such that var(µ k ) → ∞ while δ(µ k ) → 0. The moral of Theorem 3 is, that if cov(µ) Id n , then the deficit δ(µ) controls the distance of cov(µ) to the identity. For example, a weaker bound which can be deduced from (9) using 1 x − 1 + log x ≥ 1 2 (x − 1) 2 for x ∈ (0, 1] is, δ(µ) ≥ 1 4 cov(µ) − Id n 2 HS , where the norm on the right hand side is the Hilbert-Schmidt norm. On the other hand, if the covariance of µ is not a priori bounded by the identity, then one can have an arbitrarily small deficit with arbitrarily large variance. Remark 3. Theorem 3 can also be phrased as a statement about minimizing the deficit subject to a covariance constraint. For simplicity let us consider the one-dimensional situation. Fix a scalar σ > 0. Of all distributions µ with variance σ, which one minimizes δ(µ)? Theorem 3 shows that the answer is dramatically different depending on whether or not σ is greater than 1. (If σ = 1 then obviously µ = γ minimizes δ(µ).) If σ < 1, then the minimizer is the Gaussian measure with variance σ. On the other hand, if σ > 1, then by taking µ to be an appropriate mixture of Gaussians, we can make δ(µ) smaller than the Gaussian with variance σ. The Gaussian mixtures in Theorem 3 served as counterexamples to stability estimates in terms of the distance of cov(µ) from the identity. In fact, such mixtures show the impossibility of many other stability estimates: Theorem 4. For m ∈ R n let γ m,Id be the Gaussian measure centered at m with identity covariance matrix. There exists a sequence (µ k ) of probability measures on R, each of which is a mixture of two Gaussian measures of variance 1, satisfying δ(µ k ) → 0 and lim k→∞ inf m∈R {W 1 (µ k , γ m,Id )} = +∞. Additionally, there exists a sequence of dimensions n(k) ↑ +∞ and a sequence (µ k ) of isotropic (i.e. centered with identity as covariance) measures on R n(k) , satisfying δ( µ k ) = O(n(k) −1/3 ) → 0 and inf m∈R n(k) {W 2 (µ k , γ m,Id )} = Ω(n(k) 1/6 ) → +∞. The first statement shows that the log-Sobolev inequality is unstable for W 1 , even in dimension 1. The second statement shows that even for isotropic measures, there is no dimension free stability result for W 2 . Note however that our second counterexample does not work for W 1 ; as far as we know it could still be the case that δ(µ) ≥ cW 1 (µ, γ) 2 for every isotropic µ on R n . (Recall that by Jensen's inequality we have W 1 (µ, ν) ≤ W 2 (µ, ν).) Explicit counterexamples to stability were discussed recently in the literature, see [19]. These examples however are complicated and require a lot of tedious computations while ours are completely elementary. We just observe that Gaussian mixtures have small log-Sobolev deficit, see Proposition 5 below. Similar Gaussian mixture examples can be found in the context of stability of the entropy power inequality, see [9] and references therein. Proposition 5. Let p be a discrete measure on R n and let S(p) = − p(x) log p(x) be its Shannon entropy. Then δ(p * γ) ≤ S(p). Decompositions into mixtures. If we take stock of the results in the preceding sections, we see that while a result of the form δ(µ) ≥ c n W 4 2 (µ, γ) holds under the assumption that E µ [\|x\| 2 ] ≤ n, we cannot replace the right hand side by c ′ √ n W 3 2 (µ, γ), let alone c ′′ W 2 2 (µ, γ). (These bounds increase in strength since W 2 2 (µ, γ) ≤ 4n under the assumption E µ [\|x\| 2 ] ≤ n.) As we saw, mixtures of Gaussians pose counterexamples to such bounds. Our next result shows that in a certain sense, these counterexamples are the only obstacles. Theorem 6. Let µ be a probability measure on R n . Then there exists a measure ν on R n such that δ(µ) ≥ 1 15 W 3 2 (µ, ν * γ) √ n ,(10) and so that ν is a Dirac point mass whenever δ(µ) = 0. In fact, that a small deficit implies that µ is close to being a mixture of Gaussians, is an implication which comes out naturally from our stochastic proof technique as we will see below. The relation between approximate equality in the log-Sobolev inequality and proximity to mixtures of product measures, appears in a recent work of Austin [2] in a more abstract setting of product spaces. Given Theorem 6 and Proposition 5 we pose the following question. Question 7. Given a probability measure µ on R n , is it true that there exists a discrete probability measure p on R n satisfying S(p) ≤ C δ(µ) and W 2 2 (µ, p * γ) ≤ C δ(µ), where C is a universal constant? Note that both sides of the inequality above behave additively when taking tensor products. In particular, if µ is the n-th power of a one-dimensional measure then both sides are of order n. The inequality is thus completely dimension-free, which is our main motivation for it. Remark 4. While the Wasserstein distance is a bona fide distance between probability measures, in the context of the log-Sobolev inequality it seems more natural to work with lower bounds which are expressed in terms of relative entropy and relative Fisher information. Thus one may wonder, whether it is possible to replace the lower bound on the deficit in Question 7 by the relative entropy or Fisher information between µ and a mixture of Gaussians. We focus on the Wasserstein two-distance distance since by the log-Sobolev and Talagrand's inequalities such results are weaker. Moreover, our decomposition results are easier to prove for the Wasserstein distance. As a step towards answering this question, we prove that an estimate similar in spirit does indeed hold. We show that a random vector distributed like µ, can be written as the sum of two random vectors which are orthogonal in expectation, one of which is close to a Gaussian in a dimension-free way. Theorem 8. Let µ be a probability measure on R n and let X ∼ µ. There exists a decomposition X D = Y + W with the property that E[ Y, W ] = 0, such that δ(µ) ≥ 1 2 W 2 2 (ν, γ) where Y ∼ ν. Theorem 8 can be seen as an improvement on (5). Indeed, assume that E µ [\|x\| 2 ] ≤ n. The theorem implies that W 2 (µ, γ) ≤ W 2 (µ, ν) + W 2 (ν, γ) ≤ E[\|W \| 2 ] 1/2 + 2δ(µ). Moreover, since E[ Y, W ] = 0, we have E[\|W \| 2 ] = E[\|X\| 2 ] − E[\|Y \| 2 ] ≤ n − E[\|Y \| 2 ]. If δ(µ) ≥ Cn, then (5) holds trivially, so we can assume additionally that δ(µ) = O(n). Then by the theorem W 2 (ν, γ) = O( √ n) and thus E[\|Y \| 2 ] ≥ n − C √ n W 2 (ν, γ) ≥ n − C 2nδ(µ). Putting everything together, we get W 2 (µ, γ) ≤ 2δ(µ) + C ′ n 1/4 δ(µ) 1/4 ≤ C ′′ n 1/4 δ(µ) 1/4 , which is (5). 1.5. Methods. We provide two sets of proofs for Theorems 1 and 3. The first set of proofs proceeds by establishing Theorem 1 via a scaling argument, and then deduces the first part of Theorem 3 from Theorem 1 via the Cramér-Rao bound. The second set of proofs uses a stochastic process known as the Schrödinger bridge, or the Föllmer process, depending on the context. This process is entropy-minimizing and is thus suitable for the logarithmic Sobolev inequality. For example, it is used in [22] to give a simple proof of the log-Sobolev inequality (see section 3), and in [14] to obtain a reversed form (see also [15]). We use this process to prove Theorems 6 and 8 as well. Some of our arguments are essentially semigroup proofs (see [20]), phrased in a stochastic language, which uses the semigroup of the Föllmer process rather than the more common heat or Ornstein-Uhlenbeck semigroups. A key point in our proofs is that we essentially compute two derivatives of the entropy rather than one. This gives us more precise information about the log-Sobolev inequality. The stochastic formulation allows for relatively simple computations. We go however an additional step beyond semigroup techniques, and also analyze pathwise behavior of the Föllmer process. This analysis provides us with a natural way of decomposing the measure µ (see the proofs of Theorem 6 and Theorem 8). 1.6. Organization of paper. In section 2 we give the first set of proofs of Theorems 1 and 3. In section 3 we define the Föllmer process and analyze its properties. This analysis provides us with ways of decomposing µ. Section 4 contains the second set of proofs of Theorems 1 and 3 via the Föllmer process, and section 5 contains the proofs of Theorems 6 and 8. Finally, the counterexamples to stability (and the proof of Theorem 4) are discussed in section 6. 1.7. Acknowledgments. We are grateful to Ramon van Handel for his enlightening comments, in particular, the observation that the first part of Theorem 3 can be deduced from Theorem 1 via the Cramér-Rao bound, is due to him. Our original proof of the first part of Theorem 3 can be found in section 4. We would also like to thank Max Fathi, Michel Ledoux and Dan Mikulincer for discussions and suggestions. In addition we would like to acknowledge the hospitality of MSRI and to thank the organizers of the program on Geometric Functional Analysis and Applications in the fall 2017 where part of this work was done. self-improvements of the log-Sobolev inequality In this section we show how Theorem 1 and the first part of Theorem 3 follow from scaling the log-Sobolev inequality appropriately and the Cramér-Rao bound. Recall that the log-Sobolev inequality can be rewritten H(µ \| L) − H(γ \| L) ≤ 1 2 (I(µ \| L) − n) .(11) Let Σ be an n × n symmetric positive definite matrix and let µ Σ be the law of ΣX where X ∼ µ. Easy computations show that H(µ Σ \| L) = H(µ \| L) − log det Σ and I(µ Σ \| L) = Σ −1 I(µ \| L)Σ −1 . In particular I(µ Σ \| L) = Tr Σ −2 I(µ \| L) . Applying (11) to µ Σ thus yields H(µ \| L) − H(γ \| L) ≤ 1 2 Tr Σ −2 I(µ \| L) − n + log det Σ 2 . The right-hand side of the inequality is minimal when Σ = I(µ \| L). This choice of Σ yields the desired inequality (3). Note that the scaling proof of (2) amounts to considering only diagonal matrices of the form Σ = σId n for some scalar σ > 0. The first part of Theorem 3 follows from Theorem 1 via the Cramér-Rao bound: cov(µ) −1 I(µ \| L).(12) Indeed, recall that {λ i } and {α i } denote the eigenvalues of cov(µ) and I(µ \| L), respectively. Since the map x → 1 {x>1} (x − 1 − log x) is increasing on [0, +∞), inequality (12) imply that 1 2 n i=1 1 {λ i <1} (λ −1 i − 1 + log λ i ) ≤ 1 2 n i=1 1 {α i >1} (α i − 1 − log α i ) ≤ 1 2 n i=1 (α i − 1 − log α i ). By Theorem 1 this is upper bounded by the deficit δ(µ) and we obtain the first statement of Theorem 3. The second part of the theorem follows from a straightforward computation which shows that δ(γ λ ) = 1 2 n i=1 1 λ i − 1 + log λ i , see section 6 below. The third part is also proved in section 6. The Föllmer Process Given an absolutely continuous probability measure µ on R n we consider a stochastic process (X t ) which is as close as possible to being a Brownian motion while having law µ at time 1. Namely X 1 has law µ, and the conditional law of X given the endpoint X 1 is a Brownian bridge. Equivalently, the law of X has density ω → f (ω 1 ) with respect to the Wiener measure, where f is the density of µ with respect to γ and ω is an element of the classical Wiener space. In particular the process X minimizes the relative entropy with respect to the Wiener measure among all processes having law µ at time 1. This process was first considered by Schrödinger who was interested in the problem of minimizing the entropy with endpoint constraints, see [26] and the survey [23] where a nice historical account on the Schrödinger problem is given as well as the connection with optimal transportation. It was first observed by Föllmer [17] that the process (X t ) solves the following stochastic differential equation: dX t = dB t + ∇ log P 1−t f (X t ) dt where (B t ) is a standard Brownian motion, and (P t ) is the heat semigroup, defined by P t h(x) = E[h(x + B t )] for every test function h. We call the process (X t ) the Föllmer process and the process (v t ) given by v t := ∇ log P 1−t f (X t ), the Föllmer drift. Below we recall some basic properties of this process, and we repeat the proof from [22] of the log-Sobolev inequality based on the Föllmer process. We then prove more refined properties of the bridge which are needed for our stability results. Roughly, the properties (i),(ii) below correspond to the first derivative of entropy along the process while the further properties (iii),(iv),(v) correspond to the second derivative. Finally we show how the Föllmer process leads to natural decompositions of µ. From now on we assume that the measure µ has finite Fisher information I(µ \| γ) = R n \|∇ log f \| 2 dµ < +∞. Proposition 9. The Föllmer drift (v t ) has the following properties: (i) The relative entropy of µ with respect to γ satisfies H(µ \| γ) = 1 2 E 1 0 \|v t \| 2 dt .(13)(ii) The Föllmer drift (v t ) is a square integrable martingale. The proof of this proposition can be found in [22]. As was noticed in [22], the log-Sobolev inequality follows immediately from these properties once it is realized that E[\|v 1 \| 2 ] = I(µ \| γ).(14) Indeed, as (v t ) is a martingale, (\|v t \| 2 ) is a sub-martingale so H(µ \| γ) = 1 2 E 1 0 \|v t \| 2 dt ≤ 1 2 E[\|v 1 \| 2 ] = 1 2 I(µ \| γ). In particular we obtained the following expression for the deficit. Proposition 10. Let µ be a probability measure on R n with finite Fisher information and let (v t ) be the associated Föllmer drift. Then δ(µ) = 1 2 E 1 0 \|v 1 − v t \| 2 dt . Proof. Since (v t ) is a square integrable martingale we have E[ v 1 , v t ] = E[\|v t \| 2 ] so E \|v 1 \| 2 − \|v t \| 2 = E \|v 1 − v t \| 2 . Combining this with (13) and (14) yields the result. The above proof of the log-Sobolev inequality utilizes information about the first derivative of the entropy, that is, the fact that the derivative \|v t \| 2 is a sub-martingale. In order to obtain stability estimates for the log-Sobolev inequality we need to look at the second derivative of the entropy. This is the role of the next proposition. In what follows (F t ) is the natural filtration of the process (X t ) and cov(X 1 \| F t ) := E[X ⊗2 1 \| F t ] − E[X 1 \| F t ] ⊗2 denotes the conditional covariance of X 1 given F t . Proposition 11. Set Q t = ∇ 2 P 1−t f (X t ), then (iii) v t = t 0 Q s dB s for all t.( iv) At least for t < 1 the following alternative expressions for Q t hold true Q t = cov(X 1 \| F t ) (1 − t) 2 − Id n 1 − t (15) = E[∇ 2 log f (X 1 ) \| F t ] + cov(v 1 \| F t ).(16) (v) The process (Q t + t 0 Q 2 s ds) is a martingale. Proof. The computation of dv t is a straightforward application of Itô's formula. For (iv) recall that P 1−t f is the convolution of f with some Gaussian. Putting derivatives on the Gaussian we get after some computations ∇ 2 log P 1−t f = − Id n 1 − t + 1 (1 − t) 2 P 1−t (f (x)x ⊗2 ) P 1−t f − 1 (1 − t) 2 P 1−t (f (x)x) P 1−t f ⊗2 . On the other hand, for every test function u, the following change of measure formula holds true E[u(X 1 ) \| F t ] = P 1−t (uf )(X t ) P 1−t f (X t ) . This follows from the explicit expression that we have for the law of (X t ). Plugging this into the previous display yields (15). The proof of (16) is similar, only we put the derivatives on f rather than the Gaussian when computing ∇ 2 log P 1−t f . To get (v) observe that by (16) Q t = martingale − v t ⊗ v t . Now since v t = Q t dB t we have d(v t ⊗ v t ) = d(martingale) + Q 2 t dt. Hence the result. Note that since X 1 = B 1 + 1 0 v t dt and since the expectation of v t is constant over time, the expectation of v t coincides with that of µ. In addition, it follows from (15) that E[Q t ] = E µ [∇ 2 log f + (∇ log f ) ⊗2 ] − E[v t ⊗ v t ] for every t. Integrating by parts yields the following: Proposition 12. For every time t we have E[v t ] = E µ [x] and E[Q t ] = E µ [x ⊗ x] − Id n − E[v t ⊗ v t ] = cov(µ) − Id n − cov(v t ). Other than facilitating an immediate proof of the log-Sobolev inequality, the Föllmer process provides a canonical decomposition of the measure µ which we now describe. Recall that E[h(X 1 ) \| F t ] = P 1−t (hf )(X t ) P 1−t f (X t ) , for every test function h. This allows to compute the density of the conditional law of X 1 given F t . Namely, let µ t be the conditional law of X 1 −Xt √ 1−t given F t . Then µ t (dx) = f ( √ 1 − t x + X t ) P 1−t f (X t ) γ(dx).(17) Lemma 13. We have X 1 = 1 0 cov(µ t )dB t(18) almost-surely, and δ(µ) ≥ 1 0 E [δ(µ t )] dt.(19)Proof. Again E[X 1 \| F t ] = P 1−t (xf )(X t )/P 1−t f (X t ). So d E[X 1 \| F t ] = ∇ P 1−t (xf ) P 1−t f (X t ) dB t . Arguing as in the proof of (iv) we get ∇ P 1−t (xf ) P 1−t f (X t ) = cov(X 1 \| F t ) 1 − t = cov(µ t ), which proves (18). For the inequality (19), observe that by (17) δ (µ t ) = 1 − t 2 E[\|∇ log f (X 1 )\| 2 \| F t ] − E[log f (X 1 ) \| F t ] + log P 1−t f (X t ). Also, by Itô's formula d log P 1−t f (X t ) = v t dB t + 1 2 \|v t \| 2 dt. Putting everything together we get E [δ(µ t )] = 1 2 1 t E \|v 1 − v s \| 2 ds. Thus, by Proposition 10 1 0 E [δ(µ t )] dt = 1 2 1 0 s E[\|v 1 − v s \| 2 ] ds ≤ δ(µ). Remark 5. At this stage it maybe worth noticing that the measure-valued process (µ t ) coincides with a simplified version of the stochastic localization process of the first named author [12]. Comparison theorems In this section we prove Theorems 1 and 3 via the Föllmer process. Proof of Theorem 1. Because the result is invariant by scaling we can assume without loss of generality that cov(µ) is strictly smaller than the identity. Let m(t) = −E[Q t ]. We know from Proposition 12 that m(t) = −cov(µ) + Id n + cov(v t ).(20) This shows in particular that m(t) is positive definite. Item (v) of Proposition 11 shows that d dt m(t) m(t) 2 . Since m(t) is positive definite this amounts to d dt m(t) −1 −Id n . We use this information to compare m(t) with m(1). We get m(t) m(1) −1 + (1 − t)Id n −1 .(21) Letf be the density of µ with respect to the Lebesgue measure and observe that m(1) = −E[Q 1 ] = −E µ [∇ 2 log f ] = −E µ [∇ 2 logf ] − Id n = I(µ \| L) − Id n . Taking the trace in (21) and using Proposition 12 thus gives −E µ [\|x\| 2 ] + n + E[\|v t \| 2 ] ≤ n i=1 1 (α i − 1) −1 + 1 − t , where the α i are the eigenvalues of I(µ \| L). Integrating between 0 and 1 and applying item (i) of Proposition 9 yields −E µ [\|x\| 2 ] + n + 2 H(µ \| γ) ≤ n i=1 log(α i ). Lastly, a straightforward computation shows that the left hand side equals 2H(µ \| L) − 2H(γ \| L). Proof of Theorem 3. Consider the orthogonal decomposition cov(µ) = n i=1 λ i u ⊗2 i where u i are unit orthogonal vectors, and again let m(t) = −E[Q t ]. Recall (20), which shows in particular that m(0) = −cov(µ) + Id n . Fix i ∈ [n] such that λ i < 1 and denote θ = u i . Note that θ, m(0)θ = 1 − λ i > 0. Moreover, we have d dt θ, m(t)θ ≥ θ, m(t) 2 θ ≥ θ, m(t)θ 2 . Since the function g(t) = 1 1/c−t solves the ordinary differential equation d dt g(t) = g(t) 2 with the boundary condition g(0) = c, an application of Grônwall's inequality gives θ, m(t)θ ≥ 1 θ, m(0)θ −1 − t = 1 (1 − λ i ) −1 − t . Summing up over all i such that λ i < 1, we have d dt E[\|v t \| 2 ] = Tr m(t) 2 ≥ n i=1 1 {λ i <1} ((1 − λ i ) −1 − t) 2 . Integrating this between t and 1, we obtain E[\|v 1 \| 2 ] − E[\|v t \| 2 ] ≥ n i=1 1 {λ i <1} 1 λ i − 1 − 1 (1 − λ i ) −1 − t . Now we integrate between 0 and 1 and we use Proposition 10. We get δ(µ) ≥ 1 2 n i=1 1 {λ i <1} 1 λ i − 1 + log λ i , which is the desired inequality. Decompositions into mixtures In this section we prove Theorems 6 and 8. Proof of Theorem 6. The idea of the proof is to show that for any t, the transportation distance between X 1 and the sum of the independent random vectors E[X 1 \| F t ] + (B 1 − B t ) can be controlled by the deficit. Optimizing over t yields the theorem. The map t → E \|v 1 \| 2 − \|v t \| 2 is a non-increasing function since (\|v t \| 2 ) is a sub-martingale. Hence by Proposition 10 and as (v t ) is a martingale, δ(µ) = 1 2 E 1 0 \|v 1 − v t \| 2 dt = 1 2 1 0 E \|v 1 \| − \|v t \| 2 dt ≥ t 2 E \|v 1 − v t \| 2(22)for every t ∈ [0, 1]. Let Y t = E[X 1 \| F t ] + B 1 − B t and note that since B 1 − B t is independent of F t , the random vector Y t has law ν t * γ 0,1−t where ν t is the law of E[X 1 \| F t ]. Hence since X 1 has law µ and E[X 1 \| F t ] = X t + (1 − t)v t , we get by Jensen's inequality that W 2 2 (µ, ν t * γ 0,1−t ) ≤ E \|X 1 − Y t \| 2 = E 1 t (v s − v t )ds 2 ≤ (1 − t) 1 t E \|v s − v t \| 2 ds ≤ (1 − t) 2 E[\|v 1 − v t \| 2 ] ≤ E[\|v 1 − v t \| 2 ]. Combining this with (22) yields W 2 2 (µ, ν t * γ 0,1−t ) ≤ 2 t δ(µ). This inequality gives the distance between µ and a mixture of Gaussians but with the wrong covariance. To remedy that we must pay a dimensional price. By the triangle inequality for W 2 and the fact that W 2 2 (γ 0,1−t , γ 0,1 ) ≤ (1 − √ 1 − t) 2 n ≤ t 2 n, we get W 2 (µ, ν t * γ) ≤ W 2 (µ, ν t * γ 0,1−t ) + W 2 (ν t * γ 0,1−t , ν t * γ) ≤ W 2 (µ, ν t * γ 0,1−t ) + W 2 (γ 0,1−t , γ) ≤ 2δ(µ) t + √ nt. If δ(µ) ≤ n, choosing t = δ(µ) n 1 3 in the previous display gives W 2 (µ, ν t * γ) ≤ ( √ 2 + 1)n 1 6 δ(µ) 1 3 , which in turn yields the desired inequality (10). If on the contrary δ(µ) ≥ n, inequality (10) holds with ν = µ, simply because W 2 (µ, µ * γ) = √ n. If δ(µ) = 0 the argument shows that µ = ν 0 * γ, where ν 0 is the Dirac point mass at E[X 1 ]. Proof of Theorem 8. The starting point of the proof is identity (18): X 1 = 1 0 cov(µ t )dB t . The idea is then to extract from this identity two processes (Y t ), (Z t ) close to each other in transportation distance such that Z 1 ∼ γ. We then write X 1 = Y 1 + W for some random vector W and show that E[ Y, W ] = 0. The requirement Z 1 ∼ γ is enforced by ensuring that the quadratic variation of (Z t ) satisfies [Z] 1 = Id n . We start with some notation. Let M be an n×n symmetric matrix and let M = n i=1 κ i u i ⊗u i be its eigenvalue decomposition. We then set M + := n i=1 max(κ i , 0) u i ⊗ u i and similarly max(M, Id n ) = n i=1 max(κ i , 1) u i ⊗ u i . Using Theorem 3 together with the fact that 1 x − 1 + log(x) ≥ 1 2 (x − 1) 2 for all x ∈ (0, 1], we conclude that for every measure ν, one has δ(ν) ≥ 1 2 Tr (Id n − cov(ν)) 2 + . Using this bound and inequality (19) we get, δ(µ) ≥ 1 2 E 1 0 Tr (Id n − cov(µ t )) 2 + dt .(23) Next we will write the right-hand side above as roughly the difference in transportation distance between the random vectors Y 1 and Z 1 mentioned above. For convenience, define A t := cov(µ t ). We now define a random process (C t ) taking values in the set of symmetric matrices as follows. Set C 0 = 0 and dC t = max(A 2 t , Id n ) dt, t ∈ [0, τ 1 ) where τ 1 is the first time the largest eigenvalue of C t hits the value 1. Notice that tId n C t on [0, τ 1 ), so τ 1 ≤ 1. If C τ 1 = Id n , which implies that τ 1 < 1, we let O 1 be the orthogonal projection onto the range of C τ 1 − Id n , and set dC t = O 1 max(A 2 t , Id n )O 1 dt, t ∈ [τ 1 , τ 2 ) where τ 2 is the first time the largest eigenvalue of O 1 C t O 1 hits the value 1. If C τ 2 = Id n we let O 2 be the projection onto the range of C τ 2 − Id n and proceed similarly, and so on, until the first time τ k such that C τ k = Id n . On [τ k , 1] we let dC t = 0 and thus C t = Id n . To sum up, the matrix C t satisfies 0 C t Id n for all t ∈ [0, 1], C 1 = Id n and dC t = L t max(A 2 t , Id n )L t dt, where L t is the orthogonal projection onto the range of C t − Id n . Next, consider the processes (Y t ), (Z t ) defined by Y 0 = Z 0 = 0, dY t = L t A t dB t , dZ t = L t max (A t , Id n ) dB t . and note that [Z] t = t 0 L s max A 2 s , Id n L s ds = C t . This implies that [Z] 1 = Id n almost surely so Z 1 ∼ γ. On the other hand, we have by (23) and Itô's isometry, E[\|Y 1 − Z 1 \| 2 ] = E 1 0 Tr L t (max (A t , Id n ) − A t ) 2 L t dt ≤ E 1 0 Tr (max (A t , Id n ) − A t ) 2 dt = E 1 0 Tr (Id n − A t ) + 2 dt ≤ 2δ(µ). Letting ν be the law of Y 1 , we thus get W 2 2 (ν, γ) ≤ 2δ(µ). Now define the random vector W : (18), X 1 = Y 1 + W . It remains to show that E[ Y, W ] = 0. This is again a consequence of Itô's isometry: = 1 0 (A t − L t A t ) dB t so byE[ Y, W ] = E 1 0 Tr L t A t (A t − L t A t ) T dt = E 1 0 Tr L t A 2 t − L t A 2 t L t dt = 0 since L t = L 2 t . This completes the proof. Counterexamples to stability In this section we provide simple counterexamples to the stability of the logarithmic Sobolev inequality with respect to the Wasserstein distance, thus proving Theorem 4 as well as the third part of Theorem 3. The standard Gaussian on R is denoted by γ and for a ∈ R, s ≥ 0 we let γ a,s be the Gaussian centered at a with variance s. Our counterexamples are nothing more than Gaussian mixtures. For such measures, the following two lemmas provide a lower bound on the Wasserstein p-distance to translated Gaussians, and an upper bound on the log-Sobolev deficit. The combination of these two lemmas will prove Theorems 4 and 3. We start with the upper bound on the log-Sobolev deficit. Lemma 14. Let a, b ∈ R, and σ, t ∈ [0, 1]. Then δ ((1 − t)γ a,σ + tγ b,σ ) ≤ 1 4 σ −1 − 1 2 − (1 − t) log(1 − t) − t log t. Proof. Let ϕ(t) = t log t + (1 − t) log(1 − t) and µ, ν be probability measures on R. The lemma follows immediately by combining the estimates δ ((1 − t)µ + tν) ≤ (1 − t)δ (µ) + tδ (ν) − ϕ(t)(24) and δ(γ 0,σ ) ≤ 1 4 σ −1 − 1 2 .(25) (when σ ≤ 1) and using the fact that δ is invariant under translations. The validity of (24) follows immediately from the combination of the convexity estimates I ((1 − t)µ + tν \| γ) ≤ (1 − t)I(µ \| γ) + tI(ν \| γ) and H ((1 − t)µ + tν \| γ) ≥ (1 − t)H(µ \| γ) + tH(ν \| γ) + ϕ(t). The convexity of the Fisher information is a well-known fact, it is a direct consequence of the convexity of the map (x, y) → y 2 /x on (0, ∞) × R. For the second inequality, let f and g be the respective densities of µ and ν with respect to γ and use the fact that the logarithm is increasing to write ((1 − t)f + tg) log ((1 − t)f + tg) ≥ (1 − t)f log ((1 − t)f ) + tg log(tg). Integrating with respect to γ yields the result. For the estimate (25), a direct computation shows that H(γ 0,σ \| γ) = 1 2 (σ − 1 − log σ) and I(γ 0,σ \| γ) = (σ − 1) 2 /σ, so that δ(γ 0,σ ) = 1 2 σ −1 − 1 + log(σ) . We conclude using x − 1 − log x ≤ (x − 1) 2 /2 for x ≥ 1. Proof of Proposition 5. When σ = 1, Lemma 14 can be rewritten δ(p * γ) ≤ S(p) for any probability measure p in R which is a combination of two Dirac point masses. The argument can easily be generalized to any discrete probability measure p, and to any dimension, proving Proposition 5. Proof of the third part of Theorem 1. Note that var((1 − t)γ a,1 + tγ b,1 ) = 1 + t(1 − t)(b − a) 2 . Set µ k = 1 − 1 k γ 0,1 + 1 k γ k 2 ,1 . Then var(µ k ) → ∞. On the other hand δ(µ k ) → 0 by Lemma 14. Next we move on to the lower bound on the Wasserstein distance. Lemma 15. Let a, b ∈ R, σ ∈ (0, 1], t ∈ [0, 1] and let µ = (1 − t)γ a,σ + tγ b,σ . Suppose that min(t, 1 − t) ≥ 2 exp − (b − a) 2 32 .(26) Then, for every p ≥ 1 inf m∈R W p p (µ, γ m,1 ) ≥ min(t, 1 − t) \|b − a\| p 4 p+1 . Proof. Let m ∈ R and suppose without loss of generality that \|a − m\| ≤ \|b − m\| and that b > a. Define z = m + 2 log 2 t and note that the assumption (26) Therefore, in order to transport γ m,1 to µ, at least t/4 unit of mass to the left of z should move to the right of b. As a result W p p (µ, γ m,1 ) ≥ t 4 (b − z) p ≥ t 4 \|b − a\| 4 p , which yields the result. Proof of Theorem 4. For the first part of the theorem we shall work in dimension 1 but the result extends to any dimension by taking the tensor product of the one dimensional example by a standard Gaussian. Consider the sequence of measures (µ k ) given by µ k = 1 − 1 k γ 0,1 + 1 k γ k 2 ,1 . Lemmas 14 and 15 imply that δ(µ k ) → 0 and W 1 (µ k , γ) → ∞. 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[ "Photonic Properties of Metallic-Mean Quasiperiodic Chains", "Photonic Properties of Metallic-Mean Quasiperiodic Chains" ]	[ "Stefanie Thiem \nInstitut für Physik\nTechnische Universität Chemnitz\nD-09107ChemnitzGermany\n", "Michael Schreiber \nInstitut für Physik\nTechnische Universität Chemnitz\nD-09107ChemnitzGermany\n" ]	[ "Institut für Physik\nTechnische Universität Chemnitz\nD-09107ChemnitzGermany", "Institut für Physik\nTechnische Universität Chemnitz\nD-09107ChemnitzGermany" ]	[]	The light propagation through a stack of two media with different refractive indices, which are aligned according to different quasiperiodic sequences determined by metallic means, is studied using the transfer matrix method. The focus lies on the investigation of the influence of the underlying quasiperiodic sequence as well as the dependence of the transmission on the frequency, the incidence angle of the light wave and different ratios of the refractive indices. In contrast to a periodically aligned stack we find complete transmission for the quasiperiodic systems for a wide range of different refraction indices for small incidence angles. Additional bands of moderate transmission occur for frequencies in the range of the photonic band gaps of the periodic system. Further, for fixed indices of refraction we find a range of almost perfect transmission for angles close to the angle of total reflection, which is caused by the bending of photonic transmission bands towards higher frequencies for increasing incidence angles. Comparing with the results of a periodic stack the quasiperiodicity seems to have only an influence in the region around the midgap frequency of a periodic stack.	10.1140/epjb/e2010-00226-y	[ "https://arxiv.org/pdf/1212.6605v1.pdf" ]	118,654,841	1212.6605	61fe959bcf81466ede7521d15f9a1b71c638ce37	Photonic Properties of Metallic-Mean Quasiperiodic Chains Stefanie Thiem Institut für Physik Technische Universität Chemnitz D-09107ChemnitzGermany Michael Schreiber Institut für Physik Technische Universität Chemnitz D-09107ChemnitzGermany Photonic Properties of Metallic-Mean Quasiperiodic Chains EPJ manuscript No. (will be inserted by the editor) the date of receipt and acceptance should be inserted later The light propagation through a stack of two media with different refractive indices, which are aligned according to different quasiperiodic sequences determined by metallic means, is studied using the transfer matrix method. The focus lies on the investigation of the influence of the underlying quasiperiodic sequence as well as the dependence of the transmission on the frequency, the incidence angle of the light wave and different ratios of the refractive indices. In contrast to a periodically aligned stack we find complete transmission for the quasiperiodic systems for a wide range of different refraction indices for small incidence angles. Additional bands of moderate transmission occur for frequencies in the range of the photonic band gaps of the periodic system. Further, for fixed indices of refraction we find a range of almost perfect transmission for angles close to the angle of total reflection, which is caused by the bending of photonic transmission bands towards higher frequencies for increasing incidence angles. Comparing with the results of a periodic stack the quasiperiodicity seems to have only an influence in the region around the midgap frequency of a periodic stack. Introduction Since the discovery of quasicrystals by Shechtman et al. in 1984 [1], many efforts have been conducted to understand the physical properties of these aperiodic materials, which possess a long range order without having a translational symmetry. Hence, quasicrystals are regarded to have a degree of order intermediate between crystals and disordered systems. Quasicrystalline systems have been extensively studied e.g. with respect to their structure, which shows uncommon rotational symmetries [1,2,3], and their electronic states, which were found to show a Cantor-set spectrum in one dimension [4,5,6,7]; but also phonons and magnetic properties of these materials have been investigated [8,9,10,11,12]. Further, the photonic properties are of special interest because the complex symmetries in quasicrystals make them suitable for the application in several optical devices such as single-mode light-emitting diodes, polarization switching and microelectronic devices that are based on photons rather than on electrons, which potentially can be the electromagnetic analogue to semiconductors [13,14,15,16,17]. For the theoretical study of quasiperiodic systems one often applies the concept of aperiodic mathematical sequences / tilings. Especially, the photonic properties of one-dimensional systems have been extensively analyzed with this approach, not only because transfer matrix methods can be applied but also because such systems can be relatively easily produced in reality. There are examples based upon the Fibonacci sequence, the Thue-Morse sea e-mail: [email protected] quence or Cantor sequences [14,18,19,20], and also systems for negative refractive indices have been studied [21,22,23]. Further, one can find a close resemblance of the theoretical and the experimental results [24,25]. However, most of the investigations do not deal with the angle of incidence, but rather assume that the incident light wave is perpendicular to the layers of the stack, and do not investigate the influence of the structure of the quasiperiodic sequence on the transmission. In this paper we discuss the photonic properties of quasiperiodic systems consisting of a sequence of layers made of two different refractive indices n A and n B (cp. Figure 1), where the configuration of the layers is given according to the so-called metallic-mean sequences. In particular, we focus on the transmission of the light wave in dependence on the incidence angle ϑ and investigate the influence of the underlying construction rule for different metallic-mean quasicrystals on the photonic properties as well. We find that additional transmission bands occur for frequencies corresponding to the photonic band gap of the periodic system. Further, for small incident angles we obtain almost complete transmission for the quasiperiodic systems for a wide range of ratios of refractive indices n A /n B in contrast to a system with periodically stacked layers, and for angles close to the angle of total reflection there is also a region of relatively high transmission. The configuration of the layers of these so-called metallic-mean quasicrystals can be constructed by the inflation rule Fig. 1. Transmission of light through a quasiperiodic stack consisting of the materials A and B with different indices of refraction. The light wave E 0 r is incident on the surface with an angle ϑ. Parts of the light wave are transmitted through the stack (E N r ) and other parts are reflected (E 0 l ). P a = B → A A → ABA a−1 ,(1)C A B A A B A B C E N r E 0 r ϑ j = N j = 0 E 0 l iterated m times starting from the symbol B. For instance, this yields for the parameters a = 1, 2, 3 the approximants B P1 −→ A P1 −→ AB P1 −→ ABA P1 −→ ABAAB P1 −→ . . . B P2 −→ A P2 −→ ABA P2 −→ ABAAABA P2 −→ . . . (2) B P3 −→ A P3 −→ ABAA P3 −→ ABAAAABAAABAA P3 −→ . . . . We refer to the resulting sequence after m iterations as the mth order approximant C a m with the length f m given by the recursive equation f m = f m−2 + af m−1 and f 0 = f 1 = 1. Depending on the parameter a the inflation rule generates different metallic means, i.e., the lengths of two successive sequences satisfy the relation lim m→∞ f m f m−1 = τ (a) ,(3) where τ (a) is an irrational number with the continued fraction representation τ (a) = [ā] = [a, a, a, ...]. Also, the number f A m of layers A and number f B m of layers B in an approximant C a m are related by these metallic means according to lim m→∞ f A m f B m = τ (a) .(4) For example, a = 1 yields the well-known Fibonacci sequence related to the golden mean τ Au = [1] = (1+ √ 5)/2, the case a = 2 corresponds to the octonacci sequence with the silver mean τ Ag = [2] = 1 + √ 2, and for a = 3 one obtains the bronze mean τ Bz = [3] = (3 + √ 13)/2 [20,26]. In general the relation τ (a) = (a + √ a 2 + 4)/2 holds [27]. Due to the recursive inflation rule (1), these quasiperiodic chains possess a hierarchical structure, which is more clearly visible by using the alternative construction rule C a m = C a m−1 C a m−2 (C a m−1 ) a−1 ,(5) yielding the same quasiperiodic sequences for m ≥ 2 setting C a 0 = B and C a 1 = A. Likewise, there is a corresponding expression for the recursive construction of the transfer matrix as given in Sec. 2. The outline of this paper is as follows: In Sec. 2 we give an introduction to the transfer matrix method used for the calculations and in Sec. 3 we present our results and discuss them. This is followed by a brief conclusion in Sec. 4. Transfer Matrix Method The propagation of light through a layered system as shown in Figure 1 is commonly investigated by means of transfer matrix methods [18,19,21,23]. Regarding the geometry we consider the propagation of linearly polarized light with the electric field perpendicular to the plane of the light path (transverse electric waves) [28]. Within one layer the electric field E = Eê y can be described as a superposition of a right and a left traveling plane wave E = E j r e ikj x−iωt + E j l e −ikj x−iωt ,(6) where k j = n j k denotes the wave number of the light in the jth layer made either of medium A or B with refractive index n j , k denotes the wave number in the vacuum and ω denotes the frequency of the light. Note, that each amplitude E j r and E j l actually represents the resultant of all possible waves traveling in the particular direction. The propagation of the waves through the stack is given on the one hand by their refraction at the interfaces between the layers and on the other hand by their propagation through the layers. Addressing the first point, the boundary conditions at the interfaces between the layers require the tangential component of the electric field E and the magnetic field H = n ck × E to be continuous across the boundaries. This yields for an incidence angle ϑ α , the emergence angle ϑ β , and the corresponding indices of refraction n α and n β the relations (cp. Figure 2) E j r + E j l = E j+1 r + E j+1 l (7) (E j r − E j l )n α cos ϑ α = (E j+1 r − E j+1 l )n β cos ϑ β .(8) Applying the notation used by Kohmoto et al. [18,19] with the variables E + = E r + E l and E − = −i (E r − E l ) ,(9) the transfer of the light wave from medium α to medium β can be described by E + E − j = T αβ E + E − j+1(10) with the interface matrix T αβ = T −1 βα = 1 0 0 n β cos ϑ β nα cos ϑα .(11) Thereby, the incidence and emergence angles are related by Snell's law [18] sin ϑ α sin ϑ β = n β n α . Fig. 2. Electric and magnetic field vectors E and H at the interface between layer j and j + 1 with different indices of refraction n α and n β . (12) y x z k j r ϑ α H j+1 r k j+1 r ϑ β n α n β E j+1 r E j l H j l H j r H j+1 l k j l E j+1 l k j+1 l E j r On the other hand, the propagation of the light waves within the layers results in a phase difference, which is comprised by the propagation matrix T γ = cos ϕ γ sin ϕ γ − sin ϕ γ cos ϕ γ .(13) The phase difference for the wave length λ = 2π/k of the light and a layer thickness d γ amounts to (cp. Figure 3) [28] ϕ γ = k[n γ (OP + P Q) − n α OR] = 4π λ d γ n γ cos ϑ γ . (14) Note, that the paper by Kohmoto et al. [18] and some other related papers as e.g. [20,25] only state the phase shift, which the waves undergo by traversing the stack, and not the phase difference between adjacent waves. The transfer matrix E + E − 0 = M C a m E + E − N(15) of the system is then given as a combination of the different interface and propagation matrices according to a certain quasiperiodic sequence. Thereby, the amplitude of the incident light is denoted by E 0 r , of the reflected light by E 0 l and of the transmitted light by E N r (cp. Figure 1). For instance for the Fibonacci sequence ABAAB . . . B one obtains M C Au = T CA T A T AB T B T BA T A T A T AB T B . . . T B T BA T AC .(16) In general the recursive equation M C a m = M C a m−1 M C a m−2 {M C a m−1 } a−1(17) is applicable for m ≥ 2 with M C a 0 = T AB T B T BA and M C a 1 = T A , which has the same structure as the inflation rule of equation (5). Here we assume that either the surrounding medium has the same index of refraction as medium A or one has to add the corresponding transfer matrices T CA and T AC at the edges, respectively. In particular, we are interested in the calculation of the transmission coefficient T (also known as transmittance) of the light through the stack, which is defined as T = \|E N r \| 2 /\|E 0 r \| 2 and can be derived from equation (15) by eliminating E 0 l . Assuming that there is no incident light from the right, i.e., E N l = 0, one obtains T = 4 \|M \| 2 + 2 det M ,(18) where \|M \| 2 corresponds to the sum of the squares of all four matrix elements of M . Further, for the complete stack it can be shown that det M = 1, which simplifies the expression for the transmission coefficient T . Results In this section we comprise the results for several metallicmean quasicrystals. Especially, we focus on the change of the transmission coefficient T in dependence on the used inflation rule P and the incidence angle ϑ. For the choice of the wave length of the incident light λ 0 , we use the commonly applied quarter wave length condition n A d A = n B d B = λ 0 /4, which leads for an incidence angle ϑ = 0 to an identical optical wave path for the light in the two materials A and B of the stack [23,24,25]. Hence, equation (14) results in ϕ γ = λ 0 λ π cos ϑ γ = ω ω 0 π cos ϑ γ .(19) In Figure 4 the results for the transmission coefficients T (ω) are shown for different approximants of the octonacci chain C Ag m for an incidence angle ϑ = 0. Due to the structure of the interface and propagation matrices only the ratio of the indices of refraction u = n A /n B is important. Here we choose a fixed ratio u = 2/3, and n C = n A for the surrounding medium. Results are only displayed in the interval [0, ω/ω 0 ] because the pattern of the transmittance T repeats every ω/ω 0 for ϑ = 0 (cp. Figure 5). In general, the transmission spectrum of metallic-mean sequences or Cantor sequences is assumed to show either complete transmission or complete reflectance for infinite systems [19,29] similar to the singular continuous energy spectrum of quasiperiodic systems [4,6,7]. This characteristic becomes more and more visible for larger system sizes f m . However, especially for small approximants there still occurs a significant transmittance T (ω) > 0 in the photonic band gaps since the waves can tunnel through the stack to a certain extent. Likewise, T (ω) = 1 is often not reached within the bands. When varying the incidence angle, we have to keep in mind that total reflection occurs for n B < n A at the incidence angles ϑ > ϑ total = arcsin u −1 and hence we can only expect to obtain transmission coefficients for incidence angles ϑ < ϑ total . The transmittance T (ϑ, ω) in dependence on the reduced frequency ω/ω 0 and the incidence angle ϑ < ϑ total is shown in Figure 5 and Figure 6 for different metallicmean quasicrystals. Figure 5 displays the transmission for the 8th approximant of the octonacci sequence for a broad range of frequencies for two examples corresponding to the cases n A < n B and n B > n A . As mentioned, for ϑ = 0 the transmission spectrum is periodic with respect to the frequency. However, for all angles ϑ > 0 the photonic transmission bands bend towards higher frequencies for increasing incidence angles and this effect becomes stronger for higher frequencies. Further, for u > 1 one obtains a relatively large range of frequencies with nearly complete transmission for large incidence angles and for u < 1 there are certain frequencies with nearly complete transmission over a large range of angles due to the bending of the transmission bands. In Figure 6 the transmittance T (ϑ, ω) is displayed for different metallic-mean sequences. Comparing with the results obtained for a periodic alignment of the layers A and B in Figure 7, one can see that there is a typical photonic band gap at ω = ω 0 /2 for the periodic chain for ϑ = 0. Also for ω/ω 0 = b + 1/2 (b ∈ N 0 ) such gaps occur and with increasing incidence angles these photonic band gaps bend also towards higher frequencies. We denote the corresponding frequencies of these photonic band gaps as midgap frequencies. For the quasiperiodic sequences this band gap appears to be much wider, but is intersected by an increasing number of new lines of moderately high transmission for increasing a of the inflation rule. For ω/ω 0 = b (b ∈ N 0 ) the characteristics of the transmittance T (ϑ, ω) hardly change for the different construction rules. The behavior at the photonic band gap is in consistency with the observation that for ω = ω 0 /2 (ϑ = 0) the quasiperiodicity is most effective [30]. For u > 1 the narrow transmission bands caused by the quasiperiodicity do not bend as strongly as the transmission bands already present for the periodic case. Therefore, for large incidence angles ϑ the bands corresponding to the periodic case intersect the photonic band gaps and complete transmission occurs. In this case the transmission spectrum for the quasiperiodic and the periodic case look quite similar. For u < 1 all bands bend nearly equally strong but also form regions of almost complete transmission at certain frequencies for a large range of incidence angles (cp. Figure 5(b)). Additionally, in both cases we obtain that the additional transmission bands are more uniformly distributed with increasing a of the inflation rule and hence the photonic band gaps become smaller. The dependency of the transmission bands on the incidence angle can be explained by the difference of the optical light paths in the layers A and B for ϑ > 0. In this case we expect that the transmission coefficients T (ω, ϑ = 0) and T (ω , ϑ > 0) yield almost the same results if the overall phase difference for the stack is equal. At first, we consider only the behavior arising from the change of the phase difference within one layer. Hence, for layer B the relation ϕ B (ω, ϑ = 0) ! = ϕ B (ω , ϑ B > 0)(20) has to be fulfilled, which yields a behavior according to ω (ϑ) (19) = ω(ϑ = 0) cos ϑ B = ω(ϑ = 0) cos (arcsin (u sin ϑ)) . Note that for n A = n C the relation ϑ A = ϑ holds. Likewise, the relation for layer A can be derived with ω (ϑ) (19) = ω(ϑ = 0) cos ϑ . However, the actual bending of the transmission bands is given as a superposition of equations (21) and (22). For a periodic stack with the same number of layers A and B the total phase difference ϕ = (ϕ A + ϕ B )f m /2 results in a dependency ω (ϑ) = 2ω(ϑ = 0) cos ϑ + cos (arcsin (u sin ϑ)) . (23) respectively. Plot (c) shows these limit functions as well but additionally the bending for quasiperiodic sequences in dependency on the metallic mean τ according to equation (25). The Moiré patterns in (a) and (b) are caused by strong oscillations of the transmittance T , which are not adequately displayed due to the used resolution. The functions (21)- (23) are shown in comparison with the numerical results of the transmission in a periodic stack of layers A and B in Figure 7(a) and (b). The functions (23) fit the bending of the transmission bands quite well, especially for small angels. Further, we obtained that the regions of almost total transmission behave according to equation (21), which indicates that ϕ B plays a significant part for this behavior. In contradistinction the bending caused by ϕ A often coincides with regions of lower transmission coefficients. This effect becomes even more evident for systems with u 1 or u 1. For the quasiperiodic stacks the numbers of layers A and B are not identical but related to the metallic means τ by equation (4). The overall phase difference in the limit of infinite stacks is then given by ϕ = τ 1 + τ ϕ A + 1 1 + τ ϕ B f m(24) and yields a bending according to ω (ϑ) = (1 + τ )ω(ϑ = 0) τ cos ϑ + cos (arcsin (u sin ϑ)) . These functions are shown in Figure 7(c) for the two cases u = 0.5 and u = 0.2. Again, we obtain a bending in between the behavior arising from one single layer of medium A or B. In particular, the bending increases with the parameter τ (respectively a) in the case u < 1 and shows the reverse behavior for u > 1. Further, for the quasiperiodic systems the bending according to equation (25) approaches for τ → 1 the bending of a periodic stack and for τ 1 the behavior arising from one single layer B. Comparing the functions (25) with the numerical results of the transmission for the quasiperiodic layered systems with different values of τ (cp. Figure 6), we find a very good resemblance of the bending behavior for small and intermediate angles. In Figure 8 the characteristics of the transmittance T are investigated with respect to the change of the ratio of the indices of refraction u. In Figure 8(a)-(c) we study the influence of the reduced frequency ω/ω 0 and in Figure 8(d)-(f) the change of the transmittance T with respect to the incidence angle ϑ for the periodic, the golden mean and the silver mean sequence. In the first case, i.e., for the variation of the frequency, it becomes again clearly visible that the quasiperiodicity has the largest effect for ω ≈ ω 0 /2. In this region the transmittance T (ω, u) strongly varies with the used construction rule. By increasing the parameter a, more and more bands appear. In the periodic case complete transmission occurs for this frequency only for u = 1 , whereas for the quasiperiodic systems there are several bands in this region with a high transmission coefficient. Of course, complete transmission occurs always for u = 1. The results for the investigation of the influence of the incidence angle ϑ for different refraction coefficients are shown in Figure 8(d)-(f). A comparison of the plots for the periodic and the metallic-mean sequences shows that there is a similar structure of the transmittance T (ϑ, u) near the regions of total reflection. However, for small incidence angles there is a completely different behavior, i.e., we have zero transmission for u = 1 in the periodic case, whereas for the quasiperiodic sequences complete transmission occurs for a significant range of ratios u. Conclusion We investigated the transmission of light through a stack of quasiperiodically aligned materials with respect to the underlying inflation rule and for different incidence angles. One main result for the quasiperiodic systems is that moderately high or even complete transmission can occur for certain ranges of parameters, for which no transmission is found for the periodic system: e.g. we obtained additional bands of moderate transmission for frequencies near the midgap frequency (b + 1/2)ω 0 (b ∈ N 0 ) with the number of bands increasing depending on the parameter a of the inflation rule, and for a whole range of ratios of refraction indices u even complete transmission occurs. Further, for all considered systems almost complete transmission occurs for incidence angles ϑ close to the angle of total reflection. This is caused by the bending of the transmission bands due to the different optical light paths in mediums A and B. Additionally, the quasiperiodicity shows only an influence on the transmission coefficients around the region of the midgap frequencies (b + 1/2)ω 0 , whereas in between the patterns appear to be similar for different quasiperiodic generation rules as well as for the periodic system. Fig. 3 . 3Phase difference between two waves originating from traversing a layer with index of refraction n γ and thickness d γ . Fig. 4 . 4Transmittance T (ω) for different approximants of the octonacci chain with incidence angle ϑ = 0 and indices of refraction n A = n C and u = n A /n B = 2/3: (a) C Ag 5 with f 5 = 41 layers, (b) C Ag 7 with f 7 = 239 layers, (c) C Ag 9 with f 9 = 1393 layers, and (d) C Ag 11 with f 11 = 8119 layers. Fig. 5 . 5Transmittance T (ϑ, ω) of light through a stack of the materials A and B, which are arranged according to the octonacci sequence C Ag 8 with f 8 = 577 layers. Results are shown in dependence on the reduced frequency ω/ω 0 versus the incidence angle ϑ for n A = n C and (a) u = 3/2 and (b) u = 2/3. Fig. 6 . 6Transmittance T (ϑ, ω) for different quasiperiodic inflation rules: (a) the Fibonacci sequence C Au 14 with f 14 = 610 layers, (b) the octonacci sequence C Ag 8 with f 8 = 577 layers and (c) the bronze mean quasiperiodic sequence C Br 6 with f 6 = 469 layers. The red lines show the expected limit behavior of the bending according to equation(25). The ratio of the indices of refraction is u = 2. Fig. 7 . 7Transmittance T (ϑ, ω) for a periodic sequence of alternating layers ABAB . . . with length 500 for the ratios of the indices of refraction (a) u = 2 as inFigure 6and (b) u = 0.5. 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[ "Quantum search on graphene lattices", "Quantum search on graphene lattices" ]	[ "Iain Foulger \nSchool of Mathematical Sciences\nUniversity of Nottingham\nNG7 2RDUniversity Park, NottinghamUK\n", "Sven Gnutzmann \nSchool of Mathematical Sciences\nUniversity of Nottingham\nNG7 2RDUniversity Park, NottinghamUK\n", "Gregor Tanner \nSchool of Mathematical Sciences\nUniversity of Nottingham\nNG7 2RDUniversity Park, NottinghamUK\n" ]	[ "School of Mathematical Sciences\nUniversity of Nottingham\nNG7 2RDUniversity Park, NottinghamUK", "School of Mathematical Sciences\nUniversity of Nottingham\nNG7 2RDUniversity Park, NottinghamUK", "School of Mathematical Sciences\nUniversity of Nottingham\nNG7 2RDUniversity Park, NottinghamUK" ]	[]	We present a continuous-time quantum search algorithm on a graphene lattice. This provides the sought-after implementation of an efficient continuous-time quantum search on a two-dimensional lattice. The search uses the linearity of the dispersion relation near the Dirac point and can find a marked site on a graphene lattice faster than the corresponding classical search. The algorithm can also be used for state transfer and communication.	10.1103/physrevlett.112.070504	[ "https://arxiv.org/pdf/1312.3852v1.pdf" ]	19,703,038	1312.3852	5ae427ea74f8ce8dd7c05c5baeed92f1a106e86a	Quantum search on graphene lattices (Dated: December 18, 2013) Iain Foulger School of Mathematical Sciences University of Nottingham NG7 2RDUniversity Park, NottinghamUK Sven Gnutzmann School of Mathematical Sciences University of Nottingham NG7 2RDUniversity Park, NottinghamUK Gregor Tanner School of Mathematical Sciences University of Nottingham NG7 2RDUniversity Park, NottinghamUK Quantum search on graphene lattices (Dated: December 18, 2013)numbers: 0367Hk0365Sq0367Lx7280Vp We present a continuous-time quantum search algorithm on a graphene lattice. This provides the sought-after implementation of an efficient continuous-time quantum search on a two-dimensional lattice. The search uses the linearity of the dispersion relation near the Dirac point and can find a marked site on a graphene lattice faster than the corresponding classical search. The algorithm can also be used for state transfer and communication. Introduction.-Quantum walks [1,2] can provide polynomial and even exponential speed-up compared to classical random walks [3][4][5][6] and may serve as a universal computational primitive for quantum computation [7]. This has led to substantial interest in the theoretical aspects of this phenomenon, as well as in finding experimental implementations [8][9][10][11][12][13]. One of the most fascinating applications of quantum walks is their use in spatial quantum search algorithms first published for the search on the hypercube in [14]. Like Grover's search algorithm [15,16] for searching an unstructured database, quantum walk search algorithms can achieve up to quadratic speed-up compared to the corresponding classical search. For quantum searches on d-dimensional square lattices, certain restrictions have been observed, however, depending on whether the underlying quantum walk is discrete [3] or continuous [17]. While effective search algorithms for discrete walks have been reported for d ≥ 2 [18,19], continuous-time quantum search algorithms on square lattices show speed-up compared to the classical search only for d ≥ 4 [20]. This problem has been circumvented in [21], however, at the conceptual cost of adding internal degrees of freedom (spin) and a discrete Dirac equation. Experimental implementations of discrete quantum walks need time stepping mechanisms such as laser pulses [8][9][10][11]13]. It is thus in general simpler to consider experimental realizations with continuous time evolution. However, in the absence of internal degrees of freedom, no known search algorithm on lattices exists up to now in the physically relevant regime d = 2 or 3. Finding such an algorithm is highly topical due to applications in secure state transfer and communication across regular lattices as demonstrated in [22]. We will show in the following that continuous-time quantum search in 2D is indeed possible! We will demonstrate that such a quantum search can be performed at the Dirac point in graphene. This is potentially of great interest, as graphene is now becoming available cheaply and can be fabricated routinely [23,24]. Performing quantum search and quantum state transfer on graphene provides a new way of channeling energy and information across lattices and between distinct sites. Graphene sheets have been identified as a potential single-molecule sensor [26,27] being very sensitive to a change of the density of states near the Dirac point. This property is closely related to the quantum search effect described in this paper. Continuous-time quantum search algorithms take place on a lattice with a set of N sites interacting via hopping potentials (usually between nearest neighbors only). Standard searches work at the ground state energy which, due to the periodicity of the lattice, is related to quasimomentum k = 0. After introducing a perturbation at one of the lattice sites, the parameters are adjusted such that an avoided crossing between the localized 'defect' state and the ground state is formed. The search is now performed in this two-level sub-system [28]. Criticality with respect to the dimension is reached when the gap at the avoided crossing and the eigenenergy spacing near the crossing scale in the same way with N . Continuous-time quantum walks (CTQW) [17] operate in the position (site) space. If the states \|j represent the sites of the lattice, the Schrödinger equation governing the probability amplitudes α j (t) = j\| ψ(t) is given by d dt α j (t) = −i N l=1 H jl α l (t)(1) where the Hamiltonian H = D I + vA is of tight-binding type where A is the adjacency matrix of the lattice and I is the identity matrix, D is the on-site energy and v is the strength of the hopping potential. In [20], the walk Hamiltonian was set to be the discrete Laplacian where v = −1 and D is the coordination number of the lattice. A marked site is then introduced by altering the on-site energy of that site. The system is initialized at t = 0 in the ground state of the unperturbed lattice leading to an effective search for d ≥ 4. For the search based on the discretized Dirac operator [21], an additional spin degree of freedom is introduced. This gives optimum search times for lattices with dimension d ≥ 3 and a search time of O( √ N ln N ) for d = 2 recovering the results for discrete time walks [18]. We note that d = 2 is the critical dimension in the discrete case independent of the lattice structure; one thus finds an O( √ N ln N ) also for discrete time walks on graphene [19]. The lack of speed-up for continuous search algorithms in two dimensions can be overcome by making two adjustments: i. the avoided crossing on which the search operates is moved to a part of the spectrum with a linear dispersion relation; ii. the local perturbation is altered in order to couple a localized perturber state and the lattice state in the linear regime. The first point is addressed by considering graphene lattices with the well-known linear dispersion curves near the Dirac point. The perturbation at the marked site is achieved by locally changing the hopping potential (instead of changing the on-site energy as in [20,21]). We start by giving an introductory account of basic properties of the graphene lattice and its band structure [23,24]. Review of graphene.-The graphene (or honeycomb) lattice is bipartite with two triangular sublattices, labeled A and B. The position of a cell in the lattice is denoted by R = αa 1 + βa 2 where α and β are integers and a 1 (2) are basis vectors of the lattice (see Fig. 1). States on the two sites within one cell will be denoted by \|R A(B) ≡ \|α, β A(B) . The corners of the Brillouin zone (see Fig. 1) are denoted K and K and the primitive cell contains two of these points. The solution for the tight-binding Hamiltonian on graphene as described above is well-known [23,24] and leads to the dispersion relation (k) = D ±(2) v 1 + 4 cos 2 k x a 2 + 4 cos k x a 2 cos √ 3k y a 2 depicted in Fig. 2 for an infinite graphene lattice. It is indeed linear near the Dirac points K and K at the energy D where the conduction and valence bands meet. Around the Dirac points the dispersion relation (k) can be approximated by (k) ≈ D ± va √ 3 2 δk 2 x + δk 2 y = D ± va √ 3 2 \|δk\| . (3) In the following, we will consider finite graphene lattices with periodic boundary conditions, i.e. \|Ψ = m α=1 n β=1 ψ A α,β \|α, β A + ψ B α,β \|α, β B with ψ A(B) α,β = ψ A(B) α+m,β = ψ A(B) α,β+n . This simplifies the analysis allowing us to focus on the relevant features of the search by avoiding boundary effects. The general description does not change for other boundary conditions, the localization amplitude on the marked site, however, becomes site dependent in a non-trivial way. Understanding this dependency is not essential in the context of this paper. We denote S = ma 1 + na 2 the vector describing the spatial dimensions of the lattice. Using Bloch's theorem [25], the momentum is quantized as k x = 2πp ma k y = 1 √ 3 4πq na − k x(4) where p ∈ {0, 1, . . . m − 1} , q ∈ {0, 1, . . . n − 1} and the spectrum (2) becomes discrete. In what follows, for simplicity, we have assumed that our lattice is square in the number of cells, that is, that m = n = N 2 . Four-fold degenerate states with energy D and wave numbers exactly on the Dirac points K and K exist if m and n are some multiples of 3. We assume this in the following for simplicity. In the general case, one needs to consider the states closest to the Dirac energy which gives a more complex theoretical analysis while essential signatures do not change. Quantum search.-Setting up a continuous-time search by changing the on-site energy of the marked site as done in [20] does not work for graphene. Using the ground state as the starting state fails for the same reason as it fails for rectangular lattices in d = 2 or 3 as the dispersion relation is quadratic near the ground state, see Fig. 2. Alternatively, moving the search to the Dirac point implies constructing an avoided crossing between a localized perturber state and a Dirac state. As the Dirac energy coincides with the on-site energy D , this leads to the condition, that the on-site energy perturbation must vanish at the crossing, which brings us back to the unperturbed lattice. We therefore mark a given site by changing the hopping potentials between the site and its nearest neighbors. Focusing on a symmetric choice of the perturbation and setting D = 0 for convenience, we obtain the (search-) Hamiltonian H γ = −γA + W.(5) Here, W denotes the perturbation changing the hopping potential to and from the marked site (α 0 , β 0 ) A which has been chosen to be on the A lattice, that is, W = √ 3 \|α 0 , β 0 A \| + √ 3 \| α 0 , β 0 \| A .(6) The state \| denotes the symmetric superposition of the three neighbors of the marked site, that is, \| = 1 √ 3 \|α 0 , β 0 B + \|α 0 , β 0 − 1 B + \|α 0 + 1, β 0 − 1 B . (7) At γ = 1, the perturbation corresponds to a hopping potential v = 0 between the site (α 0 , β 0 ) A and its neighbors, effectively removing the site from the lattice. It is this perturbation strength which is important in the following. Experimentally, such a perturbation is similar to graphene lattices with atomic vacancies as they occur naturally in the production process [29]; in microwave analogs of graphene as discussed in [30], this can be realized by removing single sites from the lattice. The effect of marking (or perturbing) the graphene Hamiltonian can be seen numerically in the parametric behavior of the spectrum of H γ as a function of γ, see Fig. 3 for the case n = m = 12. Note that W is a rank two perturbation which creates two perturber states. These states start to interact with the spectrum of the unperturbed graphene lattice from γ ≈ 0.5 onwards working their way through to a central avoided crossing at γ = 1, E = 0. Below we will show, how the avoided crossing can be used for searching; note, that the parameter dependence of the avoided crossing (γ = 1) is evident from the tight-binding Hamiltonian H in (5). In a realistic set-up, the perturbation needs to be fine-tuned in general to be in resonance with an eigenstate of the (unperturbed) system near the Dirac point. At the avoided crossing there are altogether six states close to the Dirac energy: the two perturber states and the four degenerate Dirac states \|K A(B) = 2 N α,β e i 2π 3 (α+2β+2σ) \|α, β A(B) \|K A(B) = 2 N α,β e i 2π 3 (2α+β) \|α, β A(B)(8) where σ = 1 (σ = 0) for states on the B (A) lattice and N = 2nm is the number of sites in the lattice. One finds directly W \|K B = W \|K B = 0, that is, Dirac states on the B lattice do not interact with an A-type perturbation for all γ. Furthermore, at γ = 1, the marked state \|α 0 , β 0 A is an eigenvector of H γ=1 with eigenvalue E = 0 -the marked site is disconnected from the lattice. Thus, the avoided crossing involves only the two Dirac states \|K A , \|K A and one perturber state ˜ . Neglecting the interaction of the perturbation with the rest of the spectrum at the avoided crossing, we set ˜ ≈ \| (see H = 6 N   0 0 e −i 2π 3 (αo+2βo) 0 0 e −i 2π 3 (2αo+βo) e i 2π 3 (αo+2βo) e i 2π 3 (2αo+βo) 0  (9) with eigenvaluesẼ ± = ±2 3 N ,Ẽ 0 = 0, and eigenvectors ψ ± = 1 2 e −i 2π 3 (αo+2βo) \|K A + e −i 2π 3 (2αo+βo) \|K A ± √ 2 \|(10)ψ 0 = 1 √ 2 e −i 2π 3 (αo+2βo) \|K A − e −i 2π 3 (2αo+βo) \|K A .(11) For searching the marked site (α 0 , β 0 ) A , the system is initialized in a delocalized starting state involving a superposition of Dirac states. This state will then rotate into a state localized on the neighbors of the marked site. The search is initialized in the optimal starting state \|s = 1 √ 2 ψ + + ψ − (12) = e −i 2π 3 (α0+2β0) √ 2 \|K A + e −i 2π 3 (α0−β0) \|K A which still depends on the perturbed site. Lack of knowledge of (α 0 , β 0 ) leads, however, only to an N independent overhead, see the discussion below. Letting \|s evolve in time with the reduced Hamiltonian (9) we obtain \|ψ (t) = e −iHt \|s = 1 √ 2 e −iẼ+t ψ + + e −iẼ−t ψ − = cos Ẽ + t \|s − i sin Ẽ + t \| ,(13) that is, the system rotates from \|s to \| in time graphene. This, together with the linear dispersion relation near the Dirac point, where the spacing between successive eigenenergies scales like O(1/ √ N ), makes the search on this 2D lattice possible. In contrast to the algorithms described in [20,21], the system localizes here on the neighbors of the marked site, the marked site can be found by three additional direct queries. Furthermore, the initial starting state is not the uniform state here, but the state \|s in (12). To construct this initial state uniquely requires some information about the site that is being searched for. Without this knowledge, one has three possible optimal initial states for an A-type perturbation as can be seen from Eqn. (12). The same applies for marking a B-type site, so in total there are six possible optimal starting states. As these states are not orthogonal, this increases the number of runs for a successful search by a factor of 4. The additional overhead is independent of N, and thus does not alter the scaling with system size. In an experiment one may have little control about how the system is excited at the Dirac energy, so the initial state will be in a more or less arbitrary superposition of all four Dirac states. The search is then not optimal but runs with a success probability that is, on average, again reduced by a factor 1/4. Fig. 4 shows a numerically obtained quantum search initialized in \|s and evolving under the full search Hamiltonian. As expected from the analysis on the reduced Hamiltonian, the state localizes on the three neighboring sites with a probability of about 45% which is two orders of magnitude larger than the average probability 100/N , here roughly 0.5%. The search does not reach 100% due to the fact that the actual localized state \|˜ extends beyond the nearest neighbors of the marked site, so ˜ \| = O(1) < 1. Our reduced model neglects contributions from the rest of the spectrum; like for other discrete and continuous time walks at the critical dimension [17][18][19][20][21], these contributions give ln N corrections (such as the O 1/ N ln (N ) scaling of the gap at the avoided crossing shown in the inset of Fig. 3). These logarithmic corrections have been derived in the appendix by going beyond the reduced three-state model, see also [20]. The relevant exact eigenenergies E + = −E − at the avoided crossing satisfy the resolvent condition F (E ± ) = √ 3 N k 1 E ± − (k) + 1 E ± + (k) = 0 ,(14) with (k) > 0 the eigenenergies of the unperturbed system at quasi-momenta k given in (4). Expanding We note in passing that our search algorithm can -like all quantum searches -be used for quantum communication and state transfer. Following [22], our continuous time search can be used to send signals between different sites by adding an additional perturbation to the lattice. The quantum system is then initialized in a state localized on one of the perturbed sites and the system oscillates between states localized on the perturbations. We find that the mechanism works best when both perturbations are on the same sublattice. Due to the nature of the coupling between the A and B sublattice and the fact that the localized perturber states live (mostly) on one sublattice, signal propagation between perturbations on different sublattices takes place over a much longer timescale. F (E + ) = 4 √ 3/(N E + ) − ∞ n=1 I 2n E 2n Discussion.-Continuous-time quantum search can be performed effectively on a 2D lattice without internal degrees of freedom by running the search at the Dirac point in graphene. We find that our search succeeds in time For simplicity of the analysis, we have focused here on perturbations which alter the hopping potential to all three nearest-neighbors symmetrically. Efficient search algorithms can also be obtained using other types of perturbations such as a single-bond perturbation or perturbing the lattice by adding additional sites. In all cases, it is important to fine-tune the system parameters in order to operate at an avoided crossing near the Dirac point. Given the importance of graphene as a nano-material, our findings point towards applications in directed signal transfer, state reconstruction or sensitive switching. This opens up the possibility of a completely new type of electronic engineering using single atoms as building blocks of electronic devices. We present here supplementary notes related to the article Quantum search on graphene lattices. There, we proposed an implementation of an efficient continuoustime quantum search on a two-dimensional graphene lattice. The notes here provide additional details -not presented in the main text -on the scaling relation of the search time and the success probability. We derive in particular the log N correction terms. The search dynamics is defined through a tight-binding Hamiltonian H consisting of the Hamiltonian of the unperturbed (finite) graphene lattice, H 0 , and a perturbation term changing the hopping potential between a single marked site \|α o , β o and its three neighbors. The perturbation has the effect of decoupling the marked site from its neighbors. The Hamiltonian is of the form H = H 0 + √ 3 \|α o , β o \| + √ 3 \| α o , β o \| (A1) and we assume periodic boundary conditions on the unperturbed graphene Hamiltonian H 0 , see the main text for details. Here, \| is a uniform superposition of the three neighboring vertices adjacent to \|α o , β o . We derive in what follows the search time t = T , that is, the time at which the amplitude at \| reaches a maximum. The amplitude squared is interpreted as the success probability for the search to succeed. The amplitude is determined by evaluating \| e −iHT \|start = \|ψa \| ψ a ψ a \| start e −iEaT , (A2) with \|ψ a , E a , the eigenstates and eigenenergies of the perturbed lattice. Note that \|α o , β o is itself an eigenstate with H \|α o , β o = 0 and \| α o , β o = 0, so that it does not contribute to the sum (A2). Without loss of generality, we choose our initial state on the A sublattice of graphene, that is \|start = 1 √ 2 A K + A K (A3) with A K , A K being degenerate eigenenergies of the unperturbed lattice at the Dirac energy and on the Alattice. (If the perturbation is on the B-lattice, the search is not successful and we repeat the search on the B sublattice). In these notes, we will justify our simple reduced model in the main text showing that our algorithm effectively takes place in a two-dimensional subspace of the full Hilbert-space spanned by combinations of the energy-states near the Dirac point and a localized perturber state. Our analysis follows the treatment in Ref. [S1] adjusted to include symmetry properties of graphene around the Dirac point. For any eigenstate \|ψ a of the perturbed system such that the corresponding eigenenergy E a is not in the spectrum of H 0 we may rewrite the perturbed eigenequation in the form \|ψ a = √ 3R a E a − H 0 \|α o , β o ,(A4) where √ R a = \| ψ a (where we chosen the phase of \|ψ a such that \| ψ a ≥ 0). If E a is in the spectrum of H 0 then √ R a = 0. This is trivial for the state \|ψ a = \|α o , β o -otherwise it can be shown as follows. Let ψ 0 a be an unperturbed eigenvector such that H 0 ψ 0 a = E a ψ 0 a . Projecting the eigenvalue equation H \|ψ a = E a \|ψ a onto ψ 0 a yields ψ 0 a α o , β o \| ψ a + ψ 0 a α o , β o \| ψ a = 0. Clearly, α o , β o \| ψ a = 0; in addition for a given E a , we can always find at least one corresponding ψ 0 a such that ψ 0 a α o , β o = 0. From that it follows that \| ψ a = 0 ≡ R 2 a , which is what we wanted to show. Note that due to high degeneracies in the unperturbed system and due to the fact, that the perturbation is of rank two, one will have many states whose eigenenergies do not change under the perturbation and for which (A4) is a priori not well defined. The property R a = 0 allows us to remove these states from all sums that involve (E a − H 0 ) −1 . Let us now use (A4) to derive a condition for an energy E a = 0 to be a perturbed eigenenergy of H -perturbed eigenenergy here implies that it is not in the spectrum of H 0 . As \|α o , β o is a known eigenstate of the perturbed lattice, we have α o , β o \| ψ a = 0 and thus (A4) implies 3R a α o , β o \| (E a − H 0 ) −1 \|α o , β o = 0 .(A5) Expressing α 0 , β 0 \| in terms of the eigenstates of the unperturbed Hamiltonian H 0 , we may write this as a quantization condition F (E a ) =0 F (E) = √ 3 N k 1 E − (k) + 1 E + (k) .(A6) Here, N , is the total number of sites and (k) are the positive eigenenergies of the unperturbed Hamiltonian H 0 . (Note that the spectrum of H 0 as well as H is symmetric around E = 0. This constitutes the main difference to the treatment considered in [S1]. ) We may choose \|ψ a to be normalized ψ a \| ψ a = 1 -(A4) then implies 3R a α o , β o \| (E a − H 0 ) −2 \|α o , β o = 1,(A7) which allows R a to be rewritten as R a = 1 √ 3\|F (E a )\| .(A8) We may now rewrite the amplitude (A2) in the form \| e −iHT \|start = a:Ra =0 R a ψ a \| start e −iEaT = α o , β o \| start a e −iEaT E a \|F (E a )\| (A9) where we have used the adjoint of (A4) and have removed the restrictions on the summation in the last line. This is no longer necessary as \|F (E a )\| → ∞ when E a is in the unperturbed spectrum. In the main text, we show that by adding a perturbation which creates a localized state energetically close to the Dirac point one can construct an efficient search algorithm. Consequently, we concentrate on evaluating the time-evolution involving the eigenstates of the perturbed Hamiltonian closest to the Dirac point; we denote these states \|ψ ± in what follows. We estimate the corresponding eigenenergies, E ± where E + = −E − > 0 and we will focus on E + in the following. Using the sum in Eq. (A6), we will also derive a leading order expression for F (E + ). Separating out the contribution to F (E + ) from the Dirac points where (K) = K = 0, and expanding the remaining contribution to the sum in (A6) at E = 0, one obtains F (E + ) = 4 √ 3 N E + − ∞ n=1 I 2n E 2n−1 + . (A10) The sums I n are given by I n = √ 3 N k =K,K 1 [ (k)] n + 1 [− (k)] n .(A11) Due to the symmetry of the unperturbed spectrum only those I n with even n are non-zero. The non-vanishing I 2k coefficients obey the following rigorous estimates I 2 =O (ln N ) ,(A12)lim N →∞ I 2k N k−1 =4 √ 3 Z 2 (S K , k) + Z 2 (S K , k) for k ≥ 2,(A13) where the estimate (A12) is sharp (I 2 is logarithmically bounded from above and below) and Z 2 (S, x) is the Epstein zeta-function Z 2 (S, x) = 1 2 (p,q)∈Z 2 \(0,0) S 11 p 2 + 2S 12 pq + S 22 q 2 −x( Here the sums over integers p and q is over a rectangular region L of the lattice Z 2 which is centered at (0, 0) and has side lengths proportional to √ N -the center (0, 0), corresponding to the relevant Dirac point, is omitted from the sum. For k > 1 the corresponding sums converge which proves (A13). For k = 1 we will establish constant C 1 and C 2 such that C 1 ln N < (p,q)∈L 1 S K,11 p 2 + 2S K,12 pq + S K,22 q 2 < C 2 ln N (A17) which then directly leads to (A12). To establish C 1 note that because each term in the sum (A17) is positive its value decreases by restricting it to a square region −a 1 √ N ≤ p ≤ a 1 √ N , −a 1 √ N ≤ q ≤ a 1 √ N which is completely contained in L. Up to an error of order one the sum over a square region can in turn be written as a sum over eight terms of the form a1 √ N p=1 p q=1 1 S K,11 p 2 ± 2S K,12 pq + S K,22 q 2 . (A18) For fixed p we can find q max such that p q=1 1 S K,11 p 2 ± 2S K,12 pq + S K,22 q 2 > p S K,11 p 2 ± 2S K,12 pq max + S K,22 q 2 max . (A19) We may choose q max = b 1 p for some constant b 1 ≥ 0, so a1 √ N p=1 p q=1 1 S K,11 p 2 ± 2S K,12 pq + S K,22 q 2 > c √ N p=1 1 p (A20) which diverges as ln N . Establishing C 2 and the corresponding logarithmic bound from above follows the same line by first extending the sum to a square of side length 2a 2 √ N that completely contains L and then establishing p q=1 1 S K,11 p 2 ± 2S K,12 pq + S K,22 q 2 < p S K,11 p 2 ± 2S K,12 pq min + S K,22 q 2 min (A21) with q min = b 2 p. Let us note in passing that estimates based on Poisson summation reveal more detail, i.e. I 2 =A ln N + O(1),(A22)I 2k =2 √ 3N k−1 Z 2 (S K , k) + Z 2 (S K , k) + + O(N k−2 ) for k ≥ 2,(A23) where A > 0 is a constant. A rigorous treatment of the O(1) estimate is more involved, however. In fact, (A12)and (A13) are sufficient in the context of this paper. We note that each term in Z 2 S K , k is smaller than the corresponding term in Z 2 S K , 2 for k > 2, and so it follows that Z 2 (S K , k) < Z 2 (S K , 2) for k > 2. This property of the Epstein zeta function and the estimate (A13) imply ∞ n=2 I 2n E 2n−1 + < C N E+ ∞ n=2 (N E 2 + ) n i. e. the infinite sum in (A10) converges for E + < 1/ √ N . Let us now show that one indeed finds a solution of F (E + ) = 0 using the expansion (A10) inside the convergence radius. We start with the estimate that is obtained by truncating the sum in (A10) at n = 1, i.e. N I2 -for sufficiently large N this is in the radius of convergence of the complete expansion (A10). We will now show rigorously that this estimate gives the leading order correctly. First of all, the estimate implies that a zero inside the radius of convergence exists. Moreover since I 2n > 0 all terms of (A10) that have been neglected in the estimate enter with the same sign. So the true value E + > 0 has to be smaller than the estimate, which we write as E 2 + = 4 √ 3 N I 2 − ∆ > 0 (A24) with ∆ > 0. We will show rigorously that ∆/E + → 0 as N → ∞. Indeed one may rewrite F (E + ) = 0 in the form 4 √ 3 N − I 2 E 2 + = ∞ n=2 I 2n E 2n + (A25) ⇒ I 2 ∆ = ∞ n=2 I 2n E 2n + .(A26) Following the same arguments as used above for the calculation of the convergence radius and using the already established fact that E + is inside the convergence radius for sufficiently large N we get the following inequality 0 < N I 2 ∆ < C Applying our previous results for the states nearest to the Dirac point, we see that \| start\| ψ ± \| ≈ 1 √ 2 + O 1 ln 2 N . (A30) Our starting state is thus a superposition of the perturbed eigenstates, \|ψ ± , which also facilitate the search algorithm, see the main text. This now allows us to investigate the running time and success amplitude of the algorithm by looking at the time-evolution, that is, \| e −iHt \|start (A31) ≈ 1 √ 2 e −iE+t \| ψ + − e iE+t \| ψ −( FIG. 1 . 1Left: Graphene with lattice vectors a 1/2 , translation vectors δ i and unit cell (dashed lines). Right: Reciprocal lattice with basis vectors b 1/2 , symmetry points Γ, K, K , M and first Brillouin zone (hexagon). online) Dispersion relation for infinite graphene sheet ( D = 0). FIG. 3 . 3Spectrum Hγ in Eq. 5 as a function of γ for a 12 × 12 cell torus (N = 288). The spectrum is symmetric around D = 0. Inset: Scaling of the gap ∆ =Ẽ+ −Ẽ− (dots) and curves c1/ √ N (solid blue), c2/ √ N log N (dashed red) for comparison. ( 7 ) 7) and use this to reduce the full Hamiltonian locally in terms of the 3-dimensional basis {\|K A , \|K A , \| }. The reduced Hamiltonian takes the form t = π 4 N 3 .FIG. 4 . 434We find a √ N speed-up for the search on Search on 12 × 12 cell graphene lattice with starting state \|s . For tori with m = n the dynamics at each neighboring site is the same so only one is shown. − 1 + 1, one finds I 2 = O(lnN ) and I 2k = O(N k−1 ) for k ≥ 2, see the appendix for details. The scaling of the gap follows then directly. The localization time scales inversely proportional to the gap, that is, T = O N ln (N ) ; one also obtains that the return amplitude drops like O 1/ √ ln N . T = O N ln (N ) with probability O (1/ ln N ). This is the same time complexity found in [18, 19] for discretetime and in [21] for continuous-time searches. To boost the probability to O (1), O (ln N ) repetitions are required giving a total time T = O √ N ln 3 2 N . Amplification methods [31-33] may be used to reduce the total search time further. S A14) for a real positive definite real symmetric 2 × 2 matrix S [S2]. The matrices S K = S K = 4π 2 spectrum close to the Dirac points. The linear dispersion behaviour near the Dirac points K and K is the same and so are the matrices S K and S K . Before moving on let us derive the estimates (A12) and (A13). It is clear that the dominant contributions come from the vicinity of the Dirac points. K ,11 p 2 + 2S K ,12 pq + S K ,22 q us to show that only two states have an overlap O(1) with the starting state and are thus the relevant states to be considered in the time-evolution of the algorithm. Using the definitions of \|ψ a and R a in Eqs. (A4) and (A8), the inner product of the starting state and the perturbed eigenvectors can be expressed as o , β o . clear from our earlier results for E ± and I 2 that our algorithm localizes on the neighbor state \| in timeT = π 2E+ = O √ N ln N with probability amplitude O 1/ √ ln N .[S1] Spatial search by quantum walk, Andrew M.Childs and Jeffrey Goldstone, Phys. Rev. A. 70:022314 (2004). [S2] On Advanced Analytic Number Theory, C.L. Siegel, Tata Institute of Fundamental Research (1961). ACKNOWLEDGMENTSThis work has been supported by the EPSRC network 'Analysis on Graphs' (EP/I038217/1). Helpful discussions with Klaus Richter are gratefully acknowledged. R , Quantum Walks and Search Algorithms. SpringerR. 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[ "USING ARTIFICIAL INTELLIGENCE TO DETECT CHEST X-RAYS WITH NO SIGNIFICANT FINDINGS IN A PRIMARY HEALTH CARE SETTING IN OULU, FINLAND", "USING ARTIFICIAL INTELLIGENCE TO DETECT CHEST X-RAYS WITH NO SIGNIFICANT FINDINGS IN A PRIMARY HEALTH CARE SETTING IN OULU, FINLAND" ]	[ "Keski-Filppula Tommi \nFaculty of Medicine\n-Research Unit of Medical Imaging, Physics and Technology\nUniversity of Oulu\nOuluFinland\n", "Nikki Marko \nDepartment of Diagnostic Radiology\nOulu University Hospital\n90029 OYS 3 -Oxipit UABP.O. Box 10Oulu, VilniusFinland., Lithuania\n", "Haapea Marianne \nDepartment of Diagnostic Radiology\nOulu University Hospital\n90029 OYS 3 -Oxipit UABP.O. Box 10Oulu, VilniusFinland., Lithuania\n", "Ramanauskas Naglis ", "Tervonen Osmo \nFaculty of Medicine\n-Research Unit of Medical Imaging, Physics and Technology\nUniversity of Oulu\nOuluFinland\n\nDepartment of Diagnostic Radiology\nOulu University Hospital\n90029 OYS 3 -Oxipit UABP.O. Box 10Oulu, VilniusFinland., Lithuania\n", "Keski-Filppula Corresponding ", "Tommi " ]	[ "Faculty of Medicine\n-Research Unit of Medical Imaging, Physics and Technology\nUniversity of Oulu\nOuluFinland", "Department of Diagnostic Radiology\nOulu University Hospital\n90029 OYS 3 -Oxipit UABP.O. Box 10Oulu, VilniusFinland., Lithuania", "Department of Diagnostic Radiology\nOulu University Hospital\n90029 OYS 3 -Oxipit UABP.O. Box 10Oulu, VilniusFinland., Lithuania", "Faculty of Medicine\n-Research Unit of Medical Imaging, Physics and Technology\nUniversity of Oulu\nOuluFinland", "Department of Diagnostic Radiology\nOulu University Hospital\n90029 OYS 3 -Oxipit UABP.O. Box 10Oulu, VilniusFinland., Lithuania" ]	[]	ObjectivesTo assess the use of artificial intelligence-based software in ruling out chest X-ray cases, with no significant findings in a primary health care setting.MethodsIn this retrospective study, a commercially available artificial intelligence (AI) software was used to analyse 10 000 chest X-rays of Finnish primary health care patients. In studies with a mismatch between an AI normal report and the original radiologist report, a consensus read by two board-certified radiologists was conducted to make the final diagnosis.ResultsAfter the exclusion of cases not meeting the study criteria, 9579 cases were analysed by AI. Of these cases, 4451 were considered normal in the original radiologist report and 4644 after the consensus reading. The number of cases correctly found nonsignificant by AI was 1692 (17.7% of all studies and 36.4% of studies with no significant findings).After the consensus read, there were nine confirmed false-negative studies. These studies included four cases of slightly enlarged heart size, four cases of slightly increased pulmonary opacification and one case with a small unilateral pleural effusion. This gives the AI a sensitivity of 99.8% (95% CI= 99.65-99.92) and specificity of 36.4 % (95% CI= 35.05-37.84) for recognising significant pathology on a chest X-ray.3ConclusionsAI was able to correctly rule out 36.4% of chest X-rays with no significant findings of primary health care patients, with a minimal number of false negatives that would lead to effectively no compromise on patient safety. No critical findings were missed by the software.Key point• Artificial intelligence can reliably and safely rule out significant pathology in a chest X-ray.	10.48550/arxiv.2205.08123	[ "https://arxiv.org/pdf/2205.08123v1.pdf" ]	248,834,067	2205.08123	06d6d3274bdfd53ea93be6d2d2726bbc3fbe5ddd	USING ARTIFICIAL INTELLIGENCE TO DETECT CHEST X-RAYS WITH NO SIGNIFICANT FINDINGS IN A PRIMARY HEALTH CARE SETTING IN OULU, FINLAND Keski-Filppula Tommi Faculty of Medicine -Research Unit of Medical Imaging, Physics and Technology University of Oulu OuluFinland Nikki Marko Department of Diagnostic Radiology Oulu University Hospital 90029 OYS 3 -Oxipit UABP.O. Box 10Oulu, VilniusFinland., Lithuania Haapea Marianne Department of Diagnostic Radiology Oulu University Hospital 90029 OYS 3 -Oxipit UABP.O. Box 10Oulu, VilniusFinland., Lithuania Ramanauskas Naglis Tervonen Osmo Faculty of Medicine -Research Unit of Medical Imaging, Physics and Technology University of Oulu OuluFinland Department of Diagnostic Radiology Oulu University Hospital 90029 OYS 3 -Oxipit UABP.O. Box 10Oulu, VilniusFinland., Lithuania Keski-Filppula Corresponding Tommi USING ARTIFICIAL INTELLIGENCE TO DETECT CHEST X-RAYS WITH NO SIGNIFICANT FINDINGS IN A PRIMARY HEALTH CARE SETTING IN OULU, FINLAND 1Artificial intelligenceRadiography, ThoracicNatural Language ProcessingPrimary Health CareRadiographic Image Interpretation, Computer-Assisted ObjectivesTo assess the use of artificial intelligence-based software in ruling out chest X-ray cases, with no significant findings in a primary health care setting.MethodsIn this retrospective study, a commercially available artificial intelligence (AI) software was used to analyse 10 000 chest X-rays of Finnish primary health care patients. In studies with a mismatch between an AI normal report and the original radiologist report, a consensus read by two board-certified radiologists was conducted to make the final diagnosis.ResultsAfter the exclusion of cases not meeting the study criteria, 9579 cases were analysed by AI. Of these cases, 4451 were considered normal in the original radiologist report and 4644 after the consensus reading. The number of cases correctly found nonsignificant by AI was 1692 (17.7% of all studies and 36.4% of studies with no significant findings).After the consensus read, there were nine confirmed false-negative studies. These studies included four cases of slightly enlarged heart size, four cases of slightly increased pulmonary opacification and one case with a small unilateral pleural effusion. This gives the AI a sensitivity of 99.8% (95% CI= 99.65-99.92) and specificity of 36.4 % (95% CI= 35.05-37.84) for recognising significant pathology on a chest X-ray.3ConclusionsAI was able to correctly rule out 36.4% of chest X-rays with no significant findings of primary health care patients, with a minimal number of false negatives that would lead to effectively no compromise on patient safety. No critical findings were missed by the software.Key point• Artificial intelligence can reliably and safely rule out significant pathology in a chest X-ray. INTRODUCTION The use of artificial intelligence (AI) and machine learning in radiology has been under investigation in recent years. In Finland, approximately 700 000 chest X-rays were performed in 2018, making it the most frequently utilised radiologic study after the dental X-ray [1]. Prior research indicates that AI can perform as well as, or in some cases, even better than, a radiologist, at recognising certain findings in a chest X-ray [2][3][4][5]. Currently, the most promising results against a radiologist are on detecting isolated findings, such as lung nodules, or pneumothoraces [2,6]. AI can indeed perform well, compared against a radiologist, in recognising isolated findings, but few studies thus far have compared AI to a radiologist in a setting where the radiologist has access to the patient's clinical information, as would be the case in real life [7]. One practical angle of approach is to ask how AI can help a radiologist. Two recent studies demonstrated that a comprehensive, deep-learning model significantly improved the interpretation of a chest X-ray by a radiologist [8,9] and was well received [8]. Another promising real-life use case of AI in the context of chest X-rays is triaging to reduce report turnaround times for critical findings [10,11]. In this study, we explore one possible pathway for AI utilisation: ruling out normal chest X-ray studies in a primary health care setting, where a significant number of studies contain no significant findings to begin with. Our hypothesis is that AI can reliably recognise a significant percentage of normal chest X-rays with a very small number of false negatives. Furthermore, we hypothesise that, in a clinical setting, if the chest X-ray is interpreted as normal by AI, no radiologist report is needed unless the clinician requests it. This could help optimize healthcare resources by reducing the workload of chest X-ray reporting. To our knowledge, one previous study with a setting like ours has been conducted. In a recent study, Dyer et al. demonstrated that a deep learning algorithm was able to classify 15% of unselected chest X-rays as normal with a precision of 97.7% and an error rate of 0.33%, removing 24.9% of radiologist-confirmed normal studies from the workflow [12]. 5 MATERIALS & METHODS The source of data Oulu is a city of 209 000 inhabitants, located in the North Ostrobothnia region of Finland. There are several primary health care centres and one hospitalthe Oulu University Hospital. Primary health care centres mostly take care of patients with stable chronic illnesses and acutely ill patients, who do not require hospital-level care. More severely ill patients are referred to the hospital. Our data consist of all chest X-rays taken of primary health care patients residing in the city of Oulu, Finland, from the years 2019-2020, up to 10 000 images. The following are the exclusion criteria for chest X-rays in our study: children (under 18 years of age), pregnant women, images with technical issues (poor image quality, rotation, incompletely recorded lung fields) and other than a posterior-anterior (PA)-position of the image. AI-based commercial software (ChestLink®, Oxipit, Vilnius, Lithuania) was used to analyse the X-rays. At the time of the data analysis, only frontal X-ray images were used, as lateral view analysis was not yet enabled in the software. The AI software has been developed and optimised heavily towards maximising sensitivity in order to identify the studies without any abnormalities with high sensitivity. The AI has been tested with approximately one million chest X-rays, and the fraction of normal chest X-ray studies, for which a normal report is generated depending on the dataset, has ranged from 10 to 55 %. The software analyses both frontal and lateral chest X-ray images. A normal report by the software can only be generated for the studies which are as follows: a) Erect PA (posterioranterior radiographs) b) performed for adult patients (>= 18 years). Data classification and processing The original radiologist report, and consensus read as necessary, was considered a ground truth in our study. The content of the original radiologist report was analysed using a natural language processing (NLP) software provided by Oxipit. The software can recognise 75 radiological findings in radiologist reports and for each finding, a special rule-based system is used to capture whether it was mentioned in the radiologist report or not. The software can handle multiple ways of mentioning the given findings in a report as well as handle the negations to minimise any false positives. For non-English reports, the software utilises a translation service as a part of the text mining pipeline. If AI assessed the study as normal, and the NLP system recognised the radiologist report as normal, then the study was considered a software-true negative / normal. If AI assessed the study as normal, but the NLP system recognised the radiologist report as abnormal, the study was considered an initial software false negative. We then excluded purely NLPrelated errors, which left us with 156 cases, where there was a discrepancy between the AI and radiologist reports. Two board-certified radiologists (MN and OT) then evaluated the images, and the decision was made by consensus. We classified the study as normal if the finding originally commented in the radiologist report was considered insignificant, over-7 diagnosed or miscellaneous in the consensus read. There were 147 of such cases, including healed rib fractures, slight pleural thickening or pleural adhesions and small linear pulmonary atelectases. Statistical analysis We used sensitivity, specificity, negative predictive value, positive predictive value and accuracy with their 95% confidence intervals (CI) to evaluate and represent the performance of AI. All parameters were calculated using an online MedCalc Diagnostic test evaluation calculator: https://www.medcalc.org/calc/diagnostic_test.php. RESULTS True negatives Of the 9579 images, 4451 (46.4%) were reported as normal by a radiologist, as recognised by the NLP system ( Figure 2). Using the original radiologist report as ground truth, AI correctly assessed 1499 images (33.6% of original radiologist negatives) as normal. When added in studies assessed normal by AI, where the NLP software incorrectly recognised the radiologist report as abnormal (N=46), and where the originally reported abnormal findings were considered insignificant in the consensus reading (N=147), there was a total number of 1692 AI true normal studies (17.7% of all studies and 36.4% of studies that a radiologist originally classified as normal). False negatives There were 202 cases where AI classified the case as normal, but the original radiologist report described abnormality, as recognised by the NLP system. After excluding NLPrelated errors (N=46), the initial comparison between the AI and radiologist report showed 156 false negative studies. 147 of these studies were classified as containing nonsignificant, over-diagnosed or otherwise unremarkable findings in the consensus reading. These findings are presented in Figure 3. To rule out the possibility of false negative studies in the true normal group, we manually examined the radiologist reports on all AI true negative studies (N = 1499). There were 8 cases where the NLP system had incorrectly recognised the radiologist report as normal. These radiologist reports included one case with a slightly prominent right hilum, three cases of suspected peribronchovascular consolidation, one case of suspected rib fracture and three cases of suspected unspecific opacities. All these cases were classified as normal or insignificant in the consensus read. These cases are also included in Figure 3. Of the 202 initial false negatives, nine were considered potentially significant, "confirmed false negatives". Four of these studies had a slightly increased heart size, four had slightly increased pulmonary opacification and one had a small unilateral pleural effusion. One Figure 3. The findings in the chest X-ray image not depicted by AI but depicted by a radiologist and further considered as nonsignificant, over-diagnosed or otherwise unremarkable in the consensus read. The x-axis represents the absolute number of findings. It is notable that there is overlap: multiple findings can be commented in the original radiologist report of a single study. The unspecific small opacities included for example pleural plaques and mamillary shadows. 10 study of each category is presented in Figure 4. Full-sized images are available in the electronic supplementary material. The other three cases of both slightly enlarged heart size and increased pulmonary opacification were comparably similar. DISCUSSION In this study, we demonstrated that AI can provide a reliable way to rule out significant pathology in chest X-rays. The number of confirmed false negatives was small. There were four cases with a slightly enlarged heart size, one case with a small amount of pleural fluid and four cases with slightly increased pulmonary opacification. The potential effect of these misdiagnoses on patient safety is negligible and none of them would lead to immediate threat to the patient. A slightly enlarged heart size is usually not an acute finding, except in the rare case of pericardial fluid, and does not require immediate treatment. A small amount of pleural fluid has little clinical relevance. The four cases with a slightly increased pulmonary opacification might indicate mild bronchopneumonia, but the finding is controversial, and interobserver variability is high among radiologist in reporting this finding. While the X-ray finding is not unequivocal, the clinical diagnosis and treatment is mostly based on the clinical condition of the patient. No lobar consolidates indicating bacterial pneumonia were missed by the software. The number of nonsignificant findings in a chest X-ray is generally high, and the main task in the interpretation is to depict the clinically relevant ones. An AI software should be able to reliably depict the clinically relevant findings. In this study, there were many chest Xray findings which were not depicted by AI and depicted by a radiologist, and they were further evaluated as nonsignificant by the consensus reading ( Figure 3). Our approach to classifying these as nonsignificant was clear and consensus-based: in the context of primary health care, the purpose of a chest X-ray is to rule out treatable disease. The prevalence of spinal degeneration in asymptomatic people is high especially in the elderly population and does not represent disease [13]. Pulmonary linear atelectases are common, usually self-resolving, most commonly caused by hypoventilation [14]. Some of the findings, such as suspected pulmonary opacification or slightly increased interstitial markings, represent the subjectivity of chest X-ray reporting: inter-reader agreement varies considerably in a clinical setting [15][16][17]. In our study, the final diagnosis was made by consensus by two experienced, board-certified radiologists. Comparing the results of this study to the study by Dyer et al. in 2021, our results are on a similar trajectory. However, in their study, 4 out of 584 high-confidence normal studies 13 showed critical findings missed by the software: one lung nodule, two lung masses and one pulmonary consolidation. In our study, no critical findings were missed. Using AI to rule out normal studies could provide one way to optimise the use of resources in healthcare. With the increasing volumes of CT and MRI, the radiologist workforce could thus be directed to more complicated studies. Further research in a prospective clinical setting is needed to gather real-life clinical data and use the experience of autonomous AI-based reporting of chest X-rays. Limitations The main limitation of this study is the ground truth against which the AI analysis is compared. The ground truth in our study was the original radiologist report and further consensus-read when necessary. This does represent a real-life scenario, where radiologist reports are used to guide patient care, but does potentially create uncertainty regarding borderline findings, as a definitive diagnosis based on a chest X-ray is often not possible. Also, we did not have computed tomography or follow-up to confirm the chest X-ray findings, but the evaluation was based on the consensus reading. NLP software was used to analyse the initial radiologist reports, which may cause one source of error. To minimise the effect of NLP on the results, we manually consensus-read all the software false negative studies and checked all the original radiologist reports in the true normal group. This eliminates NLP-related errors regarding the false negative studies. Another limitation of this study is the study population. While the study population should represent the general patient base of our primary health care centres, an accurate description of the study population is not possible, as we did not have access to patient demographic information. This study was based on frontal chest X-ray images only, as the functionality to analyse lateral images was not available at the time of the study. This may lead to underdiagnoses 14 of findings best appreciated in the lateral view, such as vertebral body changes or blunting of the posterior costophrenic angles. Conclusions Based on the results of this study, AI can reliably remove 36.4% of normal chest X-rays from a primary health care population data set with a minimal number of false negatives, leading to effectively no compromise on patient safety. No critical findings, such as lung masses, pneumothoraces or lobar consolidations were missed by the software. Figure 1 . 1An illustration of the AI software used in the study (ChestLink®). 6 Some of the findings, such as vertebral changes or posterior costophrenic angle blunting, could only be appreciated on lateral images. As lateral image analysis was not enabled at the time of analysis, lateral-view-only findings were classified as insignificant. Figure 2 . 2Flowchart representing data processing and relevant results. NLP = Natural language processing. Figure 4 . 4A representation of the confirmed false negative AI studies.A) Bilateral basal peribronchovascular opacification, potentially indicating bronchopneumonia. B) A slightly enlarged heart size. C) A small left-sided pleural effusion. AI specificity of 36.4 % (95% CI = 35.05-37.84) and a sensitivity of 99.8 % (95% CI = 99.65-99.92) for recognising significant pathology on a chest X-ray.(Table 1.)Table 1. Performance of AI in recognising significant pathology in a chest X-ray.Percentages were rounded to two-decimal accuracy. CI = confidence interval.Statistic Value (%) 95 % CI (%) Sensitivity 99.82 99.65-99.92 Specificity 36.43 35.05-37.84 Positive predictive value 62.53 62.02-63.04 Negative predictive value 99.47 98.99-99.72 Accuracy 69.09 68.15-70.01 ACKNOWLEDGEMENTSThe authors state that this work has not received any funding. 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[ "BILIMITS ARE BIFINAL OBJECTS", "BILIMITS ARE BIFINAL OBJECTS" ]	[ "Andrea Gagna ", "Yonatan Harpaz ", "Edoardo Lanari " ]	[]	[]	We prove that a (lax) bilimit of a 2-functor is characterized by the existence of a limiting contraction in the 2-category of (lax) cones over the diagram. We also investigate the notion of bifinal object and prove that a (lax) bilimit is a limiting bifinal object in the category of cones. Everything is developed in the context of marked 2-categories, so that the machinery can be applied to different levels of laxity, including pseudo-limits.	10.1016/j.jpaa.2022.107137	[ "https://arxiv.org/pdf/2103.16394v2.pdf" ]	232,417,584	2103.16394	7bfc5b07b751a8103bc225da90e14a6a077f6727	BILIMITS ARE BIFINAL OBJECTS 26 Apr 2022 Andrea Gagna Yonatan Harpaz Edoardo Lanari BILIMITS ARE BIFINAL OBJECTS 26 Apr 2022arXiv:2103.16394v2 [math.CT] We prove that a (lax) bilimit of a 2-functor is characterized by the existence of a limiting contraction in the 2-category of (lax) cones over the diagram. We also investigate the notion of bifinal object and prove that a (lax) bilimit is a limiting bifinal object in the category of cones. Everything is developed in the context of marked 2-categories, so that the machinery can be applied to different levels of laxity, including pseudo-limits. Introduction The theory of limits and colimits sits at the core of category theory and it has become an essential tool to express natural constructions of interest to many areas of mathematics. As the formalism of category theory matured, it became clear that some important phenomena are better understood when framed in a 2-categorical setting. It was therefore natural to pursue a generalization of the useful concept of (co)limits in such a context. The notions of 2-limit and 2-colimit were first formulated independently by Auderset in [5], where the Eilenberg-Moore and the Kleisli category of a monad are recovered as a 2-limit and 2-colimit, and by Borceux-Kelly [6], who introduced the notion of enriched limits and colimits and so in particular limits and colimits enriched in the cartesian closed category of small categories. These notions were further studied and developed by Street [24], Kelly [19,18] and Lack [22], who also introduced and investigated the lax and weighted versions. More recently two papers by clingman and Moser appeared in the literature, namely [9] and [10], where they investigate whether the well-known result that limits are terminal cones extends to the 2-dimensional framework. The first one proves that the answer is negative, i.e. terminal cones are no longer enough to capture the correct universal property, no matter what flavour of slice category one uses. In the second paper, the authors leverage on results from double-category theory on representability of Cat-valued functors to show that being terminal still captures the notion of limit, provided one is willing to work with an alternative 2-category than that of cones, there denoted by mor(F ) for a given diagram F . Motivations. The main goal of this paper is to clarify, with its main result (Theorem 4.10), that a natural characterization of lax bilimits in terms of cones is still possible. The use of a marking on the domain of the 2-functor of which we want to study the bilimit addresses in a fundamental way all the possible levels of laxity of the bilimit (pseudo, lax or anything in between). We remark that this further level of generality is necessary from a technical viewpoint to coherently interpolate laxity from pseudo to lax in all the 2-categorical constructions. For example, a similar technique is also needed in the development of flat pseudo-functors, and therefore in the theory of 2-categorical filteredness as well as in the theory of 2-topoi, carried out by Descotte, Dubuc and Szyld in [13]. Another reason that drove us to write this paper was filling the gap in the literature as concerns final objects in 2-category theory. Although some results on final 2-functors already appeared in [1] by Abellán García and Stern (and in the (∞, 2)-categorical context in [14] by the authors), we consider establishing the connections between final objects, contractions and bilimits long overdue. Finally, further motivation comes from the realm of homotopy theory and in particular of (∞, 2)-category theory. On the one hand, we wanted to build some lowdimensional intuition based on the (∞, 2)-categorical treatment of (lax, weighted) (co)limits we developed in [14]. On the other hand, we are convinced that classical 2-category theory and (∞, 2)-category theory can benefit significantly from each other techniques and results. Indeed, the results of this paper have been written with a fibrational point of view typical of weak higher categories. From this perspective, the use of marked edges is all the more natural. Conversely, 2-category theory encompasses many structures and results that would be useful to generalize to (∞, 2)-categories, as they are needed in derived algebraic geometry. Consider for instance the theory of 2-(co)filtered 2-categories and 2-Ind/2-Pro constructions. These are worked out by Descotte and Dubuc in [12] for 2-categories and their (∞, 2)-categorical counter-part is a useful tool in geometry, see for instance [23, §A.3] of Porta and Sala. Profiting to a greater extend of such an interactive relationship will be the subject of further investigations. Technical summary. In this work, we insist on keeping the category of cones as the object of interest, but we claim that the notion of "terminality" is not the correct one to consider (hence the no-go theorem of [9]). Instead, we focus our attention on bifinal objects, and we make use of the formalism of marked 2categories and contractions to characterize (lax) bilimits as bifinal objects in the 2-category of cones. The reason why this is not needed in the 1-dimensional case is that in such context final objects coincide with terminal ones. That is to say, an object c of a category C is terminal if and only if the inclusion functor {c} ↪ C is a final functor or equivalently if the projection C c → C has a section mapping c to the identity 1 c . It is therefore natural to investigate the meaning of finality of a 2-functor {c} ↪ C, with C a 2-category with a set of marked 1-cells, and to inspect well-behaved sections of the projection 2-functor C c → C for a pseudo/lax version of the slice 2-category. By doing so, one is led to consider conditions of local terminality, by which we mean choices of terminal objects in the hom-categories C(a, c), for all objects a of C. Exploiting this point of view, we show that the appropriate 2-categorical generalizations of these characterizations of terminality still coincide and are equivalent to a "coherent" notion of local terminality known as contraction. An archetypical example of terminal object in category theory is given by the identity arrow 1 x , thought as an object of C x for C a category and x an object of C. Armed with the understanding of 2-categorical terminal conditions as contractions or as suitable final 2-functor from a singleton, one can examine the (lax) 2-slice C x for a 2-category C and an object x of C. This is the 2-categorical equivalent of the slice, but where the triangles of C with tip x defining the 1-cells C x are filled with a 2-cell of C. A first interesting observation concerns the local terminal objects: for an arrow f ∶ a → x of C, the terminal objects of the hom-category C x (f, 1 x ) are precisely the cartesian edges of the projections C x → C. These cartesian edges are simply the triangles C with edges f and 1 x such that the 2-cell filling it is invertible. Said otherwise, such a cartesian edge is the data of a 1-cell g∶ a → x of C together with an invertible 2-cell between f and g. But there are potentially other cartesian edges of C x from f ∶ a → x to another arrow g∶ b → x of C, which are just 1-cells h∶ a → b together with an invertible 2-cell between f and hg. Adopting a marking of these cartesian edges, which acts as a book-keeping device, we show that 1 x satisfies a finer finality property with respect to all the cartesian edges. In particular, we find that 1 x is biterminal in the sub-2-category where the objects are the same as in C x and the hom-categories are the full sub-categories spanned by the cartesian edges. So a more classical terminality property can be obtained, if we are willing to restrict to these triangles filled with invertible 2-cells. But this biterminal property does not characterise the stronger local terminality in the full (lax) 2-slice. Once we turn to (lax) bilimits, keeping track of all the cartesian edges proves to be the right approach. Given 2-functor F ∶ J → C and an object ℓ of C, a (lax) 2-cone over F with tip ℓ is a (lax) 2-natural transformation from the constant 2functor on ℓ to F . This is the 2-categorical equivalent of a cone over F , where the triangles constituting the 2-cone are filled with 2-cells, which can be invertible or not according to the laxity prescribed. The (lax) 2-cones over F can be canonically organised to form a 2-category of (lax) 2-cones C F . Again this is similar to the classical category of cones over a functor, but now every triangle is filled with a 2-cell of C. A pair (ℓ, λ) is said to be a (lax) bilimit of the functor F if the representable hom-category C(x, ℓ) is equivalent to the category of (lax) 2-cones over F with tip x; that is, the pair (ℓ, λ) 2-represents the (lax) 2-cones over F . The main result of the paper (Theorem 4.10, together with Proposition 4.6) states that the pair (ℓ, λ) is a (lax) bilimit of F if and only if the pair (ℓ, λ), thought as an object of the 2-category C F of (lax) 2-cones over F , is final with respect to the class of cartesian edges of the projection C F → C. The local terminality that we get is of the following kind: If (ℓ, λ) is a (lax) bilimit of the 2-functor F ∶ J → C and (x, α) is another (lax) 2-cone over F , then every morphism of (lax) 2-cones (f, σ) from (x, α) to (ℓ, λ), i.e., any object of the hom-category C F (x, α), (ℓ, λ) , with σ an invertible "2-cell" is terminal; here f ∶ x → ℓ is a morphism of C, α is a (lax) cone of x over F and σ is a modification from λ precomposed (component-wise) by f to α. Said otherwise, any such morphism of (lax) cones over F , morally a triangle over F , filled by an invertible 2-cell is terminal in the appropriate hom-category. These morphisms of (lax) 2-cones are precisely the cartesian edges of the projection C F → C with target the (lax) 2-cone (ℓ, λ). In particular, if we restrict to the hom-categories spanned by the cartesian edges we get that (ℓ, λ) is biterminal. Structure of the article. The paper is structured as follows. After a preliminary section where we fix the notation, we briefly recall the necessary background on 2-categories and relevant constructions, namely joins, slices and the Grothendieck construction for fibrations of 2-categories. We then move on to Section 2, where we introduce lax marked bilimits and contractions (following Descotte, Dubuc and Szlyd [13]), and we prove in Proposition 2.2.11 that final objects can be characterized in several ways, one of which involves contractions. Next, Section 3.1 is an investigation on representable fibrations and the properties of the corresponding representing objects. By looking at the cartesian morphisms of the projection p∶ C F → C we are able to highlight the difference from the simpler case of ordinary 1-categories, drawing a parallelism in Proposition 3.2.4. Moreover, we carve out a full subcategory of the category of maps over a given cone on a diagram, that we prove in Proposition 3.3.8 to be equivalent to the slice over the tip of the given cone. Note that this is a low-tech version of Corollary 5.1.6 from [14]. Finally, in the last section we blend all together, culminating in the main result recorded as Theorem 4.10, which characterizes (lax, marked) bilimits as limiting bifinal cones. Furthermore, we prove that such bilimits are also terminal in the appropriate subcategory of cones obtained by restricting to cartesian morphisms between them. 1.1.1. Recall that a 2-category C is composed by a class Ob(C) of objects or 0-cells, a class of 1-cells, a class of 2-cells, identities for objects and 1-cells, source and target functions for 1-cells and 2-cells, together with a "horizontal" composition * 0 for 1-cells and 2-cells and a "vertical" composition * 1 for 2-cells. The identity of an object or 1-cell x will be denoted by 1 x . We will denote the composition of 1-cells just by juxtaposition. Given a 1-cell f ∶ x → y and a 2-cell α∶ g → g ′ ∶ y → z, we will denote by α * 0 f the whiskering, by which we mean the composition α * 0 1 f . For any pair of objects x, y, we will denote by C(x, y) the category whose objects are 1-cells of C from x to y and the morphisms are the 2-cells between these. The category of small 2-categories and strict 2-functors will be denoted by 2-Cat, while its full subcategory spanned by 1-categories will be denoted by Cat. The 2-category of small 2-categories, pseudo-functors and pseudo-natural transformations will be denoted by 2-Cat ps . By pseudo-functor we will always mean normal pseudofunctors, i.e., unit preserving. D 0 = • , D 1 = 0 → 1 , D 2 = 0 1 the 2-categories corepresenting, respectively, the free-living object, 1-cell and 2-cell in 2-Cat. 1.1.4. Given a 2-category C, one can consider the 2-category C op obtained from C by reversing all the 1-cells, i.e., formally swapping source and target. By reversing the 2-cells of C, we get another 2-category which we denote by C co . Thus, for every pair of objects x, y, we have C(x, y) op = C co (x, y). Combining these two dualities, we get a 2-category C op co , where both 1-cells and 2-cells of C have been reversed. 1.1.5. A 1-cell f ∶ x → y of a 2-category C is an equivalence if we can find a 1-cell g∶ y → x together with invertible 2-cells gf → 1 x and f g → 1 y . A biequivalence of 2-categories is a 2-functor F ∶ C → D such that C(x, y) → D(F x, F y) is an equivalence of categories for every pair of objects x, y of C, and for every object z of D we can find an object c of C equipped with an equivalence F c → z. Biequivalences of 2-categories are the weak equivalences for a model category structure of 2-categories established by Lack [20,21]. In particular, we shall use the fact that the class of biequivalences enjoys the 2-out-of-3 property. Furthermore, in order to check whether a 2-functor F ∶ C → D is a biequivalence it is sufficient to check that: • the 2-functor F is surjective on objects, • the 2-functor F is full on 1-cells, i.e., lifts 1-cells with prescribed lifts of the boundary 0-cells (i.e., objects), • the 2-functor F is fully faithful on 2-cells, i.e., it uniquely lifts 2-cells with prescribed lifts of the boundary 1-cells. In fact, such a 2-functor is a trivial fibration in the above-mentioned model category structure. In detail, the first condition corresponds to a lifting property of the form: ∅ C D 0 D. The second one is encoded by the following lifting property: D 0 ∐ D 0 C D 1 D. (s,t) F Here we used the globular notation, so that s, t∶ D 0 → D 1 are the functors mapping the unique object of D 0 to 0, resp. 1. Finally, the third condition merges together two lifting properties, depicted below: D 1 ∐ D0 ∐ D0 D 1 C D 2 ∐ D1 ∐ D1 D 2 C D 2 D D 2 D. (s,t) F (Id,Id) F An analogous characterisation holds if F ∶ C → D is instead a pseudo-functor (see [21]). Definition 1.1.6. A marked 2-category is a pair (C, E) where C is a 2-category and E is a class of 1-cells in C containing the identities. A marked 2-functor between marked 2-categories F ∶ (C, E C ) → (D, E D ) is a 2-functor F ∶ C → D that maps every marked 1-cell in E C to a marked 1-cell in E D , i.e., F (E C ) ⊆ E D . We denote by 2-Cat + the category of marked 2-categories and marked 2-functors. Whenever we will consider a marked 2-category, we will only mention the nontrivial marked 1-cells, i.e., those which are not identities. 1.1.7. We say that an object c of a 2-category C is quasi-terminal, or that C admits c as a quasi-terminal object, if for any object x of C the category C(x, c) has a terminal object. The notion of quasi-terminal object was introduced in [17] by Jay under the name of locally terminal object. The term quasi-terminal appears also in [3], where Ara and Maltsiniotis extend this to strict n-categories, for 1 ≤ n ≤ ∞. 1.1.8. LetJ = (J, E) be a marked 2-category, C be a 2-category and F, G∶ J → C be 2-functors. A lax E-natural transformation α∶ F → G consists of the following data: • a 1-cell α i ∶ F i → Gi of C for any i in Ob(J), • a 2-cell α k ∶ G(k) * 0 α i → α j * 0 F (k) of C, that we depict as follows F i F j Gi Gj, F (k) αi αj G(k) α k for every 1-cell k∶ i → j of J, satisfying the following conditions: identity: we have α 1i = 1 αi for all objects i of J; marking: the 2-cell α k is invertible for all k in E; compositions: we have α lk = (α l * 0 F k) * 1 (Gl * 0 α k ), for all k∶ i → i ′ and l∶ i ′ → i ′′ 1-cells in J, compatibility: we have α l * 1 (Gδ * 0 α i ) = (α j * 0 F δ) * 1 α k , for every 2-cell δ∶ k → l∶ i → j of J. We will denote by [J, C] E the 2-category of 2-functors from J to C, lax E-natural transformations and modifications. Remark 1.1.9. In the notation of the previous paragraph, if E consists only of the identity 1-cells of J (resp. of all the 1-cells of J), we recover the notion of lax natural transformation (resp. pseudo natural transformation). Ara and Maltsiniotis [4] to the category of strict ∞-categories (also known as ω-categories). By truncating, their notion also provides a definition for strict n-categories (see [4, Ch. 8]). In fact, there are at least two sensible notions of join that one can give for strict higher categories, due to the choice of the variance of the higher cells: the lax and the oplax join (cf. [4, Remark 6.37]). The two joins collapse to the classical 1-categorical join once we truncate with respect to the higher cells. This choice of variance is important in relation to the kind of slice one intends to consider. We shall mainly consider slices over a 2-functor, and in this case the lax variance enjoys better formal properties. We start by recalling the general formalism that allows us to get the generalised (op)lax slices. 1.2.2. Given two 2-categories A and B, their lax join A * B is the following 2category: • the objects are those of A ∐ B, that we denote by a ⋆ ∅ and ∅ ⋆ b, for a ∈ Ob(A) and b ∈ Ob(B); • to the 1-cells of the coproduct 2-category A ∐ B, that we denote by f ⋆ ∅ and ∅ ⋆ g, for any f ∶ a → a ′ in A and any g∶ b → b ′ in B, we add a 1-cell a ⋆ b∶ a ⋆ ∅ → ∅ ⋆ b for every pair of objects (a, b) in Ob(A) × Ob(B), and then closing by composition in the obvious way; • to the 2-cells of the coproduct 2-category A ∐ B, that we denote by α ⋆ ∅ and ∅ ⋆ β, for any α∶ f ⇒ f ′ in A and every β∶ g ⇒ g ′ in B, we add 2-cells ∅ ⋆ b a ⋆ ∅ ∅ ⋆ b ′ ∅⋆g a⋆b a⋆b ′ a⋆g and a ⋆ ∅ ∅ ⋆ b a ′ ⋆ ∅ a⋆b a ′ ⋆b f ⋆∅ f ⋆b for any f ∶ a → a ′ in A and g∶ b → b ′ in B. We close by composition in the obvious way, but imposing the following relations. • For any a in Ob(A) and b in Ob(B), the 2-cells a⋆1 b and 1 a ⋆b are identities; • for any α∶ f ⇒ f ′ in A and b in Ob(B), we impose f ′ ⋆ b * 1 (a ′ ⋆ b * 0 α ⋆ ∅) = f ⋆ b; • for any a in Ob(A) and β∶ g ⇒ g ′ in B, we impose a ⋆ g ′ * 1 (∅ ⋆ β * 0 a ⋆ b) = a ⋆ g; • for any f ∶ a → a ′ in A and any g∶ b → b ′ in B, we impose a ⋆ ∅ a ′ ⋆ ∅ ∅ ⋆ b ∅ ⋆ b ′ f ⋆∅ a⋆b ∅⋆g a⋆b a⋆b ′ f ⋆b a⋆g a ⋆ ∅ a ′ ⋆ ∅ ∅ ⋆ b ∅ ⋆ b ′ f ⋆∅ a⋆b ∅⋆g a ′ ⋆b ′ a⋆b ′ f ⋆b ′ a ′ ⋆g This gives a monoidal category structure on 2-Cat , the category of small 2-categories, whose unit is the empty 2-category. This monoidal structure is not closed, but it is locally closed in the sense that the functors 2-Cat → 2-Cat A 2-Cat → 2-Cat B B ↦ (A * B, i A ∶ A → A * B) A ↦ (A * B, i B ∶ B → A * B), both have a right adjoint, where A A * B B i A i B are the canonical inclusions. These adjoint functors are denoted by 2-Cat A → 2-Cat 2-Cat B → 2-Cat (C, u∶ A → C) ↦ C u (C, v∶ B → C) ↦ C v and called the (generalized) lax slice functors. The 2-categories C u and C v correspond to the 2-category of lax cocones under u and the 2-category of lax cones over v, respectively. If A and B are both the terminal (2-)category D 0 and u and v correspond to the object c of C, then the generalized lax slices over and under the object c will simply be denoted by C c and C c . 1.2.3. We now provide an explicit description of the 2-categorical lax slice over a point. Let C be a 2-category and c be an object of C. • The objects of C c are pairs (x, α), with x in Ob(C) and α∶ x → c a 1-cell of C. They correspond to 2-functors D 0 * D 0 → C and D 0 * D 0 ≅ D 1 . • A 1-cell of C c from (x, α) to (y, β) is given by a 2-functor D 1 * D 0 → C such that its restriction to {0} * D 0 is (x, α) (resp. to {1} * D 0 is (y, β)). Explicitly, this means that such a 1-cell is a pair (f, γ), where f ∶ x → y is a 1-cell and γ∶ βf → α is a 2-cell of C, that we depict by x y c f α β γ • A 2-cell of C c from (f, γ) to (g, δ) is given by a 2-functor D 2 * D 0 → C such that the appropriate restrictions to D 1 * D 0 are (f, γ) to (g, δ). Explicitly, this means that such a 2-cell is given by a 2-cell Ξ∶ f → g of C satisfying 1.2.4. Based on the description of the slice over an object of a 2-category, we provide the description of the lax slice of a 2-category C over a 2-functor F ∶ J → C. δ * 1 (β * 0 Ξ) = γ, • The objects of C F are pairs (x, α), where x is an object of C, and α consists of a family of 1-cells (α i ∶ x → F i) i∈Ob(J) of C, and a family of 2-cells x F i F j αi αj F (k) α k indexed by 1-cells k∶ i → j of J, such that for every 2-cell Λ∶ k → k ′ in J the following relation is satisfied (1) α k = α k ′ * 1 (F Λ * 0 α i ). Furthermore, this assignment has to respect identities and be functorial. More precisely, α 1i = 1 αi for all objects i of J and given another 1-cell h∶ i ′ → i ′′ of J, we have α hk = α h * 1 (F h * 0 α k ). They correspond to 2-functors D 0 * J → C such that the restriction to the 2-functor ∅ * J → C is F , i.e., lax cones over F . • A 1-cell of C F from (x, α) to (y, β) is given by a 2-functor D 1 * J → C such that its restriction to {0} * D 0 is (x, α) (resp. to {1} * D 0 is (y, β)). It is a lax morphism of lax cones over F . Explicitly, this means that such a 1-cell is a pair (f, µ), where f ∶ x → y is a 1-cell of C and µ i ∶ β i f → α i is a 2-cell of C for all i in Ob(J), that we depict by x y F i f αi βi µi That is, for every object i of J the pair (f, µ i ) is a 1-cell in C F i , which corresponds to the restriction D 1 * {i} → C. Moreover, the set of 2-cells µ must satisfy the following relations: (2) α k * 1 (F k * 0 µ i ) = µ j * 1 (β k * 0 f ) for every 1-cell k∶ i → j of J, that we can depict as the following commutative diagram x y F i F j f βi F k αi αj µi α k x y F i F j f βi F k βj αj µj β k in C and corresponds to D 1 * {k∶ i → j} → C. • A 2-cell of C F from (f, µ) to (g, δ) is given by a 2-functor D 2 * J → C whose restrictions to D 1 * J are (f, µ) to (g, δ) . Explicitly, this means that such a 2-cell is given by a 2-cell Ξ∶ f → g of C satisfying δ i * 1 (β i * 0 Ξ) = µ i for all i in Ob(J), i.e., it provides a 2-cell from (f, µ i ) to (g, δ i ) in the slices of C over the objects F i. Remark 1.2.5. From the explicit definition, it is straightforward to check that an object in C F corresponds to a pair (x, α), where x is an object of C and α∶ ∆x → F is a lax natural trasformation, where ∆x is the constant 2-functor on x from J to C. A 1-cell of C F corresponds to a pair (f, µ), where f ∶ x → y is a 1-cell of C and µ∶ β ⋅ ∆f → α is a modification. Remark 1.2.6. For a 1-cell (f, µ)∶ (x, α) → (y, β) of C F we have not imposed any equations to be satisfied for a 2-cell Γ∶ k → k ′ of J. This is because they are implied by the relations already present. To be more precise, let us consider D 1 * D 2 as a strict ∞-category, in fact a strict 4-category, and let us describe its 2-categorical truncation in detail. There are four "generating" 3-cells in D 1 * D 2 (generating here is meant in the sense of polygraphs or computads, see for instance [4, 1.4]). These are sent via (f, µ)∶ D 1 * D 2 → C to the identities witnessing: • the relation (1) for x and Λ, that we denote by x * Λ; • the relation (1) for y and Λ, that we denote by y * Λ; • the relation (2) for f and k, that we denote by f * k; • the relation (2) for f and k ′ , that we denote by f * k ′ . These are all relations that we have by assumption on (x, α) and (y, β) or by the requirement for 1-cells of J that (f, µ) must satisfy. The unique 4-cell of D 1 * D 2 has as source (resp. target) an appropriate whiskered composition of the preimages of x * Λ and f * k (resp. of f * k ′ and y * Λ). Since truncation identifies source and target 2-cells of every 3-cell, we see that the condition for (what we can denote by) f * Λ is already satisfied. A similar argument explains why a 2-cell Ξ of C F needs no further coherence constraints than those corresponding to objects of J, the ones with respect to 1-cells and 2-cells of J being automatically satisfied. 1.2.7. LetJ = (J, E) be a marked 2-category and F ∶ J → C a 2-functor. As with the natural transformations, we can use the marking E in order to impose invertibility conditions on the slice C F . We denote by C F E , or simply by C F when E is clear from the context, the full sub-2-category of C F spanned by the objects (x, α) for which the 2-cell α k ∶ F k * 0 α i → α j is invertible for all k∶ i → j in E. If E consists of just the identity 1-cells of J, the 2-categories C F E and C F coincide. If E consists of all the 1-cells of J then we get the pseudo slice of C over F . 1.2.8. ForĀ = (A, E A ) andB = (B, E B ) two marked categories, it is possible to define their marked joinĀ * B. This is the 2-category A * B where we formally invert all the 2-cells of the kind ∅ ⋆ b a ⋆ ∅ b ′ ∅⋆g a⋆b a⋆b ′ a⋆g and a ⋆ ∅ b a ′ ⋆ ∅ a⋆b f ⋆∅ a ′ ⋆b f ⋆b for f an element of E A and g an element of E B . This operation defines a functor 2-Cat + × 2-Cat + → 2-Cat, which agrees with the standard join on minimally marked 2-categories. The functors 2-Cat + → 2-Cat A 2-Cat + → 2-Cat B B ↦ (Ā * B, i A ∶ A →Ā * B)Ā ↦ (Ā * B, i B ∶ B →Ā * B), both have a right adjoint functor. These adjoint functors are denoted by 2-Cat A → 2-Cat + 2-Cat B → 2-Cat + (C, u∶ A → C) ↦ C u E (C, v∶ B → C) ↦ C v F and we will often omit the marking in the notation of the slice, when it is clear from the context, coherently with the notation of the previous paragraph. Notice that, by adjunction, a 1-cell (f, µ) of C u E (resp. of C v F ) is marked if and only if it corresponds to a 2-functorĀ * D ♯ 1 → C (resp. to a 2-functor D ♯ 1 * B → C), where D ♯ 1 is the category D 1 with its unique non-trivial 1-cell marked. More explicitly, this means that for every object a ofĀ, (resp. b ofB) the triangle x u(a) y f µ f ≃ , resp. x v(b) y f µ f ≃ , is filled with an invertible 2-cells µ f . We stress the fact that we consider the output of the marked join as a bare 2-category, without marking. Even though it inherits a natural marking from the two inputs, it will play no role in what follows. 1.2.9. Following the notation of the previous paragraph, letĀ be the terminal marked 2-category D 0 , so that a marked 2-functor D 0 → C simply corresponds to an object c of C. The marked 2-category C c (resp. C c ) has C c (resp. C c ) as underlying 2-category (cf. 1.2.3) and the marked edges are the triangles filled with an invertible 2-cell. These marked 2-categories appear in [1] where Abellán García and Stern denote them by C † c↙ . 1.3. Bilimits and bicolimits. In this section we recall the notions of E-bilimit of a 2-functor with source a marked 2-category (J, E). 1.3.1. LetJ = (J, E) be a marked 2-category, C a 2-category and F ∶ J → C a 2-functor. For a given object x of C, we denote by ∆x the constant 2-functor on x from J to C. Given a 2-functor F ∶ J → C, an E-lax F -cone α with vertex x, that we shall often denote by (x, α), is an object of [J, C] E (∆x, F ). This means that for every object i of J we have a 1-cell α i ∶ x → F i of C, for every 1-cell k∶ i → j of J we have a 2-cell x x F i F j αi αj F k α k invertible whenever k is in E. Moreover, this assignment has to respect the identities and be functorial. Finally, for every 2-cell Γ∶ k → k ′ of J we have the relation α k = α k ′ * 1 (F Γ * 0 α i ). Hence, an E-lax F -cone with vertex x corresponds to an object (x, α) of the 2-category C F , that is the E-lax slice of C over F . Remark 1.3.2. Notice that the assignment of a 2-functor F ′ ∶ D 0 * J → C corresponds to the data of the 2-functor F together with an E-lax cone λ∶ ∆x → F over F , where x is the image via F ′ of the unique object of D 0 . The 2-category D 0 * J will be denoted byJ ◁ , and we will useJ ▷ for the dual case of cocones. 1.3.3. Let F ∶ J → C be a 2-functor, withJ = (J, E) a marked 2-category, and α∶ ∆x → F an E-lax F -cone. For every object z of C there is a canonical functor α * = α ⋅ ∆(−)∶ C(z, x) → [J, C] E (∆z, F ) given by post-composition with α. More explicitly, for f ∶ z → x a 1-cell of C we define α ⋅f to be α i f ∶ z → F i for every i in Ob(J) and α k * 0 f for every 1- cell k∶ i → j of J. A 2-cell ζ∶ f → g∶ z → x defines an evident modification α ⋅ ζ∶ α ⋅ f → α ⋅ g. Definition 1.3.4. For a 2-functor F ∶ J → C, withJ = (J, E) a marked 2-category, an E-lax F -cone (ℓ, λ) is said to be an E-bilimit cone if the canonical functor (3) λ * = λ ⋅ ∆(−)∶ C(x, ℓ) → [J, C] E (∆x, F ) is an equivalence of categories for every x in Ob(C). We also say that (ℓ, λ) is the 1.4. Grothendieck construction for 2-categories. Fibrations of 2-categories were initially introduced by Hermida in his paper [16], but his definition is not powerful enough to obtain a Grothendieck construction for such fibrations. This notion was later perfected by Buckley, who gave the correct definition in [7] and proved the corresponding (un)straightening theorem. In what follows we present a concise summary of the main results which are relevant for our treatment. We start by recalling the standard notion of cartesian edge and cartesian fibration for the 1-categorical setting. E-bilimit of F . The dual definition gives the notion of E-bicolimit of F , which is just the E op - bilimit of the 2-functor F op ∶ J op → C op .Definition 1.4.1. Let p∶ E → B be a functor between categories. An arrow f ∶ x → y of E is p-cartesian if the square E(a, x) E(a, y) B(pa, px) B(pa, py) f ○− pa,x pa,y p(f )○− is a pullback square of sets for any a in Ob(E). The functor p∶ E → B is a cartesian, or Grothendieck, fibration if for any object e ∈ E and any arrow f ∶ b → p(e) in B there exists a p-cartesian arrow h∶ a → e in E with p(h) = f . Now we recall the notion of cartesian 1-cell and 2-cell for the 2-categorical setting. Definition 1.4.2. Let p∶ E → B be a 2-functor between 2-categories. • A 1-cell f ∶ x → y in E is p-cartesian ifE(a, x) E(a, y) B(pa, px) B(pa, py) f ○− pa,x pa,y p(f )○− • A 2-cell α∶ f ⇒ g∶ x → y in E is p-cartesian if it is a p x,y -cartesian 1-cell, with p x,y ∶ E(x, y) → B(px, py). 1.4.3. We spell out explicitly the pullback defining a p-cartesian 1-cell f ∶ x → y, since we will need it in the following sections. The pullback gives existence and uniqueness property both at the level of 1-cells and of 2-cells. 1-cells: for every 1-cell g∶ a → y of E and every 1-cell k∶ pa → px in B such that pf * 0 k = pg, there is a unique 1-cellk∶ a → x such that fk = g and pk = k. (1) for every object e ∈ E and every 1- 2-cells: for every 2-cell α∶ g → g ′ ∶ a → y of E and every 2-cell τ ∶ k → h∶ pa → px in B such that pf * 0 τ = pα, there is a unique 2-cell β∶k →h such that f * 0 β = α and pβ = τ .cell f ∶ b → p(e) in B there exists a p- cartesian 1-cell h∶ a → e in E with p(h) = f . (2) for every pair of objects x, y in E, the map p x,y ∶ E(x, y) → B(px, py) is a Cartesian fibration of categories. (x, y, z) in E, the functor ○ x,y,z ∶ E(y, z) × E(x, y) → E(x, z) sends p y,z × p x,y -cartesian 1-cells to p x,z ones. In this case we call E the total 2-category of the fibration p, and B is said to be the base 2-category. Remark 1.4.6. Observe that condition (3) can be rephrased by requiring that given 1-cells in E of the form f ∶ w → x and g∶ y → z, the whiskering functors −○f ∶ E(x, y) → E(w, y) and g ○ −∶ E(x, y) → E(x, z) preserve cartesian 1-cells. This follows from the fact that horizontal composition can be obtained from vertical composition and whiskerings. Definition 1.4.7. Given a cartesian 2-fibration p∶ E → B we will denote by E cart the sub-2-category of E spanned by all objects and the p-cartesian edges between them. More precisely, E cart is the 2-category whose objects are the elements of Ob(E), and such that, for every pair of objects x and y, the hom-category E cart x, y is the full subcategory of E x, y spanned by p-cartesian edges. The motivation for introducing 2-fibrations is that they are a convenient way to encode functors into 2-Cat or 2-Cat ps . More precisely, we have the following result, which is a combination of Theorems 2.2.11 and 3.3.12 in [7]. The reason why we need to introduce pseudo-functoriality and pseudo-naturality is that the quasi-inverse of a strict 2-functor F ∶ A → B that is a biequivalence is not in general a strict 2-functor, but instead a pseudo-functor (see [20,Example 3.1]). In turn, via this biequivalence this also reflects the fact that given a 2-natural transformation between two 2-functors F, G∶ A → 2-Cat that is object-wise a biequivalence, then the transformation built up using the object-wise quasi-inverses is not a strict 2-natural transformation in general but instead a pseudo-natural one, even if the quasi-inverses are all strict 2-functors. This phenomenon is already evident for 2natural transforformations between 2-functors with values in the 2-subcategory Cat of 2-Cat that are object-wise equivalences; in fact, this will be the case of interest for us. Notation 1.4.10. By considering fibrations over B op , B co or B op co we obtain four different variants of 2-fibrations, corresponding to the four possible types of variance for 2-functors B → 2-Cat . Instead of using four different names, we will adopt the name fibration for each of these cases, and specify the variance when needed. 1.4.11. In the same paper [7], Buckley proves several weakening of Theorem 1.4.8, by looking at fibrations without a choice of lifts (which correspond to pseudofunctors) and fibrations of bicategories. We content ourself with the strict case as this is the level of generality needed for this work. We will only be using the part of Theorem 1.4.8 that deals with 1-fibrations, that we now record. Given the explicit description of the biequivalence in Theorem 1.4.8 provided by Buckley in his paper, we can detail the action on objects of the biequivalence of the previous corollary. [J, C] E (∆−, F )∶ C op → Cat . Indeed, thanks to Proposition 1.4.15 we have to compute the pullback E Cat D0 op C Cat op . p U [J,C] E (∆−,F ) An object of E is a pair (x, α) , where x is an object of C and α an object of [J, C] E (∆x, F ). That is, (x, α) is a E-lax F -cone with vertex x, which by what we observed in paragraph 1.3.1 corresponds precisely to an object (x, α) in C F . A 1-cell in E from (x, α) to (y, β) is a pair (f, µ), where f ∶ x → y is a 1-cell in C and µ∶ β ⋅ ∆f → α is a modification. For every object i in J, this corresponds to a commutative diagram x y F i f αi βi µi in C, i.e., the pair (f, µ) is a 1-cell of C F . A 2-cell in E from (f, µ) to (f ′ , µ ′ ) corresponds to a 2-cell Ξ of C which is a modification from µ to µ ′ . That is, for every i in J we have a commutative diagram x y F i f f ′ αi βi Ξ µi = x y F i f αi βi µ ′ i This is precisely a 2-cell Ξ∶ (f, µ) → (f ′ , µ ′ ) in C F . Bifinality In this section we motivate and provide the definition of contraction and bifinal object and we study some of their basic properties. Contractions and local terminality. 2.1.1. Recall that a category C has a terminal object c if and only if the canonical projection functor C c → C has a section mapping c to 1 c . This is straightforward, since such a section provides for each object x of C a morphism γ x ∶ x → c and moreover for every morphism f ∶ x → c the triangle • the image of the center c is equal to 1 c ; and • for every x in Ob(C) the 2-cell γ(γ(x)) (that we will denote by γ 2 x ) is the identity of γ x . We will also say that the object c is bifinal if there is a contraction on C having c as center. • For every x in Ob(C) we have a 1-cell γ x ∶ x → c, such that γ c = 1 c . • For every 1-cell f ∶ x → y of C we have a 2-cell x y c f γx γy γ f such that γ γx = Id γx . • For every composable pair of 1-cells f, g the relation γ g * 0f = γ f * 1 (γ g * 0 f ) holds. • For every 2-cell α∶ f → g of C the relation γ f = γ g * 1 (γ y * 0 α) holds. From this explicit description it is easy to see that a contraction on C with center c also corresponds to the datum of an object γ in [C, C](1 C , ∆c) where γ c = 1 c and such that the collection of the 1-cells of the form γ x are mapped by γ to identities. A contraction uniquely determines a set E c of 1-cells consisting of (the identities together with) the 1-cells of the form γ x . With this at hand, it is clear that such a contraction determines an object of [C, C] Ec (1 C , ∆c). 2.1.4. The notion of contraction in the more general case of strict ∞-categories is due to Ara and Maltsiniotis [3] and it also appears under the name of "initial/terminal structure" in an unpublished text of Burroni [8]. Beware that what we call here contraction for simplicity is in fact the dual of the notion given in [3]: our notion of contraction corresponds to what they call dual contraction. Notice also that their definition is equivalent but stated differently. In fact, they use the notion of (lax) Gray tensor product in the definition of contraction. This is a closed monoidal structure that was first introduced by Gray on 2-categories [15] and was later generalized to strict higher categories by Al-Agl and Steiner [2] and alternatively by Crans in his thesis [11]. With the Gray tensor product ⊗∶ 2-Cat × 2-Cat → 2-Cat at our disposal, we can define a contraction on C with center c to be a 2-functor γ∶ D 1 ⊗ C → C such that • the restriction to γ {0}×C is the identity on C and the restriction to γ {1}×C is constant on c, • the image of {0 → 1}, {c} is the identity of c, and • for every x in Ob(C) the image of γ x = γ {0 → 1}, x is the identity 2-cell. Remark 2.1.5. A priori, one might want to weaken the notion of contraction, according to the homotopy coherent approach of (weak) higher categories. Indeed, for a contraction γ on C with center c, we might request the 2-cell γ γx to be an invertible 2-cell, while we require the stronger condition of being the identity of γ x . However, in our 2-categorical framework this stronger condition follows from the relations we impose on 2-cells. More precisely, the 2-cell γ γx ∶ γ x → γ x ∶ x → c must satisfy the equality γ γx = γ γx * 1 γ γx . As γ γx is an invertible 2-cell, this implies that it must be the identity 2-cell of γ x . 2.1.6. If c is a bifinal object of a 2-category C, then the contractions we can associate to it are essentially unique. Indeed, let α and β be two contractions on C with center c. For every object x of C we have two 1-cells α x , β x ∶ x → c. Applying α to β x and β to α x we get two 2-cells of C. Finally, applying once more α and β to β αx and α βx respectively we get the relations α βx * 1 β αx = 1 αx and β αx * 1 α βx = 1 βx . Hence, α βx and β αx are inverses of one another. Similarly, for every 1-cell f ∶ x → y of C one gets (4) α βx * 1 β f = α f * 1 (α βy * 0 f ) so that α f and β f can be obtained one from the other via compositions with invertible 2-cells. In particular, equation (4) provides a pseudo natural transformation ρ∶ α → β∶ C → C c defined by ρ x = (1 x , α βx ) and for which the square α x α y β x β y α f ρx ρy β f in C c commutes for every 1-cell f ∶ x → y of C. The above paragraph shows that if a contraction with center c exists then it is essentially unique. The following result concerns the existence of such a contraction, relating it to the existence of an appropriate supply of terminal arrows with target c. Proposition 2.1.7. Let C be a 2-category and c ∈ Ob(C) an object. Let f a ∶ a → c be a choice of a morphism from each object a ∈ Ob(C) to c. Then the following are equivalent: (1) There exists a contraction γ with center c such that γ a = f a for every a. (2) Every f a is terminal in C(a, c) and f c = 1 c . Proof. Assume given a contraction γ with center c and γ a = f a for every a, so that in particular f c = 1 c by definition. Then for every g∶ a → c we have a 2-cell γ g ∶ g → γ a = f a . We claim that this is the unique 2-cell from g to γ a . Indeed, Given another 2-cell β∶ g → γ a , we can apply γ to it, resulting in the equality γ g = γ γa * 1 β. Since γ γa is an identity 2-cell, we get that β and γ g must coincide. On the other hand, suppose that f c = 1 c and that each f a is terminal in C(a, c). We define a contraction γ as follows. On objects we define γ a = f a and on 1-cells g∶ a → b we set γ g to be the unique 2-cell γ b * 0 g → f a . We have to check that this assignment defines a contraction. For every pair x f → y g → z of composable 1-cells of C, we have two parallel 2-cells γ g * 0f and γ f * 1 (γ g * 0 f ) from γ z * 0 g * 0 f to γ x . As the target γ x is a terminal object in C(x, c), it follows that these 2-cells must be equal. For the same reason, for every 2-cell Ξ∶ f → g of C the relation γ f = γ g * 1 (γ y * 0 Ξ) holds. Finally, γ γa ∶ γ a ⇒ γ a is a 2-cell from a terminal 1-cell to itself and is hence the identity. x ∈ E, the induced map [D, E] E D (F, ∆x) → [C, E] E C (F u, ∆x) is an equivalence of categories. A terminal object c in a category C can be characterized by the fact that the functor c∶ D 0 → C is final. Indeed, c is terminal if and only if the identity functor 1 C ∶ C → C admits c as its colimit. Our goal in this section is discuss the analogous situation in the setting of final 2-functors D 0 → C, for C a 2-category. For this, we will need to extend the notion of contraction to the framework of marked 2categories. Definition 2.2.4. Let (C, M ) be a marked 2-category and γ a contraction on C with center c. We say that γ is an M -contraction if the following two conditions hold: (1) For every x ∈ Ob(C) the edge γ x of C is marked. (2) For every marked edge f of (C, M ), the 2-cell γ f of C is invertible. If such an M -contraction exists, we will say that c is M -bifinal. Remark 2.2.5. Any contraction γ with center c is automatically an M γ -contraction, where M γ denotes the collection of edges γ a for a ∈ Ob(C) (together with the identities). On the other hand, the combination of the above conditions is clearly equivalent to the statement that for each a ∈ Ob(C) there exist marked edges a → c, and that these are all terminal in their respective hom categories. In particular, conditions (1) and (3) hold in this case. In addition, since all identities are marked (2) is implied, and when the collection of marked edges is closed under composition we also get that (4) holds. The following result shows that the notions of finality for an object that we have introduced so far coincide. we have that f a is terminal in C(a, c) for every a ∈ Ob(C). By Proposition 2.1.7 there exists a contraction γ such that γ a = f a for every a ∈ Ob(C). To see that γ is an M -contraction we now note that γ a is marked by construction, and for g∶ a → b a marked 1-cell, condition (4) entails that γ g ∶ γ b * 0 g → γ a is a 2-cell between two terminal 1-cells, and is hence invertible. This shows that γ is an M -contraction. (ii) ⇒ (iii). Let c be an M -bifinal object and γ an M -contraction with center c such that γ a = f a for every a ∈ Ob(C). We need to show that for any 2-functor F ∶ C → D, and any d ∈ D, the evaluation at c map • c ∶ [C, D] M (F, ∆d) → D(F c, d), is an equivalence of categories. We construct an explicit inverse γ * ∶ D(F c, d) → [C, D] M (F, ∆d) to this functor by means of post-composition with γ. More precisely, given f ∶ F c → d, we can consider the family of 1-cells f F (γ x )∶ F x → d x∈Ob(C) , and that of 2-cells f F (γ h )∶ f F (γ x ) → f F (γ y )F (h) h∈C(x,y) . It is clear that these data all organize into an M -lax natural transformation γ * (f ) from F to ∆d and one can similarly obtain a modification γ * (α)∶ γ * (f ) → γ * (g) from a 2-cell α∶ f → g. We now claim that γ * is inverse to • c . Indeed, on the one hand, the composition • c ○ γ * is the identity on D (F c, d). In the other direction, we need to show that the M -lax transformations ρ and γ * (ρ c ) are isomorphic objects in [C, D] M (F, ∆d), naturally in ρ. For x an object of C, we get γ * (ρ c ) x = ρ c F γ x . This gives us a triangle F x F c d F γx ρx ρc βx in D, where the 2-cell β x is ρ γx . As γ x is in M the 2-cell ρ γx is invertible by assumption. We need to check that the collection of the 2-cells β x , for all objects x of C, organizes into an invertible modification between ρ and γ * (ρ c ). For this we must check that given any 1-cell f ∶ x → y of C, the equation ρ f * 1 (ρ γy * 0 F f ) = ρ γx * 1 (ρ c * 0 F γ f ). holds. This is precisely the relation satisfied by the action of ρ on the 2-cell it then follows that the identity 2-functor 1 C ∶ (C, M ) → C admits an M -bicolimit cocone γ such that γ c = 1 c . Now by assumption every object a ∈ Ob(C) admits a marked edge f a ∶ a → c. Since γ is an M -lax cocone the 2-cell γ f ∶ f a ⇒ γ a is invertible, and hence f a and γ a are isomorphic 1-cells in C(a, c). We now observe that for a general morphism g∶ a → c the condition that γ g is invertible is closed under isomorphism in C(a, c). It then follows that γ γa is invertible as well, so that γ constitutes a contraction with center c. By Proposition 2.1.7 we now get that each γ a is terminal in C(a, c), and since γ c = Id c this means that that c is pre-final. In addition, Condition (3) holds by assumption, and since γ is an M -lax cocone we have that for every marked edge g∶ a → b the 2-cell γ g is invertible, and so γ a ≅ g * 0 γ b . We thus have that pre-composition C(b, c) → C(a, c) with any marked edge g∶ a → b preserves terminal objects, which shows (4). Proof of Proposition 2.2.7. This is a particular case of the implication (ii) ⇔ (iii) of Proposition 2.2.11 applied to M c . Slice fibrations In this section we further study the slice fibrations p∶ C F → C associated to a marked 2-categoryJ = (J, E) and a 2-functor F ∶ J → C as in Paragraph 1.2.7. We begin in §3.1 by giving an explicit description of the p-cartesian edges in C F , and show that they coincide with the marked edges with respect to the marking of Paragraph 1.2.8. In §3.2 we then focus on the particular case where J = D 0 , so that C F = C F ( * ) is a representable fibration, and show that in this case the object 1 F ( * ) is the center of a contraction relative to the collection of p-cartesian edges. Finally, in §3.3 we construct a modified 2-category of cones which projects to C F , and show that this projection is a biequivalence if and only if F admits an E-bilimit. Cartesian edges in slice fibrations. 3.1.1. Let us fix a marked 2-categoryJ = (J, E), a 2-functor F ∶ J → C, and consider the associated projection 2-functor p∶ C F → C. We wish to describe the p-cartesian morphisms of C F . Consider an object (x, λ)∶ D 0 * J → C of C F , that we can represent by x F i F j λi λj F h λ h for h∶ i → j in J, with λ h invertible whenever h is in E. We claim that every 1-cell (f, σ)∶ D 1 * J → C from (z, α) to (x, λ): z x F j f αj λj σj ≅ such that the 2-cell σ j of C is invertible for every object j of J, is a p-cartesian lift of f ∶ z → x. In the notations of Paragraph 1.2.8, we are claiming that (f, σ) is a p-cartesian lift whenever it is represented by a 2-functor D ♯ 1 * J → C. One possible such choice is given by the precomposition of (x, λ) with f , i.e., where σ j is the identity of λ j f for all j in Ob(J). Indeed, let (f ′ , µ)∶ (z ′ , α ′ ) → (x, λ) be a 1-cell of C F and g∶ z ′ → z a 1-cell of C such that f g = f ′ . We wish to find an edge (g,μ)∶ (z ′ , α ′ ) → (z, α) such that (5) (f, σ) * 0 (g,μ) = (f ′ , µ) . Now (f, σ) * 0 (g,μ) = (f g,μ • σ), where we set (μ • σ) j =μ j * 1 (σ j * 0 g) so that necessarily f * 0 g = f ′ and µ =μ•σ. As p(g,μ) = g, we must have (g,μ) = (g, µ•σ −1 ), where we set (µ • σ −1 ) j = µ j * 1 (σ −1 j * 0 g) for all objects j in J. Said otherwise, if such a 1-cell (g,μ) of C F exists, then for any object j of J the 2-cellμ of C must be equal to µ j * 1 (σ −1 j * 0 g). This shows that if (g,μ) satisfying (5) exists, then it is unique. As for its existence, it is easy to check that the following diagram z ′ z F i F j g αi F h α ′ i α ′ j µi α ′ h z ′ z F i F j g αi F h αj α ′ j µj α h in C commutes for every edge h∶ i → j in J, so thatμ∶ α ′ → α defines a 1-cell in C F . This concludes the existence and uniqueness property for 1-cells. Let Ξ∶ (f ′ , µ) → (f ′′ , µ ′ ) be a 2-cell of C F . In particular, for any object i of J we have a commutative diagram z ′ x F i f ′ f ′′ α ′ i λi Ξ µ ′ i = z ′ x F i f ′ α ′ i λi µi in C. Fix τ ∶ g → g ′ a 2-cell of C such that p(Ξ) = Ξ = f * 0 p(τ ) . We know from the argument above that there are unique lifts (g,μ)∶ (z ′ , α ′ ) → (z, α) , (g ′ , µ ′ )∶ (z ′ , α ′ ) → (z, α) such that (f, σ) * 0 (g,μ) = (f ′ , µ) and (f, σ) * 0 (g ′ , µ ′ ) = (f ′′ , µ ′ ). We wish to find a unique 2-cellΞ∶ (g,μ) → (g ′ , µ ′ ) of C F such that p(Ξ) = τ and 1 (f,σ) * 0 τ = Ξ. in C F . The condition p(Ξ) = τ forces the uniqueness. Since 1 (f,σ) * 0 τ = 1 f * 0 τ = Ξ, the existence part follows once we show that τ is a 2-cell from (g,μ) to (g ′ , µ ′ ). For every i in Ob(J) we have µ ′ i * 1 (α i * 0 τ ) = µ ′ i * 1 (σ −1 i * 0 g) * 1 (α i * 0 τ ) = µ ′ i * 1 (λ i * 0 f * 0 τ ) * 1 (σ −1 i * 0 g ′ ) = µ ′ i * 1 (λ i * 0 Ξ) * 1 (σ −1 i * 0 g ′ ) = µ i * 1 (σ −1 i * 0 g ′ ) =μ i , where the second equality is given by the interchange law, the third and fourth by the assumptions and the first and the last one are the definitions of µ ′ andμ, respectively. We have thus shown that the 1-cell (f, σ) satisfies the properties of p-cartesian 1-cell detailed in paragraph 1.4.3. Proof. We have shown in the paragraphs above that the edges of C F of this form are indeed p-cartesian. If (g, µ)∶ (z, β) → (x, α) is a p-cartesian edge, then there exists a unique 1-cell (1 z , µ ′ )∶ (z, α ⋅ g) → (z, β) such that (g, µ) * 0 (1 z , µ ′ ) = (g, 1 α⋅g ), which is equivalent to the condition µ ′ i * 0 µ i = 1 αi * 0g for all i in Ob(J). But µ ′ i is an invertible 2-cell of C, since by [7, Proposition 2.1.4(2)] the 1-cell (1 z , µ ′ ) is an isomorphism of C F and so in C F . Hence, µ i is an invertible 2-cell of C for all objects i of J. 3.1.3. Proposition 3.1.2 yields an explicit description of the sub-2-category (C F ) cart of C F spanned by all objects and the p-cartesian edges between them (see Definition 1.4.7). Indeed, the 1-cells of (C F ) cart are given by z z ′ F i g α α ′ σi ≃ with σ i an invertible 2-cell of C for every i in Ob(J). Representable fibrations. In this section we focus on the particular case of slice fibrations where J = D 0 , that is, a 2-functor F ∶ J → C simply corresponds to an object x ∈ C. In this case C x is equipped with a designated object 1 x , whose finality properties we wish to understand. We start by recalling the classical 1-categorical scenario. The 1-categorical case. Let p∶ E → B be a cartesian fibration of 1-categories and suppose that it is represented by an object x of B. By this we mean that the functor Unlike the 1-categorical case, the objectx of E is not biterminal in general (x biterminal means that for every object z in E the category E(z,x) is equivalent to D 0 ). Instead, we have the following: B(−, x)∶ B op → Set is classified by p∶ E → B, i.e.,y x x β β ′ α 1x Ξ ζ ′ = y x x β α 1x ζ , that is ζ ′ * 1 Ξ = ζ. We claim that the object (α, 1 x ). Moreover, it is clearly the unique going from (β, ζ) to (α, 1 α ), as any such morphism Ξ must satisfy 1 x * 1 Ξ = Ξ = ζ. In addition, taking α = 1 x we also get that the identity on 1 1 α ) of B x (α, 1 x ) is terminal. Indeed, for every other object (β, ζ) in B x (α, 1 x ) we have that ζ∶ (β, ζ) → (α, 1 α ) is a morphism in B x (α,x is terminal in B x (1 x , 1 x ). We have thus established that 1 x is a pre-final object of B x . To show that it is M cart -final we now point out that by Proposition 3.1.2 each of the arrows (α, 1 α )∶ α → 1 x just contracted are p-cartesian, so that (3) holds, and that any other p-cartesian edge α → 1 x is of the form y x x β α 1x ≃ and hence isomorphic in B x (α, 1 x ) to (α, 1 α ). This means that all p-cartesian edges α → 1 x are in fact terminal, and since the collection of p-cartesian edges is also closed under isomorphism we have that these are exactly the terminal edges in B x (α, 1 x ). Condition (4) is then a consequence of the fact that p-cartesian edges are closed under composition. 3.2.2. Combining Proposition 3.2.1 and Proposition 2.1.7 we obtain a contraction γ on B x with center 1 x . This contraction can be explicitly described as follows: • To a given object (y, α) we assign the 1-cell (α, 1 α ), that we can depict by y x x α α = • For every 1-cell (f, ζ)∶ (y, α) → (z, β) we have to provide a 2-cell going from γ(z, β) * 0 (f, ζ) to γ(y, α). By definition, γ(z, β) * 0 (f, ζ) = (βf, ζ) and γ(y, α) = (α, 1 α ). We set this 2-cell γ(f, ζ) to be ζ. This assignment can be depicted as follows: Proof. Consider any object α∶ y → x of (B x ) cart . We wish to show that the category (B x ) cart (y, α), (x, 1 x ) is equivalent to the terminal category D 0 . First of all, this category is non-empty, since it has α, 1 α as an object. Given a 1-cell (β, ζ)∶ (y, α) → (x, 1 x ), the composition 1 x * 0 β = β must be the source of ζ and moreover ζ∶ β → α must be an invertible 2-cell of B by Proposition 3.1.2. For every pair of 1-cells (β, ζ), (β ′ , ζ ′ )∶ (y, α) → (x, 1 x ), a 2-cell τ ∶ (β, ζ) → (β ′ , ζ ′ ) must satisfy the relation ζ ′ * 1 τ = ζ. It follows that such a 2-cell is invertible and in fact it always exists and it is unique, namely it is given by τ = (ζ ′ ) −1 * 1 ζ. This shows that for every object (y, α) in (B x ) cart the category (B x ) cart (y, α), (x, 1 x ) is the chaotic category on the objects of the form (β, ζ), with ζ∶ β → α invertible, and it is thus equivalent to the terminal category D 0 , i.e., (x, 1 x ) is biterminal. Remark 3.2.5. In the general case of a 2-functor F ∶ J → C and a marking E on J, the 2-category C F does not necessarily have a quasi-terminal object and similarly (C F ) cart does not necessarily have a biterminal object. 3.3. The modified 2-category of cones. LetJ = (J, E) be a marked 2-category, F ∶ J → C be a 2-functor, ℓ an object of C, λ∶ ∆ℓ → F an E-cone over F and F ◁ ∶J ◁ → C the corresponding 2-functor (cf. Remark 1.3.2). In this section we introduce an auxiliary 2-category of cones that is always biequivalent to C ℓ = C ℓ and that is biequivalent to C F if and only if (ℓ, λ) is an E-bilimit of F . Definition 3.3.1. We define C F ◁ as the 2-full sub-2-category of C F ◁ whose objects are the 2-functors D 0 * D 0 * J → C such that their restrictions to D 0 * D 0 * {j} → C, for any object j of J, determine diagrams x ℓ F (j) h αj λj σj ≃ in C with σ j ∶ λ j h → α j an invertible 2-cell of C. 3.3.2. We denote an object of C F ◁ by (x, h, α, σ), where we mean that: • h∶ x → ℓ is a 1-cell of C; • the pair (x, α) is an E-cone over F , so that in particular for every 1-cell κ∶ i → j in J we have a triangle x F i F j αi αj F κ α h , with α κ invertible whenever κ ∈ E. • for every i in Ob(J), σ i ∶ λ i h → α i is an invertible 2-cell of C, that we can depict as x ℓ F (i) h αi λi σi ≃ . Notice that by Remark 1.2.5 we have that σ∶ λ ⋅ h → α is an invertible modification. The objects of the form (x, h, λ ⋅ h, ι), with ι i the identity 2-cell of λ i h for all i in Ob(J), are particularly simple and we will make use of them in the following proofs in order to simplify the coherences appearing in explicit form of the cells of C F ◁ . We shall also commit the notational abuse of denoting by ι the appropriate trivial modification, regardless of the 1-cell of C involved; for instance, for g∶ y → ℓ another 1-cell of C we shall write (y, g, λ ⋅ g, ι) where we mean that here ι i is the identity of λ i g for all i ∈ Ob(J). A 1-cell from (x, h, α, σ) to (y, g, β, τ ) is given by a triple (f, ζ, µ), where f ∶ x → y is a 1-cell in C, ζ∶ g * 0 f → h a 2-cell of C and µ is the modification satisfying µ * 1 (τ * 0 f ) = σ * 1 λ ⋅ ζ. Since τ is an invertible modification, we actually have that µ is defined as σ * 1 λ ⋅ ζ * 1 (τ −1 * 0 f ). Notice that a 1-cell from (x, h, λ ⋅ h, ι) to (y, g, λ ⋅ h, ι) must be of the form (f, ζ, λ ⋅ ζ). In light of Lemma 3.3.3 below it will suffice to describe the 2-cells of C F ◁ whose 0-dimensional source and target are of the form (x, h, λ ⋅ h, ι) and (y, g, λ ⋅ h, ι). Here and in what follows, we will always commit the abuse of denoting by ι the appropriate trivial modification. In this case, if we are given 1-cells (f, ζ, λ ⋅ ζ) and (f ′ , ζ ′ , λ ⋅ ζ ′ ) from (x, h, λ ⋅ h, ι) to (y, g, λ ⋅ g, ι), then we have that a 2-cell of C F ◁ is precisely a 2-cell of C ℓ , that is a 2-cell Ξ∶ f → f ′ ∶ x → y of C satisfying ζ ′ * 1 (g * 0 Ξ) = ζ. Indeed, if for instance k∶ i → j is a 1-cell of J, then by the coherences imposed by the 1-cells of C F ◁ we have λ j ζ ′ * 1 (λ k g * 0 f ′ ) * 1 (F k * 0 λ i g * 0 Ξ) = λ k h * 1 (F k * 0 λ i ζ ′ ) * 1 (F k * 0 λ i g * 0 Ξ) = λ k h * 1 (F k * 0 λ i ζ), and similarly for a 2-cell of J. Morally, the coherences of a 2-cell of C F ◁ are encoded by higher cells of D 2 * D 0 * D m , with m = 0, 1, 2, but since we are truncating at the 2-dimensional level, many of these higher cells are trivial (cf. Remark 1.2.6) and moreover we are choosing specific objects (and 1-cells) of C F ◁ which further simplify some of the coherences involved with identities. We shall now prove that the 2-functor q∶ C F ◁ → C ℓ is a biequivalence, following the strategy outlined in 1.1.5. The proof is subdivided in few steps and a preliminary auxiliary lemma, that we shall use to simplify some of the coherences involved. Proof. The 1-cell (1 x , σ) of C F is an isomorphism and it is such that (h, ι) * 0 (1 x , σ) = (h, σ). This is equivalent to say that (1 x , 1 h , σ −1 ) is an isomorphism from (x, h, h ⋅ λ, ι) to (x, h, α, σ) in C F ◁ . 3.3.4 (Surjectivity on objects). Consider an object h∶ x → ℓ of C ℓ . Then the object (x, h, h ⋅ λ, ι) of C F ◁ maps to (x, h). (Fullness on 1-cells) . Let (f, ζ)∶ (x, h) → (y, g) be a 1-cell in C ℓ , that we can depict as x y ℓ f h g ζ Given two objects (x, h, α, σ) and (y, g, β, τ ) of C F ◁ that lift the source and target (respectively) of (f, ζ), we want to find a 1-cell D 1 * D 0 * J → C from (x, h, α, σ) to (y, g, β, τ ) such that its projection via q is the 1-cell (f, ζ) of C ℓ . According to paragraph 3.3.2, the triple (f, ζ, µ) is a 1-cell of C F ◁ with the correct boundary, with µ = σ * 1 λ ⋅ ζ * 1 (τ −1 * 0 f ), and it is clearly mapped to (f, ζ). Notice that this lifting requires no additional data, but just commutativity conditions, so that such a lift must be unique. We also remark that if we choose (x, h, λ⋅h, ι) and (y, g, λ⋅g, ι) as objects lifting the source and target of (f, ζ), then the 1-cell (f, ζ, λ ⋅ ζ) between these two objects of C F ◁ maps to (f, ζ). A simple verification shows that the square (6) (x, h, α, σ) (y, g, β, τ ) (x, h, λ ⋅ h, ι) (y, g, λ ⋅ ζ, ι) (f,ζ,µ) (1x,1 h ,σ) (1y,1g ,τ ) (f,ζ,λ⋅ζ) of C F ◁ is commutative, where the vertical 1-cells are (the inverses of) those of Lemma 3.3.3. In particular, this means that it was actually enough to find the (unique) lift (f, ζ, λ ⋅ ζ) of (f, ζ) with (x, h, λ ⋅ h, ι) and (y, g, λ ⋅ g, ι) as source and target, since then the composition (1 y , 1 g , τ −1 ) * 0 (f, ζ, λ ⋅ ζ) * 0 (1 x , 1 h , σ) is a lift of (f, ζ) with (x, h, α, σ) and (y, g, β, τ ) as source and target. (Fullness on 2-cells). Let Ξ∶ (f, ζ) → (f ′ , ζ ′ )∶ (x, h) → (y, g) be a 2-cell in C ℓ , that we can depict as x y ℓ f f ′ h g Ξ ζ ′ = x y ℓ f h g ζ Given two objects (x, h, α, σ) and (y, g, β, τ ) of C F ◁ , and lifts of (f, ζ, µ) and (f ′ , ζ ′ , µ ′ ) to C F ◁ , we wish to find a 2-cell D 2 * D 0 * J → C from (f, ζ, µ) to (f ′ , ζ ′ , µ ′ ) such that its projection via q is the 2-cell Ξ of C ℓ . Thanks to Lemma 3.3.3 and the observation at the end of the previous point, we can again assume that α = λ ⋅ h, β = λ ⋅ g and σ i and τ i are the identity 2-cells for all objects i of J. Indeed, if we find a lift Ξ ′ of Ξ with (x, h, λ ⋅ h, ι) and (y, g, λ ⋅ g, ι) as source and target, then the 2-cell (1 y , 1 g , τ −1 ) * 0 Ξ ′ * 0 (1 x , 1 h , σ) is a lift of Ξ having (x, h, α, σ) and (y, g, β, τ ) as source and target. By the discussion in paragraph 3.3.2, we know that Ξ is a 2-cell of C F ◁ with correct source and target and that it clearly lifts Ξ. (Faithfulness on 2-cells). By the observation in paragraph 3.3.2. a 2-cell of C F ◁ from (f, ζ, λ ⋅ ζ) to (f ′ , ζ ′ , λ ⋅ ζ ′ ) simply amounts to a 2-cell Ξ∶ f → f ′ of C satisfying ζ ′ * 1 (g * 0 Ξ) = ζ, which is the same thing as a 2-cell of C ℓ from (f, ζ) to (f ′ , ζ ′ ). Therefore the map C F ◁ (f, ζ, λ ⋅ ζ), (f ′ , ζ ′ , λ ⋅ ζ ′ ) → C ℓ (f, ζ), (f ′ , ζ ′ ) is a bijection. Consider now two generic parallel 1-cells (f, ζ, µ) and (f ′ , ζ ′ , µ ′ ) of C F ◁ , say from (x, h, α, σ) to (y, g, β, τ ). The commutativity of diagram (6) implies that we have an isomorphism of categories C F ◁ (x, h, α, σ), (y, g, β, τ ) ≅ C F ◁ (x, h, λ ⋅ h, ι), (y, g, λ ⋅ g, ι) mapping (f, ζ, µ) to (f, ζ, λ ⋅ ζ), so that in particular we get a bijection C F ◁ (f, ζ, µ), (f ′ , ζ ′ , µ ′ ) ≅ C F ◁ (f, ζ, λ ⋅ ζ), (f ′ , ζ ′ , λ ⋅ ζ ′ ) . Hence, the 2-functor q∶ C F ◁ → C ℓ is faithful on 2-cells. Putting together the previous four points, we get the following result. and only if the corresponding 2-functor C ℓ → C F depicted below is a biequivalence. Proposition 3.3.8. The canonical projection q∶ C F ◁ → C ℓ is a biequivalence.C ℓ C F C Now, the triangle C F ◁ C ℓ C F ≃ ≅ of 2-categories commutes up-to an invertible 2-natural transformation given by Lemma 3.3.3 (see also diagram (6)). Hence, applying Proposition 3.3.8 together with the 2-out-of-3 property of biequivalences and the stability of biequivalences by 2-natural isomorphisms to the previous triangle we deduce that the 2-functor C ℓ → C F is a biequivalence if and only if C F ◁ → C F is one. Bilimits and bifinal cones 4.1. We fix a marked 2-categoryJ = (J, E) and a 2-functor F ∶ J → C, and we denote by p∶ C F → C the projection 2-functor. In this section we show that a cone (ℓ, λ) over F is an E-bilimit cone if and only if it is the center of a limiting contraction H on C F , as defined below. Notation 4.2. In what follows, we will use the letter H to denote a contraction, in contrast with the use of γ and other greek letters in previous sections. In fact, we will confine our use of greek letters to denote cones on diagrams. Definition 4.3. Let H be a contraction on C F . We say that H is a limiting contraction if it is an M cart -contraction, where M cart is the collection of p-cartesian arrows in C F . A cone (ℓ, λ) over C F will be called limiting bifinal if it is the center of a limiting contraction on C F . 4.4. More explicitly, the condition of being an M cart -contraction means that (1) For every cone (x, α) over F , the 1-cell H(x, α) is p-cartesian. (2) For every p-cartesian edge (f, µ)∶ (x, α) → (y, β), the 2-cell H(f, µ) of C F is invertible. The first condition then means that for all objects i of J the 2-cell Proposition 4.6. C F admits a limiting contraction with center (ℓ, λ) if and only if every cone (ℓ ′ , λ ′ ) admits a p-cartesian edge (ℓ ′ , λ ′ ) → (ℓ, λ) and the 2-functor {(ℓ, λ)} → (C F , M cart ) is final. Given a contraction H on C F with center (ℓ, λ), we may consider the canonical projection C F ◁ → C F , where F ◁ ∶ D 0 * J → C is the cone determined by (ℓ, λ). Proposition 4.7. If (ℓ, λ) is limiting bifinal then the associated projection 2-functor C F ◁ → C F is a biequivalence. Proof. Call H the M cart -contraction. We will prove that the projection C F ◁ → C F is a trivial fibration. Surjectivity on objects: For a cone (x, α) over F , the 1-cell H(x, α) is by definition an object of C F ◁ that lifts (x, α). Fullness on 1-cells: Let (x, h, α, σ) and (y, g, β, τ ) be objects of C F ◁ and fix a 1-cell (f, µ)∶ (x, α) → (y, β) of C F . We wish to lift (f, µ) to C F ◁ . Applying the contraction to (f, µ) gives a triangle (x, α) (y, β) (ℓ, λ) (f,µ) H(x,α) H(y,β) Γ in C F ,of C F . The composite 2-cell ζ = H(h, σ) −1 * 1 H(f, µ) * 1 H(g, τ ) * 0 (f, µ) of C F fills the triangle (x, α) (y, β) (ℓ, λ) (f,µ) (h,σ) (g,τ ) ζ therefore providing a 1-cell (f, ζ, µ, )∶ (x, h, α, σ) → (y, g, β, τ ) of C F ◁ lifting the 1-cell (f, µ) of C F . Fullness on 2-cells: Let (f, ζ, µ) and (f ′ , ζ ′ , µ ′ ) be two parallel 1-cells of C F ◁ from (x, h, α, σ) to (y, g, β, τ ). We wish to find a lift to C F ◁ for any 2-cell Ξ∶ (f, µ) → (f ′ , µ ′ ) of C F . Using the description of 3.3.2, we may view Ξ itself as such a lift: for this, we have to check that the identity (8) ζ ′ * 1 (g * 0 Ξ) = ζ is satisfied in C F . Applying the constraint of the contraction H to the 2-cell ζ∶ (gf, τ ⋅ µ) → (h, σ)∶ (x, α) → (ℓ, λ), where (gf, τ ⋅ µ) = (g, τ ) * 0 (f, µ), of C F we get the relation H(gf, τ ⋅ µ) = H(h, σ) * 1 ζ. By the functoriality of the contraction, the left-hand side of this equation is equal to H(f, µ) * 1 H(g, τ ) * 0 (f, µ) and moreover the 2-cell H(h, σ) is invertible, as (h, σ) is p-cartesian by assumption. This implies that we can write ζ = H(h, σ) −1 * 1 H(f, µ) * 1 H(g, τ ) * 0 (f, µ) and similarly ζ ′ = H(h, σ) −1 * 1 H(f ′ , µ ′ ) * 1 H(g, τ ) * 0 (f ′ , µ ′ ) . Hence, equation (8) is satisfied if and only if we have H(f ′ , µ ′ ) * 1 H(g, τ ) * 0 (f ′ , µ ′ ) * 1 (g, τ ) * 0 Ξ = H(f, µ) * 1 H(g, τ ) * 0 (f, µ) . Notice that by the interchange rule for (x, α) (y, β) (ℓ, λ) (f,µ) (f ′ ,µ ′ ) (g,τ ) H(y,β) Ξ H(g,τ ) we actually have H(g, τ ) * 0 (f ′ , µ ′ ) * 1 (g, τ ) * 0 Ξ = H(y, β) * 0 Ξ * 1 H(g, τ ) * 0 (f, µ) and the 2-cell H(g, τ ) is invertible, since (g, τ ) is p-cartesian by assumption, so that equation (8) is satisfied if and only if the following equation is: (9) H(f ′ , µ ′ ) * 1 H(y, β) * 0 Ξ = H(f, µ). Now, using the coherence for 2-cells given by the contraction H applied to the 2-cell on the left-hand side of the previous equation and using the constraints for which H(H(x, α)) = 1 H(x,α) and H(ℓ, λ) = 1 (ℓ,λ) , we finally get that equation (9) is indeed satisfied. Faithfulness on 2-cells: This is clear by the explicit description of the 2-cells of C F ◁ and C F , which are just 2-cells of C satisfying some coherence conditions. We now prove the converse of Proposition 4.7. For convenience, we first record the following observation about bilimits. Proposition 4.8. If (ℓ, λ) is an E-bilimit of the 2-functor F ∶ J → C, then C F admits a limiting contraction with center (ℓ, λ). Proof. Let (ℓ, λ) be an E-bilimit of F . Our goal is to construct a limiting contraction H with center (ℓ, λ). Since (ℓ, λ) is an E-bilimit the functor λ ⋅ (−)∶ C(x, ℓ) → [J, C] E (∆x, F ) is an equivalence for every x ∈ Ob(C). In particular, λ ⋅ (−) is essentially surjective, so that we can find, for any α ∈ [J, C] E (∆x, F ), a 1-cell h∶ x → ℓ of C and an isomorphism σ∶ α ≅ → λ ⋅ h. The data of α, h and σ then determine a p-cartesian 1-cell of C F depicted by the triangle (x, α) (x, λ ⋅ h) (ℓ, λ) (1x,σ) (h,σ) (h,1 λ⋅h ) We define our contraction on the level of objects by associating to (x, α) the pcartesian edge H(x, α) ∶= (h, σ), where we may assume without loss of generality that we have picked H(ℓ, λ) to be 1 (ℓ,λ) = (1 ℓ , 1 λ ). Now consider a 1-cell (f, µ)∶ (x, α) → (y, β) in C F , that can be seen as a 1-cell µ of [J, C] E (∆x, F ) from α to β ⋅ f (that is, a modification between the E-lax natural transformations α and β ⋅f ). Let (h, σ) = H(x, α) and (g, τ ) = H(y, β) be the 1-cells constructed above. The composite ρ = σ −1 * 1 µ * 1 (τ * 0 ∆f ) then gives a morphism from λ ⋅ (gf ) to λ ⋅ h in the category [J, C] E (∆x, F ). Since λ ⋅ (−) is fully faithful this morphism lifts to a unique morphism Γ∶ gf → h in C(x, ℓ), which we can write as a 2-cell x y ℓ f h g Γ , in C. One immediately checks, by applying λ ⋅ (−), that we have σ i * 1 (λ i * 0 Γ) = µ i * 1 (τ i * 0 f ) for all i ∈ Ob(J). We may thus consider Γ as a 2-cell of C F with source H(y, β) * 0 (f, µ) and target H(x, α). We then extend our contraction to the level of 1morphisms by setting H(f, µ) = Γ. Explicitly, the 2-cell H(f, µ) is given by the pasting Γ in C F , where the left-most and the right-most triangles are commutative by definition. With this description at hand, and given that Γ is uniquely determined by the fully faithfullness of λ ⋅ (−), it is easy to verify that the assignment H is compatible with composition of 1-cells, that is H((f ′ , µ ′ ) ○ (f, µ)) = H(f, µ) * 1 (H(f ′ , µ ′ ) * 0 (f, µ)) for any pair of composable 1-cells (f, µ), (f ′ , µ ′ ) of C F . In addition, since λ ⋅ Γ = σ −1 * 1 µ * 1 (τ * 0 ∆f ) with τ and σ invertible and λ⋅(−) is an equivalence we have that Γ is invertible whenever µ is invertible. We conclude that H sends p-cartesian 1-cells to invertible 2-cells, and consequently that H(H(x, α)) = 1 H(x,α) by Remark 2.1.5. To show that H constitutes a limiting contraction it will now suffice to show that for any 2-cell Ξ∶ (f, µ) → (f ′ , µ ′ )∶ (x, α) → (g, β) of C F , the relation (11) H(f, µ) = H(f ′ , µ ′ ) * 1 (H(y, β) * 0 Ξ) is satisfied. We begin by noticing that since Ξ is a 2-cell of C F , by definition we have µ i = µ ′ i * 1 (β i * 0 Ξ) for every object i of J. Using the previous part of the proof, we know that there exist 1-cells h∶ x → ℓ and g∶ y → ℓ, isomorphic modifications σ∶ λ ⋅ h → α and τ ∶ λ ⋅ g → β and a unique 2-cells Γ∶ gf → h and Γ ′ ∶ gf ′ → h of C such that µ = σ * 1 λ ⋅ Γ * 1 (τ −1 * 0 ∆f ), µ ′ = σ * 1 λ ⋅ Γ ′ * 1 (τ −1 * 0 ∆f ′ ). So the modification µ is equal to σ * 1 λ ⋅ Γ ′ * 1 (τ −1 * 0 ∆f ′ ) * 1 (β * 0 ∆Ξ), which using the interchange law can be rewritten as σ * 1 λ ⋅ Γ ′ * 1 (λ ⋅ g * 0 ∆Ξ) * 1 (τ −1 * 0 ∆f ). Since σ and τ are invertible by construction, we obtain the relation (12) λ ⋅ Γ = λ ⋅ Γ ′ * 1 (λ ⋅ g * 0 ∆Ξ). Observe that since (1 x , σ −1 )∶ (x, λ ⋅ h) → (x, α) and (1 y , τ )∶ (y, β) → (y, λ ⋅ g) are invertible 1-cells of C F , equation (11) But the equality between these 2-cells of C F is expressed precisely by equation (12), which is satisfied by assumption. This complete the definition of a limiting contraction H with center (ℓ, λ), thereby finishing the proof. We are now ready to prove the converse of Proposition 4.7. Corollary 4.9. Let (ℓ, λ) be an E-lax cone over the 2-functor F ∶ J → C. If the projection 2-functor C F ◁ → C F is a biequivalence then (ℓ, λ) is limiting bifinal. Proof. This follows from Corollary 3.3.9 and Proposition 4.8. We have all the elements to finally state and prove our main theorem. Theorem 4.10. Let (ℓ, λ) be an E-lax cone of the 2-functor F ∶ J → C. The following statements are equivalent: (1) the E-lax F -cone (ℓ, λ) is an E-bilimit of F ; (2) the induced 2-functor C F ◁ → C F is a biequivalence; (3) the object (ℓ, λ) of C F is limiting bifinal. Proof. The equivalence between statements (1) and (2) is provided by Corollary 3.3.9. By virtue of Proposition 4.7 and its converse Corollary 4.9, C F ◁ → C F is a biequivalence if and only if (ℓ, λ) is limiting bifinal in C F . This proves the equivalence between statements (2) and (3), thereby concluding the proof of the theorem. Remark 4.11. Given a 2-functor F ∶ J → C, a marking E on J and a weight W ∶ J → Cat, we observed in Remark 1.3.6 that the W -weighted E-bilimit can be expressed as the E W -bilimit of the functor El F p → J F → C, where p∶ El F → J is the fibration classifying the weight 2-functor W and E W is a canonical marking on El F detailed in [13, Definition 2.1.4] by Descotte, Dubuc and Szyld. In particular, a W -weighted E-bilimit of F is equivalent to a limiting bifinal object (ℓ, λ) in C F p . Restricting to the cartesian edges of C F as a fibration over C (see 3.1.3) and using the analysis on cartesian edges performed in §3.1, we can deduce the following statement, which already appears as Proposition 5.4 in [9] by clingman and Moser. Corollary 4.12. An E-bilimit (ℓ, λ) of the 2-functor F ∶ J → C is a biterminal object in (C F ) cart . Proof. It follows from the previous theorem that the 2-functor C ℓ → C F induced by (ℓ, λ) is a biequivalence, that we think of as a morphism of fibrations over C. This biequivalence of fibrations induces a 2-functor (C ℓ ) cart → (C F ) cart , since biequivalences preserve and create cartesian edges, and it is clear that such a 2functor is still a biequivalence. By virtue of Proposition 3.2.4 the object (ℓ, 1 ℓ ) is a biterminal object in C ℓ and therefore its image (ℓ, λ) is a biterminal object in (C F ) cart . Basic notions and notations. Remark 1.1.10. The notion of E-natural transformation was introduced in [13, Definition 2.1.1] by Descotte, Dubuc and Szyld, where they are called σ-natural transformation and the marking is denoted by Σ. 1.2. Join and slices of 2-categories. 1.2.1. The notions of join and slice were generalized by cells is inherited by C. The 2-category C c admits a similar description, and it is canonically isomorphic to the 2-category C op c op . Remark 1.3.5. The notions of E-cones and E-bilimits were introduced in [13, Definition 2.4.3], and called σ-cones and σ-colimits by Descotte, Dubuc and Szyld where the marking E is there denoted by Σ, and emerge naturally in the study of flat pseudo-functors. Remark 1.3.6. The case of weighted E-bilimits can be recovered from that of Ebilimits, as proven in[13, Theorem 2.4.10]. In fact, given a 2-functor F ∶ J → C and a weight W ∶ J → Cat , if we denote by p∶ El W → J the 2-fibration associated with the 2-functor W (cf. Proposition 1.4.15), then the E-bilimit of F weighted by W is precisely the E W -bilimit of F ○ p∶ El W → C, where the marking E W is described in[13, Definition 2.4.8]. ( 3 ) 3cartesian 2-cells are closed under horizontal composition, i.e., for every triple of objects There exists a biequivalence of 2-categories between 2Fib s (B) ps and [B op co , 2-Cat ps ] ps , the former being the 2-category of 2-fibrations over B equipped with a choice of cartesian lifts compatible with composition, pseudo-functors preserving 1-cartesian and 2-cartesian cells and making the obvious triangle commute up-to-isomorphism and pseudo-natural transformations satisfying the obvious property, while the latter is the 2-category of (strict) 2-functors into 2-Cat ps , pseudonatural transformations and modifications. Remark 1.4.9. In the proof of Theorem 1.4.8 (specifically in the proof of [7, Theorem 2.2.11]), Buckley exhibits a "Grothendieck construction" 2-functor [B op co , 2-Cat] → 2Fib s (B) (in fact, a 3-functor, but we will not need this additional structure) which is surjective on objects up-to-isomorphism. In particular, such a biequivalence of 2-categories preserves and reflects equivalences (cf. 1.1.5); the equivalences of 2Fib s (B) are simply the cartesian preserving pseudo-functors that are biequivalences of 2-categories, while the equivalences of [B op co , 2-Cat ] are the pseudo-natural transformations that are object-wise biequivalences of 2-categories. Remark 1.4.12. If every 2-cell in the total 2-category E of a fibration p∶ E → B is p-cartesian, then the fibers E x are (2, 1)-categories for every x ∈ B, i.e., 2-categories where every 2-cell is invertible. Furthermore, there is at most a 2-cell between each pair of 1-cells f, g in E x , so we can view these fibers as 1-categories, by quotienting out these invertible 2-cells. We shall call 1-fibrations this class of fibrations.Example 1.4.13. One of the canonical examples of a 1-fibration is given by slice projections. Given a 2-category B and an object x ∈ B, it is easy to see that the projection p∶ B x → B from the lax slice of B over x to B is a 1-fibration. It corresponds to the hom-2-functor B(−, x)∶ B op → Cat. In particular, every 2-cell in B x is p-cartesian, which is the reason why the associated 2-functor factors through the inclusion Cat ↪ 2-Cat. Corollary 1.4.14. There exists a biequivalence of 2-categories between 1Fib s (B) ps and [B op co , Cat ] ps , the former being the 2-category of 1-fibrations equipped with a choice of cartesian lifts compatible with composition, pseudo-functors preserving 1cartesian cells and making the obvious triangle commute up-to-isomorphism and pseudo-natural transformations satisfying the obvious property, while the latter is the 2-category of (strict) 2-functors into Cat, pseudo-natural transformations and modifications. . Let C be a 2-category and c an object of C. A contraction on C with center c is section γ∶ C → C c of the canonical projection C c → C, i.e., a 2-functor C → C c making the triangle C C c C of 2-categories commute, such that: 2. 1 . 3 . 13Let us spell out the content of the definition of contraction. Suppose we have such a contraction γ on C with center c. Remark 2.1.8. Proposition 2.1.7 implies in particular that if there exists a contraction with center c then c is quasi-terminal. This fact is proven in much greater generality (for strict ∞-categories) by Ara and Maltsiniotis in [3, Proposition B.13]. 2.2. Marked bifinality and final 2-functors. In category theory, a functor u∶ J → C is final if for every functor F ∶ C → D the colimit lim → F exists if and only if the colimit lim → F u does and, whenever they exist, the canonical morphism lim → F u → lim → F is an isomorphism. We now give the appropriate 2-categorical generalization. Definition 2.2.1. A 2-functor u∶ (C, E C ) → (D, E D ) of marked 2-categories is said to be final if for every 2-functor F ∶ D → E with E a 2-category, and for every Remark 2.2. 2 . 2It follows directly from the 2-categorical Yoneda lemma that if u∶ (C, E C ) → (D, E D ) is a final 2-functor and F ∶ D → E is a 2-functor then F admits an E D -bicolimit if and only if F u admits an E C -bicolimit, and when these two equivalent conditions hold, the canonical 1-cell lim → F u → lim → F is an equivalence. Remark 2 .2. 3 . 23The definition of final 2-functor given above is equivalent to the one studied in[1] by Abellán García and Stern, as can be verified by comparing the explicit description of marked cocones in [1, Definition 6.1.3] with the explicit description of E-lax cocones in Paragraph 1.3.1 above. Remark 2.2.6. Let (C, M ) be a marked 2-category, and γ, ρ two contractions with center c. By Paragraph 2.1.6, we have that γ and ρ are isomorphic as contractions. It then follows that when the collection of 1-cells M is closed under isomorphism of 1-cells then γ is an M -contraction if and only if ρ is an M -contraction. Proposition 2.2.7. Let c ∈ C be an object and for each a ∈ Ob(C) fix a morphism f a ∶ a → c in such a way that f c = 1 c . Then c is bifinal with a contraction γ satisfying γ a = f a if and only if the inclusion {c} ↪ (C, M c ) is a final 2-functor, where M c = {f a } a∈Ob(C) . The previous proposition is a particular case of a more general result characterizing the final 2-functors of the form {c} → (C, M ), for a given marking M of C containing M c . Definition 2.2.8. Let (C, M ) be a marked 2-category and c an object of C. We say that c is pre-final if (1) for every a in Ob(C) the category C(a, c) has a terminal object. (2) the identity 1-cell of c is terminal in C(c, c). Moreover, we say that the pre-final object c is M -final if (3) for every a in Ob(C) there exists a marked edge a → c in (C, M ); (4) for every marked edge f ∶ a → b in M , the induced functor f * ∶ C(b, c) → C(a, c) preserves terminal objects. Remark 2.2.9. If c is M -final in (C, M ) then every marked edge a → c with target c is terminal in C(a, c). To see this, note that since 1 c is terminal in C(c, c) and pre-composition with marked edges preserves terminal objects it follows that every marked edges a → c is terminal in C(a, c). On the other hand, such a marked terminal edge exists in C(a, c) by condition (3), and hence any other terminal edge in C(a, c) is isomorphic to it. In particular, if the collection M is closed under isomorphisms of 1-cells then the marked edges in C(a, c) are exactly the terminal ones. Remark 2.2.10. In [13, §3.3] Descotte, Dubuc and Szyld work out a theory of Mfinal 2-functor in the case where the source is an M -filtered 2-category. For a marked 2-functor {c} → (C, M ) the definition of loc. cit. is equivalent to c being M -final in the sense of Definition 2.2.8 above, at least when the collection of marked edges M is closed under composition of 1-cells (as it is assumed in loc. cit.). Indeed, their definition adapted to the case where the source is D 0 states that the following conditions are satisfied: C0 for every a in Ob(C) there exists a 1-cell a → c in M ; C1 for every object a of C and every pair of parallel 1-cells f, g∶ a → c with g in M , there is a 2-cell α∶ f → g; C2 for every object a of C, every pair of parallel 1-cells f, g∶ a → c with g in M , the 2-cell α∶ f → g is unique. These conditions all hold when c is M -final by Remark 2.2.9. Proposition 2.2.11. Let (C, M ) be a marked 2-category, c an object of C, and f a ∶ a → c a choice of a marked 1-cell for every a ∈ Ob(C) such that f c = 1 c . Then the following statements are equivalent:(i) c is M -final; (ii) c is M -bifinal with contraction γ such that γ a = f a ; (iii) the inclusion {c} ↪ (C, M ) is final.Warning 2.2.12. In Proposition 2.2.11 we implicitly assume in advance that a marked edge a → c exists for every a ∈ Ob(C). This assumption automatically holds if c is M -final or M -bifinal, but not necessarily if one only assumes that the inclusion {c} ↪ (C, M ) is final. In particular, while Proposition 2.2.11 shows that for a given object c ∈ Ob(C) the properties of being M -final and M -bifinal are equivalent, these conditions are only shown to imply the finality of the inclusion {c} ↪ (C, M ), and to be implied by it if (C, M ) possesses sufficiently many marked edges. Proof of Proposition 2.2.11. (i) ⇒ (ii). Assume that c is M -final. By Remark 2.2.9 . Hence, the functor • c is an equivalence.(iii) ⇒ (i). Suppose that c∶ D 0 → (C, M ) is a final 2-functor. The object c with the cocone 1 c is clearly the bicolimit of the 2-functor c. In light of Remark 2.2.2 . The p-cartesian edges (f, σ)∶ (z, β) → (x, α) of C F are all of the form with σ i an invertible 2-cell of C for all i in Ob(J). In particular, the p-cartesian edges of C F are precisely its marked edges, as detailed in Paragraph 1.2.8. Proposition 3.2.1. Let p∶ E → B be a cartesian 2-fibration represented by x ∈ Ob(B) and letx ∈ E be the associated lift of x. Let M cart denote the collection of p-cartesian edges. Thenx is M cart -final in E.Proof. Without loss of generality we may assume that E = B x andx = 1 x . The objects of the category B x (α, 1 x ) are of the form (β, morphisms Ξ∶ (β, ζ) → (β ′ , ζ ′ ) are of the form on 2-cells is given by the identity on the 2-cells of B joint with a coherence, which is trivially satisfied. Notice that here we are identifying the 2-cells of B x (and therefore (B x ) 1x , too) with some 2-cells of B and we are saying that the assignment of 2-cells of the contraction is the identity, but with the 2-cells seen in different 2-categories.3.2.3. We give an explicit description of the 2-category (B x ) cart (see Definition 1.4.7). The objects are the elements of Ob(B x ). For every pair of objects α∶ y → x and α ′ ∶ y ′ → x, the hom-category (B x ) cart (y, α), (y ′ , α ′ ) is the full subcategory of B x ((y, α), (y ′ , α ′ ) spanned by p-cartesian edges. By Proposition 3.1.2, these are the 1-cells (β, σ) of B x of the form σ an invertible 2-cell of B. Proposition 3 .2. 4 . 34The object (x, 1 x ) is biterminal in (B x ) cart . . Let (x, h, α, σ) be an object of C F ◁ . Then it is isomorphic to the object (x, h, λ ⋅ h, ι), where ι i is the identity 2-cell of λ i h for all i in Ob(J). Corollary 3.3. 9 . 9Let (J, E) be a marked 2-category, F ∶ J → C a 2-functor, and (ℓ, λ) ∈ C F an E-lax cone over F . Then (ℓ, λ) is an E-bilimit cone if and only if the projection C F ◁ → C F is a biequivalence. Proof. By definition, (ℓ, λ) is an E-bilimit cone if and only if the canonical 2-natural transformation λ ⋅ (−)∶ C(−, ℓ) → [J, C] E (∆−, F ) of functors C op → Cat, is an equivalence. Since the fibrations classifying these two 2-functors are C ℓ and C F , respectively (cf. Example 1.4.16), by virtue of Theorem 1.4.8 (see also Remark 1.4.9), this 2-natural transformation is an equivalence if C, where H(x, α) = (h(x, α), σ(x, α)) and H(y, β) = (h(y, β), σ(y, β)), whenever the 2-cells µ i are invertible for all i ∈ Ob(J), the 2-cell H(f, µ) is invertible. Equivalently, the 2-cell H(f, µ) is invertible whenever the 2-cells λ i * 0 H(f, µ) are invertible for all i ∈ Ob(J). Remark Example 1.1.2. The canonical example of a 2-category is given by Cat, where natural transformations play the role of 2-cells. 1.1.3. We will denote by the following square is a pullback of categories for every a in Ob(E): 1.4.4. The notion of cartesian fibration for 2-categories amounts to the existence of enough cartesian lifts, as in the 1-dimensional case, but it also requires an additional property: cartesian 2-cells must be closed under horizontal composition. Note that, by definition and by the obvious fact that cartesian 1-cells are closed under composition, cartesian 2-cells are automatically closed under vertical composition. Definition 1.4.5. A 2-functor between 2-categories p∶ E → B is called a 2-fibration if it satisfies the following properties: The fibration p∶ E → B associated with a 2-functor F ∶ B → Cat is obtained by forming the pullback displayed below.Proposition 1.4.15. The following facts hold true: (1) The fibration corresponding to the identity 2-functor Id∶ Cat → Cat is given by the forgetful functor U∶ Cat D0 → Cat from the lax slice of Cat under the terminal category to Cat . (2) E Cat D0 B Cat. p U F Example 1.4.16. The slice fibration C F → C classifies the 2-functor The 2-categorical case. Let p∶ E → B be a cartesian 2-fibration and suppose that it is represented by an object x of B. That is, the 2-functor B(−, x)∶ B op → Cat is classified by p∶ E → B, meaning that by Proposition 1.4.15 we have a pullback squarewe have a pullback square E Set { * } op B Set op p of categories. In this case, E is isomorphic to B x , and we may consider the object x ∈ Ob(E) corresponding under this isomorphism to 1 x ∈ B x . The objectx ∈ Ob(E) can then be internally characterized inside E as being a terminal object. E Cat D0 op B Cat op p of 2-categories. In this case, E is isomorphic to B x , and we may consider the object x ∈ Ob(E) corresponding under this isomorphism to 1 x ∈ B x . 4.5. It follows from Remark 2.2.6 and the fact that M cart is closed under isomorphisms of 1-cells that if H and K are two contractions on C F with center (ℓ, λ) then H is limiting if and only if K is limiting. Applying Proposition 2.2.11 to the present case yields: where we have set Γ = H(f, µ). By assumption, the 1-cells (h, σ)∶ (x, α) → (ℓ, λ) and (g, τ )∶ (y, β) → (ℓ, λ) of C F are p-cartesian. Applying to these 1-cells the limiting contraction H, by point (2) we get two invertible 2-cells H(h, σ)∶ (h, σ) → H(x, α) and H(g, τ )∶ (g, τ ) → H(y, β) holds if and only if it is whiskered by these two cells. 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Institute of MathematicsInstitute of Mathematics, Czech Academy of Sciences,Žitná 25, 115 67 Praha 1, Czech Republic Email address: [email protected] URL: https://sites.google.com/view/andreagagna/home Université Paris 13, 99 avenue Jean-Baptiste Clément, 93430 Villetaneuse, France Email address: [email protected]. Institut Galilée, Czech Academy of Sciences. Žitná 25, 115 67 Praha 1, Czech Republic Email address: edoardo.lanari.el@gmailInstitut Galilée, Université Paris 13, 99 avenue Jean-Baptiste Clément, 93430 Villeta- neuse, France Email address: [email protected] URL: https://www.math.univ-paris13.fr/~harpaz Institute of Mathematics, Czech Academy of Sciences,Žitná 25, 115 67 Praha 1, Czech Republic Email address: [email protected] URL: https://edolana.github.io/	[]
[ "ON NON-SEPARABLE GROWTHS OF ω SUPPORTING MEASURES", "ON NON-SEPARABLE GROWTHS OF ω SUPPORTING MEASURES" ]	[ "Piotr Borodulin-Nadzieja ", "And Tomaszżuchowski " ]	[]	[]	We present several ZFC examples of compactifications γω of ω such that their remainders γω\ω are nonseparable and carry strictly positive measures.	10.1090/proc/13804	[ "https://arxiv.org/pdf/1604.03568v1.pdf" ]	119,148,549	1604.03568	8e381c723c8cf437b093285732c6c5e4eb0e5bc9	ON NON-SEPARABLE GROWTHS OF ω SUPPORTING MEASURES 12 Apr 2016 Piotr Borodulin-Nadzieja And Tomaszżuchowski ON NON-SEPARABLE GROWTHS OF ω SUPPORTING MEASURES 12 Apr 2016arXiv:1604.03568v1 [math.LO] We present several ZFC examples of compactifications γω of ω such that their remainders γω\ω are nonseparable and carry strictly positive measures. Introduction A compact space K is called a growth of ω if there is a compactification γω of a countable discrete space ω such that K is homeomorphic to the remainder γω\ω. It is well-known that every separable compact space is a growth of ω. Moreover, every such space carries a strictly positive (regular probability Borel) measure, i.e. measure that is positive on every nonempty open subset. Stone spaces of subalgebras of P(ω)/fin are natural examples of growths of ω. Recall that due to Parovičenko theorem every Boolean algebra of size at most ω 1 can be embedded in P(ω)/fin. In particular, under Continuum Hypothesis the Lebesgue measure algebra Bor([0, 1])/ λ=0 can be embedded in P(ω)/fin. Since the Stone space of measure algebra is nonseparable and supports a measure, under CH there is a nonseparable growth of ω supporting a measure. On the other hand, under Open Coloring Axiom the measure algebra cannot be embedded in P(ω)/fin (see [DH00]). Therefore, it is natural to ask the following question: Problem 1.1 ( [DP15]). Is there a ZFC example of a non-separable growth of ω supporting a measure? In terms of Boolean algebras we ask if there is (in ZFC) a non-σ-centered Boolean algebra supporting a measure (loosely speaking, a non-trivial piece of measure algebra) which can be embedded in P(ω)/fin. Notice that compact spaces supporting measures are ccc. Even constructing a ZFC example of a ccc nonseparable growth of ω is not a trivial task -in [vM79] van Mill offered a bottle of Jenever for such example 1 . The award apparently went to Bell (see [ 1 See [vM79] for the definition of Jenever. Unfortunately the authors are not aware of any offer of this kind for finding a ZFC example of such space which additionally supports a measure. a nonseparable growth of ω carrying a strictly positive measure under the assumption b = c. We present three ZFC constructions of compactifications of ω providing a positive answer to the raised question: Theorem 1.2. There exists a compactification γω of ω such that its remainder γω\ω is not separable and supports a measure. Although all of the constructions are the Stone spaces of some Boolean subalgebras of P(ω)/fin, they are of quite different nature. The first example (see Section 3) is a subalgebra of Bor(2 ω ) generated by all clopen subsets of 2 ω and some sequence {U α : α < c} of open sets, where each U α serves to kill certain candidates for a countable dense set in the resulting Stone space. It is proved that this algebra has required properties with the Lebesgue measure λ being strictly positive on it and, moreover, it can be embedded into P(ω)/fin in such way that λ is transferred to the asymptotic density. This in a sense strengthen the result of Frankiewicz and Gutek that under CH one can construct an embedding of the Lebesgue measure algebra into P(ω)/fin with such property (see [FG81]). The second example (see Section 4) is constructed using methods from [Tod00, Theorem 8.4]. The non-σ-centerdness is a result of certain maximality property of the algebra. Contrary to the case of the first example, although it supports a strictly positive measure, it is not clear how this measure looks like. Its existence is deduced from the Kelley criterion based on the notion of an intersection number. This example appears also in [BNI] although there the existence of measure is proved using forcing methods. It is worth to mention that this example can be easily modified to obtain some additional properties, e.g. in [BNI] it is proved that under add(N ) = non(M) we can additionally require that this Boolean algebra does not contain an uncountable independent family and its Stone space has a countable π-character. Ironically, after carrying out the above constructions, we have realized that Bell's example of ccc non-separable growth of ω mentioned above also supports a measure. Bell constructed a non-σ-centered Boolean subalgebra of P(ω)/fin. We show (see Section 5) that it can embedded to Bor([0, 1])/ λ=0 . It is, however, not clear if it can be done in a way that the asymptotic density is transferred to the Lebesgue measure. Preliminaries In the sequel, we shall consider Boolean subalgebras of the quotient algebra P(ω)/fin. For every such algebra B, its Stone space ULT(B) (of ultrafilters on B) is a continuous image of ULT(P(ω)/fin) ≃ βω\ω, where by βω we denote theČech-Stone compactification of ω. Therefore, in that case ULT(B) is a remainder γω\ω of some compactification γω of ω. Recall that ULT(B) is separable iff B is σ-centered. By B + we understand the family of all positive elements of B. If B is a Boolean algebra and A ∈ B, then by A = {x ∈ ULT(B) : A ∈ x} we denote the corresponding clopen subset of ULT(B). By a measure µ on a Boolean algebra B we mean a finitely additive probability measure. We call it a strictly positive measure if µ(B) > 0 for each B ∈ B + (in this case we say also that B supports the measure µ). If µ is a measure on B, then µ defined by µ( A) = µ(A) is a finitely additive measure on the algebra of clopen subsets of ULT(B). The measure µ can be uniquely extended to a (regular Borel probability) σ-additive measure µ on ULT(B). Note that if µ is strictly positive, then µ is a strictly positive measure on ULT(B). Therefore, Theorem 1.2 is an immediate consequence of the following statement. Theorem 2.1. There exists a non-σ-centered Boolean algebra B ⊆ P(ω)/fin that supports a measure. Recall some standard notions concerning the Cantor space 2 ω . For any ϕ ∈ 2 <ω by [ϕ] = {x ∈ 2 ω : ϕ ⊆ x} we denote the basic subset of 2 ω corresponding to ϕ. We shall consider the Lebesgue measure λ on 2 ω , defined as the unique Borel probability measure on 2 ω such that λ([ϕ]) = 2 −k for any k ∈ ω and ϕ ∈ 2 k . We say that a subset A ⊆ 2 ω depends on a set of coordinates I ⊆ ω, which we denote A ∼ I, if A = B × 2 ω\I for some B ⊆ 2 I . We will consider also the asymptotic density defined on elements of the algebra P(ω)/fin. It is usually defined on subsets of ω as a function d(A) = lim n→∞ \|A ∩ n\| n provided the limit exists. As d(A) = 0 for any finite A ⊆ ω, we can naturally transfer this function to the asymptotic density on P(ω)/fin, which we will also denote by d. Construction using an almost disjoint family Fix an enumeration {P α : α < c} of all countable subsets of 2 ω and let {B α : α < c} be an almost disjoint family in P(ω). Denote A 0 = Clop(2 ω ). For α < c let P α = {t α n : n ∈ ω} ⊆ 2 ω and B α = {m α i : i ∈ ω} ⊆ ω, where m α i < m α j for i < j. Define a sequence (ϕ α n ) n of elements of 2 <ω in the following way: ϕ α 0 = t α 0 \| {m α 0 } , ϕ α 1 = t α 1 \| {m α 1 ,m α 2 } , ϕ α 2 = t α 2 \| {m α 3 ,m α 4 ,m α 5 } , etc. Define U α = i∈ω [ϕ α i ] and observe that U α ⊆ 2 ω is an open set. Let F α = U c α . Finally, let A = alg (A 0 ∪ {U α : α < c}). Theorem 3.1. The Boolean algebra A has the following properties • A is not σ-centered, • A supports λ, • there is a Boolean embedding Ψ : A → P(ω)/fin such that λ(A) = d(Ψ(A)) for each A ∈ A. We will prove the above theorem in a sequence of propositions and lemmas. We begin with an easy observation. Remark 3.2. For each α < c we have P α ⊆ U α , as U α contains a basic neighbourhood of t α n for every n ∈ ω. Proposition 3.3. The Boolean algebra A is not σ-centered. Proof. We will switch to topological language and we will prove that the Stone space of A is not separable. Consider an arbitrary {x n : n ∈ ω} ⊆ ULT(A). For every n ∈ ω let t n = x n \| A 0 ∈ ULT(A 0 ) ≃ 2 ω , which means that {A ∈ Clop(2 ω ) : t n ∈ A} = {A ∈ Clop(2 ω ) : A ∈ x n }. There exists an α < c such that {t n : n ∈ ω} = P α . By Remark 3.2 we have F α ∩ P α = ∅, so t n / ∈ F α for each n ∈ ω. As 2 ω is a metric 0-dimensional space, for every n ∈ ω there exists A n ∈ Clop(2 ω ) satisfying t n ∈ A n and A n ∩ F α = ∅. Therefore, we have A n ∈ x n , hence F α / ∈ x n for every n ∈ ω. Thus we have shown that F α ∩ {x n : n ∈ ω} = ∅, which means that {x n : n ∈ ω} is not dense in ULT(A). For each α < c and i ∈ ω let C α i ⊆ ω be the minimal set with the property [ϕ α i ] ∼ C α i (so, e.g. C α 0 = {m α 0 } and so on). Remark 3.4. For every α < c we have U α ∼ B α (thus also F α ∼ B α ). Moreover, C α n ⊆ B α \n for α < c and n ∈ ω. To prove that A supports the measure λ, we need the following lemma. Lemma 3.5. Assume that U ⊆ 2 ω is an open set and the set {α 1 , . . . , α m } ⊆ c is such that U ∩ F α 1 ∩ . . . ∩ F αm = ∅. Then λ U ∩ F α 1 ∩ . . . ∩ F αm > 0. Proof. Assume that U and {α 1 , . . . , α m } are as above. Without loss of generality we can assume that U = [τ ], where τ ∈ 2 M for some M ∈ ω. For every n ∈ ω define X n = [τ ] m j=1 n i=0 [ϕ α j i ]. Let I n = M ∪ m j=1 n i=0 C α j i and observe that I n is the minimal set with property X n ∼ I n . According to our assumption [τ ] m j=1 i∈ω [ϕ α j i ]. Hence, for each n the set X n is open and nonempty and so λ(X n ) > 0. Claim 3.6. There exists N ∈ ω such that the family {I N } ∪{C α j i : i > N, j ∈ {1, . . . , m}} is pairwise disjoint. Proof. Since the family {B α : α < c} is pairwise almost disjoint, there exists N ≥ M such that for every different i, j ∈ {1, . . . , m} we have B α i ∩ B α j ⊆ N. Therefore, by Remark 3.4 we get C α i k ∩ C α j l = ∅ for every different i, j ∈ {1, . . . , m}, k > N and arbitrary l ∈ ω. If i ∈ {1, . . . , m} and k = l, then C α i k ∩ C α i l = ∅. Moreover, again using Remark 3.4 we have C α j k ⊆ ω\M for every k > N and j ∈ {1, . . . , m}. Now we prove that the set U ∩ F α 1 ∩ . . . ∩ F αm = [τ ] m j=1 i∈ω [ϕ α j i ] is λ-positive: λ [τ ] m j=1 i∈ω [ϕ α j i ] = λ X N ∩ m j=1 ∞ i=N +1 [ϕ α j i ] c ⋆ = λ(X N ) m j=1 λ ∞ i=N +1 [ϕ α j i ] c ≥ λ(X N ) 1 − ∞ i=N +1 λ([ϕ α j i ]) m = λ(X N ) 1 − 2 −(N +1) m > 0, where the equality (⋆) holds because by Claim 3.6 the sets X N and [ϕ Proof. Every element of A is a finite union of elements of the form C ∩ U β 1 ∩ . . . ∩ U βn ∩ F α 1 ∩ . . . ∩ F αm for some C ∈ Clop(2 ω ) and α 1 , . . . , α n , β 1 , . . . , β m < c. Every such element is equal to U ∩ F α 1 ∩ . . . ∩ F αm for some open U ⊆ 2 ω , hence using Lemma 3.5 we get that it is λ-positive (provided it is non-empty). Therefore, every nonzero A ∈ A is λ-positive. We are going to find a Boolean embedding of the algebra A into P(ω)/fin transferring the Lebesgue measure to the asymptotic density. First, it is easy to see that there is an embedding Ψ 0 : A 0 → P(ω)/fin with the above property. For example, it can be induced by f : 2 <ω → P (ω) such that f (σ) = {k ∈ ω : k =σ mod 2 n }, where n is the length of σ andσ is the natural number represented by σ in the binary system (such f sends [0] to even numbers, [01] to numbers equal 1 modulo 4 and so on). We want now to extend Ψ 0 to a Boolean embedding Ψ : A → P(ω)/fin. To define such Ψ, we need only to define Ψ(U α ) for any α < c. We will use the following Lemma 3.8 proved as Theorem 1.1 in [Buc53]: Lemma 3.8. If A 0 ⊆ * A 1 ⊆ * A 2 . . . ⊆ ω is a sequence of sets having an asymptotic density, then there exists a set A ⊆ ω having an asymptotic density such that A n ⊆ * A for all n ∈ ω and d(A) = sup n∈ω d(A n ). Fix α < c. Recall that U α = i∈ω [ϕ α i ] and let B n ⊆ ω be a representative of the equivalence class of Ψ 0 n i=0 [ϕ α i ] for n ∈ ω. The sequence (B n : n ∈ ω) meets the requirements of Lemma 3.8, thanks to properties of Ψ 0 . Let B be the set, whose existence is proved by Lemma 3.8 for the sequence (B n ) n . We extend Ψ 0 by defining Ψ 0 (U α ) = B • . Proposition 3.9. The function Ψ 0 defined above on A 0 ∪ {U α : α < c} can be extended to a Boolean homomorphism Ψ : A → P(ω)/fin. Proof. By Sikorski's Extension Criterion (see [Kop89,Theorem 5.5]) and the definition of A we need only to prove that (3.1) C ∩ n i=1 U β i ∩ m j=1 F α j = ∅ =⇒ Ψ 0 (C) ∩ n i=1 Ψ 0 (U β i ) ∩ m j=1 Ψ 0 (U α j ) c = 0 for every C ∈ Clop(2 ω ) and α 1 , . . . , α m , β 1 , . . . , β n < c. We shall proceed by induction on n. For n = 0 the assumption C ∩ m j=1 F α j = ∅ means that C ⊆ m j=1 U α j , so by the definition of U α 's we get C ⊆ m j=1 i∈ω [ϕ α j i ]. As C is compact, there exists N ∈ ω such that C ⊆ m j=1 N i=0 [ϕ α j i ], hence by the properties of Ψ 0 on Clop(2 ω ) we have Ψ 0 (C) ≤ m j=1 Ψ 0 N i=0 [ϕ α j i ] . By Lemma 3.8 we know that Ψ 0 n i=0 [ϕ α i ] ≤ Ψ 0 (U α ) for each α < c and n ∈ ω. Therefore, we get Ψ 0 (C) ≤ m j=1 Ψ 0 (U α j ) and so Ψ 0 (C) ∩ m j=1 Ψ 0 (U α j ) c = 0. Assume now that we have proved (3.1) for every element of the form C ∩ n i=1 U β i ∩ m j=1 F α j , where n ≤ N. Take C ∈ Clop(2 ω ) and α 1 , . . . , α m , β 1 , . . . , β N +1 < c such that (3.2) C ∩ N +1 i=1 U β i ∩ m j=1 F α j = ∅. Claim 3.10. There exists such K ∈ ω that the sets [ϕ β N+1 K ] and C ∩ N i=1 U β i ∩ m j=1 F α j depend on disjoint sets of coordinates. Proof. We have C ∼ l for some l ∈ ω. As {B α : α < c} is an almost disjoint family, there exists K ≥ l such that B β N+1 \K is disjoint from N i=1 B β i ∪ m j=1 B α j . By Remark 3.4 we have [ϕ β N+1 K ] ∼ C β N+1 K ⊆ B β N+1 \K and C ∩ N i=1 U β i ∩ m j=1 F α j ∼ l ∪ N i=1 B β i ∪ m j=1 B α j , thus the proof is complete. Fix such K ∈ ω. As [ϕ β N+1 K ] ⊆ U β N+1 , by (3.2) we get C ∩ N i=1 U β i ∩ m j=1 F α j ∩ [ϕ β N+1 K ] = ∅. If two sets depending on disjoint sets of coordinates have empty intersection, then at least one of them has to be empty. As [ϕ β N+1 K ] = ∅, we get C ∩ N i=1 U β i ∩ m j=1 F α j = ∅. Now, by inductive assumption Ψ 0 (C) ∩ N i=1 Ψ 0 (U β i ) ∩ m j=1 Ψ 0 (U α j ) c = 0, hence Ψ 0 (C) ∩ N +1 i=1 Ψ 0 (U β i ) ∩ m j=1 Ψ 0 (U α j ) c = 0. Thus, we have proved (3.1) and we are done. Now we shall prove that the homomorphism Ψ : A → P(ω)/fin transfers the Lebesgue measure to the asymptotic density. In this proof we will not use any specific properties of U α 's, but rather the properties of Ψ 0 on A 0 and the definition of Ψ(U α ) for α < c. Proposition 3.11. For every A ∈ A we have λ(A) = d(Ψ(A)). Proof. Let A ′ = {A ∈ A : λ(A ∩ C) = d(Ψ(A ∩ C)) for each C ∈ Clop(2 ω )}. Claim 3.12. If A ∈ A ′ , then A ∩ U α ∈ A ′ and A ∩ F α ∈ A ′ for each α < c. Proof. Let A ∈ A ′ and α < c. We will only show that A ∩ U α ∈ A ′ . The proof for F α is analogous. Fix C ∈ Clop(2 ω ). Denote C n = n i=0 [ϕ α i ], so that we have U α = ∞ n=0 C n , where the union is increasing. Since A ∈ A ′ and C ∩ C n is clopen for each n we have (3.3) λ A ∩ C ∩ U α = sup n∈ω λ A ∩ C ∩ C n = sup n∈ω d Ψ A ∩ C ∩ C n . Denote A ′ = A ∩ C. As Ψ is a homomorphism, for every n ∈ ω we have Ψ(A ′ ∩ C n ) ≤ Ψ(A ′ ∩ U α ), and thus (3.4) sup n∈ω d Ψ A ′ ∩ C n ≤ d Ψ A ′ ∩ U α . Also, A ′ ∩ U α = (A ′ ∩ C n ) ∪ A ′ ∩ (U α \C n ) for each n ∈ ω and hence (3.5) Ψ(A ′ ∩ U α ) = Ψ(A ′ ∩ C n ) ∪ Ψ(A ′ ) ∩ Ψ(U α \C n ) . By the definition of Ψ we get d(Ψ(U α )) = sup n∈ω d(Ψ(C n )) for every α < c and so d Ψ(U α \C n ) n→∞ − −− → 0. Therefore, we have (3.6) d Ψ(A ′ ) ∩ Ψ(U α \C n ) n→∞ − −− → 0. Finally, by (3.4), (3.5), (3.6) and the definition of A ′ we get d Ψ C ∩ A ∩ U α = sup n∈ω d Ψ C ∩ A ∩ C n , and so by (3.3) we have (3.7) d Ψ C ∩ (A ∩ U α ) = λ C ∩ (A ∩ U α ) . As C was arbitrary, A ∩ U α ∈ A ′ . We can show that A∩F α ∈ A ′ , using the fact that λ A∩C∩F α = inf n∈ω λ A∩C∩D n for every C ∈ Clop(2 ω ) and D n = C c n for each n and proceeding in the similar manner as above. Now, subsequently using the above claim we can show that if A = C ∩ U β 1 ∩ . . . ∩ U β k ∩ F α 1 ∩ . . . ∩ F αm , where C ∈ Clop(2 ω ) and α 1 , . . . , α m , β 1 , . . . , β k < c, then A ∈ A ′ and so, in particular, λ(A) = d(Ψ(A)). Finally, as every element of A can be written as a disjoint union of elements of the form as above, we prove the statement of Proposition 3.11 by the fact that the Lebesgue measure and the asymptotic density are finitely additive. Corollary 3.13. The homomorphism Ψ : A → P(ω)/fin is injective and so it is a Boolean embedding. Proof. Assume that A ∈ A satisfies Ψ(A) = 0. We have d(Ψ(A)) = 0, hence by Proposition 3.11 we get λ(A) = 0, which implies, by Proposition 3.5, that A = ∅. Construction using slaloms. In this section we present another construction announced in the introduction. It is motivated by [Tod00,Theorem 8.4]. In [BNI] the authors showed that the constructed Boolean algebra supports a strictly positive measure using the following theorem due to Kamburelis: Theorem 4.1. A Boolean algebra A supports a measure if and only if there is a measure algebra M such that M "Ǎ is σ-centered". We will prove the existence of strictly positive measure using a more classical tool: Kelley's criterion. Define the intersection number of A by κ(A) = inf{κ(s) : s ∈ A <ω }. Theorem 4.3 (Kelley's criterion). Let A be a Boolean algebra. The following conditions are equivalent: • A supports a measure, • A + = n C n , where κ(C n ) > 0 for each n. We will consider a slightly stronger property than the above (see [DP08]). Definition 4.4. We call a Boolean algebra A appoximable if for every 0 < δ < 1 there is a family (C n ) n such that A + = C n and κ(C n ) > δ. A slalom is a set S ⊆ ω × ω such that S(n) ⊆ 2 n and \|S(n)\| < 2 n for every n ∈ ω. Denote by S the family of all slaloms. For a slalom S denote S \|n = S ∩ (n × 2 n ). Let Ω = {(S, n) : n ∈ ω, S ∈ S, S ⊆ (n × 2 n )}. For each S ⊆ ω × ω define It will be convenient to make the following simple observations available. Let K be the Stone space of T/fin. Since Ω is countable, T/fin can be embedded in P(ω)/fin and so K is a growth of ω. (1) A ⊆ B if and only if T B ⊆ T A , (2) T (A∪B) = T A ∩ T B , (3) if F is finite and F ∈F T F is finite, then there is k ∈ ω such that F ∈F F (k) = 2 k , (4) if A is such that {T A : A ∈ A}/fin is centered, then A ∈ S. Theorem 4.6. The Boolean algebra T/fin is an approximable non-σ-centered Boolean algebra. Consequently, K is a growth of ω supporting a measure. Let X = {f ∈ ω ω : ∀n f (n) < 2 n }. Proposition 4.7. T/fin is not σ-centered. Proof. Suppose towards contradiction that T/fin is σ-centered. In particular W = W n , where {T W : W ∈ W n }/fin is centered for each n. By Lemma 4.5 (4) W n = W n ∈ S for each n. Now, pick f ∈ X to be such that f (n) / ∈ W n (n). Clearly, f ∈ W but f / ∈ W n for every n. A contradiction. Proposition 4.8. T/fin is approximable. Proof. Let 0 < δ < 1. For (S, n) ∈ Ω define W δ (S,n) = {W ∈ W : W \|n = S and k>n \|W (k)\|/2 k < 1 − δ}. It is easy to see that W = (S,n)∈Ω W δ (S,n) . We claim that even something stronger is true. Claim 4.9. For each infinite A ∈ T there is (S, n) ∈ Ω and V ∈ W δ (S,n) such that T V ∩ T (S,n) ⊆ A. Proof. It is sufficient to consider only elements of the form A = T c V 0 ∩ T c V 1 ∩ · · · ∩ T c V l ∩ T V l+1 ∩ · · · ∩ T V L . Also, thanks to Lemma 4.5(2) A = T c V 0 ∩ T c V 1 ∩ · · · ∩ T c V l ∩ T V , where V = V l+1 ∪ · · · ∪ V L ∈ W. Now, since A is infinite, V i V for each i ≤ l. Let n be big enough so that V i ∩ (n × 2 ω ) V \|n . Let S = V \|n . It is plain to check that T V ∩ T (S,n) ⊆ A. Claim 4.10. κ({T W ∩ T (S,n) : W ∈ W δ (S,n) }/fin) > δ for each (S, n) ∈ Ω. Proof. Fix (S, n) ∈ Ω. For W ∈ W δ (S,n) let A W = {f ∈ X : f (k) / ∈ W (k) for k > n}. Of course λ(A (i,j) ) = 1 − 1/2 i for i > k. Since A W = (i,j)∈W,i>n A (i,j) and i>n \|W (i)\|/2 i < 1 − δ we have that λ(A W ) > δ. Let (V i ) i<k be a sequence of elements of W δ (S,n) . There is I ⊆ k such that \|I\| ≥ δ · k and there is f ∈ I A V i . Therefore, I V i ∈ W, just because f (k) / ∈ I V i (k) for k > n. By Lemma 4.5(3) I T V i is infinite. Moreover, V i ∩ (n × 2 n ) = S for every i ≤ l and so I T V i ∩ T (S,n) is infinite and the claim is proved. For n ∈ ω let C (S,n) = {A ∈ A : ∃W ∈ W δ (S,n) T W ∩ T (S,n) ⊆ * A}/fin. Claim 4.9 implies that (C (S,n) ) (S,n)∈Ω is a fragmentation of A. By Claim 4.10 κ(C (S,n) ) > δ for each (S, n) ∈ Ω. Bell's construction In this section we will describe Bell's construction of a ccc non-separable growth of ω from [Bel80] (mentioned in the introduction) and we will show that it supports a measure. Let P = {f ∈ ω ω : f (n) ≤ n + 1 for each n ∈ ω} and N = {f ↾ n : f ∈ P, n ∈ ω}. Denote T = {π ∈ N ω : π(n) ∈ ω n+1 for each n ∈ ω}. For each s ∈ N define C s = {t ∈ N : s ⊆ t} and for every π ∈ T let C π = n∈ω C π(n) . Finally, let B = alg ({C π : π ∈ T }). Since N is a countably infinite set, B/fin can be embedded to P(ω)/fin and so the Stone space of B/fin is a growth of ω. It is not difficult to see that B/fin is not σ-centered (see [Bel80]). It is also ccc. In fact Bell proved that this space is σ-n-linked for each n ∈ ω, i.e. for every n we have B + = i C i where C i is n-linked for every i (i.e. F = ∅ whenever F ∈ [C i ] n ). Plainly, σ-n-linked spaces are ccc. We will show that Bell's space supports a measure. More precisely, we will show that B/fin is isomorphic to a certain subalgebra of Bor([0, 1])/ λ=0 . This implies σ-nlinkedness (see also [DS94]), so our theorem generalizes Bell's result. Endow X = n∈ω {0, . . . , n + 1} with the product topology and notice that X is homeomorphic to the Cantor set. For each s ∈ N let [s] = {t ∈ X : s ⊆ t} (the basic open subset of X corresponding to s). For every π ∈ T define V π = n∈ω [π(n)]. Finally, let C = alg ({V π : π ∈ T }). Proposition 5.1. The Boolean algebra B/fin is isomorphic to C. Proof. Define f : {C π : π ∈ T } → {V π : π ∈ T } by f (C π ) = V π for π ∈ T . We claim that such f can be extended to a functionf : B → C inducing a Boolean isomorphism of B/fin and C. By the Sikorski's Extension Criterion we only need to prove that n i=1 C π ′ i ∩ m j=1 C c π j is finite ⇐⇒ n i=1 V π ′ i ∩ m j=1 V c π j = ∅ for every π 1 , . . . , π m , π ′ 1 , . . . , π ′ n ∈ T . First, assume that there exists t ∈ n i=1 V π ′ i ∩ m j=1 V c π j for some π 1 , . . . , π m , π ′ 1 , . . . , π ′ n ∈ T . As n i=1 V π ′ i is an open subset of X, there exists k ∈ ω such that [t\|k] ⊆ n i=1 V π ′ i , thus t\|k ∈ n i=1 C π ′ i . Since t does not extend any of π j (i) for j = 1, . . . , m and i ∈ ω, for each l ≥ k we have t\|l ∈ C t\|k ∩ m j=1 C c π j ⊆ n i=1 C π ′ i ∩ m j=1 C c π j , hence the latter set is infinite. On the other hand, assume that D = n i=1 C π ′ i ∩ m j=1 C c π j is infinite for some π 1 , . . . , π m , π ′ 1 , . . . , π ′ n ∈ T . Claim 5.2. There is t ∈ X and M ∈ ω such that t\|l ∈ D for each l ≥ M. Proof. We will construct t inductively by finding the sequence of its initial segments (t k ) k≥M . First, notice that as D is infinite, it must contain arbitrarily long sequences from N. In particular, there is M > m and t M ∈ ω M ∩ D. Now, assume that we have t k of length k, such that t M ⊆ t k and t k ∈ D. We will show that there is p ≤ k + 1 such that t kˆp ∈ D and that will finish the proof. Clearly, t kˆr ∈ n i=1 C π ′ i for every r ≤ k + 1. So, we need only to find p ≤ k + 1 such that t kˆp ∈ m j=1 C c π j . Notice that if t kˆr does not extend π j (k) for some j ≤ m, then t kˆr ∈ C c π j . Hence, as k > m, there is p ≤ k + 1 such that t kˆp ∈ m j=1 C c π j . Let t be the unique element of X such that t k ⊆ t for each k ≥ M. Let t be as in Claim 5.2. Then t ∈ n i=1 V π ′ i , since t extends π ′ i (l) for each i = 1, . . . , n and l ∈ ω. Also, t ∈ m j=1 V c π j . Otherwise there would be j ∈ {1, . . . , m} and l ∈ ω such that t extends π j (l). But then t\|r would extend π j (l) for some r > M, a contradiction with Claim 5.2. So, n i=1 V π ′ i ∩ m j=1 V c π j = ∅ and we are done. The above lemma allows us to look for a strictly positive measure on C instead of B/fin. Denote by λ the standard product measure on X (defined by λ([s]) = 1 (n+1)! for s ∈ N ∩ ω n ). V c π j : s ∈ N, π 1 , . . . π m ∈ T } forms a π-base for C. So it is enough to show that every nonempty element of P is λ-positive or, equivalently, that λ [s] ∩ m j=1 V π j < λ([s]) for s ∈ N, π 1 , . . . π m ∈ T such that [s] ∩ m j=1 V c π j = ∅. Suppose towards contradiction that (5.1) λ [s] ∩ m j=1 V π j = λ([s]) for some s ∈ ω K ∩ N, where K ∈ ω, and π 1 , . . . π m ∈ T satisfying [s] ∩ m j=1 V c π j = ∅. For each n ≥ 1 denote A n = m j=1 n−1 i=0 [π j (i)] and notice that A n is a disjoint union of sets of the form [t], t ∈ ω n . Therefore, for each n > K the set [s] ∩ A c n is also a disjoint union of such basic subsets. By the assumption ∅ = [s] ∩ m j=1 V c π j and so the set [s] ∩ A c n is nonempty. Therefore, for every n > K we have (5.2) λ [s] ∩ A c n ≥ 1 (n + 1)! . For every l ∈ ω the set A l+1 \A l is a disjoint union of at most m subsets of form [t], where t ∈ ω l+1 , thus (5.3) λ A l+1 \A l ≤ m · 1 (l + 2)! . Lemma 5.4. For every n > 3m we have m · ∞ l=n 1 (l + 1)! < 1 n! . Proof. The series ∞ l=n 1 (l+1)! is a remainder term at x = 1 of the n-th order Taylor polynomial of the function g(x) = e x at 0. The Lagrange form of the remainder gives us ∞ l=n 1 (l + 1)! = e x (n + 1)! for some x ∈ [0, 1], thus m · ∞ l=n 1 (l + 1)! ≤ e · m n · 1 n! , and we are done. Fix n > max(K, 3m). Using (5.2), (5.3) and Lemma 5.4 we get λ [s] ∩ m j=1 V π j ∩ A c n = λ [s] ∩ ∞ l=n (A l+1 \A l ) ≤ m · ∞ l=n 1 (l + 2)! < 1 (n + 1)! ≤ λ [s] ∩ A c n , which contradicts the assumption (5.1). Acknowledgements We would like to thank the participants of the BF-seminar in Wroc law, Grzegorz Plebanek and Piotr Drygier, for helpful discussions concerning the subject of this paper. Bel80, Example 2.1] and [vM82, Example 3.2]) and later on another examples were found (see e.g. [Tod00, Theorem 8.4]). Recently, Drygier and Plebanek ([DP15]) have constructed 2010 Mathematics Subject Classification. Primary 03E75, 28A60, 28E15, 54D40. The first author was partially supported by National Science Center grant no. 2013/11/B/ST1/03596 (2014-2017). i ], for j ∈ {1, . . . , m} and i > N, depend on pairwise disjoint sets of coordinates. Proposition 3 . 7 . 37The measure λ is strictly positive on A. Definition 4 . 2 . 42Let A be a Boolean algebra and let A ⊆ A + be nonempty. For every finite sequence s = (A 1 , . . . , A n ) of elements of A let κ(s) = max{\|I\| : I ⊆ {1, . . . , n} and i∈I A i = 0} n . T S = {(T, n) ∈ Ω : S \|n ⊆ T }. For (S, n) ∈ Ω let T (S,n) = {(T, m) ∈ Ω : m ≥ n, T \|n = S}. Lemma 4 . 5 . 45Let A, B ∈ S. Then Let W = {S ∈ S : \|S(n)\|/2 n < ∞}. Now, we are ready to define the main object of this section. Let T = alg {T A : A ∈ W} ∪ {T (S,n) : (S, n) ∈ Ω} . Proposition 5 . 3 . 53The measure λ is strictly positive on C.Proof. Notice that the set of nonempty elements of Compact ccc nonseparable spaces of small weight. Murray G Bell, The Proceedings of the 1980 Topology Conference. Birmingham, Ala5Univ. AlabamaMurray G. Bell. Compact ccc nonseparable spaces of small weight. In The Proceedings of the 1980 Topology Conference (Univ. Alabama, Birmingham, Ala., 1980), volume 5, pages 11-25 (1981), 1980. Piotr Borodulin-Nadzieja and Tanmay Inamdar. Measures and slaloms. preprintPiotr Borodulin-Nadzieja and Tanmay Inamdar. Measures and slaloms. preprint. Generalized asymptotic density. Robert Creighton Buck, Amer. J. Math. 75Robert Creighton Buck. Generalized asymptotic density. Amer. J. Math., 75:335-346, 1953. The measure algebra does not always embed. Alan Dow, Klaas Pieter Hart, Fund. Math. 1632Alan Dow and Klaas Pieter Hart. The measure algebra does not always embed. Fund. Math., 163(2):163-176, 2000. Strictly positive measures on Boolean algebras. Mirna Džamonja, Grzegorz Plebanek, J. Symbolic Logic. 734Mirna Džamonja and Grzegorz Plebanek. Strictly positive measures on Boolean algebras. J. Symbolic Logic, 73(4):1416-1432, 2008. Nonseparable growth of the integers supporting a measure. Piotr Drygier, Grzegorz Plebanek, Topology Appl. 191Piotr Drygier and Grzegorz Plebanek. Nonseparable growth of the integers supporting a mea- sure. Topology Appl., 191:58-64, 2015. The σ-linkedness of the measure algebra. Alan Dow, Juris Steprāns, Canad. Math. Bull. 371Alan Dow and Juris Steprāns. The σ-linkedness of the measure algebra. Canad. Math. Bull., 37(1):42-45, 1994. Some remarks on embeddings of Boolean algebras and the topological spaces. Ryszard Frankiewicz, Andrzej Gutek, I. Bull. Acad. Polon. Sci. Sér. Sci. Math. 299Ryszard Frankiewicz and Andrzej Gutek. Some remarks on embeddings of Boolean algebras and the topological spaces. I. Bull. Acad. Polon. Sci. Sér. Sci. Math., 29(9-10):471-476, 1981. Handbook of Boolean algebras. Sabine Koppelberg, Amsterdam. J. Donald Monk and Robert BonnetNorth-Holland Publishing Co1Sabine Koppelberg. Handbook of Boolean algebras. Vol. 1. North-Holland Publishing Co., Am- sterdam, 1989. Edited by J. Donald Monk and Robert Bonnet. Chain-condition methods in topology. Stevo Todorčević, Topology Appl. 1011Stevo Todorčević. Chain-condition methods in topology. Topology Appl., 101(1):45-82, 2000. Weak P -points in compact F -spaces. Jan Van Mill, The Proceedings of the 1979 Topology Conference. Athens, Ohio4Ohio Univ.Jan van Mill. Weak P -points in compact F -spaces. In The Proceedings of the 1979 Topology Conference (Ohio Univ., Athens, Ohio, 1979), volume 4, pages 609-628 (1980), 1979. Weak P -points inČech-Stone compactifications. Jan Van Mill, Trans. Amer. Math. Soc. 2732Jan van Mill. Weak P -points inČech-Stone compactifications. Trans. Amer. Math. Soc., 273(2):657-678, 1982. Uniwersytet Wroc lawski E-mail address: [email protected]. Instytut Matematyczny, plInstytut Matematyczny, Uniwersytet Wroc lawski E-mail address: [email protected]	[]
[ "Multiband counterparts of two eclipsing ultraluminous X-ray sources in M 51", "Multiband counterparts of two eclipsing ultraluminous X-ray sources in M 51" ]	[ "R Urquhart \nInternational Centre for Radio Astronomy Research\nCurtin University\nGPO Box U19876845PerthWAAustralia\n", "R Soria \nInternational Centre for Radio Astronomy Research\nCurtin University\nGPO Box U19876845PerthWAAustralia\n\nNational Astronomical Observatories\nChinese Academy of Sciences\n100012BeijingChina\n\nSydney Institute for Astronomy\nSchool of Physics A28\nThe University of Sydney\n2006SydneyNSWAustralia\n", "H M Johnston \nSydney Institute for Astronomy\nSchool of Physics A28\nThe University of Sydney\n2006SydneyNSWAustralia\n", "M W Pakull \nUMR 7550\nObservatoire astronomique\nUniversité de Strasbourg\nCNRS\n11 rue de l'Université67000StrasbourgFrance\n", "C Motch \nUMR 7550\nObservatoire astronomique\nUniversité de Strasbourg\nCNRS\n11 rue de l'Université67000StrasbourgFrance\n", "A Schwope \nLeibniz-Institut für Astrophysik Potsdam\nAn der Sternwarte 1614482PotsdamGermany\n", "J C A Miller-Jones \nInternational Centre for Radio Astronomy Research\nCurtin University\nGPO Box U19876845PerthWAAustralia\n", "G E Anderson \nInternational Centre for Radio Astronomy Research\nCurtin University\nGPO Box U19876845PerthWAAustralia\n" ]	[ "International Centre for Radio Astronomy Research\nCurtin University\nGPO Box U19876845PerthWAAustralia", "International Centre for Radio Astronomy Research\nCurtin University\nGPO Box U19876845PerthWAAustralia", "National Astronomical Observatories\nChinese Academy of Sciences\n100012BeijingChina", "Sydney Institute for Astronomy\nSchool of Physics A28\nThe University of Sydney\n2006SydneyNSWAustralia", "Sydney Institute for Astronomy\nSchool of Physics A28\nThe University of Sydney\n2006SydneyNSWAustralia", "UMR 7550\nObservatoire astronomique\nUniversité de Strasbourg\nCNRS\n11 rue de l'Université67000StrasbourgFrance", "UMR 7550\nObservatoire astronomique\nUniversité de Strasbourg\nCNRS\n11 rue de l'Université67000StrasbourgFrance", "Leibniz-Institut für Astrophysik Potsdam\nAn der Sternwarte 1614482PotsdamGermany", "International Centre for Radio Astronomy Research\nCurtin University\nGPO Box U19876845PerthWAAustralia", "International Centre for Radio Astronomy Research\nCurtin University\nGPO Box U19876845PerthWAAustralia" ]	[ "MNRAS" ]	We present the discovery and interpretation of ionized nebulae around two ultraluminous X-ray sources in M 51; both sources share the rare property of showing X-ray eclipses by their companion stars, and are therefore prime targets for follow-up studies. Using archival Hubble Space Telescope images, we found an elongated, 100-pc-long emission-line structure associated with one X-ray source (CXOM51 J132940.0+471237; ULX-1 for simplicity), and a more circular, ionized nebula at the location of the second source (CXOM51 J132939.5+471244; ULX-2 for simplicity). We observed both nebulae with the Large Binocular Telescope's Multi-Object Double Spectrograph. From our analysis of the optical spectra, we argue that the gas in the ULX-1 bubble is shock-ionized, consistent with the effect of a jet with a kinetic power of ≈2 ×10 39 erg s −1 . Additional X-ray photo-ionization may also be present, to explain the strength of high-ionization lines such as He ii λ4686 and [Ne v] λ3426. On the other hand, the emission lines from the ULX-2 bubble are typical for photoionization by normal O stars suggesting that the nebula is actually an H ii region not physically related to the ULX but is simply a chance alignment. From archival Very Large Array data, we also detect spatially extended, steep-spectrum radio emission at the location of the ULX-1 bubble (consistent with its jet origin), but no radio counterpart for ULX-2 (consistent with the lack of shock-ionized gas around that source).	10.1093/mnras/sty014	[ "https://arxiv.org/pdf/1801.01889v1.pdf" ]	119,059,685	1801.01889	63b667d30aa609ee9255c0f15d305fe3199315ec	Multiband counterparts of two eclipsing ultraluminous X-ray sources in M 51 2017 R Urquhart International Centre for Radio Astronomy Research Curtin University GPO Box U19876845PerthWAAustralia R Soria International Centre for Radio Astronomy Research Curtin University GPO Box U19876845PerthWAAustralia National Astronomical Observatories Chinese Academy of Sciences 100012BeijingChina Sydney Institute for Astronomy School of Physics A28 The University of Sydney 2006SydneyNSWAustralia H M Johnston Sydney Institute for Astronomy School of Physics A28 The University of Sydney 2006SydneyNSWAustralia M W Pakull UMR 7550 Observatoire astronomique Université de Strasbourg CNRS 11 rue de l'Université67000StrasbourgFrance C Motch UMR 7550 Observatoire astronomique Université de Strasbourg CNRS 11 rue de l'Université67000StrasbourgFrance A Schwope Leibniz-Institut für Astrophysik Potsdam An der Sternwarte 1614482PotsdamGermany J C A Miller-Jones International Centre for Radio Astronomy Research Curtin University GPO Box U19876845PerthWAAustralia G E Anderson International Centre for Radio Astronomy Research Curtin University GPO Box U19876845PerthWAAustralia Multiband counterparts of two eclipsing ultraluminous X-ray sources in M 51 MNRAS 0002017Accepted XXX. Received YYY; in original form ZZZPreprint 5 November 2018 Compiled using MNRAS L A T E X style file v3.0accretion, accretion disks -stars: black holes -X-rays: binaries We present the discovery and interpretation of ionized nebulae around two ultraluminous X-ray sources in M 51; both sources share the rare property of showing X-ray eclipses by their companion stars, and are therefore prime targets for follow-up studies. Using archival Hubble Space Telescope images, we found an elongated, 100-pc-long emission-line structure associated with one X-ray source (CXOM51 J132940.0+471237; ULX-1 for simplicity), and a more circular, ionized nebula at the location of the second source (CXOM51 J132939.5+471244; ULX-2 for simplicity). We observed both nebulae with the Large Binocular Telescope's Multi-Object Double Spectrograph. From our analysis of the optical spectra, we argue that the gas in the ULX-1 bubble is shock-ionized, consistent with the effect of a jet with a kinetic power of ≈2 ×10 39 erg s −1 . Additional X-ray photo-ionization may also be present, to explain the strength of high-ionization lines such as He ii λ4686 and [Ne v] λ3426. On the other hand, the emission lines from the ULX-2 bubble are typical for photoionization by normal O stars suggesting that the nebula is actually an H ii region not physically related to the ULX but is simply a chance alignment. From archival Very Large Array data, we also detect spatially extended, steep-spectrum radio emission at the location of the ULX-1 bubble (consistent with its jet origin), but no radio counterpart for ULX-2 (consistent with the lack of shock-ionized gas around that source). INTRODUCTION Ultraluminous X-ray sources (ULXs) are off-nuclear, compact accreting X-ray sources with luminosities greater than those of Galactic stellar-mass black holes (see Roberts 2017 andSoria 2011 for reviews). They exceed the Eddington limit for a standard 10 M black hole (Lx 10 39 erg s −1 ) and are thought to be the result of either super-critical accretion onto stellar-mass compact objects or sub-critical accretion onto intermediate-mass black holes (IMBHs). Some of the brightest ULXs may contain IMBHs (Farrell et al. 2009); however, the consensus has moved towards super-Eddington stellar-mass accretors for most ULXs. The discovery of three neutron star ULXs (Bachetti et al. 2014;Israel et al. 2016;Fürst et al. 2016Fürst et al. , 2017Israel et al. 2017) has strengthened this interpretation. Emails: [email protected] (RU), [email protected] (RS) Being highly energetic sources, ULXs have a significant impact on their surroundings. Their extreme X-ray luminosity photo-ionizes the surrounding gas. Examples of ULX nebulae with a dominant X-ray photo-ionized component are Holmberg II X-1 (Pakull & Mirioni 2002) and NGC 5408 X-1 (Soria et al. 2006;Kaaret et al. 2003). The ionizing X-ray luminosities inferred from the optical line emission of these nebulae match (within a factor of a few) the apparent X-ray luminosities of the compact objects, suggesting that at least those ULXs are not strongly beamed and their luminosities are intrinsically high (e.g. Pakull & Mirioni 2002, 2003Kaaret & Corbel 2009). In addition to high photon luminosity, there is theoretical and observational evidence of powerful outflows (winds or jets) in ULXs (Ohsuga et al. 2005;Ohsuga & Mineshige 2011;Pinto, Middleton & Fabian 2016), consistent with their nature as super-critical accretors. The kinetic power of the outflows produces shockionized nebulae (ULX bubbles), such as those around IC 342 X-1 or Holmberg IX X-1 (Miller 1995;. Tell-tale signs of the presence of shock-ionized gas are the expansion velocity of the bubble (∼100-300 km s −1 ) and the high [S II]/Hα line ratio; in fact, some ULX bubbles were previously mistakenly identified as supernova or hypernova remnants. In most cases, photo-ionized and shock-ionized gas co-exist in a ULX bubble; the origin of the ionizing photons may be both the direct emission from the central object and the forward shock precursor. In a small number of shock-ionized ULX bubbles there is also direct evidence of collimated jets instead of (or in addition to) wide-angle outflows. The evidence is the elongated morphology of the bubble with symmetrical hot spots or lobes/ears. In particular, this is seen in NGC 7793-S26 Soria et al. 2010) and M 83-MQ1 . In both cases, the X-ray luminosity from the central source is less than the ULX threshold (≈10 39 erg s −1 ), either because our direct view of the source is mostly obscured, or because the instantaneous accretion rate is currently low. It is their kinetic power (Pjet > 10 39 erg s −1 ), inferred from the expansion velocity of the shocked gas and from the optical/IR line emission of the bubble , that puts those three sources in the class of super-critical accreting stellar-mass objects. Another well-known ULX bubble, MF16 in NGC 6946 (Roberts & Colbert 2003;van Dyk et al. 1994), has an elongated morphology consistent with the propagation of a symmetrical jet into the interstellar medium (ISM); in that case, the direct X-ray luminosity of the central source does exceed the ULX threshold, but may still be only a small part of the bolometric luminosity, mostly reprocessed in the UV band (Kaaret et al. 2010). Evidence of powerful jets has also been found in a few other ULXs that do not have optically bright, shock-ionized hot spots/lobes. In the ultraluminous supersoft source M 81 ULS-1, the jet is revealed by the red and blue Doppler shifts of its Balmer emission lines, corresponding to v ≈ 0.17c (Liu et al. 2015). The M 81 source is classified as an ultraluminous supersoft source, because of its thermal spectrum with blackbody temperature of ≈80 eV (Swartz et al. 2002). The presence of a collimated jet in that source suggests that jets can co-exist with thick winds, which are likely to be the hallmark of the supersoft class (Soria & Kong 2016;Urquhart & Soria 2016a;Poutanen et al. 2007); however, no evidence of jets has been found so far in other sources of that type. More often, it is bright radio emission with 5-GHz luminosity νLν > ∼ 10 34 erg s −1 that indicates the presence of a jet or at least of fast outflows. The reason is that some of the bulk kinetic energy of the outflow is used to accelerate relativistic electrons at the reverse shock; compression of the magnetic field lines in the shocked ISM leads to electron cooling via synchrotron radiation (Begelman, Blandford & Rees 1984). In Holmberg II X-1, the jet is revealed by a flaring radio core and radio-bright "knots" of synchrotron emission either side of the core (Cseh et al. 2014(Cseh et al. , 2015. The three extragalactic sources with elongated optical bubbles mentioned above all have strong optically-thin synchrotron emission. By analogy, the equally strong, optically-thin radio nebula (diameter ≈ 40 pc; Lang et al. 2007) around NGC 5408 X-1 is also interpreted as evidence of a jet. The strong radio source associated with NGC 5457 X-9 (Mezcua et al. 2013) could be another example, although in that case it is difficult to disentangle the ULX contribution from the surrounding HII region. In the Milky Way, SS 433 very likely belongs to the same class of jetted, super-critical sources with extended bubbles and mechanical power > ∼ 10 39 erg s −1 (Farnes et al. 2017;Goodall, Alouani-Bibi & Blundell 2011;Brinkmann et al. 2007;Fabrika 2004;Zealey, Dopita & Malin 1980); in that source, tell-tale signatures of the large-scale jet impact onto the ISM are the radio "ears" protruding out of the W 50 bubble (Dubner et al. 1998) and the characteristic optical line emission from filaments of shock-ionized gas (Boumis et al. 2007). Not every optically-bright, shock-ionized ULX bubble has a radio counterpart (NGC 1313 X-2 is a classical example of a large, optically bright, radio-quiet ULX bubble), and not every ULX with a bright radio counterpart is surrounded by an optically-bright, shock-ionized bubble (as we said for Holmberg II X-1). For a given kinetic power in the jet, the composition of the jet (leptonic or baryonic), the collimation angle, the nature of the compact object, the gas density and magnetic field strength in the ISM, and the age of the bubble may affect the fraction of kinetic power that goes into synchrotron cooling. In other cases, such as the radio flares detected in the transient M 31 ULX (Middleton et al. 2013), the jet may be active only for a short fraction of time during state transitions; this is in contrast with sources such as NGC 7793-S26 where the average jet power is ∼ several 10 40 erg s −1 over ∼10 5 yrs . We are conducting a study of ULXs associated with ionized nebulae in nearby galaxies, trying to determine which of them have evidence of jets, and whether the presence of a jet correlates with some X-ray spectral properties of the central source. We have recently identified (Urquhart & Soria 2016b) two ULXs in the same spiral arm (Figure 1) of the interacting galaxy M 51 (d = 8.58 ± 0.10 Mpc; McQuinn et al. 2016). The most intriguing property of those two sources is that both of them show sharp X-ray eclipses, likely due to occulations by their respective companion stars. For the first source, M 51 ULX-1, we were able to determine that the period is either ≈6-6.5 days, or ≈12-13 days; we did not detect enough eclipses to identify the period of the second source, M 51 ULX-2, but we estimated P ∼ 10 days based on its eclipse fraction (Urquhart & Soria 2016b). Both ULXs have similar luminosities (LX ≈ a few 10 39 erg s −1 ), must be viewed at high inclination (being eclipsing sources), and are likely to be located in ISM regions with similar properties (as they are only ∼ 400 pc apart along the same arm); however, they have different X-ray spectral characteristics (Urquhart & Soria 2016b). In this paper, we show that both ULXs are surrounded by ionized gas. In one case (M 51 ULX-1), the ULX is coincident with an elongated "jet-like" bubble. The other ULX (M 51 ULX-2) is surrounded by a quasi-circular bubble. We will argue that in ULX-2 the nebula is consistent with an ordinary H II region photo-ionized by young stars, while in ULX-1, the gas is shock-ionized and the morphology suggests a jet origin. We will also show that the candidate jet source has a radio counterpart, while the other ULX does not. Blue represents the F435W filter, green represents F555W and red represents F814W. The 15 ×15 white box contains ULX-1 and ULX-2 and shows the region in which we performed PSF photometry. Bottom panel: Chandra/ACIS X-ray colour image, on the same scale as the HST image. Red = 0.3-1 keV, green = 1-2 keV, and blue = 2-7 keV. Inside the box, ULX-1 is the lower source (softer colour) while ULX-2 is the upper source (harder colour). nel (WFC). This dataset includes three broadband filters (F435W, F555W, F814W) and the narrowband filter F658N, which covers Hα and the two adjacent [N II] lines (central wavelength of 6584Å and effective width of 75Å). The total exposure time for the F814W and F555W bands was 8160 s, while for F435W it was 16320 s and for F658N 15640 s. The archival data were processed "on-the-fly" using the standard ACS pipeline (CALACS: Hack & Greenfield 2000), correcting for flat-fielding, bias and dark current. We then built a normalized, weighted average of the images in the F814W and F555W bands, and subtracted it from the F658N image, in order to remove the continuum stellar contribution from the narrow-band filter. Henceforth, when we refer to the F658N image, we mean the continuum-subtracted image unless stated otherwise. OBSERVATIONS AND We performed aperture and point-spread function (PSF) photometry on the F435W, F555W and F814W filters, using the daophot package (Stetson 1987) embedded within the Image Reduction and Analysis Facility (iraf) software Version 2.16 (Tody 1993). The daofind task was used to identify point-like sources within a 50 ×50 region centred on the mid-point between ULX-1 and ULX-2 (a concentric region approximately four times larger than the white box in Figure 1). Aperture photometry was then performed on these stars using phot. Due to the crowded nature of the field (Figure 2), a circular source aperture with a radius of 3 pixels (0 .15) was used; the encircled energy fraction within that radius, for a point-like source, is ≈80%. For the PSF photometry, several bright, isolated sources were manually selected and inspected (with the tasks pstselect and psf); each candidate PSF star was manually inspected and those that appeared slightly extended were rejected. The average PSF created from those stars was then used as input to the photometry task allstar. PSF photometry was conducted on all stars within a 15 ×15 box centred on the mid-point of ULX-1 and ULX-2 (white box overplotted in Figure 1 and displayed in its entirety in Figure 2). In addition to the photometry of point-like sources, we measured the total background-subtracted count rates of the emission nebulae around the two ULXs. Finally, count rates of stellar and nebular sources were converted to physical magnitudes and fluxes in the various bands, using the zeropoint tables for ACS-WFC available on the STScI web site 2 . New LBT spectra Optical spectroscopic data together with the corresponding calibration exposures (dark and bias frames, flat fields, arcs) were taken with the Large Binocular Telescope's (LBT) Multi-Object Double Spectrograph (MODS) (Pogge et al. 2010). The targets were observed from 2016-05-10 UTC 06:31:53.064 (MJD 57518.272142) for a total of 3×900=2700 seconds, on each of the two MODS, and for each of two slit positions. We used a 0 .8-wide, segmented long slit with gratings G400L and G670L for the blue and red beam channels, respectively. The blue G400L grating has a resolution of 1850 at 4000Å and a nominal dispersion of 0.5Å per pixel, while the red G670L has a resolution of 2300 at 7600Å and nominal dispersion of 0.8Å per pixel. In the first observing configuration (OB1), the slit was oriented along the jet-like structure of ULX-1 (position angle PA = 80 • , measured from North to East); in the second configuration (OB2), the slit was oriented to contain both the (candidate) point-like optical counterparts of ULX-1 and ULX-2 ( Figure 3), at PA = 322.5 • . The raw data were bias-subtracted and flat-fielded with the reduction package modsCCDRed 3 Version 2.0.1, provided by Ohio State University. Spectral trimming and wavelength calibration were done with the Munich Image Data Analysis System (midas; Warmels 1992). We then used standard iraf tasks for further analysis. We extracted onedimensional background-subtracted spectra with the iraf task apall. We flux-calibrated the red and blue spectra using the response files provided by the MODS instrumenta- In the first observation, the slit was aligned along the jet-like structure near ULX-1, including also the candidate optical counterpart of the ULX. This corresponded to an instrument celestial position angle of 80 • East of North. The second slit was positioned so as to observe the candidate optical counterparts of both ULX-1 and ULX-2, along with some of the diffuse emission around the two X-ray sources. This corresponded to a position angle of 322.5 • . The yellow circles represent the Chandra positions of ULX-1 and ULX-2 with 0 .4 uncertainty. tion team. Finally, we analysed the spectra using the iraf task splot; in particular, for each emission line, we measured equivalent width (EW), full width at half maximum (FWHM), and central wavelength (by fitting a Gaussian profile). One of the main parameters we were interested in was the total Hα flux from the elongated nebula around ULX-1. The LBT slit was placed and oriented in order to cover as much of that region as possible ( Figure 3) but some of the flux (about 20 percent) falls out of the slit. However, we can measure the total emission of that nebula from the HST F658N image, which includes the flux from Hα plus [N ii] λ6548 and [N ii] λ6583. We combined the information on the Hα/[N ii] line ratio obtained from the LBT spectra, with the measurement of the total Hα + [N ii] flux from the HST image. This gave us an absolute measurement of the total Hα luminosity of the nebula, regardless of what fraction of it falls on the LBT slit. After calibrating the Hα flux, we then obtained the fluxes of all other emission lines in both the red and the blue spectrum, using their ratios to Hα. (In other words, we assumed as a first-order approximation that the same fraction of total flux was collected in the red and blue spectra.) We applied a similar procedure to determine the total Hα flux (and by extension, the fluxes in the other lines) of the nebula around ULX-2, which could only be partly covered by the 0 .8 LBT slit ( Figure 3). Finally, we de-reddened the spectra assuming line-of-sight extinction AV = 0.095 mag corresponding to E(B − V ) = 0.031 mag (Schlafly & Finkbeiner 2011), using the extinction curves of Cardelli, Clayton & Mathis (1989). We chose to correct only for the line-of-sight Galactic extinction as a simple, model-independent, standard reference value. We are aware that there must additional extinction intrinsic to M 51, but it is difficult to quantify it. The total dust reddening of the nebulae is likely less than or similar to the value indirectly inferred from X-ray spectral fitting of the two ULXs inside the nebulae (Urquhart & Soria 2016b), that is E(B − V )int < ∼ 0.1 mag; a plausible guess is that the intrinsic extinction through the M 51 halo is similar to the extinction through the Milky Way halo, that is the total AV is about twice the Galactic line-of-sight value. An additional caveat is that MODS does not have an Atmospheric Dispersion Corrector, so the spectral data suffer from slit losses due to atmospheric dispersion, when the slit is not fixed at or near the parallactic angle. Normalizing the Hα line flux to the value inferred from the HST image allows us to sidestep this problem, at least for the red spectra. We used the Ohio State University's online calculator 4 to estimate the effect that differential atmospheric refraction may have on our MODS spectra. The observations with the slit in the OB1 configuration were taken at an hour angle ≈ 0 h .9-1 h .2; the differential refraction between Hα and Hβ is < ∼ 0 .2. For the OB2 configuration, the spectra were taken at hour angles ≈ 2 h .2-2 h .5, corresponding to differential refractions ≈0 .3 between Hα and Hβ and ≈0 .4 between Hα and Hγ. This shift is already of the same size as the typical spatial scale of structural inhomogeneities (brighter and fainter clumps) in the two nebulae; therefore, in principle, a fraction of nebular emission may be on the slit at Hα but fall outside the slit at bluer wavelengths. We will argue a posteriori, from our results, that the uncertainty on the total extinction and on the effect of differential refraction are small and do not affect our interpretation. Archival VLA data Radio maps of M 51 were provided by G. Dumas (private communication); see Dumas et al. (2011) for a detailed discussion of the instrumental setup. In summary, observations were taken from 1998 to 2005 using the Very Large Array (VLA) at 1.4 GHz (20 cm) in the A, B, C and D configurations, at 4.9 GHz (6 cm) in the B, C and D configurations, and at 8.4 GHz (3.6 cm) in the C and D configurations. A total integration time of ∼ 170 hr over all three frequencies was achieved. The 8.4 and 4.9 GHz observations had been combined with data from the Effelsberg 100-m telescope Dumas et al. (2011), to detect extended emission regions that are resolved out in the VLA maps. The synthesised beam sizes for the 8.4 GHz, 4.9 GHz and 1.4 GHz are 2 .4, 2 .0 and 1 .5 respectively. Radio fluxes were determined using the Astronomical Imaging Processing System (aips; Greisen 2003). We fitted elliptical Gaussian profiles in the image plane, using the task imfit to determine the integrated flux of the detected sources; we also calculated peak fluxes and rootmean-square noise levels in the field around the ULXs. We placed 3-σ upper limits on non-detections. Astrometric alignment In order to identify any optical and/or radio candidate counterparts of ULX-1 and ULX-2, we first needed to verify and, if necessary, improve the astrometric alignment between the Chandra, HST, and VLA images. For our first iteration, we took the default Chandra astrometry of the re-processed and stacked archival data. The X-ray data were processed and analyzed with standard tasks within the Chandra Interactive Analysis of Observations (ciao) Version 4.7 (Fruscione et al. 2006); for a description of the Chandra observations, see Urquhart & Soria (2016b). The Chandra/ACIS-S astrometry is known to be accurate within ≈0 .5 for 68% of the observations, and within ≈0 .7 for 90% of the observations 5 . The HST astrometry was aligned onto the 2MASS catalogue (Skrutskie et al. 2006), which provides an accuracy within 0 .3. The VLA astrometry was assumed absolutely correct by default. With this first set of astrometric solutions, we looked for coincidences between the X-ray, optical, and radio sources (not including the ULXs that were the subject of our study), within ≈0 .5. The positions of the X-ray sources were determined with the ciao task wavdetect; the optical centroids of pointlike sources with profile fitting in the ds9 package (Joye & Mandel 2003); the radio positions with the aips package imfit. We found seven coincidences between X-ray and optical point-like sources, four coincident optical and radio sources, and four X-ray/radio coincidences. From a comparison of the optical/X-ray and optical/radio coincidences, we noticed that the HST astrometry was systematically offset by ≈0 .1 in both RA and Dec; therefore, we corrected the HST astrometry by that amount. The revised HST astrometry is still within the 0 .3 uncertainty of the 2MASS catalog. The X-ray positions (based on only 4 coincidences) were already within 0 .1 of the radio position and also within 0 .02 of the average of the improved HST positions. The resulting relative astrometry can be seen in Figure 4, with the average offsets between the three bands marked by stars. We did not find it necessary to make any further systematic corrections to any band. The residual discrepancies between the bands are simply random scatter of the individual sources. This is mostly due to the fact that the point spread function of point-like sources a few arcmin away from the Chandra/ACIS aimpoint becomes very extended and elongated, which increases the uncertainty in their central positions. In summary, we adopt the following X-ray positions for our ULXs: RA (J2000) = 13 h 29 m 39 s .960, Dec. (J2000) = +47 • 12 36 .86 for ULX-1; RA (J2000) = 13 h 29 m 39 s .444, Dec. (J2000) = +47 • 12 43 .60 for ULX-2. For both positions, we estimate a 3σ error radius of 0 .4 arcsec, due to their off-axis positions and elongated PSF shape. HST and VLA/Effelsberg sources. Chandra and HST coincidences are plotted as blue squares; Chandra and radio coincidences as red triangles; HST and radio coincidences as green circles. Stars represent the average offset between each pair of bands; systematic offsets are <0 .1 between any pair of bands. The residual discrepancies (<0 .4) are due to random scatter, mostly because of poor centroid determination of X-ray sources located far from the ACIS aimpoint. 23.14 ± 0.03 23.33 ± 0.03 23.59 ± 0.03 ULX-2 (south) 23.60 ± 0.04 23.77 ± 0.02 23.87 ± 0.04 Hα nebula ( Figure 3). For ULX-2, there are two potential optical counterparts within the Chandra error circle (Figure 2, bottom right), separated by ≈0 .25, with comparable brightness and colours. The relatively low precision of the Chandra position makes it impossible to rule out either candidate at this stage. The Vegamag brightnesses of the three sources in the HST bands (de-reddened for a line-of-sight extinction corresponding to AV = 0.095 mag, with Cardelli extinction law) are summarized in Table 1. At a distance modulus of 29.67 mag, the absolute magnitudes and colours of all three sources are consistent with those of bright giants (luminosity class II) or blue supergiants (class Ib); if two of those three sources are the true counterparts of ULX-1 and ULX-2, they are likely to contain also a contribution from disk emission. For simplicity, we investigate two extreme scenarios; first, purely emission from the stellar companion and second, purely emission from the irradiated disk. It is likely that the true emission is some combination of both processes, and thus we investigate the upper limits of stellar brightness and irradiation in this section. MAIN RESULTS Point-like optical counterparts To constrain the ages of the three sources and of the surrounding stellar population, we used colour-magnitude diagrams in the three HST broadband filters ( Figure 5); we plotted all stars with reliable brightness values within the 15 ×15 field shown in Figure 2. The black arrow in each plot indicates the effect of a hypothetical additional intrinsic extinction AV = 1 mag. We compared the data with a set of theoretical isochrones 6 (Bressan et al. 2012;Chen et al. 2015) for a metallicity Z = 0.015 ( Figure 5, top panels) and Z = 0.040 ( Figure 5, bottom panels). The choice of those two alternative metal abundances reflects the two discrepant classes of values generally quoted in the literature for the disk of M 51; see, e.g., Croxall et al. (2015) and Bresolin, Garnett & Kennicutt (2004) for the argument in favour of slightly sub-solar metallicities, and Moustakas et al. (2010) and Zaritsky, Kennicutt & Huchra (1994) in favour of supersolar values. We will return to this metallicity discrepancy in Section 3.2.2. Comparing the colour-magnitude plots, we notice that the inferred ages of the optical companion of ULX-1 are not consistent: ≈30 Myr in the (F435W − F555W) versus F555W diagram, and ≈10-20 Myr in the (F555W − F814W) versus F814W diagram, regardless of metal abundance. The two ages cannot be made consistent with the addition of intrinsic extinction; the star is ≈0.15 mag brighter in the F555W band than predicted by stellar isochrones. The reason for this small discrepancy is unclear. We note that the source is embedded in the most luminous part of the optical nebula, with its strong [O iii] λ4959,5007 emission lines falling into the F555W band, which makes the subtraction of the local background emission more challenging and may be responsible for the small excess in that band. Alternatively, some of the optical emission from that point-like source is coming from the accretion disk or the outflow around the ULX, so that its colours are slightly different from a pure stellar spectrum. Distinguishing between the optical emission from donor star and accretion flow is a notoriously difficult task in ULXs (Heida et al. 2014;Gladstone et al. 2013;Soria et al. 2012;Grisé et al. 2012;Tao et al. 2012Tao et al. , 2011; we will return to this issue later in this Section. Keeping those caveats in mind, we conclude that the optical counterpart for ULX-1 has a most likely age of ∼20 Myr, if dominated by a single donor star. We also plot stellar density bands (blue and green shaded regions in Figure 5). These shaded regions indicate regions of the parameter space where stars have a stellar density consistent with a Roche-lobe-overflow binary of period 6 or 12 days respectively. These two alternative values of the period come from the recurrent eclipses found in the X-ray light-curves for ULX-1 (Urquhart & Soria 2016b); the 6-day period is a solution suitable for mass ratios 4 < ∼ q < ∼ 10, while the 12-day period is the solution corresponding to mass ratios 0.5 < ∼ q < ∼ 1, where q ≡ M2/M1. Those two characteristic periods are then converted to stellar densities via the period-density-q relation (Eggleton 1983). The current optical data (taking into account also the possible contamination from disk and Table 2. Extinction-corrected emission line fluxes for the nebulae around ULX-1 and ULX-2, inferred from our LBT spectra and scaled by the HST images. Fluxes were corrected for a line-ofsight extinction A V = 0.095 mag. All fluxes were re-normalized so that the Hα + [N ii] fluxes match the corresponding line fluxes measured from the HST image. Fluxes are displayed in terms of their strength relative to the Hβ line (100F λ /F Hβ ). Line nebular emission) are not sufficient to test whether the candidate star sits in either band. Thus we cannot yet use this method to rule out either of the proposed orbital periods of ULX-1, and constrain its mass ratio. However, we illustrate this method as a proof of concept, and to highlight the fact that the candidate optical star is at least approximately consistent with the kind of Roche-lobe-filling donor star required for ULX-1. 100 × (f λ /f Hβ ) ULX-1 ULX-2 3426 [Ne v] 16 ± 1 3728 [O ii] (λ3726 + λ3729) 605 ± 28 88 ± 6 3726 [O ii] a 243 For ULX-2, both candidate optical counterparts have ages consistent with ≈10 Myr, and mass M ≈ 18M , if we correct for Galactic line-of-sight extinction only. If we also include additional dust reddening within M 51 equal to the Galactic value, the inferred age of both stars is instead ≈8 Myr, and their masses M ≈ 22-23 M . Stars at this evolutionary stage have a characteristic radius ∼ 30 R and a characteristic density ρ ∼ 3 × 10 −4 g cm −3 . This would imply a characteristic binary period of 24 days for q ≈ 1 (i.e. for the case of a canonical black hole primary), or a period of 10 days for q ≈ 17 (i.e. for the case of a canonical neutron star primary). The period of ULX-2 could not be directly measured from its X-ray light-curve (Urquhart & Soria 2016b) and so we do not have enough observational constraint to test this prediction yet; however, there is at least a plausible hint, from the average eclipse fraction, that the period is ∼10 days (Urquhart & Soria 2016b), consistent with the neutron star scenario. From Figure 5, we can see that both potential ULX-2 optical counterparts appear to be outliers, much brighter and bluer than the surrounding stars. We considered and dismissed the possibility that they are the counterparts of two luminous X-ray sources, unresolved in the Chandra image. The reason this is unlikely is that ULX-2 shows sharp, deep eclipses, during which the residual luminosity is L ≈ 3 × 10 37 erg s −1 . Barring implausible coincidences, this must be the upper limit to the luminosity of any potential second X-ray source. Such a luminosity would not be enough to contribute significantly to the optical emission, via reprocessing of X-ray irradiation, given that both optical sources have MV ∼ −6 mag. Thus, we conclude that at least one of the two blue point-like sources in the Chandra error circle for ULX-2 is an intrinsically bright star, while the other source may either be dominated by irradiation, or also be an intrinsically bright donor star. Next, we try to quantify whether X-ray irradiation can significantly contribute to the optical counterpart of ULX-1 and to one of the two blue stars at the location of ULX-2. The X-ray spectrum of ULX-1 is well fitted by an irradiated disk model (Urquhart & Soria 2016b), for example the XSPEC model diskir (Gierliński, Done & Page 2009). In Figure 6, we plot the combined X-ray spectrum and optical data points, together with a sequence of diskir models with varying levels of reprocessing (fout = 0.1, 0.05, 0.01 and 0.005). If we want to explain the optical flux as the intercepted and reprocessed component of the X-ray flux, we need a reprocessing fraction fout ≈ 0.075. This is an order of magnitude higher than predicted (Dubus et al. 1999;King, Kolb & Szuszkiewicz 1997;de Jong, van Paradijs & Augusteijn 1996;Vrtilek et al. 1990) or observed (Russell et al. 2014;Gierliński, Done & Page 2009;Hynes et al. 2002) in Galactic X-ray binaries with sub-Eddington standard disks, although Gandhi et al. (2010) found that the reprocessing fraction could be as high as 20% in the Galactic black hole binary GX 339-4. However, much less is known about the reprocessing fraction in ULXs. For a transient ULX in M 83, Soria et al. (2012) estimated a reprocessing fraction of f ≈ 5 × 10 −3 ; for other ULXs, reprocessing fractions of a few 10 −2 were inferred (Sutton, Done & Roberts 2014). Super-critical accreting sources have geometrically thicker disks in the inner region, but even so, the disk alone cannot directly intercept such a high fraction of the X-ray emission; however, if the ULX has a broad disk outflow and polar funnel geometry, some of the X-ray photons emitted along the funnel may be scattered by the wind and irradiate the outer disk (Sutton, Done & Roberts 2014;Narayan, Sadowski & Soria 2017). Indeed, we argued (Urquhart & Soria 2016b) that ULX-1 may have a strong wind, because of its soft X-ray spectral residuals, which in other ULXs are associated with outflows (Middleton et al. 2015;Pinto, Middleton & Fabian 2016). Nonetheless, we also know that ULX-1 must have a fairly large donor star, because of its eclipses, and we have already shown that the inferred size of the star is roughly consistent with the observed optical flux. Alternatively, the non-simultaneity of the X-ray and optical data may contribute to the unphysically high reprocessing fraction that is required. However, studying the long-and short-term Xray variability of ULX-1 (and ULX-2) from Chandra, XMM-Newton and Swift 7 , we find that the source only varies by a factor of ≈ 2. Any change in the optical flux (assuming irradiation of the disk) must be less than that amount, because the optical emission comes from the Rayleigh-Jeans tail of the disk spectrum and it includes an intrinsic non-variable contribution from the stellar companion. We can use a similar argument for ULX-2. In that case, the best-fitting xspec model to the X-ray spectrum reported by (Urquhart & Soria 2016b) is a diskpbb model (slim disk) rather than diskir, so we cannot directly constrain the reprocessing fraction from the model parameters. However, forcing a fit of the same spectrum with diskir we obtain a reprocessing fraction ≈ 0.2; this is because the observed X-ray flux is approximately the same as from ULX-1, while the optical flux is three times brighter. That makes it even more difficult to explain the optical emission as dominated by an irradiated disk rather than by a supergiant donor star. Ionized nebulae The HST/ACS image in the continuum-subtracted F658N filter clearly shows the presence of line emission around both ULXs (Figures 2 and 3). The nebula around ULX-1 is elongated, with a length of ≈2 .5 ≈100 pc and a width of ≈0 .7 ≈30 pc. The nebula around ULX-2 is more circular, with an outer radius of ≈1 .6 ≈65 pc. There is also another region of line emission ≈200 pc to the northwest of ULX-2, of similar size and brightness, but without any bright X-ray source inside. In the absence of other evidence, we assume that this nebula is unrelated to ULX-2, rather than (for example) being part of a double-lobed structure powered by the ULX. We defined source and background regions suitable for the two nebulae, and extracted their net count rates. We obtain rates of 21.52 ± 0.10 ct s −1 and 12.79 ± 0.25 ct s −1 for the ULX-1 and ULX-2 nebulae, respectively, in the F658N band. We then used the HST/ACS Zeropoint tables available online 8 (see also Bohlin 2016) to convert count rates to fluxes. A count rate of 1 ct s −1 corresponds to a flux of ≈1.95 ×10 −18 erg cm −2 s −1Å−1 in that band. The effective width of the filter is 75Å. Thus, the integrated line fluxes from the two nebulae are (3.39 ± 0.02) × 10 −15 erg cm −2 s −1 and (2.01 ± 0.04) × 10 −15 erg cm −2 s −1 , respectively, where we have also applied a de-reddening correction for line-ofsight extinction. At a distance of 8.6 Mpc, this corresponds to line luminosities of ≈3.0 ×10 37 erg s −1 and ≈1.8 ×10 37 erg s −1 , respectively, in the F658N filter. The filter bandpass covers [N ii] λ6548, Hα λ6563, and [N ii] λ6583. Those three lines are redshifted by ∆λ ≈ 9Å with respect to their rest-frame wavelengths, because of the systemic velocity of M 51. This ensures that the flux from the [N ii] λ6548 line is entirely included in the F658N bandpass, given the moderate line widths that we will discuss later in this Section. Of course the HST image alone cannot tell us how the line flux is split between Hα and the [N ii] lines. For this, we need to examine the LBT spectra. The LBT spectra across the ULX-1 and ULX-2 nebulae are displayed in Figures 7 and 8 respectively. The emission lines significantly detected in the two spectra, and their flux ratios to Hβ, are listed in Table 2. Those values are the total emission from the whole of each nebula. We recall that we obtain the total flux of each line by rescaling the Hα + [N ii] flux measured in the LBT spectrum to the total flux measured from the HST image in the F658N filter, and then rescaling the LBT fluxes of every other line according to their ratios to Hα. We also recall that ULX-1 was observed both in the OB1 (slit along the bubble) and OB2 (slit across the bubble) configurations. To a first approximation, the two spectra are identical; in this paper, we use the spectra for the OB1 slit, because they provide a better approximation to the average emission of the whole nebula. A more comprehensive comparison of the two spectra and of the internal structure of the ULX-1 nebula is left to further work. From the line ratios (Table 2), we find Balmer decrements Hα:Hβ = 2.99 ± 0.14, Hγ:Hβ = 0.46 ± 0.03 in the ULX-1 nebula; in the ULX-2 nebula, Hα:Hβ = 3.07 ± 0.15, Hγ:Hβ = 0.30 ± 0.02. The canonical values for Case B recombination are 2.87:1:0.47 at 10,000 K, and 2.75:1:0.48 at 20,000 K (Osterbrock & Ferland 2006). However, the Balmer decrement is slightly steeper for radiative shocks: for solar metallicity and a shock velocity of 150 km s −1 , the ratios are 3.06:1:0.46, while for a shock velocity of 500 km s −1 , the Balmer decrement is 2.92:1:0.47 (Allen et al. 2008). Thus, the observed Balmer decrement in the ULX-1 nebula is perfectly consistent with shock-ionized gas. Instead, for the ULX-2 nebula, the observed decrement is steeper than expected. Part of the reason for the discrepancy may be that a few percent of the flux in the bluer part of the spectrum is lost (compared with the flux in the red spectrum) because of differential refraction, as we mentioned in Section 2.2; this affects particularly Hγ and particularly the ULX-2 spectrum because of the slit orientation in the OB2 configuration (Section 2.2). In addition, we have only corrected for line-of-sight extinction; if we add an intrinsic dust reddening comparable to the Galactic reddening (Section 2.2), the extinction-corrected Hα:Hβ ratio is reduced by 4% and the Hγ:Hβ ratio is increased by 3%. Overall, those small uncertainties do not affect our main results and interpretation of the spectra. In the rest of this section, we focus on the most significant properties of the two nebulae, and their main differences. (Mathewson & Clarke 1973;Blair & Long 1997;Gordon et al. 1998). Based on those ratios alone, the bubble could be classified as a typical supernova remnant; however, its very elongated morphology and the F555W) versus F555W bands. The red square represents the candidate optical counterpart of ULX-1. The blue triangle represents the northern candidate counterpart for ULX-2 while the upsidedown blue triangle represents the southern candidate counterpart. Black datapoints show the surrounding stars, with the black error bars on the left indicating their average uncertainties for each F555W magnitude interval. The black arrow shows the effect of one magnitude of extinction in F555W. Overlaid are Padova stellar isochrones for metallicity Z = 0.015. The isochrones have been corrected for a line-of-sight extinction A V = 0.095 mag. The green and blue vertical bands drawn across the isochrones represent the expected brightness and colour of Roche-lobe-filling stars with stellar densities consistent with a binary period of 6 or 12 days respectively, as seen in the X-ray light-curve of ULX-1 (see Section 3.1). Top right panel: as in the top left panel, but for F555W − F814W versus F814W. Bottom left panel: as in the top left panel, but with Padova stellar isochrones corresponding to a metallicity Z = 0.040. Bottom right panel: as for the top right panel, but for Z = 0.040. ULX-1 nebula presence of a ULX in the middle make it more likely that the gas is shocked by a jet emitted from the compact object, as is the case in other ULX bubbles. The ULX-1 bubble also shows He ii λ4686 (Figure 7) and [N v] λ3426 (Table 2) emission. Both features are consistent with shock-ionized gas and/or X-ray photo-ionized gas, but are not consistent with stellar photo-ionization. In order to estimate the electron temperature and density of the gas, we used standard diagnostic line ratios (Osterbrock 1989), coded in the task temden in iraf (Shaw & Dufour 1994 = 20, 000 ± 1000 K. (In general, those two temperature values do not need to agree, as they may be due to emission coming from different regions.) Those temperatures are a factor of two higher than expected for typical H ii regions, and suggest that there is a significant component of shockionized gas. For the [S ii] line (sensitive to electron density), we find an intensity ratio [S ii] I(λ6716)/I(λ6731) ≈ 1.20. Around the previously estimated values of Te, this ratio corresponds to ne = (280 ± 60) cm −3 . This is to be interpreted as the density of the compressed gas behind the shock, much higher than the density of the undisturbed interstellar medium (ISM), typically ∼1 cm −3 in the spiral arms of a galaxy. Figure 6. Stacked Chandra/ACIS-S spectrum and HST datapoints (black) for ULX-1, fitted with a diskir plus thermal plasma models (red curves). A sequence of four models has been plotted, differing only for their optical reprocessing fraction fout. A very high reprocessing fraction (almost 10%) is formally required in this model to reproduce the high optical/X-ray flux ratio. Such a high value strongly suggests that we are seeing a significant contribution to the optical continuum flux from the donor star, although we cannot separate the disk and star components with the data at hand. From the dereddened Hβ flux, at the distance of M 51, we calculate an emitted luminosity L Hβ ≈ 3.7 × 10 36 erg s −1 . Knowing the characteristic temperature and density of the emitting gas, and using the Hβ volume emissivity for Case B recombination (Osterbrock 1989), we infer an emitting volume Ve ≈ 8 × 10 56 cm 3 . This is much lower than the volume inferred from the appearance of the nebula in the HST image, that is (roughly) an elongated bubble with a size of ≈30 × 100 pc corresponding to a volume of ≈10 60 cm 3 . Such a discrepancy is expected for a shock-ionized bubble, because the line-emitting gas occupies only a very thin layer behind the shock. For the inferred density and volume of the shocked layer, the mass of currently line-emitting gas is ∼200 M . The total gas mass originally present in the 10 60 cm 3 region that has now been swept up in the bubble must have been ∼10 3 M for ISM densities ∼1 cm −3 . The cooling timescale of the shocked gas from the immediate, hot post-shock temperature (T > 10 6 K) to the warm temperature range in which most of the optical line emission is produced (T ∼ 10 4 K) is ∼10 4 years, and that is followed by a short recombination timescale (≈10 3 years) of the warm gas (Dopita & Sutherland 2003); on the other hand, typical ages of shock-ionized ULX bubbles are ∼10 5 yr. Thus, it is likely that much of the gas that passed through the shock has already cooled, even if the ULX is continually supplying radiative and kinetic power. We then investigated whether the lines are broad or narrow. From the bright night-sky lines [O i] λ5577 and Hg i λ4358, we find an instrumental full width at half maximum FWHM ins,b = 2.95Å for the blue spectrum. In the red spectrum, strong night-sky lines are seen at λ = 6329Å, λ = 6498Å, λ = 6863Å: from the aver-age width of those lines, we measure FWHMins,r = 4.76Å. The intrinsic line width FWHMint is obtained from the observed width FWHM obs , following the relation FWHMint = (FWHM 2 obs −FWHM 2 ins ) 1/2 . Gaussian fits to the Hβ and Hα line profiles give observed FWHM obs of ≈3.7 and 4.0Å, respectively, corresponding to intrinsic widths of 140 ± 20 km s −1 and 110 ± 20 km s −1 . Other low-ionization lines have similar intrinsic widths: 115 ± 20 km s −1 for [O i] λ6300, 115 ± 20 km s −1 for [N ii] λλ6548,6583, and 130 ± 20 km s −1 for [S ii] λλ6716,6731. Thus, all low-ionization lines are significantly resolved, and are indicative of a shock velocity vs ≈ FWHMint ≈ 110-130 km s −1 . Further evidence of line broadening is obtained from the intrinsic half-width at zero-intensity (HWZI) for the Balmer lines; we measure HWZI(Hα) ≈ 280 km s −1 , and HWZI(Hβ) ≈ 250 km s −1 . On the other hand, somewhat unexpectedly, high-ionization lines are slightly narrower than low-ionization ones. Both [O iii] λ4959 and [O iii] λ5007 have intrinsic FWHMs = 90 ± 5 km s −1 and HWZI ≈ 200 km s −1 . He ii λ4686 is even narrower, with an intrinsic FWHM = 60±20 km s −1 and HWZI ≈ 180 km s −1 . Finally, we examined the more characteristic line ratios. A detailed modelling of the spectrum is beyond the scope of this paper; however, here we highlight the presence and strength of the high ionization lines [Ne v] λ3426, with I(λ3426)/I(Hβ) = 0.16 ± 0.01, and He ii λ 4686, with I(λ4686)/I(Hβ) = 0.22 ± 0.01. Neither of those lines can be produced at such intensity by a relatively slow (≈120 km s −1 ) shock plus precursor. The combined presence of [Ne v] λ3426 and He ii λ 4686 is the most intriguing feature of some ultracompact dwarf galaxies (Izotov, Thuan & Privon 2012); the origin of the lines in those galaxies was unclear, but the most likely candidates were suggested to be fast radiative shocks or an AGN contribution. In M 51 ULX-1, we have the rare opportunity to identify a source of such line emission. Moreover, the emission from [Ne iii] λ3869, [O iii] λ4959,5007, and [N ii] λ6548,6583 is remarkably strong, with I(λ3869)/I(Hβ) = 0.92 ± 0.05, I(λ4959)/I(Hβ) = 2.1 ± 0.1 and I(λ6548)/I(Hβ) = 1.27 ± 0.05. If entirely due to shockionization, those line ratios are approximately consistent only with shock velocities ∼500-600 km s −1 . Based on the shock plus precursor models of Allen et al. (2008), at this range of velocity, these line ratios are only weakly dependent on the unperturbed ISM density, metal abundance and magnetic field. We do not see such high-velocity wings in the LBT spectra; howewer, this could be due to our viewing angle of the collimated outflow in the plane of the sky, with the direction of the fast outflow almost perpendicular to our line of sight. ULX-2 nebula The spectrum of the nebula around ULX-2 (Table 1 and Figure 8) is clearly different from that of the ULX-1 nebula. From an analysis of the line width, we find that all lines are consistent with the instrumental widths (see Section 3.2.1). That is evidence that the gas is photo-ionized rather than shock-ionized. temperature and density. Unfortunately, the [O iii] λ4363 line is too faint to provide meaningful temperature constraints. Instead, for the [S ii] line doublet, we find an intensity ratio [S ii] I(λ6716)/I(λ6731) ≈ 1.45 ± 0.05. For temperatures ∼10 4 K, this ratio corresponds to the low-density limit, ne < 10 cm −3 , consistent with the standard ISM density of ∼1 cm −3 . Given the close proximity and similar environment of ULX-1 and ULX-2, it is plausible to assume that was also the unperturbed ISM density around ULX-1, before the gas was compressed by the shock front. With log ([N ii] λ6583/Hα) ≈ −0.51, log ([S ii] λ6716 + λ6731/Hα) ≈ −0.55, and log ([O iii] λ5007/Hα) ≈ −0.02, the nebula sits along the sequence of H ii regions photoionized by stellar emission (Kewley & Dopita 2002;Veilleux & Osterbrock 1987;Baldwin, Phillips & Terlevich 1981). The lack of high-ionization lines such as He ii λ4686 and [Ne v] λ3426 (detected in the ULX-1 nebula, Section 3.2.1) is further evidence that most of the ionizing photons come from the stellar population rather than the ULX. Nondetection of [O i] λ6300 suggests the lack of an extended low-ionization zone, which we would expect to find if the nebula were ionized by soft X-ray photons from ULX-2. Using nitrogen and oxygen lines as metallicity indica-tors, we run into the well-known discrepancy (Kennicutt, Bresolin & Garnett 2003;Bresolin, Garnett & Kennicutt 2004) between the "direct" metallicity calibration based on oxygen temperatures, and the "indirect" calibration based on photo-ionization models. The direct Te-based measurements require detection of faint auroral lines (e.g., [O iii] λ4363; Kennicutt, Bresolin & Garnett 2003), while photoionization models constrain the metal abundance from the intensity of stronger lines, for example the ratio between [N ii] λ6584 and [O ii] λλ3726,3729 (Kewley & Dopita 2002). It was noted (Bresolin, Garnett & Kennicutt 2004) that indirect photo-ionization abundances of H ii regions in nearby spiral galaxies are systematically higher than Te-measured abundances by a factor of 2-3. For our present study, we cannot directly measure Te abundances for the ULX-2 nebula; we have to rely on stronger lines ( Prieto, Lambert & Asplund 2001). This is the same as the metal 2007)). The FWHM of the two lines is also overplotted; both the FWHM and the FWZI are higher in the ULX-1 nebula. Middle panel: same as the top panel, but for the Hβ line; again, the line from the ULX-1 nebula is broader. Bottom panel: comparison between the line profiles of He ii λ4686, [O iii] λ5007, and Hβ (plotted with the same peak intensity for convenience) for the ULX-1 nebula: the relatively small FWHM and lack of broad wings in the higher-ionization lines are inconsistent with the high shock velocity that would be required to produce such lines entirely from shocks, and suggests a significant contribution to those lines from X-ray photo-ionization. abundance directly measured for several other H ii regions in the same spiral arm of M 51, at similar radial distances from the galactic nucleus (Croxall et al. 2015;Bresolin, Garnett & Kennicutt 2004). On the other hand, based on the indirect calibration of Kewley & Dopita (2002), Kobulnicky & Kewley (2004), and Pilyugin & Thuan (2005), the same line ratios correspond to log (O/H) +12 ≈ 9.1 ± 0.1, that is Z ≈ 2.5Z . A similar discrepancy also occurs if we use the so-called R23 metallicity indicator, that is the intensity ratio between ([O ii] λλ 3726,3729 + [O iii] λλ4959,5007) over Hβ (Pagel et al. 1979;Edmunds & Pagel 1984;Kewley & Dopita 2002;Kewley & Ellison 2008). The traditional calibration of this ratio based on photo-ionization models gives log (O/H) +12 ≈ 9.1, 3 times higher than the value derived from the direct measurements of Kennicutt, Bresolin & Garnett (2003) and Bresolin, Garnett & Kennicutt (2004) (see in particular Fig. 7 in the latter paper). Previous work in the literature supporting a super-solar metallicity for the M 51 disk (Moustakas et al. 2010;Zaritsky, Kennicutt & Huchra 1994) was also calibrated on photo-ionization models. Resolving the systematic discrepancy between the two metallicity calibrations will be important for population synthesis models of ULXs, because the ratio between neutron stars and black holes produced from stellar collapses increases strongly at super-solar metallicities (Heger et al. 2003). Radio counterpart of ULX-1 The VLA+Effelsberg data show a faint radio source near ULX-1, detected both at 1.4 GHz and at 4.9 GHz, however not detected at 8.4 GHz. In the 1.4 and 4.9 GHz radio images the source is marginally resolved, elongated in the east-west direction (Figure 10), that is the same long axis of the optical nebula. The position of the peak intensity in the radio source is slightly offset, by ≈1 , to the east of the Chandra position and of the central region of the ULX bubble in the HST image. Such an offset is significantly larger than the astrometric uncertainty (Section 2.4) of the X-ray and optical images. Thus, we speculate that the peak radio emission comes from a hot spot where the eastern jet interacts with the ISM, rather than being associated with the core. From the primary-beam corrected data, we determined peak brightnesses f1.4 GHz = (44.8 ± 9.2) µJy beam −1 , f4.9 GHz = (29.8 ± 7.3) µJy beam −1 , and an 8.4-GHz upperlimit of f8.4 GHz < 54 µJy beam −1 . The integrated fluxes are f1.4 GHz = (110±23) µJy and f4.9 GHz = (52.7±12.4) µJy. At the distance of M 51, this corresponds to a 4.9-GHz luminosity 9 L4.9 GHz = (2.3±0.5)×10 34 erg s −1 . From the integrated fluxes, we determine a spectral index 10 α = −0.6 ± 0.3, consistent with the optically-thin synchrotron emission expected from hot spots. Radio flux may also come from freefree emission associated with the ionized bubble: however, based on the Balmer emission, we only expect a free-free flux density of ≈1.5 µJy at 4.9 GHz (Caplan & Deharveng 1986, Appendix A), entirely negligible. The radio emission may also be a result of a supernova remnant; however, the highly elongated morphology of the optical nebula suggests that we are looking at a strongly collimated outflow, thus more likely driven by the ULX. We do not detect any radio emission associated with ULX-2 or its surrounding nebula. We can only place 3σ upper limits on the peak flux densities: f1.4 GHz < 28 µJy beam −1 , f4.9 GHz < 22 µJy beam −1 , and f8.4 GHz < 54 µJy beam −1 . This corresponds to an upper limit to the 4.9-GHz luminosity L4.9 GHz < 10 34 erg s −1 . The expected free-free emission from the H ii region around ULX-2 is ≈2 µJy at 4.9 GHz (Caplan & Deharveng 1986), well below the detection limit. DISCUSSION Although the X-ray luminosity of ULX-1 and ULX-2 is only moderately above 10 39 erg s −1 , the two sources have interesting properties that can help us understand the behaviour of super-critical accretion. Both sources are eclipsing (Urquhart & Soria 2016b), showing sharp transitions in which the companion star occults the compact object. Eclipsing sources are very rare among ULXs and X-ray binaries in general. The presence of periodic eclipses enables a number of binary parameters to be constrained (e.g. binary period, inclination angle, mass function, size of the X-ray emitting region). This makes these objects prime targets for follow-up multiwavelength observations. Here we have shown that both ULXs are associated with ionized gas nebulae, with different optical spectral properties. We have also shown that ULX-1 (but not ULX-2) has a radio counterpart (usually taken as a signature of jet activity). We shall now discuss the physical interpretation of the sources, putting together the various clues from these multiband observations. Energetics of the two nebulae As we have shown in Section 3.2, the most obvious difference between the two nebulae is that one is dominated by shock ionization (with possible additional contribution from X-ray photo-ionization), the other by near-UV photo-ionization. We shall discuss here how much kinetic and radiative energy is required to produce the luminosity observed from the two nebulae, and whether the two ULXs are the origin of such power. Mechanical power of ULX-1 From standard bubble theory (Weaver et al. 1977), the mechanical power Pjet inflating the bubble is equal to ≈77/27 of the total radiative luminosity L rad . The luminosity in the Hβ line (L Hβ ) is between ≈4 ×10 −3 and ≈7 ×10 −3 times the total radiative luminosity, for all shock velocities 100 < ∼ vs < ∼ 500 km s −1 (from the set of pre-run shockionization mappings III models for equipartition magnetic field and either solar or twice-solar abundance; Allen et al. 2008). Thus, Pjet ≈ (400-650) × L Hβ . Knowing that L Hβ ≈ 3.7 × 10 36 erg s −1 , we infer a mechanical power Pjet ≈ (1.5-2.5) × 10 39 erg s −1 . This value is comparable to the X-ray luminosity of ULX-1 (Urquhart & Soria 2016b), taking into account the uncertainty in the viewing angle and therefore the geometric projection factor for the disk emission. It is also similar to the jet power inferred for SS 433 (Panferov 2017;Fabrika 2004). There is no direct conversion between jet power and luminosity of the radio counterpart because the latter also depends on other factors such as the ISM density and magnetic field, and the fraction of kinetic power carried by protons and ions; however, it is interesting to note that the 5-GHz luminosity L4.9 GHz ≈ 2.3 × 10 34 erg s −1 is within a factor of two of the radio luminosity of other jetted ULXs such as Holmberg II X-1 (Cseh et al. 2014(Cseh et al. , 2015 and NGC 5408 X-1 Lang et al. 2007;Soria et al. 2006). Thus, the radio luminosity of the M 51 ULX-1 nebula seems to correspond to "average" jetted ULX properties: it is a factor of 5 more radio luminous than SS 433, but an order of magnitude fainter than the most exceptional radio bubbles in the local universe, namely NGC 7793 S26 Soria et al. 2010) and IC 342 X-1 , which are also a few times larger. The radio spectral index α = −0.6 ± 0.3 confirms that the radio counterpart is dominated by optically-thin synchrotron emission, as expected for jet lobes. An alternative method to estimate the jet power (Weaver et al. 1977) is to assume that the moderate shock velocity ≈100-150 km s −1 inferred from the FWHM of the Balmer lines also corresponds to the advance speed of the jet-powered forward shock into the ISM. In that scenario, the characteristic age of the bubble is ≈3 ×10 5 v −1 100 yr and the jet power Pjet ≈ 1.0 × 10 39 v 3 100 ne erg s −1 , where v100 ≡ vs/(100 km s −1 ) and ne is the ISM number density. This is consistent with the other jet power estimate. There is clearly a strong discrepancy between the moderately low FWHM of all the lines (∼100 km s −1 ) and the high shock velocity (∼500 km s −1 ) that would be needed to produce the higher ionization lines with the observed ratios; such high shock velocities are also inconsistent with the observed HWZIs and the moderate gas temperature inferred from the [S ii] lines. Even if we account for the fact that about half of the high-ionization line emission occurs in the precursor region (low turbulent velocity), the discrepancy remains difficult to explain. One possibility is that there is a contribution from X-ray photo-ionization, from ULX-1. The source appears almost edge-on to us, but if, as expected, its polar funnel is aligned with the major axis of the bubble (i.e. along the jet), there is stronger direct X-ray irradiation and lower intrinsic absorption along that direction. The fact that high-ionization lines are slightly narrower than lowionization lines supports the idea that the former are partly enhanced by X-ray photo-ionization. Another possibility is that the shocked gas does have a faster expansion velocity (comparable with the shock velocity) along the direction of the jet (which is located approximately in the plane of the sky for us) and a slower expansion in the other two directions perpendicular to the jet axis; in that case, the line width observed by us only reflects the lateral expansion speed. Soft thermal X-ray emission is a possible test for the presence of fast shocks: a shock velocity ≈500 km s −1 corresponds to a shock temperature ≈0.3 keV. Soft thermal emission was indeed detected in the X-ray spectrum of ULX-1 (Urquhart & Soria 2016b); unfortunately, the location of the source away from the aimpoint in the Chandra observations, and the low spatial resolution of XMM-Newton, do not enable us to determine whether the thermal-plasma component comes only from the central point-like source or from the larger bubble. In the former case, the X-ray line emission would be directly linked to the disk outflow around the ULX; in the latter case, it would be from the jet/ISM interaction. New Chandra observations with the source at the aimpoint could answer this question. Radiative power of ULX-2 The optical spectrum of the nebula around ULX-2 is consistent with a typical H ii region; this would mean that it is only a coincidence that a ULX is apparently located inside the nebula, without contributing to its ionization. To determine whether this is the case, we start by measuring its extinction-corrected Hβ luminosity: L Hβ ≈ 4.0 × 10 36 erg s −1 . At Te ≈ 10, 000 K, the emission of one Hβ photon requires ≈8.5 ionizing photons above 12.6 eV (Osterbrock 1989), and below ≈0.2 keV (above which energy the ionization cross section becomes too low). Therefore, the observed nebular luminosity implies an ionizing photon flux Q(H 0 ) ≈ 8 × 10 48 photons s −1 . From the best-fitting parameters in the solar-metallicity stellar tracks, we estimate that the two blue stars in the error circle of ULX-2 have effective temperatures of ≈20,000 K and ≈17,000 K, and radii of ≈25 R and ≈36 R , respectively. Approximating the spectra of the two stars as blackbodies, we expect a combined ionizing flux of ≈1.4 ×10 48 photons s −1 above 12.6 eV. Adding a few other blue stars inside the outer boundary of the nebula still leaves us a factor of 3 short of the required ionizing flux. However, using only the bright blue supergiants for this type of photon accounting can be misleading. For example, the addition of a single main-sequence mid-type O star (O6V or O7V), with a temperature ≈35,000-37,000 K, would suffice to provide the remaining ionizing flux (Simón-Díaz & Stasińska 2008). Such O-type dwarfs have an absolute brightness MV ≈ −4.5 mag, negligible compared with the characteristic absolute brightness MV ≈ −6 mag of the B4-B6 supergiants around ULX-2. Thus, if one of the four supergiants visible around ULX-2 ( Figure 2, bottom right) is in a binary system with a mid-type O star, the supergiants will determine the optical colours and V -band luminosity, but the O star will provide most of the ionizing UV flux. Another way to tackle the problem is to use starburst99 (Leitherer et al. 1999(Leitherer et al. , 2014 simulations: we find that for a single population age of 7 Myr, a stellar mass of ≈10 4 M would be required to provide the necessary ionizing flux. Adding a dust reddening correction for the M 51 halo (as discussed in Sections 2.2 and 3.1) would increase the intrinsic ionizing luminosity of the stars by ≈50%, making it easier to meet the energy requirements for the H ii region. Estimating whether there may also be a direct photoionizing contribution from ULX-2 is not trivial. The X-ray source is seen at a high inclination angle θ, which implies that an isotropic conversion of observed high-energy flux into emitted luminosity may underestimate the disk emission; a conversion factor ≈2π d 2 / cos θ is more appropriate than a factor of 4π d 2 . Taking this into account, we can always find a sufficiently high viewing angle (in particular, θ > ∼ 80 • ) to increase the true emitted disk luminosity of the source. To boost the number of UV photons, we must also assume that the X-ray source is surrounded by a large disk that intercepts and reprocesses more than 1% of the X-ray flux. Thus, an ionizing flux Q(H 0 ) ≈ 8 × 10 48 photons s −1 is consistent with the X-ray and optical luminosity of ULX-2. However, the problem of this irradiated-disk scenario is that it also produces copious amount of photons at energies > 54 eV, which leads to the emission of He ii λ4686 via recombination of He ++ into He + . No He ii λ4686 emission is detected in the LBT spectra of ths nebula, with an upper limit on its flux of ≈10 −16 erg cm −2 s −1 , corresponding to an emitted photon flux < ∼ 2 ×10 47 λ4686 photons s −1 . At Te ∼ 10, 000 K, it takes ≈4.2 primary ionizing photons >54 eV to produce one λ4686 photon (Pakull & Angebault 1986;Osterbrock 1989): thus, the flux of E > 54 eV photons seen by the nebular gas must be < ∼ 8×10 47 photons s −1 ≈ 0.1Q(H 0 ). Typical models of irradiated disks (e.g., diskir in xspec) luminous enough to produce the required number of photons >13.6 eV will also predict too many photons >54 eV (typically, the latter are predicted to be ≈1/4 of the former, rather than < ∼ 1/10, as required by the observed spectrum). Given this evidence, we conclude that the ionized nebula projected around ULX-2 is a typical H ii region and the ULX, whether located inside the nebula or not, is not its major ionizing source. 4.2 Radio brighter and radio fainter ULX bubbles ULX bubbles are a crucial tool to estimate the emitted power (radiative and kinetic luminosity) of the compact object, averaged over the characteristic cooling timescale of the gas. Moreover, they enable us to identify compact objects with likely super-critical accretion that currently appear X-ray faint, either because they are in a low state, or because they are collimated away from us, or because the direct X-ray emission along our line of sight is blocked by optically thick material. As more ULX bubbles get discovered or recognized, we are starting to see that there is no simple correlation between the X-ray luminosity of the central source and the radio or Balmer line luminosity of the bubble. The differences between M 51 ULX-1 and ULX-2 illustrated in this paper are a clear example. This is also the case when we compare the M 51 ULX-1 bubble with ULX bubbles in other galaxies. For example, the collisionally ionized ULX bubble around Holmberg IX X-1 Abolmasov & Moiseev 2008;Pakull & Grisé 2008;Pakull & Mirioni 2002) has an Hα luminosity ≈10 38 erg s −1 (Abolmasov & Moiseev 2008;Miller 1995;Miller & Hodge 1994), an order of magnitude higher than that of the M 51 ULX-1 bubble, consistent with a mechanical power of ≈10 40 erg s −1 ; the central ULX powering the Holmberg IX bubble has an X-ray luminosity LX ≈ 2 × 10 40 erg s −1 and references therein), again an order of magnitude higher than M 51 ULX-1. However, the Holmberg IX ULX bubble has a 1.4-GHz luminosity of ≈2 ×10 34 erg s −1 (our estimate based on the radio maps of Krause, Beck & Hummel 1989), similar to the radio luminosity of M 51 ULX-1. Conversely, there are ULXs with more luminous radio nebulae (such as Holmberg II X-1 and NGC 5408 X-1; see Section 4.1.1 for references) and proof of powerful jets, but with relatively weak or no collisionally ionized gas detectable from optical lines. The different ratio of synchrotron radio luminosity over mechanical power (or radio luminosity over total accretion power) in different ULXs may be due to various factors: i) a different initial kinetic energy distribution between a collimated, relativistic jet and a slower, more massive disk wind; ii) different composition of the jet (leptonic or baryonic); iii) different amount of entrainment of ISM matter along the jet path (related to different ISM densities). A general classification of ULX bubbles in terms of their multiband luminosity ratios is beyond the scope of this work. For the ULXs that do show radio evidence of extended jets, typical 5-GHz luminosities span an order of magnitude between ≈2×10 34 erg s −1 in M 51 ULX-1 to ≈2×10 35 erg s −1 in NGC 7793-S26 ) and IC 342 X-1 ; the characteristic age of the ULX bubbles is ∼ a few 10 5 yr. This luminosity range corresponds to a range of observed fluxes f5GHz ≈ (30-300) d −2 10 µJy, where d10 is the source distance in units of 10 Mpc. The synchrotron radio flux observed from microquasar lobes during the active expansion phase, that is when the jet is still inflating them, was modelled by Heinz (2002) as f5GHz ∼ 10 n 0.45 e P 1.3 jet,39 2ϕ 3/4 /(1 + ϕ) t 0.4 5 d −2 10 mJy, where ϕ is the ratio between the magnetic pressure and the gas pressure at the base of the jet, Pjet,39 is the jet power in units of 10 39 erg s −1 , t5 is the age of the jet active phase in units of 10 5 yr, and ne is the number density in the ISM. While the Heinz (2002) scaling relation works well for the radio nebula around the neutron star microquasar Cir X-1, it overestimates the synchrotron radio luminosity of the bubble around the BH microquasar Cyg X-1 by at least two orders of magnitude Gallo et al. (2005); it also over-estimates the synchrotron radio luminosity of ULX bubbles (given the input mechanical power inferred from line-emission studies) by a similar amount. For Cyg X-1, this may be the result of a significant fraction of the jet energy being stored in baryons that do not radiate (Gallo et al. 2005;Heinz 2006). A similar explanation may apply to jet-driven ULX bubbles . CONCLUSIONS We presented the discovery and multiband study of two ionized nebulae spatially associated with the two eclipsing ULXs in M 51, using new and archival data from Chandra, the LBT, the HST, and the VLA. The nebula around ULX-1 has a very elongated morphology, indicative of a jet-driven bubble. We showed that the optical emission lines are collisionally ionized, and that there is a spatially resolved radio counterpart. Such findings are consistent with the jet interpretation. We know from X-ray photometry that ULX-1 is seen almost edge on; therefore the jet is likely propagating in the plane of the sky, which explains the lack of systemic velocity shifts observed between the eastern and western ends of the optical bubble. The mechanical power of the jet estimated from the Hβ luminosity, Pjet ≈ 2 × 10 39 erg s −1 , agrees with the power estimated from the FWHM of the lines (proportional to the shock velocity) and the characteristic age of the source (t ∼ 10 5 yr). However, high-ionization lines (specifically, [Ne v] λ3426, He ii λ4686, and a few [O iii] lines) are too strong for the shock velocity vs ∼ 100-150 km s −1 inferred from the width of the low-ionization lines. Either there is also a shock component with vs ∼ 500 km s −1 , or the high-ionization lines are boosted by X-ray photo-ionization. We favour the latter scenario because the high-ionization lines are slightly narrower than the Balmer lines. Conversely, the nebula around ULX-2 has a quasicircular appearance and its line ratios and widths are consistent with a normal H ii region, that is with stellar photoionization from one or two O-type dwarfs and a few blue supergiants. There is no radio counterpart and no other evidence of jet emission. The reason why some ULXs produce jets while others (with comparable X-ray luminosities) do not, both in the case of the two M 51 sources and in the ULX class as a whole, remains unknown. The lack of highionization lines indicative of X-ray photo-ionization suggests that ULX-2 does not have a strong direct contribution to the ionization of the nebula. It is likely that the nebula is a chance alignment and is not powered by the ULX. The X-ray and multiband flux from the two compact objects and surrounding nebulae is at the low end of the ULX class distribution; however, the peculiar property of both sources is that they show X-ray eclipses, from which one can in principle strongly constrain the binary parameters, the masses of the two compact objects, and the geometry of accretion and emission. Therefore, they are particularly suitable targets for follow-up multiband studies. The metallicity of the parent stellar population can have a direct effect on the type of ULXs and on their observational appearance. For example at super-solar metallicities, the ratio of neutron stars over black holes is strongly increased, which may be relevant to the ongoing debate over the relative proportion of neutron star and black hole accretors in the ULX population. Neutron stars with a Roche-lobefilling massive donor star are also more likely to be eclipsing systems than black holes with the same type of donor star, which may explain why two eclipsing ULXs (Urquhart & Soria 2016b) and another eclipsing X-ray binary with a luminosity just below the ULX threshold (Song et al. , in prep.) were all found in this galaxy. Our LBT spectra of the ionized gas around ULX-2 are in agreement with the line ratios observed in previous optical studies of H II regions in the disk of M 51 (as discussed in Section 3.2.2); however, the conversion between such line ratios and a metallicity scale remains a subject of debate, with alternative models suggesting either Z ≈ 0.015 (slightly sub-solar) or Z ≈ 0.04 (2.5 times super-solar). Future, more accurate, determinations of the metallicity in nearby spiral galaxies will provide an important input parameter to test population synthesis models of ULXs. ACKNOWLEDGEMENTS We thank Gaelle Dumas for the reduced VLA+Effelsberg radio data. We also thank Jifeng Liu, Michela Mapelli, Mario Spera, Song Wang for useful discussions, and the anonymous referee for their constructive comments. RU acknowledges that this research is supported by an Australian Government Research Training Program (RTP) Scholarship. RS acknowledges support from a Curtin University Senior Research Fellowship; he is also grateful for support, discussions and hospitality at the Strasbourg Observatory during part of this work. JCAM-J is the recipient of an Australian Research Council Future Fellowship (FT140101082). The International Centre for Radio Astronomy Research is a joint venture between Curtin University and the University of Western Australia, funded by the state government of Western Australia and the joint venture partners. Based on observations made with the NASA/ESA Hubble Space Telescope, obtained from the Data Archive at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. These observations are associated with program 10452. This paper used data obtained with the MODS spectrographs built with funding from NSF grant AST-9987045 and the NSF Telescope System Instrumentation Program (TSIP), with additional funds from the Ohio Board of Regents and the Ohio State University Office of Research. IRAF is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation. Figure 1 . 1Top panel: HST/ACS-WFC RGB colour image of M 51. Figure 2 . 2Top panel: zoomed-in HST/ACS image of the white box displayed inFigure 1. Blue represents the F435W band, green = F555W and red = F814W. Yellow circles represent the Chandra positions of ULX-1 and ULX-2 with 0 .4 errors. White contours represent optical line emission in the continuum-subtracted F658N band. Line emission can be seen near both ULXs; in ULX-1 there is an elongated jet-like structure while ULX-2 is inside a more spherical nebula. Contours levels are arbitrary and are simply used to indicate the morphology of the ionized gas nebulae. Bottom left panel: zoomed-in view of the ULX-1 field. Bottom right panel: zoomed-in view of the ULX-2 field. Figure 3 . 3LBT slit positions, overplotted on the continuumsubtracted F658N image. ForFigure 4 . 4ULX-1, only one bright optical candidate lies within the Chandra error circle(Figure 2, bottom left). This candidate is embedded in the brightest part of the elongated Relative positions of coincident, point-like Chandra, We see strong [N ii] and [S ii] lines relative to Hα (Figure 7), with a flux ratio ([N ii] λ6548 + [N ii] λ6583)/Hα ≈ 1.7 and ([S ii] λ6716 + [S ii] λ6731)/Hα ≈ 0.81. Such high ratios (even after accounting for a possible super-solar abundance in the disk of M 51) are traditionally good indicators of shock-ionized material Figure 5 . 5Top left panel: colour-magnitude diagram in the (F435W − Figure 7 .Figure 8 . 78Left panel: LBT MODS1 plus MODS2 spectrum (blue arm) of the nebula around ULX-1, normalized to the continuum. The spectrum comes from the OB1 slit configuration (PA = 80 • ), which is along the long axis of the jet-like nebula. Inset: zoomed-in view of the He ii λ4686 emission line. Right panel: as in left panel, but for the red side of the spectrum. (Note the different flux density scales of the two panels.) Left panel: LBT MODS1 plus MODS2 spectrum (blue arm) of the nebula around ULX-2, normalized to the continuum. The spectrum comes from the OB2 slit configuration (PA = 322 • .5); the stellar continuum includes both candidate optical counterparts of ULX-2. Right panel: as in the left panel, but for red side of the spectrum. (Note the different flux density scales of the two panels.) Figure 9 . 9Top panel: Hα line profile for the nebulae around ULX-1 (blue) and around ULX-2 (red). The shaded band indicates the systemic velocity expected in the M 51 spiral arm at the location of ULX-1 and ULX-2 (≈400-425 km s −1 , from Vlahakis et al. (2013) and Shetty et al. ( Figure 10 . 10Top panel: HST/ACS continuum-subtracted F658N image of the ULX field, with VLA radio contours overplotted. White and red contours represent the 4.9-GHz and 1.4-GHz flux densities, respectively. Both sets of contours have levels of −2 √ 2σ, 2 √ 2σ and 4σ, where σ is the respective noise level (σ 4.9 = 7.3 µJy beam −1 , σ 1.4 = 9.2 µJy beam −1 ). Beams for both frequencies are indicated in the top left corner of the image. The yellow circles represent the Chandra positions of ULX-1 and ULX-2 with 0 .4 uncertainty. Bottom panel: zoomed-in view of the ULX-1 bubble. DATA REDUCTION 2.1 Archival HST images Publicly available Hubble Space Telesecope (HST) data for M 51 were downloaded from the Hubble Legacy Archive 1 . The observations were part of the HST Mosaic of M 51 1 http://hla.stsci.edu/hlaview.html (proposal ID 10452) taken on 2005 January 12 with the Advanced Camera for Surveys (ACS), Wide Field Chan Table 1 . 1De-reddened brightness of the candidate optical counterparts. Brightness values have been corrected for a line-of-sightGalactic reddening E(B − V ) = 0.031 mag, corresponding to an extinction A F435W = 0.13 mag, A F555W = 0.10 mag and A F814W = 0.06 mag. All three optical sources are consistent with bright giants (luminosity class II) or supergiants (class Ib), of ap- proximate spectral type B5-B6 for the ULX-1 star, and B3-B4 for the ULX-2 stars. Source ID F435W F555W F814W (mag) (mag) (mag) ULX-1 24.62 ± 0.03 24.64 ± 0.05 24.87 ± 0.07 ULX-2 (north) Hβ flux (10 −16 erg cm −2 s −1 ) 4.21 ± 0.18 4.47 ± 0.17 a Deblended with iraf line-fitting tools b Slightly contaminated by [S ii] λ4076± 11 3729 [O ii] a 353 ± 16 3869 [Ne iii] 92 ± 5 3889 Hζ + He i 17 ± 2 3967 [Ne iii] + H λ3970 47 ± 2 4069 [S ii] b 22 ± 5 4104 Hδ 24 ± 2 4340 Hγ 46 ± 3 30 ± 2 4363 [O iii] 23 ± 2 4686 He ii 22 ± 1 4861 Hβ 100 100 4959 [O iii] 207 ± 10 30 ± 3 5007 [O iii] 604 ± 28 96 ± 5 5200 [N i] (λ5198 + λ5202) 15 ± 1 6300 [O i] 66 ± 3 6364 [O i] 24 ± 2 6548 [N ii] 127 ± 6 48 ± 2 6563 Hα 299 ± 14 307 ± 15 6583 [N ii] 377 ± 19 95 ± 5 6716 [S ii] 132 ± 6 51 ± 3 6731 [S ii] 111 ± 5 35 ± 3 7136 [Ar iii] 21 ± 1 7320 [O ii] 51 ± 4 9069 [S iii] 20 ± 1 9532 [S iii] 40 ± 3 10049 H i Paα 3.0 ± 0.2 12818 H i Paβ 2.2 ± 0.3 * ). For the [O iii] line (sensitive to electron temperature), we measure an intensity ratio I(λ4959+λ5007)/I(λ4363) ≈ 35, which corresponds to Te[O iii]= 22, 000 ± 2000 K. From the [O ii] line intensity ratio I(λ3726 + λ3729)/I(λ7320 + λ7330) ≈ 12, we measure Te[O ii] The [N ii] and [S ii] lines are much weaker, relative to Hα, with flux ratios ([N ii] λ6548 + [N ii] λ6583)/Hα ≈ 0.47 and ([S ii] λ6716 + [S ii] λ6731)/Hα ≈ 0.28. This is again consistent with a photo-ionized H ii region.Again we tried using standard diagnostic line ratios for3800 4000 4200 4400 4600 4800 5000 Wavelength (Å) 0 20 40 60 80 100 120 Normalised Flux Density [O II] [Ne III] Hζ H [S II] Hδ Hǫ [O III] Hβ [O III] [O III] 4680 4690 4700 0 1 2 3 4 5 He II 6400 6600 6800 7000 7200 7400 Wavelength (Å) 0 20 40 60 80 100 Normalised Flux Density [O I] [O I] [N II] Hα [N II] [S II] [S II] [Ar III] [O II] Table 2 ) 2and use ei- MNRAS 000, 1-16(2017) http://www.stsci.edu/hst/acs/analysis/zeropoints/old page/ localZeropoints MNRAS 000, 1-16(2017) http://www.astronomy.ohio-state.edu/MODS/Software/ modsCCDRed/ http://www.astronomy.ohio-state.edu/MODS/ObsTools/ obstools.html#DAR http://cxc.harvard.edu/cal/ASPECT/celmon/ MNRAS 000, 1-16 (2017) Available at http://stev.oapd.inaf.it/cgi-bin/cmd For XMM-Newton and Swift, ULX-1 and ULX-2 are unresolved and thus we can only study their combined variability. 8 https://acszeropoints.stsci.edu/ The luminosity is defined as Lν = 4πd 2 νfν . . 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[ "Quelling the concerns of EPR and Bell", "Quelling the concerns of EPR and Bell" ]	[ "K L H Bryan \nDepartment of Physics & Electronics\nRhodes University\n6140GrahamstownSouth Africa\n", "A J M Medved [email protected] \nDepartment of Physics & Electronics\nRhodes University\n6140GrahamstownSouth Africa\n" ]	[ "Department of Physics & Electronics\nRhodes University\n6140GrahamstownSouth Africa", "Department of Physics & Electronics\nRhodes University\n6140GrahamstownSouth Africa" ]	[]	We begin with a review of the famous thought experiment that was proposed by Einstein, Podolsky and Rosen (EPR) and mathematically formulated by Bell; the outcomes of which challenge the completeness of quantum mechanics and the locality of Nature. We then suggest a reinterpretation of the EPR experiment that utilizes observer complementarity; a concept from quantum gravity which allows spatially separated observers to have their own, independent reference frames. The resulting picture provides a self-consistent resolution of the situation that does not jeopardize causality nor unitarity, nor does it resort to "spooky" (nonlocal) interactions. Our conclusion is that EPR and Bell rely on an overly strong definition of locality that is in conflict with fundamental physics.	null	[ "https://arxiv.org/pdf/1401.5988v1.pdf" ]	118,409,357	1401.5988	2d34c569ae60b478fe2cc11b6f6a5218f1246422	Quelling the concerns of EPR and Bell 23 Jan 2014 K L H Bryan Department of Physics & Electronics Rhodes University 6140GrahamstownSouth Africa A J M Medved [email protected] Department of Physics & Electronics Rhodes University 6140GrahamstownSouth Africa Quelling the concerns of EPR and Bell 23 Jan 20141 We begin with a review of the famous thought experiment that was proposed by Einstein, Podolsky and Rosen (EPR) and mathematically formulated by Bell; the outcomes of which challenge the completeness of quantum mechanics and the locality of Nature. We then suggest a reinterpretation of the EPR experiment that utilizes observer complementarity; a concept from quantum gravity which allows spatially separated observers to have their own, independent reference frames. The resulting picture provides a self-consistent resolution of the situation that does not jeopardize causality nor unitarity, nor does it resort to "spooky" (nonlocal) interactions. Our conclusion is that EPR and Bell rely on an overly strong definition of locality that is in conflict with fundamental physics. Introduction Prologue Quantum mechanics is a field of study that is infested with counter-intuitive concepts, and many of our classical preconceptions are brought into question when we deal with situations in the quantum realm. At the forefront of quantum paradoxes is the thought experiment that was put forward in the famous and influential paper by Einstein, Podolsky and Rosen (EPR). The EPR paper sparked a debate that is still in progress and requires a resolution if quantum mechanics is to be validated as a theory which gives a complete description of reality in its domain of applicability. This paper examines the EPR argument as well as Bell's formulation thereof and then, using a concept brought in from quantum gravity, suggests a different tact for addressing the unease of the situation. In brief, by reworking the EPR and Bell scenario within the framework of observer complementarity, any "spooky action at a distance" is vanquished and the concerns of EPR and Bell are resolved. Where it all started The EPR thought experiment presented a challenge for quantum mechanics. It highlighted a fundamental paradox within the theory which suggested that, while the theory worked as a calculation device, it may not represent reality. In the EPR paper [1], the argument begins with the consideration of a standard quantum-mechanical concept; namely, non-commuting operators. Let us consider an experimenter, called Alice, who has access to a particle in a state given by \|Ψ . The mathematical description of Alice physically measuring a property of the particle, say momentum, is given by acting on the initial state with the relevant operator. The value that the operator retrieves is associated with the physical value that would be measured by Alice. EPR does not present a comprehensive definition of reality but, rather, remains satisfied with a "criterion of reality" by which they identify a physical reality with the corresponding physical quantity. By this criterion, the value produced by an operator acting on a state represents a real physical quantity; an element of reality. However, quantum operators are not always so compliant and the issue of non-commuting operators arises. The momentum and position of a particle form just such a pair of incompatible operators. The EPR argument then states that, if Alice had measured the position of the particle rather than its momentum, the position has physical reality but the momentum does not. The original state is changed by the measurement interaction and the momentum value of the original state is lost as a consequence of measuring the position. And so there are pairs of operators for which the corresponding physical values may be determined but having a definite value for one implies that no definite value exists for the other. 1 By this reasoning, the EPR argument concludes that we must either accept that these physical quantities relating to incompatible observables do exist but quantum mechanics does not fully describe them or that the properties associated with two non-commuting operators cannot have physical reality at the same time. Only one can be considered an element of reality. The next step of the EPR argument involves combining the EPR reality criterion with the assumption that quantum mechanics does offer a complete description. Their logic is perhaps more easily explained in terms of the spins of two particles, rather than the position and momentum of a single particle as originally used in the EPR paper. Bohm introduced this newer formulation of the EPR setup and showed that the argument's structure remained the same, regardless of which pair of non-commuting observables are being considered [2]. Bohm's scenario uses a pair of particles in a singlet state and the role of non-commuting operators is now adopted by the different components of either particle's spin vector. This state can be described, in standard Dirac notation, 2 as \|Ψ = 1 √ 2 \|a+ \|b− − \|a− \|b+ ,(1) with the two particles, α and β, having anti-correlated spins of a± and b∓, respectively. Here, + or − refers to spin up or down with respect to some chosen reference axis (typically but not exclusively, the z-axis). We now introduce two spatially separated observers, Alice and Bob. They are each sent one particle, α and β respectively, from an initial starting point. The particles, prior to their being sent off, are prepared in a singlet state as described above. All the information about the particles is provided by the relevant theory; presumably, orthodox quantum mechanics. (Later on, λ is used to denote, schematically, the encapsulation of all this information.) When the particles arrive at Alice and Bob, each observer has a choice as to which direction they choose to measure. In general, they choose directions that differ from the initial reference axis and differ from one another. Let us now focus on Alice's measurement of α. Alice can determine α's spin in one direction only, sayn a . However, the perfect anti-correlation of the particles means that, by measuring the spin in a given direction, Alice will be able to predict β's spin in that same direction. By the reality criterion of EPR, this means that β has a physical property relating to Alice's measurement value. This physical property, as EPR points out, cannot have "sprung" into existence at Bob's location as a result of Alice's measurement as this violates the condition of locality [1]. A non-local action by Alice cannot instantaneously effect Bob. It should be noted, though, that this is not a violation of causality but of the definition of locality as assumed by EPR. Although Bob's outcome can depend on the choice made by Alice (assuming that he, by chance, choosesn b =n a ), she cannot communicate with or signal Bob using only the anti-correlation of the particles (e.g., [4]). In order to signal Bob, Alice would be required to pick up a phone, write an email or physically move to Bob's location to relay any information regarding the chosen direction. The puzzle does not lie, then, in the realm of faster-than-light signaling but in the world of "spooky" dependence of the state on non-local variables [5]. EPR's contention, therefore, is that the physical quantity corresponding to the spin of β in the directionn a must have been determined before the spatial separation of the two spins. Then, remembering that Alice could have chosen any direction to measure in, EPR are lead to the conclusion that all the physical properties of β's spin must have been similarly predetermined. This, by their reality criterion, implies a physical reality for the spin of β in each and every direction. And so we must face the contradiction of having a set of non-commuting operators (the different components of spin) corresponding to simultaneously real properties. As stated above, the EPR argument comes down to the following choice: The inability of quantum mechanics to completely describe physical reality versus the properties of non-commuting observables not simultaneously existing. But, since the above argument has shown that assuming the completeness of quantum mechanics leads to the simultaneous existence of non-commuting spin directions, the conclusion apparently must be that the quantum mechanics provides an incomplete description of all elements of reality. Although the EPR paper is very clear about its conclusion, there are those who disagree with this outcome. At the forefront of the dissension is Bohr's interpretation of the EPR experiment [6]. One main concern for Bohr was EPR's assumption that hypothetical experiments could be used in conjunction with experiments that were actually done; for example, considering the value of momentum after measuring the position. Bohr's argument led him to conclude that the inability to describe certain situations is a part of reality. This assumption of EPR -that physical meaning can be ascribed to hypothetical measurements -is a large part of the disagreement between Bell's formulation of the EPR experiment and the argument that will be put forth in the current paper. The reliance of EPR and Bell's theorem on counter-factual statements about such hypothetical events make their outcomes incompatible with observer complementarity; an important consequence of quantum gravity that may (and, as later argued, does) permeate into more "conventional" physics. This will be discussed in more detail below but, first, Bell's position on the EPR experiment requires an explanation. Bell's theorem Certainly, the most significant development arising out of the prolonged debate on the EPR experiment is Bell's theorem [7,8]. However, even though its importance is never disputed, what the theorem actually proves remains a matter of controversy (see, e.g., [9]). The discord can be attributed, in part, to an "assumption" that Bell makes in his theorem; namely, the infamous "hidden variables". These being the variables that Bell includes in addition to those used in the standard description of quantum mechanics. We will, for current purposes, turn to a modern interpretation of Bell's work, as delineated in a series of articles by Norsen [10,9,11,12,13]. This account of the theorem utilizes Bell's description of locality as the primary starting point. (Bell's locality definition is clarified in Section 2.) One of Norsen's main points of emphasis is that Bell did not intend for his analysis to serve as a stand-alone argument. Instead, the EPR argument must be considered as the first of a two-step procedure, with the second step being the formulation of Bell's celebrated inequality. This is a nuance that has been missed by some, leading to a misunderstanding about the significance of Bell's "hidden variables". All the relevant variables, whether hidden or otherwise, are meant to be contained in the initial-state parametrization, λ. Therefore, the inclusion or exclusion of hidden variables is really besides the point when Bell's inequality is implemented. Consequently, what Norsen's treatment shows is how any theory, with or without hidden variables, that is able to correctly describe the behavior of quantum particles must (in some cases) violate Bell's inequality and, therefore, disobey his criteria for locality. This argument will be reviewed, in due course, with particular emphasis on the role that is played by counter-factual statements; these being ubiquitous in Bell's formulation. Such statements are, in this context, referring to hypothetical measurements that could have been performed by Alice and Bob but are never actually carried out. Following this review, we will present a different approach, from the perspective of observer complementarity, that resolves any issues regarding "spooky action at a distance". Nature's censorship of paradoxes The principle of observer complementarity arose out of horizon complementarity, which applies to Nature's seemingly exotic behavior around the horizon of a black hole [14,15,16,17]. In either case, the principle is that, although different observers can disagree on the occurrence of certain events, Nature will not allow any observer to ever experience a paradoxical situation. In order to understand the basic argument, the notion of information flowing out of a black hole must first be understood. It was Hawking who first showed that black holes slowly evaporate by emitting radiation with a nearly thermal (black-body) spectrum [18]. Importantly, this effect can be attributed to the influence of the black hole's gravitational field on quantum matter fields; that is, black holes do not radiate unless quantum effects are accounted for. Therefore, black hole radiation is realized in a situation for which both gravity and quantum theory are important. This process, however, posed problems for the quantum side as a pure state entering the black hole will later be observed in the radiation as a mixed state; information is apparently lost [19]. Hawking later revised this position [20], as the viability of information loss was challenged by ideas from string theory, the tentative theory of quantum gravity [21]. These ideas suggest that any process which results in information loss is strictly forbidden in what is a manifestly unitary theory (namely, the quantum-field-theory dual to Einstein's gravity [22]). Hawking's original calculation used only an approximation of quantum gravity because it described the black hole as a classical body. In recent work, a more rigorous version of Hawking's calculation, using a description of the black hole in quantum terms, reveals that information is not lost [23] -much in the same way that information from a burning encyclopedia can, in principle, be recovered. These arguments are beyond the scope of the present paper, but what is essential is that information can indeed be accessed from the radiation. The availability of this information to an observer outside of the black hole is the starting point of Susskind's standard argument for horizon complementarity -see [24] for a simplified account. To illustrate this argument, two observers are considered by Susskind; one falling freely into the black hole and one watching the black hole from afar. When the first observer, Alice, falls past the horizon of the black hole, 3 she is seen by the distant observer, Bob, as being vaporized before ever reaching the horizon. This is because, from Bob's perspective, the near-horizon region is extremely hot. Then, from the above description, it can be deduced that the "Hawking radiation" will eventually contain information about Alice that can be accessed by Bob. However, Alice, from her point of view, experiences no change in scenery as the space near the horizon of a large enough black hole is almost flat. 4 5 And so we are left with a situation in which one or the other experiences a violation of the laws of nature; either Alice thermalizes instead of experiencing flat space or Bob observes Alice falling in without thermalizing. Susskind's next step is to consider whether either observer must see a violation. An apparent resolution is to assume that the information (or Alice) is "cloned" at the horizon, so that one copy of Alice is radiated out and the second copy falls in. However, this requires violating a basic principle of quantum mechanics: Quantum information can not be duplicated in this way since it would be in conflict with the principle of linear superposition [27]. Considering the two observers, Susskind clarifies that the key to resolving this problem lies with the movement of information. Alice is safe from seeing any violation since she is trapped within the black hole horizon but Bob has the opportunity to gather Hawking radiation (and hence information) from the black hole and then follow Alice in. If, after entering the black hole, Bob could receive a signal from Alice, there would be a violation of Nature due to Bob having observed two cloned copies of the same information. Susskind, however, presents an argument which proves that this is not a problem. After passing through the horizon, Alice has a limited time to send a signal to Bob before she hits the singularity of the black hole (where all matter would be destroyed by immensely large tidal effects). What Susskind shows -by applying the Heisenberg uncertainty principle and the knowledge that Bob has to wait a certain time before he can retrieve a copy of the signal from the Hawking radiation [28,29] -is that Alice's signaling device would then have to be more energetic than the black hole itself. Consequently, Alice's device would no longer be able to fit inside the black hole [30], rendering the whole experiment as moot. And so we see that each observer has an individual account of events, which differ, but that neither observer can possibly compare these results and create a paradox; Nature simply does not allow it. This idea is very relevant to the EPR problem. Horizon complementarity has since been molded into observer complementarity, which promotes the former principle to one having more general applicability [17]. Since horizon complementarity arises as a consequence of the synthesis of gravity and quantum theory, it appears to be a principle of quantum gravity. As the (presumed) 4 The two disparaging descriptions are fully consistent with general relativity and can be attributed to the large gravitational red-shift between the two different perspectives. 5 We are overlooking the recently posed possibility of a sea of high-energy quanta, a "firewall", obstructing Alice's passage through the horizon [25], as the validity of this claim is still actively being contested; e.g., [26]. fundamental theory of physics, quantum gravity should be considered to be where all other theories emerge from, and so certain aspects of quantum gravity theory will apply to these emergent theories [31]. Certainly, not all aspects of quantum gravity will apply to all emergent theories, but there is no good reason not to consider the application of observer complementarity to, say, standard quantum mechanics. If an otherwise paradoxical situation can be resolved by observer complementarity, its usage can then be justified a posteriori. The above argument demonstrates that each observer of the black hole will have his or her own distinct account of the events having transpired. The expansion of this idea is that a theory describing the experiences of two or more observers must account for only one observer's results at a time. A collective description of two (or more) experiments that are not causally connected can lead to paradoxical results and, if so, each experiment must then be described individually. We will apply this very idea to the EPR problem, allowing each observer to have his or her own description of the situation. Similar stances There are some approaches in the literature which are similar to that of the current paper but with different motivations. One approach is that applied by Mermin in his so-called Ithaca interpretation of quantum mechanics [32,33]. This viewpoint places its conceptual emphasis on the correlations between the constituent subsystems of the total quantum system. What Mermin shows is that these correlations are entirely captured by the system's density matrix and can be revealed by suitable tracing procedures. He then argues that this is the correct framework for describing reality in the quantum world. 6 Our stance is similar because, as seen later, applying observer complementarity is tantamount to tracing over the inaccessible variables of the density matrix. Another such approach is that of "relational" quantum mechanics, as first presented by Rovelli [35]. This interpretation is founded on the idea of describing reality strictly in terms of relations between (quantum) observers. This is philosophically similar to but operationally distinct from observer complementarity. Indeed, Rovelli and Smerlak's resolution of the EPR paradox [35] resembles the current presentation; nonetheless, our motivation will be focused on adhering to the requirements of observer complementarity without resort- 6 Mermin originally asserted that correlations between subsystems provided a complete description of quantum reality but has since retracted this claim [34]. ing to additional assumptions and inputs from outside the realm of standard quantum mechanics. Another common link between our treatment and Rovelli's is with regard to the concept of a "super-observer". By assigning an element of reality to Alice's prediction of what Bob measures (or vice versa), EPR requires a hypothetical observer that can "see" the outcome of the prediction even if the implicated measurement never actually happens. Essentially, the predicted value must exist for some hypothetical observer who has access to all information that is held in the Universe. This element of the argument is elaborated on later in Section 4. Bell (ala Norsen) In order to better appreciate our observer-complementarity approach to the EPR paradox, we will first outline the results of Bell's mathematical treatment. The crux of Bell's argument lies in the formulation of his locality condition. Understanding this definition correctly is crucial to realizing the significance of Bell's inequality, and a diversity of locality definitions amongst authors has led to a divergence of opinions regarding the theorem and its results. Norsen, in particular, presents Bell's theorem by using a definition of locality that he refers to as "Bell locality" [10]. The condition relies mostly on Bell's original statements regarding a pair of space-like separated observables. The requirement of locality is that the probabilities associated with one of the observables, when worked out from a complete description of this one's past interactions, does not rely on the other, space-like separated observable. In other words, the information at Bob's location must be irrelevant to the probabilities being calculated at Alice's location (and vice versa). And so, in terms of the EPR experiment, this locality condition translates into P (A\|n a ,n b , B, λ) = P (A\|n a , λ) , where P is the probability of Alice's outcome being A, conditioned by the specified variables on the right-hand side of the vertical divider. Also, B refers to the value measured by Bob,n a andn b refer to Alice and Bob's respective choices of measurement direction, and λ represents a complete description of the original singlet state as prescribed by the theory under scrutiny [9]. Equation (2), the mathematical statement of Bell's criteria for locality, does not permit any non-causal action to influence the separated observations. It says that the probability of result A must be the same whether values from Bob's location are taken into account or not, as any dependence of A on Bob's outcome must be excised due to their acausal (space-like) separation. Put differently, Bob's outcomes are non-local with respect to Alice's. Adjoint to this locality definition is the requirement of separability: The joint probability for Alice and Bob must factorize into the product of two separate probabilities, one for each observer individually [10]. This factorization should, as Bell argued [8], be considered as a consequence of the locality condition rather than a new input. The conceptual motivation for this being that the locality condition (2), as well as its A ↔ B converse, requires each observer to have independently calculated probabilities at their respective locations. But the joint probability for the singlet state must still be preserved, and so this expression should be separable into a pair of independent probabilities. The mathematical description of this factorization, in terms of the Alice-Bob setup, is then P (A, B\|n a ,n b , λ) = P (A\|n a , λ) · P (B\|n b , λ) .(3) Notice that, in addition to separating the joint probability on the left, we have applied Bell locality to each observer's probability statement. Hence, each respective probability depends only on the locally chosen direction of measurement,n a orn b , and information pertaining to the initial state of the measured particle as encoded in λ. It is this relation that leads one to the inequality which Bell devised to test theories for adherence to locality. The real problems emerge when this formalism is confronted with the results from actual quantum experiments. Rather than reiterate Bell's derivation, we consider a simple example. As known from both experiment and standard quantum mechanics, the probability of measuring spin up or spin down for a member of a singlet pair, in any given direction, is 50%. Mathematically, this means for Alice that P (A =↑ \|n a , λ) = 1 2 ,(4) where λ should now be regarded as the quantum-mechanical wave function describing the singlet state. On the other hand, experimental results also tell us that Alice's measurement must adhere to the anti-correlation of the singlet, whereby P (A =↑ \|n a , λ,n b =n a , B =↓) = 1 ,(5)P (A =↑ \|n a , λ,n b =n a , B =↑) = 0 .(6) That is, when Bob's chosen measurement direction corresponds with Alice's choice,n b =n a , the spin that Alice measures must be the opposite of Bob's. When these two outcomes are compared with the condition of Bell locality, one can see a clear violation. The locality condition demands that P (A\|n a ,n b , B, λ) = P (A\|n a , λ) . Whenn a =n b , there is no such violation; for instance, the substitution of (4) and (5) into the locality condition (2) gives P (A =↑ \|n a , λ) = 1 2 = 1 2 = P (A =↑ \|n a , λ,n b =n a , B =↓) .(7) However, whenn a =n b , the same substitutions rather yield P (A =↑ \|n a , λ) = 1 2 = 1 = P (A =↑ \|n a , λ,n b =n a , B =↓) .(8) This inequality shows that, in order to respect Bell locality, quantum mechanics must include more than what is currently held in the wave function. It should be noted that the above outcome represents only one particular case of Bell's more general mathematical statement, his celebrated inequality [7]. The inequality within (8) would seem to imply that quantum mechanics is simply incomplete as a theory and, in order to fully describe reality, extra or "hidden" variables are required. However, Norsen's treatment shows that any theory which adheres to Bell locality, with or without these hidden variables, 7 cannot explain the perfect anti-correlation which is verified by experiment [9]. This is because any such variables can be included a priori in λ and, thus, lead to the very same conclusions. And so, in view of this argument, one is forced either to reject any adherence to Bell locality within quantum mechanics or to accept quantum mechanics only as a calculational device that does not fully describe reality. The crucial part of Bell's argument, as far as this paper is concerned, is the emergence of counter-factual definiteness (CFD), as demonstrated and elucidated by Norsen [9]. CFD is the claim that a statement about a measurement which was not performed can be discussed, in a meaningful way, alongside statements about actually performed experiments. This can be seen in (8), where the left-hand side of the expression assumes no knowledge of Bob's measurement. This is equivalent to Bob not yet having performed any measurement, as the left-hand side is not conditioned by any action taken by Bob. However, the right-hand side of (8) assumes Bob did perform a measurement and found a particular outcome which influences Alice's result. In order to illustrate this idea, let us consider the situation of placing a bet at a roulette wheel. A pessimistic gambler might suggest a "theory" that, if a bet is placed on red, the wheel will stop on black, otherwise it will stop on red. Perhaps, there is a particular spin of the wheel that substantiates the gambler's (flawed) assertion. However, it cannot be claimed, after the fact, that a change in the bet would have reversed the outcome on the wheel. Such a claim constitutes a discussion of what could have happened, but did not , in lieu of what actually did. This requirement of CFD within Bell's theorem is an inherent part of the overall argument and cannot be arbitrarily eliminated -it arises as a direct consequence of Bell locality. The fact that assuming Bell locality necessarily implies the use of CFD is clearly demonstrated by Norsen [9]. Moreover, the same basic claim can be made for the stochastic or probabilistic reformulation of Bell's inequality, which is known as the Clauser-Horne-Shimony-Holt inequality [37]. 8 While probabilistic theories do remove CFD, they retain a slightly weaker form, counter-factual meaningfulness (CFM), which results in the same dependence on discussing events which could have happened but did not [11]. These CFD and CFM statements are, not only inevitable in the construction of Bell's inequality, but will be a central element in our proposed resolution for alleviating the concerns of EPR and Bell. Reworking EPR and Bell We begin here with the same setup as previously considered: There is a prepared singlet state, as in (1), consisting of two particles, α and β. Each of the particles is sent to one of a pair of spatially separated observers, Alice and Bob, who are then free to measure the spin of their particles in the direction of their choosing. Alice will be measuring the spin of α, which is given by S α ·n a = ± /2 , and likewise for Bob, S β ·n b = ± /2 . Due to the properties of the singlet state, if Alice and Bob choose to measure the spin in different directions, then their results will have no correlation. For the sake of this argument, we are concerned with the case in which Alice and Bob measure along the same spin direction, as this is where the "problems" arise. We are assuming that they never plan to measure the same direction but end up doing so in the trial of immediate interest. Hence, this is not a description of a conspiracy between the observers to measure in the same direction. Let us recall the singlet state (1), \|Ψ singlet = 1 √ 2 \|a+ \|b− − \|a− \|b+ .(9) Given the coincidence of measurement directions,n a =n b , the notation can now be reinterpreted as meaning a± = S α ·n a = ± /2 and b∓ = S β ·n b = ∓ /2 . This is only a change of basis; the state has not been altered. So far, we have been describing the particle system. To talk about observer complementarity, what we really need to look at is the particle-observer system. In this regard, a subtle point is that Alice and Bob cannot directly measure the spin of their respective particle. Each of their measurements involves some (other) observable, the measurement device, with its corresponding value indicating the result of the experiment. Let us denote this observable by \|X ready and conceptualize it as a pointer that begins in the horizontal position, facing zero. Then \|X ↑ and \|X ↓ will denote a measurement of spin up or spin down, respectively. We now let \|A and \|B respectively represent the result of Alice and Bob's measurements. The system for the particles and observers (P -O) can then be described as \|Ψ P −O = 1 √ 2 \|a+ \|A ready \|b− \|B ready − \|a− \|A ready \|b+ \|B ready ,(10) where the measurements have not yet been performed. From here, we can calculate the density matrix of the combined system, ρ P −O = \|Ψ P −O Ψ P −O \| , giving ρ P −O = 1 2 \|a+ \|A ready \|b− \|B ready a + \| A ready \| b − \| B ready \| (11) −\|a− \|A ready \|b+ \|B ready a + \| A ready \| b − \| B ready \| −\|a+ \|A ready \|b− \|B ready a − \| A ready \| b + \| B ready \| +\|a− \|A ready \|b+ \|B ready a − \| A ready \| b + \| B ready \| . The density matrix provides us with a description of all the possible outcomes once the measurements occur. It should be stressed, though, that this density matrix is not attributed to any of our observers. It contains information regarding two spatially separated locations; namely Alice and Bob's measuring stations. In order to adhere to causality, we consider only local sources of information. For instance, since Alice is separated from Bob, her description must not access any information that is localized at Bob's station. So that, to find Alice's measurement outcomes, we must first determine the reduced density matrix relating to her experiment. Given a quantum density matrix ρ, it is standard operating procedure to remove what a given observer does not know by tracing out the hidden systems [38]. Mathematically, this entails calculating ρ reduced = a ′ a ′ \|ρ\|a ′ , where \|a ′ collectively represents the state kets for all the concealed systems. For Alice, this translates into ρ Alice = T r Bob [ρ P −O ] = b B b\| B\|ρ P −O \|B \|b . Therefore, Alice's results are described by 9 ρ Alice = 1 2 \|a+ \|A ready a + \| A ready \| + \|a− \|A ready a − \| A ready \| , (12) and, for Bob, we find that ρ Bob = 1 2 \|b− \|B ready b − \| B ready \| + \|b+ \|B ready b + \| B ready \| . (13) Let us briefly pause to consider the measurement process. For the sake of explanation, let us (following [39]) consider a device which measures the position of a particle. Given an initial state \|x for the particle and \|A for the measurement device, we can construct a description of the combined system as \|Ψ = \|x \|A . We can also define a unitary operator for the combined system -the interaction Hamiltonian,Ĥ int =XΠ A . Here,X is the position operator for the particle andΠ A depicts a "momentum" operator that is associated with the measuring device. Formally,Π A is the canonical conjugate toÂ and, conceptually, it is the operator which enables \|A to record the measured result. Assuming a completely efficient measuring device, we can, without describing it explicitly, denote a measurement as \|Ψ → e iĤint \|Ψ(14) = e iXΠA \|x \|A = \|x e ixΠA \|A = \|x \|A + x , whereX andΠ A are assumed to commute. And so the device has successfully registered the position of the particle. This is a very simple example. The general case for quantum mechanics involves \|x → \|x ′ . This is due to the quantum state of the particle being changed by the interaction; the particle does not necessarily end up in its original state. Applying this basic idea of measurement to Alice's reduced density matrix, we obtain ρ Alice = 1 2 \|a+ \|A ↑ a + \| A ↑ \| + \|a− \|A ↓ a − \| A ↓ \|(15) and analogously for Bob. Importantly, there is perfect correlation between the pointer direction and the spin of the particle. Notice that the information from Alice's measurement coupled with knowledge of the initial state would enable her to predict the spin of β in this same direction. But this can only be a prediction because of Alice being spatially separated from β. There would need to be some local interaction between Alice and β (or Bob) to provide an actual observable that she could measure. In order to ensure that quantum-mechanical consistency holds, we have to verify that the correlation of the singlet state is preserved when Alice and Bob's notes are compared. To do this, we follow the example of [36] and consider a third observer who checks up on Alice and Bob's results after their measurements. First, let us formulate the density matrix for the combined systems after the measurements, ρ P −O = 1 2 \|a+ \|A ↑ \|b− \|B ↓ a + \| A ↑ \| b − \| B ↓ \| (16) −\|a− \|A ↓ \|b+ \|B ↑ a + \| A ↑ \| b − \| B ↓ \| −\|a+ \|A ↑ \|b− \|B ↓ a − \| A ↓ \| b + \| B ↑ \| +\|a− \|A ↓ \|b+ \|B ↑ a − \| A ↓ \| b + \| B ↑ \| . Again, it should be pointed out that this density matrix is not attributed to any observer in our system. Rather, it is a description that could only be utilized by a super-observer having access to all the spatially separated situations. This is in direct conflict with observer complementarity and so it must be made clear that we do not consider this as a density matrix that can viably be compared with those of the different observers. We will rather use this matrix to determine the reduced density matrix for the third observer, Carol, by tracing out what she has no access to; namely, the initial states of α and β. This entails computing ρ Carol = a ′ b ′ a ′ \| b ′ \| ρ P −O \|b ′ \|a ′ ,(17) where a ′ = {a±} , b ′ = {b±} and, by tracing over the variables separately, we are not assuming any (anti-) correlations. The outcome is ρ Carol = 1 2 \|A ↑ \|B ↓ A ↑ \| B ↓ \| + \|A ↓ \|B ↑ A ↓ \| B ↑ \| ,(18) which shows that the anti-correlation is preserved from Carol's perspective. And so we see that this series of local interactions, in compliance with observer complementarity and causality, holds no paradoxes. Any "spooky action at a distance" disappears when the view of each observer is restricted to his or her own reduced density matrix. Discussion Against counter-factual definiteness The above results show that observer complementarity can account for the view of each respective observer, while still maintaining the anti-correlations between the particles as evident from Carol's perspective. On the other hand, Norsen's formulation of Bell's theorem proves that CFD is a necessary consequence of assuming Bell locality [9]. This can be weakened to CFM for a probabilistic theory but it still amounts to the same basic result. The theorem inevitably depends on discussing events that have no associated observer because, even though they could have happened, they did not actually happen. The idea of discussing, in a meaningful way, a statement about a hypothetical event is in opposition to the principle of observer complementarity. From the vantage point of this principle, an observer has no need to account for the results of an experiment that cannot be compared to his or her own findings. In other words, Alice need not account for Bob's results (and vice versa) until they can be locally compared. Similarly, results from an experiment that was not actually performed can have no meaning in the observer-complementarity framework. It follows that CFD is an assumption that cannot be made in conjunction with observer complementarity; these being antithetical constructs. To show that one is true in the domain of a given theory would be enough to rule out the other within the same realm. To this end, observer complementarity falls back on its origin. As already stated, quantum gravity can be considered as the fundamental theory from which all others are emergent. Therefore, as observer complementarity survives quantum gravity, it is the CFD assumption which ultimately must fall away. Then, in the spirit of Occam's razor, the simplest way to relieve the tension between CFD and observer complementarity is to discard CFD in any paradoxical situation. This would imply that, due to the inevitable appearance of CFD, Bell's theorem is doomed from the start. With the assumption of Bell locality and, in the case of EPR, their criterion for reality, the respective arguments set themselves up to produce results that will imply either that Nature is non-local or that quantum mechanics is incomplete. However, considering the same situation without assuming Bell locality and, instead, using observer complementarity as the motivating principle, we see that the concerns fall away without forcing a rejection of either option. No contradiction is experienced, as shown by Carol's results, and all action happens locally. This, along with the fact that observer complementarity is a consequence of quantum gravity, suggest that Bell locality and, by implication, CFD are too strong to define reality. Against conspiracy One might still be concerned as to how the particles "know" in advance which directions will be chosen by Alice and Bob. It becomes problematic if this information is not provided by the theory because the only other alternative is conspiratorial settings by Nature. That is, Nature would, somehow, be anticipating the choices of the observers and adjusting the particles accordingly but without allowing access to the determined values. To address this question, let us consider the evolution of the system through time. 10 It should be clear that the initial conditions of a system and its subsequent evolution produce the final conditions. But is the converse true? Can we not evolve the system back in time if given the final conditions? For a deterministic or classical environment, this is obviously acceptable. The quantum realm is not so deterministic, and so the answer is less clear. But, perhaps surprisingly, the proposition of evolving a quantum system back in time is valid as well [42]. Indeed, the choice of final-state boundary conditions does not violate causality nor unitarity, and so we are free to apply reversed time evolution to our previously described experiment. Let us then consider the backwards evolution from a time t 1 , when Alice has measured a + spin on her particle in the directionn a . We can evolve this state backwards to a time t 0 , just when the particles were separated. As particle α experiences no interactions during the intervening period, its time evolution is trivial, \|n a , + t0 = e iĤ f ree (t0−t1) \|n a , +(19) = (phase)\|n a , + , where we have used that the free-particle Hamiltonian commutes with all spin operators. Since, at time t 0 , the particles are known to be anti-correlated, it can be deduced that (phase)\|n a , + \|n b =n a , −(20) describes the state of the two particles. Hence, as far as Alice is concerned, there has been no conspiracy, and the same is true for Bob. But what about the initial holder of the singlet state? After all, the state (20) differs from the original singlet state (9), and we cannot appeal to observer complementarity in this case since the initial location is causally linked to both Alice nor Bob. Nevertheless, what can be appealed to is that the wave function depends on an observer's choice of "gauge", much in the same way that the electromagnetic potentials are gauge-dependent fields. That is, even causally connected observers need only agree on gauge-invariant, physically measurable quantities. Here, all observers agree that the spins are anti-correlated, which is all that can ever be known with certainty. There may also be concerns that the quantum particles do not "perceive" the arrow of time and, consequently, have the capability of time travel into the past. It is, in fact, already known that final-state selection does allow for such time travel via the process of post-state teleportation; an idea that is discussed at length [43]. While this is an area of study with many open questions, two important points will be made to assuage any discomfort at the suggestion of this type of particle behavior. (See [43] for further details and references.) Firstly, this form of time travel can be formulated in away that does not jeopardize the standard tenets of quantum mechanics, nor does it lead to any stereotypical "grandfather paradoxes". 11 And so this concept can safely be applied to quantum particles without worrying about any disorder permeating to the classical realm. Secondly, given a consistent theory of quantum gravity, it is natural if not necessary that time travel be incorporated into the quantum side of the theory at some level. This can be understood as follows: In the classical (Einstein) theory of gravity, closed time-loops or wormholes cannot be ruled out by any formal proof. For instance, the conditions around black holes produce effects which suggest that there are such time-ambiguous regions. Therefore, a consistent theory of quantum gravity requires that the quantum realm also allows for some notion of going backward in time. Post-state teleportation can be viewed as a tangible realization of this requirement. Against super-observers As alluded to earlier, the underlying premise of EPR's criterion for reality, as well as the application of Bell locality, utilizes an assumption regarding a "special" observer with the capacity to view the Universe in its totality; the super-observer. This is a concept that has been used throughout science to build theories for explaining the world [44]. A useful approach to understanding this idea is through "Laplace's demon". This is a conceptual device that is used to represent the Universe as a deterministically working model [45]. It entails a hypothetical super-intelligence having access to all laws governing the Universe as well as a precise description of the Universe for a particular moment of time [46]. So that Laplace's demon, this hyper-intelligent super-observer, would have accurate knowledge of all values for all things in the Universe. This, combined with its knowledge of the physical laws, would allow the prediction of all future occurrences as well as a retroactive prediction of all past events [45]; that is, the ability to describe the Universe, in its entirety, for all time given only one set of initial values. Such an entity is then representative of a deterministically functioning universe, where it is only our own ignorance of accurate values that prevents us from perfectly forecasting the future. And so, for every moment of time, there would be a single and complete description of the Universe and all of its constituents. This is obviously problematic for observer complementarity, which does not allow a single observer to have access to all information since he or she cannot have access to any spatially separated values. The necessity for a hypothetical super-observer has been debated over history. It was Heisenberg's uncertainty principle which first provided a strong opposition. Insofar as some information is always inaccessible, there is a fundamental limitation to the predictive capabilities of Laplace's hypothetical demon. In this way, the workings of quantum mechanics places a firm restriction on what can be known as a matter of principle. Even if given all available information at a particular time, the demon would still not be able to consistently make predictions about quantum events with certainty. The final nail in the coffin for the super-observer came from a calculation that described Laplace's demon as a computational device [47]. This paper showed that, for a device to accurately predict everything in a system, it cannot be part of the same system. Essentially, Laplace's demon cannot be situated within the Universe if it is to provide a total description of the Universal state. And so the conclusion is that, for any set of natural laws, a computational device would not be able to predict everything within a world that it is also contained in. The concept of a Laplacian demon could still hold for one living "outside" the Universe. But such a demon has no utility because, in this case, its predictive power is then literally out of reach [48]. Hence, any theory could only describe "almost everything"; there will always be certain values that are forever out of its reach. (For a recent discussion, see [49].) There is, therefore, no compelling reason to suggest that the contradiction between observer complementarity and the super-observer implies that former should be discarded. Rather, the evidence points to the dismissal of the superobserver as a part of our Universe. Conclusions In closing, the approach of this paper reveals an oversight regarding the assumptions that are used in the EPR thought experiment and Bell's formulation thereof. As made clear by Norsen and recapitulated above, the arguments rest squarely on the validity of CFD. However, as we have pointed out, the application of CFD is at odds with the principle of oberver complementarity. Then, since the latter attains its pedigree from the fundamental theory of quantum gravity, we have argued that Bell locality is too strong a definition for reality and that this is what leads to the troubling outcomes. By dismissing Bell locality and utilizing observer complementarity, we have shown that all concerns regarding locality and the completeness of quantum mechanics fall by the wayside, as the EPR scenario naturally resolves itself in a self-consistent manner. The acceptance of observer complementarity also requires one to dismiss the notion that a hypothetical super-observer is present in the Universe. This does not present a problem because, not only is there no argument for the necessity of a super-observer, there is considerable evidence against it. A super-observer may still fulfill the criteria of Laplace's demon from outside the Universe. However, any interaction of the super-observer with the Universe requires it to be part of the same; in which case, Laplace's criteria can no longer be fulfilled. Therefore, any theory explaining measurement within the Universe cannot provide a deterministically complete description. As locality and causality have been maintained by our reworked calculation, what must change is how reality is viewed; not as a single description of all subsystems but, rather, as a collection of descriptions from many different observer's points of view. We would, if it were possible, be inclined to remind Einstein that such a notion is not much different than his theory of relativity. This statement is essentially an iteration of the Heisenberg uncertainty principle. The conventions of this paper are aligned with[3]. The horizon represents the "surface of no return"; classical matter can never escape from the black hole interior after passing through this causal boundary. Bohmian quantum mechanics is one such example of a hidden-variable theory[2]. This entails that the state description be averaged over all possibilities: \|λ → ρ(λ)\|λ dλ , where ρ(λ) is a weight factor such that ρ(λ)dλ = 1 . So that, for example, P (A =↑ \|na, λ) → ρ(λ)P (A =↑ \|na, λ)dλ . We are assuming orthonormal sets of bases, even for the measuring devices. Similar sentiments to the following discussion have appeared in[40,41]. This refers to the prototypical paradox of time travel: That a person could go back in time and shoot his or her own grandfather before ever being conceived. AcknowledgmentsThe research of AJMM is supported by NRF Incentive Funding Grant 85353. Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?. A Einstein, B Podolsky, N Rosen, Phys. Rev. 47777A. Einstein, B. Podolsky and N. Rosen, "Can Quantum-Mechanical De- scription of Physical Reality Be Considered Complete?," Phys. Rev. 47, 777 (1935). D Bohm, Quantum Theory. 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[ "LAPLACIAN PRECONDITIONING OF ELLIPTIC PDES: LOCALIZATION OF THE EIGENVALUES OF THE DISCRETIZED OPERATOR ", "LAPLACIAN PRECONDITIONING OF ELLIPTIC PDES: LOCALIZATION OF THE EIGENVALUES OF THE DISCRETIZED OPERATOR " ]	[ "Tomáš Gergelits ", "Kent-André Mardal ", "ANDBjørn Fredrik Nielsen ", "Zdeněk Strakoš " ]	[]	[]	In the paper Preconditioning by inverting the Laplacian; an analysis of the eigenvalues. IMA Journal of Numerical Analysis 29, 1 (2009), 24-42, Nielsen, Hackbusch and Tveito study the operator generated by using the inverse of the Laplacian as preconditioner for second order elliptic PDEs ∇ · (k(x)∇u) = f . They prove that the range of k(x) is contained in the spectrum of the preconditioned operator, provided that k is continuous. Their rigorous analysis only addresses mappings defined on infinite dimensional spaces, but the numerical experiments in the paper suggest that a similar property holds in the discrete case. Motivated by this investigation, we analyze the eigenvalues of the matrix L −1 A, where L and A are the stiffness matrices associated with the Laplace operator and general second order elliptic operators, respectively. Without any assumption about the continuity of k(x), we prove the existence of a one-to-one pairing between the eigenvalues of L −1 A and the intervals determined by the images under k(x) of the supports of the FE nodal basis functions. As a consequence, we can show that the nodal values of k(x) yield accurate approximations of the eigenvalues of L −1 A. Our theoretical results are illuminated by several numerical experiments.Key words. Second order elliptic PDEs, preconditioning by the inverse Laplacian, eigenvalues of the discretized preconditioned problem, nodal values of the coefficient function, Hall's theorem, convergence of the conjugate gradient method	10.1137/18m1212458	[ "https://arxiv.org/pdf/1809.03790v1.pdf" ]	119,296,231	1809.03790	cb5a042861485abf6d8bbff4ef0fcf3f68f744fe	LAPLACIAN PRECONDITIONING OF ELLIPTIC PDES: LOCALIZATION OF THE EIGENVALUES OF THE DISCRETIZED OPERATOR * Tomáš Gergelits Kent-André Mardal ANDBjørn Fredrik Nielsen Zdeněk Strakoš LAPLACIAN PRECONDITIONING OF ELLIPTIC PDES: LOCALIZATION OF THE EIGENVALUES OF THE DISCRETIZED OPERATOR * AMS subject classifications 65F0865F1565N1235J99 In the paper Preconditioning by inverting the Laplacian; an analysis of the eigenvalues. IMA Journal of Numerical Analysis 29, 1 (2009), 24-42, Nielsen, Hackbusch and Tveito study the operator generated by using the inverse of the Laplacian as preconditioner for second order elliptic PDEs ∇ · (k(x)∇u) = f . They prove that the range of k(x) is contained in the spectrum of the preconditioned operator, provided that k is continuous. Their rigorous analysis only addresses mappings defined on infinite dimensional spaces, but the numerical experiments in the paper suggest that a similar property holds in the discrete case. Motivated by this investigation, we analyze the eigenvalues of the matrix L −1 A, where L and A are the stiffness matrices associated with the Laplace operator and general second order elliptic operators, respectively. Without any assumption about the continuity of k(x), we prove the existence of a one-to-one pairing between the eigenvalues of L −1 A and the intervals determined by the images under k(x) of the supports of the FE nodal basis functions. As a consequence, we can show that the nodal values of k(x) yield accurate approximations of the eigenvalues of L −1 A. Our theoretical results are illuminated by several numerical experiments.Key words. Second order elliptic PDEs, preconditioning by the inverse Laplacian, eigenvalues of the discretized preconditioned problem, nodal values of the coefficient function, Hall's theorem, convergence of the conjugate gradient method 1. Introduction. The classical analysis of Krylov subspace solvers for matrix problems with Hermitian matrices relies on their spectral properties; see, e.g., [1,15]. Typically one seeks a preconditioner which yields parameter independent bounds for the extreme eigenvalues; see, e.g., [8,18,25,14,24] for a discussion of this issue in terms of operator preconditioning. This approach has the advantage that only the largest and smallest eigenvalues (in the absolute sense if an indefinite problem is solved) must be studied, and the bounds for the required number of Krylov subspace iterations can become independent of the mesh size and other important parameters. This is certainly of great importance, but it does not automatically represent a solution to the challenge of identifying efficient preconditioning. Efficiency of the preconditioning in this approach requires that the convergence bounds based on a single number characteristics, such as the condition number, guarantee sufficient accuracy of the computed approximation to the solution within an acceptable number of iterations. Since Krylov subspace methods are strongly nonlinear in the input data (matrix and the initial residual), more information about the spectrum is needed 1 in order to capture the actual convergence behavior with its desirable superlinear character. This has been pointed out by several studies [2,3,19,20,35,29], and the acceleration of the convergence of the method of conjugate gradients (CG) has been linked with the presence of large outlying eigenvalues and clustering of the eigenvalues. Since Krylov subspace methods for systems with Hermitian matrices use short recurrences, exact arithmetic considerations must be complemented with a thorough rounding error analysis, otherwise it can in practice be misleading or even completely useless. The deterioration of convergence due to rounding errors in the presence of large outlying eigenvalues has been reported, based on experiments, already in [21]; see also [7], [19, p. 72], the discussion in [35, p. 559] and the summary in [22,Section 5.6.4,. In investigating the convergence behavior of Krylov subspace methods for Hermitian problems, we thus have to deal with two phenomena acting against each other. Large outlying eigenvalues (or well-separated clusters of large eigenvalues) can in theory, assuming exact arithmetic, be linked with acceleration of CG convergence. However, in practice, using finite precision computations, it can cause deterioration of the convergence rate. This intriguing situation has been fully understood thanks to the seminal work of Greenbaum [10] with the fundamental preceeding analysis of the Lanczos method by Paige [31,32]; see also [12,34,27,26] and the recent paper [9] that addresses the question of validity of the CG composite convergence bounds based on the so-called effective condition number. For general non-Hermitian matrices, spectral information may not be descriptive; see, e.g., [13,11] and [22,Section 5.7]. We will briefly outline the mathematical background behind the understanding of the CG convergence behavior. For Hermitian positive definite matrices (in infinite dimension, for self-adjoint, bounded, and coercive operators) CG can be associated with the Gauss-Christoffel quadrature of the Riemann-Stieltjes integral λ −1 dω(λ); see [10], [17,Section 14], [22,Section 3.5 and Chapter 5], [24,Section 5.2 and Chapter 11]. The nondecreasing and right continuous distribution function ω(λ) is given by the spectral decomposition of the given matrix B (operator) and the normalized initial residual q, B = λ l v l v * l = λ dE λ , where E λ is the spectral function representing a family of projections, [36,Chapter II,Section 7] or [37,Chapter III]. For more references on this topic, see [24,Section 5.2]. As a consequence, which has been observed in many experiments, preconditioning that leads to favorable distributions of the eigenvalues of the preconditioned (Hermitian) matrix can lead to much faster convergence than preconditioning that only focuses on minimizing the condition number. (As pointed out above, any analysis that aims at relevance to practical computations must also include effects of rounding errors). q * Bq = λ l \|v * l q\| 2 ≡ λ l ω l = λ dω(λ) , ω l = \|v * l q\| 2 , (dω(λ) = q * dE λ q); see Motivated by these facts and the results in [30], the purpose of this paper is to show that approximations of all the eigenvalues of a classical generalized eigenvalue problem are readily available. More specifically, assuming that the function k(x) is uniformly positive, bounded and measurable, we will study finite element (FE) discretizations of As mentioned above, mathematical properties of the continuous problem (1.1) are studied in [30]. In particular, the authors of that paper prove that 2 ∇ · (k(x)∇u) = λ∆u in Ω ⊂ R d , u = 0 on ∂Ω,k(x) ∈ sp(L −1 A) for all x ∈ Ω at which k(x) is continuous, where A : H 1 0 (Ω) → H −1 (Ω), Au, v = Ω k∇u · ∇v, u, v ∈ H 1 0 (Ω), (1.3) L : H 1 0 (Ω) → H −1 (Ω), Lu, v = Ω ∇u · ∇v, u, v ∈ H 1 0 (Ω). (1.4) The authors also conjecture that the spectrum of the discretized preconditioned operator L −1 A can be approximated by the nodal values of k(x). In the present text we show, without the continuity assumption on the coefficient function, how the function values of k(x) are related to the generalized spectrum of the discretized operators (matrices) in (1.2). Our main results state that: • There exists a (potentially non-unique) pairing of the eigenvalues of L −1 A and the intervals determined by the images under k(x) of the supports of the FE nodal basis functions; see Theorem 3.1 in section 3. • The function values of k(x) at the nodes of the finite element grid can be paired with the individual eigenvalues of the discrete preconditioned operator L −1 A. Furthermore, these functions values yield accurate approximations of the eigenvalues; see Corollary 3.2 in section 3. The text is organized as follows. Notation, assumptions and a motivating example are presented in section 2. Section 3 contains theoretical results. The proof of the pairing in Theorem 3.1 uses the classical Hall's theorem from the theory of bipartite graphs. Corollary 3.2 then follows as a simple consequence. The numerical experiments in section 4 illustrate the results of our analysis. Moreover, using Theorem 3.1, the discussion at the end of section 4 explains the CG convergence behavior observed in the example presented in section 2. The text closes with concluding remarks in section 5. 2. Notation and an introductory example. We consider a self-adjoint second order elliptic PDE in the form −∇ · (k(x)∇u) = f for x ∈ Ω, (2.1) u = 0 for x ∈ ∂Ω, and the corresponding generalized eigenvalue problem (1.1) with the domain Ω ⊂ R d , d ∈ {1, 2, 3} and the given function f ∈ L 2 (Ω). We assume that the real valued scalar function k(x) : R d → R is measurable and bounded, i.e., k(x) ∈ L ∞ (Ω), and that it is uniformly positive, i.e., k(x) ≥ α > 0, x ∈ Ω. Let V ≡ H 1 0 (Ω) denote the Sobolev space of functions defined on Ω with zero trace at ∂Ω and with the standard inner product. The weak formulations of the problems (1.1) and (2.1) are to seek u ∈ V , respectively u ∈ V and λ ∈ R, such that (1.4) and the function f ∈ L 2 (Ω) is identified with the associated linear functional f ∈ V # defined by Au = f, respectively Au = λLu (2.2) where A, L : V → V # , f ∈ V # are defined in (1.3) and(2.3) f, v ≡ Ω f v . Discretization via the conforming finite-element method leads to the discrete operators A h , L h : V h → V # h where the finite dimensional subspace V h is spanned by the polynomial discretization basis functions φ 1 , . . . , φ N with the local supports T i = supp(φ i ), i = 1, . . . , N. The matrix representations A h and L h are defined as [A h ] ij = A h φ j , φ i = Ω ∇φ i · k∇φ j , (2.4) [L h ] ij = L h φ j , φ i = Ω ∇φ i · ∇φ j , i, j = 1, . . . , N. (2.5) In the text below we will, for the sake of simple notation, omit the subscript h and write A ≡ A h and L ≡ L h . An example. The following example illustrates in detail the motivation outlined in section 1, i.e. that the condition number may be misleading in characterization of the convergence behavior of the CG method. Consider the boundary value problem −∇ · (k(x)∇u) = 0 in Ω , u = u D on ∂Ω , (2.6) where the domain Ω ≡ (−1, 1) × (−1, 1) is divided into four subdomains Ω i , i = 1, 2, 3, 4, corresponding to the axis quadrants numbered counterclockwise. Let k(x) be piecewise constant on the individual subdomains Ω i , k 1 = k 3 ≈ 161.45, k 2 = k 4 = 1. The Dirichlet boundary conditions are described in [28,Section 5.3]. 10 0 The numerical solution u of this problem and the linear FE discretization, using the standard uniform triangulation, are shown in the left part of Figure 2.1. The resulting algebraic problem Ax = b is solved by the preconditioned conjugate gradient method (PCG). In the right panel of Figure 2.1 we see the relative energy norm of the error as a function of iteration steps for the Laplace operator preconditioning (solid line) and for the preconditioning using the algebraic incomplete Choleski factorization of the matrix A (ICHOL) with the drop-off tolerance 10 −2 (dashed line) where the problem has N = 3969 degrees of freedom. Despite the fact that the spectral condition number λ max /λ min of the symmetrized preconditioned matrix for the Laplace operator preconditioning is an order of magnitude larger than for the ICHOL preconditioning, close to 161 and close to 16, respectively, PCG with the Laplace operator preconditioning clearly demonstrates much faster convergence. This is due to the differences in the distribution of the eigenvalues with the nonnegligible components of the initial residuals in the direction of the associated eigenvectors and effects of rounding errors. x − x k A / x − x 0 A Laplace ICHOL The spectra and distribution functions associated with the discretized preconditioned problems are given in Figure 2.2 for N = 49 degrees of freedom and in Figure 2.3 for N = 3969 degrees of freedom. Here, L = L 1/2 L 1/2 is the matrix associated with the discretized Laplace operator and CC * ≈ A is the matrix resulting from ICHOL using the drop-off tolerance 10 −2 , with the eigenvalues and eigenvectors of the associated generalized eigenvalue problems (see (1.2)) Av L i = λ L i Lv L i , i = 1, . . . , N, Av C i = λ C i CC * v C i , i = 1, . . . , N. The weights of the distribution function ω L (λ), respectively, ω C (λ), associated with the eigenvalues λ L i , respectively, λ C i , i = 1 . . . , N , related to the preconditioned algebraic systems A L (L 1/2 x) = L −1/2 b, A L = L −1/2 AL −1/2 ,A C ; (CC * ≈ A) λ L 1 λ L N -10 -5 0 λ C 1 λ C N -10 -5 0 Fig. 2.2. Top: Comparison of the spectra of the matrices A, A L and A C , N = 49 degrees of freedom. Due to a small drop-off tolerance, the eigenvalues of A and CC * are graphically indistinguishable. Therefore the right part only shows the eigenvalues of A C (using a different scale than the left part of the figure). Bottom: Comparison of the distribution functions ω L (λ) (left) and ω C (λ) (right) associated with the preconditioned problems. The vertical axes are in the logarithmic scale and λ L 1 = 1, λ L N = 161.45, λ C 1 = 0.91, λ C N = 1.07. respectively A C (C * x) = C −1 b, A C = C −1 AC − * , are given by (2.7) ω L i = \|(v L i ) * q L \| 2 , i = 1, . . . , N, ω C i = \|(v C i ) * q C \| 2 , i = 1, . . . , N. Here,v L i = L 1/2 v L i L 1/2 v L i andv C i = C * v C i C * v C i are the eigenvectors of the Hermitian and positive definite matrix A L , respectively, A C , and q L = L −1/2 b L −1/2 b , q C = C −1 b C −1 b . (We use the initial guess x 0 = 0). The distribution function ω C (λ) has its points of increase much more evenly distributed in the spectral interval [λ 1 (A C ), λ N (A C )], which leads to a difference in the PCG convergence behavior. We will return to this issue, and offer a full explanation of the observed CG convergence behavior, after proving the main results and presenting their numerical illustrations. 3. Analysis. As mentioned above, we will not only show that some function values of k(x) are related to the spectrum of L −1 A, but that there exists a one-toone correspondence, i.e., a pairing, between the individual eigenvalues of L −1 A and quantities given by the function values of k(x) in relation to the supports of the FE basis functions. The proof does not require that k(x) is continuous. If, moreover, k(x) is constant on a part of the domain Ω that contains fully the supports of one or more basis functions, then the function value of k(x) determines the associated eigenvalue exactly and the number of the involved supports bounds from below the multiplicity of the associated eigenvalue. If k(x) is slowly changing over the support of some basis function, then we get a very accurate localization of the associated eigenvalue. Our approach is based upon the intervals (3.1) k(T j ) ≡ [min x∈Tj k(x), max x∈Tj k(x)], j = 1, . . . , N, where T j = supp(φ j ). 3 We will first formulate two main results. Theorem 3.1 localizes the positions of all the individual eigenvalues of the matrix L −1 A by pairing them with the intervals k(T j ) given in (3.1). Using the given pairing, Corollary 3.2 describes the closeness of the eigenvalues to the nodal function values of the scalar function k(x). The proof of Theorem 3.1 combines perturbation theory for matrices with a classical result from the theory of bipartite graphs. For clarity of exposition, the proof will be presented after stating the corollaries of Theorem 3.1. Theorem 3.1 (Pairing the eigenvalues and the intervals k(T j ), j = 1, . . . , N ). Using the previous notation, let 0 < λ 1 ≤ λ 2 ≤ . . . ≤ λ N be the eigenvalues of L −1 A, where A and L are defined by (2.4) and (2.5) respectively (with the subscript h dropped). As in (1.1), let k(x) be measurable and bounded, i.e., k(x) ∈ L ∞ (Ω). Then there exists a (possibly non-unique) permutation π such that the eigenvalues of the matrix L −1 A satisfy (3.2) λ π(j) ∈ k(T j ), j = 1, . . . , N, where the intervals k(T j ) are defined in (3.1). The statement is illustrated in Figure 3.1. The proof of the following corollary uses the one-to-one pairing of the intervals (3.1), and therefore also the values of k(x) at the nodes of the discretization mesh, with the eigenvalues λ π(j) . Corollary 3.2 (Pairing the eigenvales and the nodal values). Using the notation of Theorem 3.1, consider any discretization mesh nodex such thatx ∈ T j . Then the associated eigenvalue λ π(j) of the matrix L −1 A satisfies (3.3) \|λ π(j) − k(x)\| ≤ max x∈Tj \|k(x) − k(x)\|. If, in addition, k(x) ∈ C 2 (T j ), then \|λ π(j) − k(x)\| ≤ max x∈Tj \|k(x) − k(x)\| ≤ĥ ∇k(x) + 1 2ĥ 2 max x∈Tj D 2 k(x) (3.4) whereĥ = max x∈Tj x −x and D 2 k(x) is the second order derivative of the function k(x). 4 Proof. Since both λ π(j) ∈ k(T j ) and k(x) ∈ k(T j ), it trivially follows that \|λ π(j) − k(x)\| ≤ max x∈Tj \|k(x) − k(x)\|. Moreover, for any x ∈ T j , the multidimensional Taylor expansion (see, e.g., [5, p. 11, where α ∈ [0, 1], with the absolute value obeying Section 1.2]) gives for k(x) ∈ C 2 (T j ) that k(x) − k(x) = ∇k(x)(x −x) + 1 2 D 2 k(x + α(x −x))(x −x, x −x) k(x) k(Tj) (3.4) (3.3) λ π(j)\|k(x) − k(x)\| ≤ ∇k(x) (x −x) + 1 2 x −x 2 D 2 k(x + α(x −x)) , giving the statement. We now give the proof of Theorem 3.1. Lemma 3.3 below and its Corollary 3.4 identify the groups of eigenvalues in any union of intervals Proof. In brief, the proof is based on the theory of eigenvalue perturbations of matrices. We locally modify the scalar function k(x) by setting it equal to a positive constant K in the union T J of the supports T j , j ∈ J . This will result, after discretization, in a modified matrixÃ J such that K is an eigenvalue of L −1Ã J of at least p multiplicity. An easy bound for the eigenvalues of (3.7) L −1 E J , where E J = A −Ã J , combined with a standard perturbation theorem for matrices, then provide a bound for the associated p eigenvalues of L −1 A. A particular choice of the positive constant K will finish the proof. Letk J (x) = K for x ∈ T J , k(x) elsewhere; with Ã J ,h u, v ≡ Ω ∇u ·k J ∇v , u, v ∈ V h , where, analogously to (2.4), [Ã J ] lj = Ã J ,h φ j , φ l = Ω ∇φ l ·k J ∇φ j . Sincek J is constant on each T j , j ∈ J , and the support of the basis function φ j is T j , it holds for any v ∈ V h that Ã J ,h φ j , v = Ω ∇φ j ·k J ∇v = Tj ∇φ i ·k J ∇v = K Tj ∇φ j · ∇v = K L h φ j , v , j ∈ J . Thus, K is an eigenvalue of the operator L −1 hÃ J ,h associated with the eigenfunctions φ j , j ∈ J , and therefore K is the eigenvalue of the matrix L −1Ã J with the multiplicity at least p. This can also be verified by construction by observing that A J e j = K Le j , j ∈ J . Consider now the eigenvalues of L −1 E J ; see (3.7). The Rayleigh quotient for an eigenpair (θ, q), q = 1, and the associated eigenfunction q = N j=1 ν j φ j , where q T = [ν 1 , . . . , ν N ], satisfies θ = q T E J q q T Lq = q T (A −Ã J )q q T Lq = (A h −Ã J ,h )q, q L h q, q = Ω ∇q · (k(x) −k J (x))∇q dx Ω ∇q 2 dx = T J (k(x) − K) ∇q 2 dx Ω ∇q 2 dx , giving (3.8) \|θ\| ≤ max x∈T J \|k(x) − K\|. Next, consider the symmetric matrices A L = L −1/2 AL −1/2 , E L = L −1/2 E J L −1/2 ,Ã L = L −1/2Ã J L −1/2 . According to a standard result from the perturbation theory of matrices, see, e.g., [33,Corollary 4.9,p. 203], we find that λ s (A L ) = λ s (Ã L + E L ) ∈ [λ s (Ã L ) + θ min , λ s (Ã L ) + θ max ], s = 1, . . . , N, where θ min and θ max are the smallest and largest eigenvalues of E L respectively. Since the matrices L −1 A, L −1 E J and L −1Ã J have the same spectrum as the matrices A L , E L andÃ L , respectively, it follows that λ s (L −1 A) = λ s (L −1Ã J + L −1 E J ) ∈ [λ s (L −1Ã J ) + θ min , λ s (L −1Ã J ) + θ max ]. Due to (3.8), θ min ≥ − max x∈T J \|k(x) − K\|, θ max ≤ max x∈T J \|k(x) − K\|, and thus, since K is at least a p-multiple eigenvalue of L −1Ã J , there exist p eigenvaluesλ 1 , . . . ,λ p of L −1 A such that Proof. (3.9)λ ∈ [ K − max x∈T J \|k(x) − K\|, K + max x∈T J \|k(x) − K\| ], = 1, . . . , p. Setting K = 1 2 ( min x∈T J k(x) + max x∈T J k(x)) givesλ ∈ [ min x∈T J k(x), Sincek(T j ) = k(T j ), for any j ∈ J , is an interval (3.1), the setk(T J ) consists of at most p intervals. We decomposek(T J ) intop mutually disjoint intervals, p ≤ p,k (T Ji ) ≡ j∈Ji k(T j ), i = 1, . . . ,p. In order to finalize the proof of Theorem 3.1, we still need to show the existence of a one-to-one pairing between the individual eigenvalues and the individual intervals k(T j ), j = 1, . . . , N . The relationship between the intervals k(T j ), j = 1, . . . , N , and the eigenvalues of L −1 A described in Lemma 3.3 and Corollary 3.4 can be represented by the following bipartite graph. Let, as above, 0 < λ 1 ≤ λ 2 ≤ . . . ≤ λ N be the eigenvalues of L −1 A. Consider the bipartite graph A subset of edges M ⊂ E is called matching if no edges from M share a common node; see [4, Section 5.1]. We will use the following famous theorem. Thus, according to Theorem 3.5, there exist a matching M ⊂ E that covers I. Since \|I\| = \|S\|, this matching defines the permutation π(i), i = 1, 2, . . . , N , such that λ π(i) ∈ k(T i ), i = 1, . . . , N, which finishes the proof. Numerical experiments. In this section we will illustrate the theoretical results by a series of numerical experiments. We will investigate how well the nodal values of k correspond to the eigenvalues and assess the sharpness of the estimates in Corollary 3.2 in a few examples, including both uniform and local mesh refinement. Furthermore, we will compute the corresponding intervals k(T j ), j = 1, . . . , N and consider the pairing in Theorem 3.1. Test problems. We will consider four test problems defined on the domain Ω ≡ (0, 1) × (0, 1) where we slightly abuse the notation above and let k = k(x, y). The first three problems use a continuous coefficient function k(x, y): (P1) k(x, y)= sin(x + y), (P2) k(x, y)= 1 + 50 exp(−5(x 2 + y 2 )), (P3) k(x, y)= 2 7 (x 7 + y 7 ). The fourth problem uses a discontinuous function k(x, y), (P4) k(x, y) = (P1) for (x, y) ∈ (0, 1) × ( 1 2 , 1), (P2) elsewhere. Numerical experiments were computed using FEniCS [23] and Matlab. 5 If not specified otherwise, we consider a triangular uniform mesh with piecewise linear discretization basis functions. Figure 4.1 we show the nodal values of k(x, y) and the corresponding eigenvalues, both sorted in increasing order on the unit square with N = 81 degrees of freedom. Clearly, there is a close correspondence between the nodal values and the eigenvalues even at this relatively coarse resolution, but there are some notable differences for (P3) and (P4) that are clearly visible. Theorem 3.1 states that there exists a pairing π such that λ π(i) ∈ k(T i ) for every i = 1, . . . , N . The proof is not constructive and it is therefore interesting to consider potential pairings. In Figure 4.2 we show the results of the previously mentioned paring of the eigenvalues and the intervals k(T i ) = k(T (x i , y i )) where the vertices (x i , y i ) have been sorted such that the nodal values k(x i , y i ) are in increasing order. The pairing appears to work quite well except for the case (P4) where in particular the eigenvalues between 30-40 are outside the intervals provided by this pairing. Illustration of Theorem 3.1 and Corollary 3.2. In In order to ensure that we employ a proper pairing, i.e., to guarantee that λ π(i) ∈ k(T i ), i = 1, . . . , N , we construct the adjacency matrix G such that By using the Dulmage-Mendelsohn decomposition 6 of this adjacency matrix G (provided by the Matlab command dmperm) we get a pairing π satisfying λ π(i) ∈ k(T i ) for every i = 1, . . . , N . Figure 4.3 illustrates the pairing π from Theorem 3.1 for (P4) and the approximation of the eigenvalues by the associated nodal values (the plots in Figure 4.3 should be compared with the lower right panels of Figures 4.1 and 4.2). (4.1) G si = 1, λ s ∈ k(T i ), 0, λ s / ∈ k(T i ). The difference between the nodal values and the corresponding eigenvalues is estimated in (3.4) and to assess the sharpness of this estimate, Figure 4.4 compares the quantities \|λ s − k π −1 (s) \| (red dots) with the first term on the right hand side of (3.4) (black stars). We observe that the first term of (3.4) in general overestimate the differences at this coarse resolution. 4.2. Effects of h-adaptivity. Corollary 3.2 states that the estimated difference \|λ s − k π −1 (s) \| improves at least linearly as the mesh is refined. Figure 4.5 shows the improvement of both the nodal value estimates of k and the associated intervals k(T π −1 (s) ) for problems (P1) and (P3) with N = 59 2 = 3481 degrees of freedom. (We would also like to note that the proof of Corollary 3.2 does not assume linear Lagrange k(T π −1 (s) ) (black vertical lines) for test problem (P3) are shown in the right panel. 4.4. Convergence of the introductory example explained. We will now finish our exposition by returning back to the motivation example presented in sec- 10 0 the fifth iteration we can identify with a remarkable accuracy the slope of the PCG convergence curves for most of the subsequent iterations, with the convergence being almost linear without a substantial acceleration. The rate of convergence is for the Laplace operator preconditioning remarkably faster than for the ICHOL preconditioning. Bottom: Problem with the ICHOL preconditioning. We do not observe yet a good approximation of any of the eigenvalues, but we can see that the extremal Ritz values approach the ends of the spectral interval. x − x k A / x − x 0 A Laplace ICHOL The convergence of the PCG method with the Laplace operator preconditioning can be completely explained using Theorem 3.1 and the results about the CG convergence behavior from the literature. Since k(x) is in the given experiment constant for most of the supports of the basis functions (being equal to one respectively to 161.45), according to Theorem 3.1 the preconditioned system matrix must have many multiple eigenvalues equal to one respectively to 161.45. This is illustrated by the computed quantities presented in Table 4.1. We see that 1922 eigenvalues are equal to one, 1922 are equal to 161.45 and the rest is spread between ≈ 28 and ≈ 134 (with the eigenvalues between 81.226 and 134 of so negligible weight (see (2.7)) that they do not contribute within the small number of iterations to the computations; they are for CG computations within the given number of iterations practically not visible; see [22,Section 5.6.4]). Assuming exact arithmetic, van der Sluis and van der Vorst prove in the seminal paper [35] that, if the Ritz values approximate (in a rather moderate way) the eigenvalues at the lower end of the spectrum, the computations further proceed with x − x k A x − x 0 A ≤ 2 κ L e − 1 κ L e + 1 k−5 , κ L e = λ L 2039 λ L 1926 = 1.02, k > 5, at least as fast as the right hand side in (4.2) suggests. The convergence is in the iterations 6-9 very fast and therefore we do not practically observe any further acceleration. At iteration 10, the convergence slows down. This is due to the effect of rounding errors that cause forming a second Ritz value that approximates the largest eigenvalue 161.45 (as mentioned above, the appearance of large outlying eigenvalues can cause deterioration of convergence due to roundoff; the detailed explanation is given, e.g., in [12], [22, Section 5.9.1; see in particular, Figures 5.14 and 5.15] and in [9]). Also for the incomplete Choleski preconditioning an analogous argumentation holds with the difference that the approximation of the five leftmost eigenvalues by the Ritz values slightly accelerate convergence. The bound (4.2) is valid with replacing κ L e by κ C e = λ C 3969 λ C 6 = 3.75; see the computed quantities in Table 4.2. We can see from Figure 4.9 that at the fifth iteration the five smallest eigenvalues are not yet approximated by the Ritz values. This needs about five additional iterations. From the tenth iteration the convergence remains very close to linear and slow because no further acceleration can take place Table 4.2 Detail of the points of increase (Ritz values) and the weights (see (2.7)) of the distribution function ω C (λ) associated with the problem with ICHOL preconditioning. The effective condition number is for the given example determined by λ C 6 and λ C 3969 ; see the bottom part of Figure 4.10. due to the widespread eigenvalues and the effects of roundoff (no further eigenvalue approximation can significantly affect the convergence behavior). The part of the spectra that practically determine the convergence rates after the fifth iteration of the Laplace operator PCG, respectively, after the tenth iteration of the ICHOL PCG are illustrated in Figure 4.10. Index Concluding remarks. We have analyzed the operator L −1 A generated by preconditioning second order elliptic PDEs with the inverse of the Laplacian. Previously, it has been proven that the range of the coefficient function k of the elliptic PDE is contained in the spectrum of L −1 A, but only for operators defined on infinitely dimensional spaces. In this paper we show that a substantially stronger result holds in the discrete case of conforming finite elements. More precisely, that the eigenvalues of the matrix L −1 A, where L and A are FE-matrices, lie in resolution dependent intervals around the nodal values of the coefficient function that tend to the nodal values as the resolution increases. Moreover, there is a pairing (possibly non-unique) of the eigenvalues and the nodal values of the coefficient function due to Hall's theory of bipartite graphs. Finally, we demonstrate that the conjugate gradient method utilize the structure of the spectrum (more precisely, of the associated distribution function) to accelerate the iterations. In fact, even though the condition number involved, for instance, with incomplete Choleski preconditioning is significantly smaller than for the Laplacian preconditioner, the performance when using Choleski is much worse. In this case, the accelerated performance of the Laplacian preconditioner can be fully explained by an analysis of the distribution functions. Av = λLv. Fig. 2 . 1 . 21Left: The solution of the problem section 2 on the background of the linear FE triangulation. Right: The relative energy norm of the PCG error as a function of the iteration steps. The Laplace operator preconditioning (solid line) is much more efficient than the incomplete Choleski preconditioning (dashed line), despite the fact that the condition numbers are 161.54 and close to 16, respectively. This can be explained by the differences in the associated distribution functions (see the end of section 4 below). Fig. 2 . 3 . 23Top: Comparison of the spectra of the matrices A, A L and A C , N = 3969 degrees of freedom. Due to a small drop-off tolerance, the eigenvalues of A and CC * are graphically indistinguishable. Therefore the right part only shows the eigenvalues of A C (using a different scale than the left part of the figure). Bottom: Comparison of the distribution functions ω L (λ) (left) and ω C (λ) (right) associated with the preconditioned problems. The vertical axes are in the logarithmic scale and λ L 1 = 1, λ L N = 161.45, λ C 1 = 7.4 × 10 −2 , λ C N = 1.16. Fig. 3 . 1 . 31Illustration of Theorem 3.1. The diameters of the dashed circles indicate the size of the intervals k(T j ), j = 1, . . . , N . The dots represent the eigenvalues λ j , j = 1, . . . , N , of the matrix L −1 A. We can find a pairing between the intervals k(T j ) and the eigenvalues λ i , but the pairing may not be uniquely determined. Fig. 3 . 2 . 32Illustration of Corollary 3.2. The relation (3.2) (indicated by the dashed blue circle) can give significantly better localization of the position of the individual eigenvalues than the bounds (3.3) (indicated by the dotted red circle) and (3.4) (indicated by the solid red circle). When k(x) is constant over T j , then k(T j ) reduces to one point λ π(j) ; see also (3.3). The bound (3.4) is weaker that (3.2) and (3.3), but the evaluation of its first term might be easier in practice. T j ), J ⊂ {1, . . . , N }. This enables us to apply Hall's theorem, see [4, Theorem 5.2] or, e.g., [16, Theorem 1], to prove Theorem 3.1. (For the sake of completeness, we have also formulated Hall's result below in Theorem 3.5.) Lemma 3.3. Using the notation introduced above, let J ⊂ {1, . . . , N } and T J = ∪ j∈J T j . Then there exist at least p = \|J \| eigenvaluesλ 1 , . . . ,λ p of L −1 A such that . . . , p. 3.3 N times with J = {1}, J = {2}, . . . , J = {N }, we see that, for the support of any basis function φ j there is an eigenvalueλ of L −1 A such thatλ ∈ k(T j ). Moreover, as an additional important consequence, for any subset J ⊂ {1, . . . , N } the associated union of intervalsk(T J ) (see (3.5)) contains at least p = \|J \| eigenvalues of L −1 A; see the following corollary. Corollary 3 . 4 . 34Let, as above, J ⊂ {1, . . . , N } and T J = ∪ j∈J T j . Then there exist at least p = \|J \| eigenvaluesλ 1 , . . . ,λ p of L −1 A such that ( 3 . 310)λ ∈k(T J ) ≡ j∈J k(T j ), = 1, . . . , p. Moreover, taking J = {1, . . . , N }, (3.10) immediately implies that any eigenvalueλ of L −1 A belongs to (at least one) interval k(T j ), j ∈ {1, . . . , N }. Lemma 3.3 then assures that each intervalk(T Ji ) contains at least \|J i \| eigenvalues of L −1 A. Summing up, at least i=1,...,p \|J i \| = \|J \| eigenvalues of L −1 A must be contained in the unionk(T J ). the sets of nodes S = I = {1, . . . , N } and the set of edges E, where {s, i} ∈ E if and only if λ s ∈ k(T i ), s ∈ S, i ∈ I. Theorem 3. 5 ( 5Hall's theorem). Let (S, I, E) be a bipartite graph. Given J ⊂ I, let G(J ) ⊂ S denote the set of all nodes adjacent to any node from J , i.e., G(J ) = {s ∈ S; ∃i ∈ J such that {s, i} ∈ E}.Then there exists a matching M ⊂ E that covers I if and only if(3.12) \|G(J )\| ≥ \|J \| for any J ⊂ I; see, e.g., [4, Theorem 5.2] and the original formulation [16, Theorem 1]. Now we are ready to finalize our argument. Proof of Theorem 3.1. Consider the bipartite graph defined by (3.11) and let G(J ) ⊂ S be the set of all nodes (representing the eigenvalues) adjacent to any node from J , J ⊂ I (representing the intervals). In other words, G(J ) represents the indices of all eigenvalues {λ s ; s ∈ G(J )} located ink(T J ) = ∪ j∈J k(T j ). Corollary 3.4 of Lemma 3.3 assures that assumption (3.12) in Theorem 3.5 is satisfied, i.e. ( 3 . 13 ) 313\|G(J )\| ≥ \|J \|. 5 FEniCS version 2017.2.0 and MATLAB Version: 8.0.0.783 (R2012b). Fig. 4 . 1 . 41Comparison of the eigenvalues λs, s = 1, . . . , N (red dots) and the increasingly sorted nodal values of k (blue circles). Top left: (P1), top right: (P2), bottom left: (P3), bottom right: (P4). As in Figure 4.2, we use the semilogarithmic scale in the lower right panel (P4). Fig. 4 . 2 . 42The eigenvalues λ 1 ≤ . . . ≤ λ N (red dots) and the associated intervals k(T P (s) ) (black vertical lines), where the pairing P is defined by the increasingly sorted nodal values of k; seeFigure 4.1. Top left: (P1), top right: (P2), bottom left: (P3), bottom right: (P4). We observe that for (P4) some of the eigenvalues are not inside the associated intervals and therefore the ordering in which eigenvalues and nodal values of k are in increasing order does not in this case conform to π from Theorem 3.1. Fig. 4 . 3 . 43Illustration of the pairing π computed by the Dulmage-Mendelsohn decomposition of the corresponding adjacency matrix G (see (4.1)) for problem (P4). Left: The eigenvalues λ 1 ≤ . . . ≤ λ N (red dots) and the associated intervals k(T π −1 (s) ) (black vertical lines). Right: The comparison of the eigenvalues and the associated nodal values k π −1 (s) (blue circles). Fig. 4 . 4 .Fig. 4 . 5 . 4445Illustration of Corollary 3.2. Comparison of the absolute difference \|λs − k π −1 (s) \| (red dots) and its estimate by the first term on the right hand side of (3.4) (black stars). Top left: (P1), top right: (P2), bottom left: (P3), bottom right: (P4).elements, but holds for any type of nodal basis functions.) Corollary 3.2 is a local estimate which allows local mesh refinement for improving Top: The eigenvalues λs, s = 1, . . . , N (red dots) and the associated intervals k(T π −1 (s) ) (black vertical lines). Bottom: Comparison of the eigenvalues λs and the nodal values k π −1 (s) (blue circles). Here we use uniform mesh with N = 3481 degrees of freedom. Left: (P1), right: (P3). We can observe a dramatic improvement of the approximation accuracy, cf.Figures 4.1 and 4.2.accuracy of the eigenvalue estimate. To see the effect of locally refined mesh on the spectrum of the preconditioned problem, we consider the test problem (P2), where we refine the mesh in the subdomain [0, 0.2] × [0, 0.2], i.e., in the area with large gradient of the function k(x, y). Figure 4 . 46 shows the discretization mesh (top), the eigenvalues with the associated intervals (middle) and the associated nodal values (bottom). As expected, we observe more eigenvalues in the upper part of the spectrum as well as their better localization; see for comparison also the top right panels ofFigures 4.1 and 4.2. 4.3. Re-entrant corner domain. The local considerations of Corollary 3.2 does not require additional regularity for the solutions of the associated PDEs and our theoretical results are valid for domains of any shape. To illuminate that no additional regularity is needed we conduct experiments on a domain with a re-entrant corner, i.e., Ω = [0, 1] × [0, 1] \ {(x, y) : x > 0.8y + 0.1 & y < 0.8x + 0.1} .The domain is shown in the left panel inFigure 4.7, while the eigenvalues λ s (red dots) with the sorted nodal values k π −1 (s) (green circles) and the associated intervals Fig. 4 . 6 . 46The influence of the locally refined mesh in the subdomain (0, 0.2) × (0, 0.2) for the test problem (P2). Left: One refinement step. Right: Three refinement steps. We use the same symbols as inFigures 4.1 and 4.2. Fig. 4 . 7 . 2 472Comparison of the eigenvalues λs (red dots) with the sorted nodal values k π −1 (s) (green circles) and the associated intervals k(T π −1 (s) ) (black vertical lines) for the test problem (P3) on the re-entrant corner domain. tion 2 and by explaining the difference in the behavior of PCG with the Laplace operator preconditioning and with the ICHOL preconditioning; see the right part ofFigure 2.1. First we present Figure 4.8, a modification of Figure Fig. 4 . 8 . 48Explanation of the PCG behavior fromFigure 1. The dotted and dash-dotted lines show the estimates of the PCG error based on the so-called effective condition number, which here (see the discussion in the text) fully describes the PCG behavior starting from the sixth iteration. Fig. 4 . 9 . 49Illustration of the Ritz values computed at the fifth PCG iteration. Top: Problem with the Laplace operator preconditioning. We observe four Ritz values approximating the eigenvalues at the lower end of the spectrum and one Ritz value very closely approximating the largest eigenvalue. Fig. 4 . 10 . 410Distribution functions: Top: Laplace operator preconditioning. Bottom: ICHOL preconditioning. Red dashed lines represent the position of eigenvalues associated with the effective condition numbers after five iterations. a rate as if the approximated eigenvalues are not present. Analysis of rounding errors in CG and Lanczos by Paige, Greenbaum and others, mentioned above in section 1, then proves that this argumentation concerning the lower end of the spectrum remains valid also in finite precision arithmetic computations. At the fifth iteration the eigenvalues 1, 28.5, 61.4, 75.3 at the lower end of the spectrum and also the largest eigenvalue 161.45 are approximated by the Ritz values; see Figure 4.9. Therefore, from then on PCG converges, using the effective condition number upper bound (4.2) Table 4 .1 4Detail of the points of increase (Ritz values) and the weights (see (2.7)) of the distribution function ω L (λ) associated with the problem preconditioned by the Laplace operator. The effective condition number is for the given example determined by λ L 1926 and λ L 2039 ; see the top part ofFigure 4.10.Index 1 -1922 1923 1924 1925 1926 Eigenvalues 1 28.508 61.384 75.324 λ L 1926 = 79.699 Total weight 9 × 10 −6 ≈ 10 −3 ≈ 10 −3 ≈ 10 −3 ≈ 10 −3 Index 1927 -1930 1931 -2039 2040 -2047 2048 -3969 Eigenvalues 80.875 -81.222 λ L 2039 = 81.224 81.226 -133.94 161.45 Total weight ≈ 10 −3 1.8 × 10 −2 8 × 10 −10 0.96 0 20 40 60 80 100 120 140 160 0.5 1 1.5 eigenvalues Ritz values, 5th it 0 0.2 0.4 0.6 0.8 1 1.2 0.5 1 1.5 eigenvalues Ritz values, 5th it Here we assume that the system matrix is Hermitian, otherwise the spectral information may not be descriptive for convergence of Krylov subspace methods; see[11,13]. The spectrum of the operator L −1 A on an infinite dimensional normed linear space is defined as sp(L −1 A) ≡ λ ∈ C; L −1 A − λI does not have a bounded inversion . If k(x) is continuous on T j , then k(T j ) coincides with the range of k(x) over T j . 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[ "RANDOMIZED SERIES AND GEOMETRY OF BANACH SPACES", "RANDOMIZED SERIES AND GEOMETRY OF BANACH SPACES" ]	[ "Han Ju Lee " ]	[]	[]	We study some properties of the randomized series and their applications to the geometric structure of Banach spaces. For n ≥ 2 and 1 < p < ∞, it is shown that ℓ n ∞ is representable in a Banach space X if and only if it is representable in the Lebesgue-Bochner L p (X). New criteria for various convexity properties in Banach spaces are also studied. It is proved that a Banach lattice E is uniformly monotone if and only if its p-convexification E (p) is uniformly convex and that a Köthe function space E is upper locally uniformly monotone if and only if its p-convexification E (p) is midpoint locally uniformly convex.Proof. In the proof, the notation E[ · \|G] means the conditional expectation with respect to the sub-σ-algebra G and we need to show that E[ϕ( S n+1 )\|F n ] ≥ 2000 Mathematics Subject Classification. 46B20,46B07,46B09.	10.11650/twjm/1500406019	[ "https://arxiv.org/pdf/0706.3740v1.pdf" ]	14,810,519	0706.3740	fff9a511029b4aeb3767694ada3d0949267481b6	RANDOMIZED SERIES AND GEOMETRY OF BANACH SPACES 26 Jun 2007 Han Ju Lee RANDOMIZED SERIES AND GEOMETRY OF BANACH SPACES 26 Jun 2007arXiv:0706.3740v1 [math.FA] We study some properties of the randomized series and their applications to the geometric structure of Banach spaces. For n ≥ 2 and 1 < p < ∞, it is shown that ℓ n ∞ is representable in a Banach space X if and only if it is representable in the Lebesgue-Bochner L p (X). New criteria for various convexity properties in Banach spaces are also studied. It is proved that a Banach lattice E is uniformly monotone if and only if its p-convexification E (p) is uniformly convex and that a Köthe function space E is upper locally uniformly monotone if and only if its p-convexification E (p) is midpoint locally uniformly convex.Proof. In the proof, the notation E[ · \|G] means the conditional expectation with respect to the sub-σ-algebra G and we need to show that E[ϕ( S n+1 )\|F n ] ≥ 2000 Mathematics Subject Classification. 46B20,46B07,46B09. Randomized series Let {r n } ∞ n=1 be a sequence of mutually independent, symmetric and integrable random variables on a probability space (Ω, F , P ) and {x n } ∞ n=0 an arbitrary sequence in a Banach space X. A randomized series S n is a vector-valued random variable defined by S n = x 0 + r 1 x 1 + · · · + r n x n , (n = 0, 1, . . .) Let F 0 be the trivial σ-algebra {∅, Ω} and F k , (k ≥ 1) the σ-algebra generated by random variables {r i } k i=1 . It is easy to see that the sequence of randomized series {S n } ∞ n=0 is a martingale with respect to the filtration {F n } ∞ n=0 . In this paper, E stands for the expectation with respect to the probability P . We begin with the basic properties of the randomized series. For more properties of random series, see [10]. ϕ( S n ) almost surely. By the independence and symmetry of {r k } we get, for almost all ω ∈ Ω, E[ϕ( S n+1 )\|F n ](ω) = Ω ϕ( S n (ω) + r n+1 (t)x n+1 ) dP (t) = Ω ϕ( S n (ω) + r n+1 (t)x n+1 ) + ϕ( S n (ω) − r n+1 (t)x n+1 ) 2 dP (t) ≥ Ω ϕ( S n (ω) ) dP (t) = ϕ( S n (ω) ), which completes the proof. Proposition 1.2. Let ϕ be a convex increasing function on [0, ∞) and x, y ∈ X. Then the function ψ on R defined by ψ(λ) = E[ϕ( x + λr 1 y )] is an increasing convex function on [0, ∞) with ψ(λ) = ψ(\|λ\|) for every λ ∈ R. Proof. By the convexity of ϕ, we get ψ(λ) = 1 2 [ψ(λ) + ψ(−λ)] ≥ ψ(0) for every real λ, which implies that λ → ψ(λ) is increasing on [0, ∞). Clearly λ → ψ(λ) is an even function on R since r 1 is symmetric. Since random variables {r i } i are independent, Proposition 1.2 shows that for any two real sequences {λ i } n i=1 and {ξ i } n i=1 satisfying \|λ i \| ≤ \|ξ i \| for i = 1, · · · , n, and for any x 0 , x 1 , . . . , x n in X, we get E[ϕ x 0 + λ 1 r 1 x 1 + · · · + λ n r n x n ] ≤ E[ϕ x 0 + ξ 1 r 1 x 1 + · · · + ξ n r n x n ]. A convex function ϕ : [0, ∞) → R is said to be strictly convex if ϕ( s + t 2 ) < ϕ(s) + ϕ(t) 2 holds for all distinct positive numbers s, t. Notice that if ϕ is strictly convex and increasing on [0, ∞) and a, b are real numbers with b = 0, then E[ϕ(\|a + r 2 b\|)] = E[ ϕ(\|a + r 2 b\|) + ϕ(\|a − r 2 b\|) 2 ] > ϕ(\|a\|). Now we state our main theorem concerning the randomized series. Theorem 1.3. Let {x i } n i=1 be a finite sequence in a Banach space X, ϕ a strictly convex, increasing function with ϕ(0) = 0 and {r i } ∞ i=1 symmetric independent random variables with r i ∞ = 1, (i = 1, 2, · · · ). Suppose that there is a constant ρ > 0 such that the following holds: sup ǫ 1 =±1,...,ǫn=±1 ǫ 1 x 1 + · · · + ǫ n x n ≥ x 1 + ρ. Then there is a constant δ = δ(ρ) > 0 such that E[ϕ( x 1 + r 2 x 2 + · · · + r n x n )] ≥ ϕ( x 1 ) + δ. In particular, if we take ρ 1 = min{ρ, 1/2}, then δ = min ϕ( ρ 1 3 ) n i=2 P {\|r i − 1\| < ρ 1 3n }, min 2≤j≤n E ϕ x 1 + r j ρ 1 3n − ϕ( x 1 ) . Proof. We adapt the argument in the proof of Proposition 2.2 in [3]. We assume that there exist 0 < ρ < 1/2 and signs ǫ 2 , . . . , ǫ n such that x 1 + ǫ 2 x 2 + · · · + ǫ n x n ≥ x 1 + ρ. Select a unit element x * in X * such that x * (x 1 ) = x 1 and let λ i = x * (x i ) for 1 ≤ i ≤ n. Now we shall consider two cases according to the size of \|λ i \|. In the first case we suppose that max 2≤i≤n \|λ i \| ≤ ρ 3n . If \|η i − ǫ i \| ≤ ρ 3n , (2 ≤ i ≤ n) , then x 1 + η 2 x 2 + · · · + η n x n ≥ x 1 + ǫ 2 x 2 + · · · ǫ n x n − ρ 3 ≥ x 1 + 2ρ 3 . Since \|λ 1 + η 2 λ 2 + · · · + η n λ n \| ≤ \|λ 1 \| + ρ 3 = x 1 + ρ 3 , we get x 1 + η 2 x 2 + · · · + η n x n ≥ \|λ 1 + η 2 λ 2 + · · · η n λ n \| + ρ 3 , if \|η i − ǫ i \| ≤ ρ 3n , (2 ≤ i ≤ n). Let F = n j=2 w ∈ Ω : \|r j (w) − ǫ j \| < ρ 3n and take T n = x 1 + r 2 x 2 + · · · + r n x n . Then we have E[ϕ( T n )] = E[ϕ( T n )χ F ] + E[ϕ( T n )χ F c ]. Since ϕ(a + b) ≥ ϕ(a) + ϕ(b), (a, b ≥ 0), E[ϕ( T n )χ F ] ≥ E[ϕ(\|λ 1 + r 2 λ 2 + · · · + r n λ n \| + ρ/3)χ F ] ≥ E[ϕ(\|λ 1 + r 2 λ 2 + · · · + r n λ n \|)χ F ] + ϕ(ρ/3)P (F ). Hence E[ϕ( T n )] ≥ E[ϕ(\|λ 1 + r 2 λ 2 + · · · + r n λ n \|)] + ϕ(ρ/3)P (F ) ≥ ϕ(λ 1 ) + ϕ(ρ/3)P (F ) = ϕ( x 1 ) + ϕ(ρ/3)P (F ). Notice that P (F ) = n i=2 P {w ∈ Ω : \|r i (w)−1\| < ρ 3n } > 0 for r i 's are independent symmetric random variables with r i ∞ = 1, (i = 1, 2, · · · ). In the second case we suppose that there exists i 0 (2 ≤ i 0 ≤ n) such that \|λ i 0 \| ≥ ρ 3n . It follows from Proposition 1.1, 1.2 and strict convexity of ϕ that E[ϕ( T n )] ≥ E[ϕ(\|λ 1 + r 2 λ 2 + · · · + r n λ n \|)] ≥ E[ϕ(\|λ 1 + r i 0 λ i 0 \|)] ≥ E ϕ λ 1 + r i 0 ρ 3n > ϕ(λ 1 ) = ϕ( x 1 ). The proof is complete. 2. Representability of ℓ n ∞ A Banach space Y is said to be representable in X if, for each λ > 1, there is a bounded linear map T : Y → X such that x ≤ T x ≤ λ x for every x ∈ Y . A Banach space Y is said to be finitely representable in X if every finite dimensional subspace of Y is representable in X. It is well-known due to the work of B. Mauray and G. Pisier [14] that c 0 is finitely representable in X if and only if c 0 is finitely representable in L p (X) for all 1 ≤ p < ∞. S. J. Dilworth considered the quatitative version of this theorem in [3], where he showed that if X is a complex Banach space, n ≥ 2 and 0 < p < ∞, then ℓ n ∞ (C) is representable in X if and only if it is representable in L p (X). As we see in the next example it is not true for real Banach spaces. Example 2.1. Let X be a nontrivial real Banach space. Then ℓ 2 ∞ is repre- sentable in L 1 ([0, 1]; X). Indeed, if we choose the Rademacher sequence {r n = sign(sin 2 n πt)} ∞ n=1 in L 1 [0, 1 ] and x 0 ∈ S X , then x = r 1 x 0 and y = r 2 x 0 are the elements of unit sphere of L 1 (X) and they satisfy x + y L 1 (X) = x − y L 1 (X) = 1, which means that ℓ 2 ∞ is representable in L 1 (X) . The subharmonicity of absolute value of holomorphic functions on C plays the crucial role in the proof in [3]. In this paper, the strict convexity of ϕ(t) = \|t\| p (1 < p < ∞) on R plays the analogous role, and thus it is shown here that ℓ n ∞ is representable in X if and only if it is representable in L p (X) for every 1 < p < ∞. The following proposition is a real version of Proposition 2.2 in [3]. Proposition 2.2. Let {r i } ∞ i=1 be symmetric independent random variables with r i ∞ = 1, (i = 1, 2, · · · ). Suppose that X is a real Banach space and that n ≥ 2. The following properties are equivalent: (1) ℓ n ∞ is not representable in X. (2) There exists ρ > 0 such that whenever x 1 , . . . , x n are unit vectors in X, then there exist signs ǫ 1 , . . . , ǫ n such that ǫ 1 x 1 + · · · + ǫ n x n ≥ 1 + ρ. (3) There exist strictly convex, increasing function ϕ on [0, ∞) with ϕ(0) = 0 and ρ > 0 such that whenever x 1 , . . . , x n are unit vectors in X then E[ϕ( x 1 + r 2 x 2 · · · + r n x n )] ≥ ϕ(1) + ρ. (4) For each strictly convex, increasing function ϕ on [0, ∞) with ϕ(0) = 0, there is ρ > 0 such that whenever x 1 , . . . , x n are unit vectors in X then E[ϕ( x 1 + r 2 x 2 · · · + r n x n )] ≥ ϕ(1) + ρ. Proof. The implications (4) ⇒ (3) ⇒ (2) ⇒ (1) are clear. To show (1) ⇒ (2), suppose that (2) fails, so for any ρ > 0 there exist unit vectors x 1 , . . . , x n in X such that for all signs ǫ 1 , . . . , ǫ n , we have ǫ 1 x 1 + · · · + ǫ n x n < 1 + ρ. It follows that for all λ 1 , . . . , λ n with 1 = \|λ i 0 \| = max 1≤i≤n \|λ i \|, we have λ 1 x 1 + · · · + λ n x n < 1 + ρ. Hence 1 = λ i 0 x i 0 ≤ 1 2 λ 1 x 1 + · · · + λ n x n + 1 2 λ 1 x 1 + · · · + λ n x n − 2λ i 0 x i 0 ≤ 1 2 λ 1 x 1 + · · · + λ n x n + 1 2 (1 + ρ). Thus, λ 1 x 1 + · · · + λ n x n ≥ 1 − ρ. Since ρ is arbitrary, it follows that ℓ n ∞ is representable in X. Now we have only to show that (2) ⇒ (4). Suppose that (2) holds and that ϕ is a strictly convex, increasing function on [0, ∞) with ϕ(0) = 0. There is 0 < ρ < 1/2 such that whenever x 1 , . . . , x n are unit vectors in X, there exist signs ǫ 2 , . . . , ǫ n such that x 1 + ǫ 2 x 2 + · · · ǫ n x n ≥ 1 + ρ. Then Theorem 1.3 shows that (2) implies (4). Notice that in the case of the Rademacher sequence {r n }, for every x 1 , . . . , x n in X, E[ϕ( r 1 x 1 + · · · + r n x n )] = E[ϕ( x 1 + r 2 x 2 + · · · + r n x n ) ]. The following theorem shows the lifting property of representability of ℓ n ∞ . Theorem 2.3. Suppose X is a Banach space, (M, M, µ) is a measure space with a measurable subset A satisfying 0 < µ(A) < ∞, and 1 < p < ∞, n ≥ 2. Then ℓ n ∞ is representable in X if and only if it is representable in L p (M, M, µ; X). Proof. One implication is clear. To prove the other implication, suppose that ℓ n ∞ is not representable in X and let {r n } be the Rademacher sequence. By Proposition 2.2, there exits 0 < ρ < 1/2 such that whenever x 1 , . . . , x n are unit vectors in X, we have E r 1 x + r 2 x 2 + · · · + r n x n p ≥ (1 + ρ) p . Suppose that f 1 , . . . , f n are unit vectors in L p (X). We define the following functions on M. For w ∈ M, let q(w) p = E r 1 f 1 (w) + · · · + r n f n (w) p , M(w) = max{ f i (w) : 1 ≤ i ≤ n}, m(w) = min{ f i (w) : 1 ≤ i ≤ n}. By Proposition 1.1, q(w) ≥ M(w) for all w ∈ M. Now the argument divides into two cases according to the relative sizes of M(w) and m(w). In the first case we suppose that (1 − ρ/3)M(w) ≥ m(w). Then 1 n n i=1 f i (w) p ≤ n − 1 n M(w) p + 1 n m(w) p ≤ 1 − ρ 3n M(w) p ≤ q(w) p − ρ 3n 1 n n i=1 f i (w) p and so q(w) p ≥ 1 + ρ 3n 1 n n i=1 f i (w) p . In the second case, we suppose that (1 − ρ/3)M(w) < m(w). Then q(w) p ≥ (1 + ρ) p m p (w) ≥ (1 + ρ) p 1 − ρ 3 p 1 n n i=1 f i (w) p ≥ 1 + ρ 2 p 1 n n i=1 f i (w) p . Hence by the Fubini theorem, E rf 1 + · · · + r n f n p L P (X) = M q(w) p dµ ≥ min 1 + ρ 2 p , 1 + ρ 3n , which shows that ℓ n ∞ is not representable in L p (X) by Proposition 2.2. The proof is completed. Applications to the convexity of Banach spaces Recall that a point x in S X is an extreme point of B X if max{ x+y , x−y } = 1 for some y ∈ X implies y = 0. A point x ∈ S X is called a strongly extreme point of B X if, given ǫ > 0, there is a δ = δ(x, ǫ) > 0 such that inf{max{ x + y , x − y } : y ≥ ǫ} ≥ 1 + δ. A Banach space is said to be strictly convex (resp. midpoint locally uniformly convex) if every point of S X is (resp. strongly) extreme point of B X . A Banach space is called uniformly convex if, given ǫ > 0, there is a δ = δ(ǫ) > 0 such that inf{max{ x + y , x − y } : y ≥ ǫ, x = 1} ≥ 1 + δ. ǫ > 0 there is a δ > 0 such that whenever y ≥ ǫ, we get E[ϕ( x + ry )] ≥ ϕ(1) + δ. (3) X is uniformly convex if and only if the modulus δ ϕ (ǫ) > 0 for every ǫ > 0, where δ ϕ (ǫ) = inf{E[ϕ( x + ry )] − ϕ(1) : x ∈ S X , y ≥ ǫ}. Proof. We prove only (3) because the proof of the others are similar. Suppose that X is uniformly convex. Given ǫ > 0, there is a ρ > 0 such that for any x ∈ S X and y ∈ X with y ≥ ǫ, we have max{ x + y , x − y } ≥ 1 + ρ. Then Theorem 1.3 shows that there is δ > 0 such that for any x ∈ S X and y with y ≥ ǫ, E[ϕ( x + ry )] ≥ ϕ(1) + δ. Conversely, suppose that δ ϕ (ǫ) > 0 for every ǫ > 0. Then given ǫ > 0 for any x ∈ S X and y ∈ X with y ≥ ǫ, we have max{ϕ( x + y ), ϕ( x − y )} ≥ max −1≤t≤1 {ϕ( x + ty )} ≥ E[ϕ( x + ry )]. Since ϕ is strictly increasing we get, for any x ∈ S X and y ∈ X with y ≥ ǫ, max{ x + y , x − y } ≥ ϕ −1 (1 + δ ϕ (ǫ)) > 1. Therefore X is uniformly convex and this completes the proof. It is worthwhile to notice that Theorem 3.1 does not hold if we consider the general increasing convex function ϕ with ϕ(0) = 0. Indeed, it is easily checked that if ϕ(t) = \|t\| and {r n } n is the Rademacher sequence then for any nontrivial Banach space X, E x + r 1 x = 1 (x ∈ S X ). Consequently, we cannot characterize the extreme point of B X with ϕ(t) = t. We shall discuss the uniform convexity of p-convexification E (p) for uniformly monotone Banach lattice E. For more details on Banach lattices, order continuity and Köthe function spaces, see [13]. For the definition of p-convexification E (p) of E and the addition ⊕ and multiplication ⊙ there, see [11,13]. A Banach lattice is said to be uniformly monotone (resp. upper locally uniformly monotone) if given ǫ > 0 M p (ǫ) = inf{ (\|x\| p + \|y\| p ) 1/p − 1 : y ≥ ǫ, x = 1} > 0 (resp. N p (ǫ; x) = inf{ (\|x\| p + \|y\| p ) 1/p − 1 : y ≥ ǫ} > 0 ) for some 1 ≤ p < ∞. It is shown in [11,12] that, given ǫ > 0 and 1 ≤ p < ∞ there is a C p > 0 such that for every ǫ > 0, (3.1) C −1 p M 1 (C −1 p ǫ p ) ≤ M p (ǫ) ≤ M 1 (ǫ). In the case when E is an order continuous Banach lattice or a Köthe function space, we also get the following relations by Lemma 2.3 in [12]: There is a C p > 0 such that every x ∈ S X and ǫ > 0, (3.2) C −1 p N 1 (C −1 p ǫ p ; x) ≤ N p (ǫ; x) ≤ N 1 (ǫ; x). Notice that relations (3.1), (3.2) show that if a Banach lattice E is uniformly monotone then E (1/p) is uniformly monotone quasi-Banach lattice for 1 < p < ∞. Similarly, if E is upper locally uniformly monotone order continuous Banach lattice or Köthe function space, then E (1/p) is also upper locally uniformly monotone quasi-Banach lattice for 1 < p < ∞ (cf. [11]). The characterizations of local uniform monotonicity of various function spaces have been discussed in [7]. In [9], H. Hudzik, A. Kamińska and M. Masty lo showed that if a Köthe function space E is uniformly monotone then its p-convexification E (p) is uniformly convex for 1 < p < ∞. A partial generalization of this result has been studied by the author in [11], where it was shown that if a Banach lattice is uniformly monotone then E (p) is uniformly convex for all 2 ≤ p < ∞. In the next theorem, the gap is completed. (1) E is uniformly monotone. (2) E (p) is uniformly convex for all 1 < p < ∞. (3) E (p) is uniformly convex for some 1 < p < ∞. Proof. Proposition 4.4 in [11] shows that uniformly convex Banach lattice is uniformly monotone. So if we assume (3), then E (p) is uniformly monotone and E is uniformly monotone. Hence (3) ⇒ (1) is proved. The implication (2) ⇒ (3) is clear. So we have only to show that (1) implies (2). We shall use Theorem 3.1 (3) with the Rademacher function \|r\| = 1. Let ǫ > 0 and let f, g ∈ E (p) with f E (p) = f 1/p E = 1 and g E (p) = g 1/p E ≥ ǫ. Recall the following well-known inequality (cf. Lemma 4.1 [11]) : for any 1 < p < ∞ there is C = C(p) such that for any reals s, t, s − t C 2 + s + t 2 2 1 2 ≤ \|s\| p + \|t\| p 2 1 p . Then applying the Krivine functional calculus to the inequality above, we get E f ⊕ (r ⊙ g) p E (p) = E[ \|f 1/p + rg 1/p \| p E ] (3.3) = E \|f 1/p + rg 1/p \| p E + E \|f 1/p − rg 1/p \| p E 2 ≥ E \|f 1/p + rg 1/p \| p + \|f 1/p − rg 1/p \| p 2 E ≥ E \|f \| 2/p + \|2g\| 2/p C 2/p p/2 E = \|f \| 2/p + \|2g\| 2/p C 2/p p/2 E . By (3.3), if 1 < p ≤ 2, then E f ⊕ (r ⊙ g) p E (p) ≥ 1 + M 2/p (2ǫ p /C). In the case of 2 < p < ∞, (3.3) shows that E f ⊕ (r ⊙ g) p E (p) ≥ \|f \| 2/p + \|2g\| 2/p C 2/p p/2 E ≥ \|f \| + \|2g\| C E ≥ 1 + M 1 (2ǫ p /C). Hence E[ f ⊕ (r ⊙ g) p E (p) ] ≥ 1 + M max{1,2/p} (2ǫ p /C) completes the proof. Now we discuss the the local version of Theorem 3.2. A point x ∈ S X in a complex Banach space X is said to be a complex strongly extreme point if there is 0 < p < ∞ such that given ǫ > 0, H p (ǫ; x) = inf 1 2π 2π 0 x + e iθ y p dθ 1/p − 1 : y ≥ ǫ > 0. It is known in [4] that x ∈ S X is a complex strongly extreme point if and only if for every ǫ > 0, H ∞ (ǫ; x) = inf{ max 0≤θ≤2π x + e iθ y − 1 : y ≥ ǫ} > 0. For more details about these moduli, see [2,3,4]. A complex Banach space X is said to be locally uniformly complex convex if every point of S X is a complex strongly extreme point. (1) E is upper locally uniformly monotone. (2) E C is locally uniformly complex convex. (3) E (p) is midpoint locally uniformly convex for all 1 < p < ∞. (4) E (p) is midpoint locally uniformly convex for some 1 < p < ∞. Proof. First we prove the equivalence of (1) and (2). We shall use a similar argument as in the proof of [1,Proposition 3.7] in the sequence space. Suppose that E is locally uniformly complex convex. Then for each x ∈ S E and ǫ > 0 there is δ = δ(x, ǫ) > 0 such that for all y ∈ X with y ≥ ǫ \|x\| + \|y\| ≥ 1 2π 2π 0 x + e iθ y dθ ≥ 1 + δ. So X is upper locally uniformly monotone. For the converse, suppose that E is upper locally uniformly monotone. Now, if we use Theorem 7.1 in [2], then we have for every pair x, y in E, 1 2π 2π 0 x + e iθ y dθ ≥ \|x\| 2 + 1 2 \|y\| 2 1/2 . Hence for every x ∈ S X and ǫ > 0, we get 1 2π 2π 0 x + e iθ y dθ ≥ 1 + N 2 (ǫ/ √ 2; x). Therefore, the upper local uniform monotonicity of E implies the local uniform complex convexity of E. For (1)⇒ (3), fix f with f E (p) = f 1/p = 1 and for any g ∈ E (p) with g E (p) = g 1/p ≥ ǫ, (3.3) holds. Hence E[ f ⊕ (r ⊙ g) p E (p) ] ≥ 1 + N max{1,2/p} (2ǫ p /C; f ), which shows that (1)⇒(3) holds. The implication (3)⇒(4) is clear. Finally assume that (4) holds. Note that every midpoint locally uniformly convex Banach lattice is upper locally uniformly monotone. Indeed, if x ∈ S X and ǫ, there is δ > 0 such that 1 + δ ≤ max{ x + y , x − y } ≤ \|x\| + \|y\| . Since the midpoint local uniform convexity of E (p) implies the upper local uniform monotonicity of E (p) , E is upper locally uniformly monotone. This completes the proof. Let X be a real Banach space and ∆ be the open unit disk in C. Let (f n ) be a sequence of continuous functions from ∆ into X and f : ∆ → X be continuous. We say that (f n ) converges to f with respect to the topology of norm uniform convergence on compact subsets of ∆ if lim n→∞ sup{ f n (z) − f (z) : z ∈ K} = 0 for all compact subsets K of ∆. We will denote by β the topology of norm uniform convergence on compact subsets of ∆. A Banach space X is said to have Kadec-Klee property with respect to topology τ (KK(τ )) if whenever (x n ) is a sequence in X and x ∈ X satisfy x n = x = 1 for all n ∈ N and τ -lim n x n = x, then lim n x n − x = 0. A function f : ∆ → X is harmonic if f is twice continuously differentiable and if the Laplacian of f is zero. It is known [8] that f : ∆ → X is harmonic if and only if x * f is harmonic for all x * ∈ X * if and only if there is a sequence {a n } n ⊂ X so that for all 0 ≤ r < 1, θ ∈ R, f (re iθ ) = ∞ n=−∞ a n r \|n\| e inθ , where the series is absolutely and locally uniformly convergent. We now define h p (∆; X) for 1 < p < ∞ by It is easy to see that on h p (∆; X), · p is β-lower semicontinuous function. It is shown by P. N. Dowling and C. J. Lennard [6] that if h p (∆; X) has KK(β), then X is strictly convex and has the Radon-Nikodým property. In fact, the following proposition is a consequence of the results in [5]. We present an easy proof. Theorem 3.4. If h p (∆; X) has KK(β), then X is midpoint locally uniformly convex. Proof. Suppose that X is not locally uniformly convex. Then applying Theorem 3.1 with r(θ) = cos θ, there exist an ǫ > 0, a sequence (x n ) in X and x ∈ S X such that x n ≥ ǫ and lim n→∞ 1 2π x + cos θx n p dθ. Then lim n f n p = 1 = f p . However Hence h p (∆; X) fails to have KK(β). The proof is complete. It is worthwhile to remark here that it has been shown in [5] that h p (∆; X) has KK(β) if and only if X has the Radon-Nikodým property and every element of S X is a denting point of B X , which is called property (G). It is easy to see that a Banach space with property (G) is midpoint locally uniformly convex. Theorem 1.3 gives the following criteria for the various convexity properties. Theorem 3 . 1 . 31Let X be a real Banach space and ϕ, a strictly convex increasing function on [0, ∞) with ϕ(0) = 0 and r, a symmetric random variable with r ∞ = 1. Then (1) A point x in S X is an extreme point of B X if and only if E[ϕ( x + ry )] = ϕ(1) for y ∈ X implies y = 0. (2) A point x in S X is a strongly extreme point of B X if and only if for every Theorem 3 . 2 . 32Let E be a Banach lattice. The following statements are equivalent. Theorem 3 . 3 . 33Let E be an order continuous Banach lattice or a Köthe function space. Then the following are equivalent: h p (∆; X) = {f : ∆ → X : f is harmonic and f p < ∞}, n +z n )x n and f : ∆ → X by f (z) = x. Then it is easy to see that β-lim n f n (z) = f ( \| cos θ\| p dθ. Acknowledgment. The author thanks A. Kamińska for useful comments. Boundaries for algebras of holomorphic functions on Banach spaces. Y S Choi, K H Han, H J Lee, Illinois J. Math. to appearY. S. Choi, K. H. Han and H. J. Lee, Boundaries for algebras of holomorphic functions on Banach spaces, Illinois J. Math. to appear. The complex convexity of quasinormed linear spaces. W Davis, D J H Garling, N Tomczak-Jagermann, J. Funct. Anal. 55W. Davis, D. J. H. Garling and N. Tomczak-Jagermann, The complex convexity of quasi- normed linear spaces, J. Funct. Anal. 55 (1984), 110-150. Complex convexity and the geometry of Banach spaces. S J Dilworth, Math. Proc. Cambridge Philos. Soc. 99S. J. Dilworth, Complex convexity and the geometry of Banach spaces, Math. Proc. Cam- bridge Philos. Soc. 99 (1986), 495-506. Complex convexity in Lebesgue-Bochner function spaces. P N Dowling, Z Hu, D Mupasiri, Trans. Amer. Math. Soc. 348P. N. Dowling, Z. Hu and D. Mupasiri, Complex convexity in Lebesgue-Bochner function spaces, Trans. Amer. Math. Soc., 348 (1996), 127-139. Geometry of spaces of vector-valued harmonic functions. P Dowling, Z Hu, M A Smith, Canad. J. Math. 462P. Dowling, Z. Hu and M. A. Smith, Geometry of spaces of vector-valued harmonic func- tions, Canad. J. Math. 46 (1994), no. 2, 274-283. Kadec-Klee properties of vector-valued Hardy spaces. P N Dowling, C J Lennard, Math. Proc. Cambridge Philos. Soc. 1113P. N. Dowling and C. J. Lennard, Kadec-Klee properties of vector-valued Hardy spaces. Math. Proc. Cambridge Philos. Soc. 111 (1992), no. 3, 535-544. Local uniform rotundity in Calderón-Lozanovskiȋ spaces. P Foralewski, P Kolwicz, PreprintP. Foralewski and P. Kolwicz, Local uniform rotundity in Calderón-Lozanovskiȋ spaces, Preprint. Hardy-Raüme vektorwertiger Funktionen. Dissertaion. W Hensgen, MunichW. Hensgen,Hardy-Raüme vektorwertiger Funktionen. Dissertaion, Munich 1986. Geometric properties of some Calderon-Lozanovskiȋ spaces and Orlicz-Lorentz spaces. H Hudzik, A Kamińska, M Masty Lo, Houston J. Math. 223H. Hudzik, A. Kamińska and M. Masty lo, Geometric properties of some Calderon- Lozanovskiȋ spaces and Orlicz-Lorentz spaces, Houston J. Math. 22 (1996), no. 3, 639-663. Some random series of functions. J P Kahane, Cambridge Studies in Advanced Mathematics. 5Cambridge University PressSecond editionJ. P. Kahane, Some random series of functions. Second edition. Cambridge Studies in Advanced Mathematics, 5. Cambridge University Press, Cambridge, 1985. Complex convexity and monotonicity in quasi-Banach lattices. H J Lee, Israel J. Math. 159H. J. Lee, Complex convexity and monotonicity in quasi-Banach lattices, Israel J. Math. 159 (2007), 57-91. Monotonicity and complex convexity in Banach lattices. H J Lee, J. Math. Anal. Appl. 307H. J. Lee, Monotonicity and complex convexity in Banach lattices, J. Math. Anal. Appl. 307 (2005), 86-101. Classical Banach spaces II. J Lindenstrauss, L Tzafriri, Springer-VerlagJ. Lindenstrauss and L. Tzafriri, Classical Banach spaces II, Springer-Verlag, 1979. Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach. B Maurey, G Pisier, Studia Math. 581B. Maurey and G. Pisier, Séries de variables aléatoires vectorielles indépendantes et pro- priétés géométriques des espaces de Banach, Studia Math. 58 (1976), no. 1, 45-90.	[]
[ "MIM: A deep mixed residual method for solving high-order partial differential equations", "MIM: A deep mixed residual method for solving high-order partial differential equations" ]	[ "Liyao Lyu es:[email protected] \nSchool of Mathematical Sciences\nSoochow University\n215006SuzhouChina\n\nChu College\nSoochow University\n215006SuzhouCWChina\n", "Zhen Zhang \nTBS Information Technology Co. Ltd\n210000Nanjing, NanjingChina\n", "Minxin Chen \nSchool of Mathematical Sciences\nSoochow University\n215006SuzhouChina\n\nMathematical Center for Interdisciplinary Research\nSoochow University\n215006SuzhouChina\n", "Jingrun Chen \nSchool of Mathematical Sciences\nSoochow University\n215006SuzhouChina\n\nMathematical Center for Interdisciplinary Research\nSoochow University\n215006SuzhouChina\n" ]	[ "School of Mathematical Sciences\nSoochow University\n215006SuzhouChina", "Chu College\nSoochow University\n215006SuzhouCWChina", "TBS Information Technology Co. Ltd\n210000Nanjing, NanjingChina", "School of Mathematical Sciences\nSoochow University\n215006SuzhouChina", "Mathematical Center for Interdisciplinary Research\nSoochow University\n215006SuzhouChina", "School of Mathematical Sciences\nSoochow University\n215006SuzhouChina", "Mathematical Center for Interdisciplinary Research\nSoochow University\n215006SuzhouChina" ]	[]	In recent years, a significant amount of attention has been paid to solve partial differential equations (PDEs) by deep learning. For example, deep Galerkin method (DGM) uses the PDE residual in the least-squares sense as the loss function and a deep neural network (DNN) to approximate the PDE solution. In this work, we propose a deep mixed residual method (MIM) to solve PDEs with high-order derivatives. Notable examples include Poisson equation, Monge-Ampére equation, biharmonic equation, and Korteweg-de Vries equation. In MIM, we first rewrite a high-order PDE into a first-order system, very much in the same spirit as local discontinuous Galerkin method and mixed finite element method in classical numerical methods for PDEs. We then use the residual of first-order system in the least-squares sense as the loss function, which is in close connection with least-squares finite element method. For aforementioned classical numerical methods, the choice of trail and test functions is important for stability and accuracy issues in many cases. MIM shares this property when DNNs are employed to approximate unknowns functions in the first-order system. In one case, we use nearly the same DNN to approximate all unknown functions and in the other case, we use totally different DNNs for different unknown functions. Numerous results of MIM with different loss functions and different choice * Corresponding authors.of DNNs are given for four types of PDEs. In most cases, MIM provides better approximations (not only for high-derivatives of the PDE solution but also for the PDE solution itself) than DGM with nearly the same DNN and the same execution time, sometimes by more than one order of magnitude. When different DNNs are used, in many cases, MIM provides even better approximations than MIM with only one DNN, sometimes by more than one order of magnitude. Numerical observations also imply a successive improvement of approximation accuracy when the problem dimension increases and interesting connections between MIM and classical numerical methods. Therefore, we expect MIM to open up a possibly systematic way to understand and improve deep learning for solving PDEs from the perspective of classical numerical analysis.	10.1016/j.jcp.2021.110930	[ "https://arxiv.org/pdf/2006.04146v1.pdf" ]	219,531,080	2006.04146	74a38581152ce936b0e2fba8bbcba723956db484	MIM: A deep mixed residual method for solving high-order partial differential equations Liyao Lyu es:[email protected] School of Mathematical Sciences Soochow University 215006SuzhouChina Chu College Soochow University 215006SuzhouCWChina Zhen Zhang TBS Information Technology Co. Ltd 210000Nanjing, NanjingChina Minxin Chen School of Mathematical Sciences Soochow University 215006SuzhouChina Mathematical Center for Interdisciplinary Research Soochow University 215006SuzhouChina Jingrun Chen School of Mathematical Sciences Soochow University 215006SuzhouChina Mathematical Center for Interdisciplinary Research Soochow University 215006SuzhouChina MIM: A deep mixed residual method for solving high-order partial differential equations Preprint submitted to Elsevier June 9, 2020 arXiv:2006.04146v1 [math.NA] 7 Jun 2020(Zhen Zhang), [email protected] (Minxin Chen), [email protected] (Jingrun Chen) In recent years, a significant amount of attention has been paid to solve partial differential equations (PDEs) by deep learning. For example, deep Galerkin method (DGM) uses the PDE residual in the least-squares sense as the loss function and a deep neural network (DNN) to approximate the PDE solution. In this work, we propose a deep mixed residual method (MIM) to solve PDEs with high-order derivatives. Notable examples include Poisson equation, Monge-Ampére equation, biharmonic equation, and Korteweg-de Vries equation. In MIM, we first rewrite a high-order PDE into a first-order system, very much in the same spirit as local discontinuous Galerkin method and mixed finite element method in classical numerical methods for PDEs. We then use the residual of first-order system in the least-squares sense as the loss function, which is in close connection with least-squares finite element method. For aforementioned classical numerical methods, the choice of trail and test functions is important for stability and accuracy issues in many cases. MIM shares this property when DNNs are employed to approximate unknowns functions in the first-order system. In one case, we use nearly the same DNN to approximate all unknown functions and in the other case, we use totally different DNNs for different unknown functions. Numerous results of MIM with different loss functions and different choice * Corresponding authors.of DNNs are given for four types of PDEs. In most cases, MIM provides better approximations (not only for high-derivatives of the PDE solution but also for the PDE solution itself) than DGM with nearly the same DNN and the same execution time, sometimes by more than one order of magnitude. When different DNNs are used, in many cases, MIM provides even better approximations than MIM with only one DNN, sometimes by more than one order of magnitude. Numerical observations also imply a successive improvement of approximation accuracy when the problem dimension increases and interesting connections between MIM and classical numerical methods. Therefore, we expect MIM to open up a possibly systematic way to understand and improve deep learning for solving PDEs from the perspective of classical numerical analysis. Introduction Solving partial differential equations (PDEs) has been the most ubiquitous tool to simulate complicated phenomena in applied sciences and engineering problems. Classical numerical methods include finite difference method [27], finite element method (FEM) [15], discontinuous Galerkin method [10], and spectral method [34], which are typically designed for low dimensional PDEs and are well understood in terms of stability and accuracy. However, there are high dimensional PDEs such as Schrödinger equation in the quantum many-body problem [11], Hamilton-Jacobi-Bellman equation in stochastic optimal control [1], and nonlinear Black-Scholes equation for pricing financial derivatives [23]. Solving these equations is far out of the capability of classical numerical methods due to the curse of dimensionality, i.e., the number of unknowns grows exponentially fast as the dimension increases. Until very recently, deep-learning based methods have been developed to solving these high-dimensional PDEs; see [13,17,14,18,32,35,24,33,2,7,16,25,3,36,38,12] for examples. Typically, there are three main ingredients (stages) of a deep-learning method for solving PDEs: (1) modeling: the loss (objective) function to be optimized; (2) architecture: the deep neural network (DNN) for function approximation; (3) optimization: the optimal set of parameters in the DNN which minimizes the loss function. By design, the number of parameters in DNNs grows at most polynomially in terms of dimension. Meanwhile, possibly high-dimensional integrals in the loss function are approximated by Monte-Carlo method. Therefore, by design, deep learning overcomes the curse of dimensionality. In practice, deep learning performs well for Schrödinger equation [17,19], Hamilton-Jacobi-Bellman equation [18,13], and nonlinear Black-Scholes equation [2,7]. Typically, deep learning solves a PDE in the following way. For the given PDE, the loss function is modeled as the equation residual in the leastsquares sense [35] or the variational form if exists [14]. ResNet is often used as the network architecture [21], which was tested to overcome the notorious problem of vanishing/exploding gradient. Afterwards, stochastic gradient descent method is used to find the optimal set of parameters in ResNet which minimizes the loss function. ResNet with the optimal set of parameters gives an approximation of the PDE solution. In this work, we propose a deep mixed residual method (MIM) for solving high-order PDEs. In the modeling stage, by rewriting a given PDE into a first-order system, we obtain a larger problem in the sense that both the PDE solution and its high-order derivatives are unknown functions to be approximated. This has analogs in classical numerical methods, such as local discontinuous Galerkin method [10] and mixed finite element method [6]. Compared to DGM, there are two more degrees of freedom in MIM: • In the loss function stage, one can choose different high-order derivatives into the set of unknown functions. Take biharmonic equation as an example. The set of unknown functions can include the PDE solution and its derivatives up to the third order, or only contain the PDE solution and its second-order derivatives, and both choices have analogs in discontinuous Galerkin method [37,9]. We then write the loss function as the sum of equation residuals in the least-squares sense, very much in the same spirit as the least-squares finite element method [5]. • In the architecture stage, one can choose the number of networks to approximate the set of unknown functions. In one case, one DNN is used to approximate the PDE solution and other DNNs are used to approximate its high-order derivatives; in the other case, the PDE solution and its derivatives share nearly the same DNN. These two degrees of freedom allow MIM to produce better approximations over DGM in all examples, including Poisson equation, Monge-Ampére equation, biharmonic equation, and Korteweg-de Vries (KdV) equation. In particular, MIM provides better approximations not only for the high-order derivatives but also for the PDE solution itself. It is worth mentioning that the usage of mixed residual in deep learning was first introduced for surrogate modeling and uncertainty quantification of a second-order elliptic equation [39] and was later adopted in a deep domain decomposition method [28]. The paper is organized as follows. In Section 2, we introduce MIM and DGM (for comparison purpose). In Section 3, numerical results for four types of high-order PDEs are provided. Conclusions and discussions are drawn in Section 4. Deep mixed residual method In this section, we introduce MIM and discuss its difference with DGM in terms of loss function and neural network structure. Loss function Consider a potentially time-dependent nonlinear PDE over a bounded domain Ω ⊂ R d      ∂ t u + Lu = 0 (t, x) ∈ (0, T ] × Ω, u(0, x) = u 0 (x) x ∈ Ω, u(t, x) = g(x) (t, x) ∈ [0, T ] × ∂Ω,(1) where ∂Ω denotes the boundary of Ω. In DGM, the loss function is defined as the PDE residual in the least-squares sense L(u) = ∂ t u + Lu 2 2,[0,T ]×Ω + λ 1 u(0, x) − u 0 2 2,Ω + λ 2 u − g 2 2,[0,T ]×∂Ω ,(2) where λ 1 and λ 2 are penalty parameters given a priori. These three terms in (2) measure how well the approximate solution satisfies the PDE, the initial condition and the boundary condition, respectively. In the absence of temporal derivatives, (1) reduces to Lu = 0 x ∈ Ω, u(x) = g(x) x ∈ ∂Ω, and the corresponding loss function in DGM becomes L(u) = Lu 2 2,Ω + λ u − g 2 2,∂Ω .(3) Equation Explicit form Loss function L(u) Table 1 lists four PDEs with their corresponding loss functions in DGM and Table 2 lists different boundary conditions, the initial condition and their contributions to loss functions in DGM and MIM. More boundary conditions can be treated in this way. Interested readers may refer to [8] for details. Poisson −∆u = f (x) ∆u + f (x) 2 2,Ω Monge-Ampére det(∇ 2 u) = f (x) det(∇ 2 u) − f (x) 2 2,Ω Biharmonic −∆ 2 u = f (u, x) ∆ 2 u + f (u, x) 2 2,Ω KdV u t + d i=1 u x i x i x i = f (x) u t + d i=1 u x i x i x i − f (x) 2 2,ΩInitial u(0, x) = u 0 (x) u − u 0 2 2,Ω In MIM, we first rewrite high-order derivatives into low-order ones using auxiliary variables. For notational convenience, auxiliary variables p, q, w represent p = ∇u, q = ∇ · p = ∆u, w = ∇q = ∇(∆u). For KdV equation, we have q = diag(∇p) instead of the second formula in (4). With these auxiliary variables, we define loss functions for four types of PDEs in Table 3. Since one can choose a subset of high-order derivatives into the set of unknown functions, there are more than one loss function in MIM. For biharmonic equation, there are two commonly used sets of auxiliary variables in local discontinuous Galerkin method and weak Galerkin finite element method: one with all high-order derivatives [37] and the other with part of high-order derivatives [9,30]. Correspondingly, if all high-order derivatives are used, we denote MIM by MIM a , and if only part of high-order derivatives are used, we denote MIM by MIM p . In Section 2.2, we will discuss how to equip different loss functions with different DNNs. In short, if only one DNN is used to approximate the PDE solution and its derivatives, we denote MIM by MIM 1 , and if multiple DNNs are used, we denote MIM by MIM 2 . In Section 3, different loss functions listed in Table 1, Table 2 and Table 3 will be tested and discussed. By default, all the penalty parameters are set to be 1. Equation Explicit form Loss function L(u, p, q, w) Poisson −∆u = f (u, x) p − ∇u 2 2,Ω + ∇ · p + f (u, x) 2 2,Ω Monge-Ampére det(∇ 2 u) = f p − ∇u 2 2,Ω + det(∇p) − f 2 2,Ω Biharmonic −∆ 2 u = f (u, x) p − ∇u 2 2,Ω + q − ∇ · p 2 2,Ω + w − ∇q 2 2,Ω + ∇ · w + f 2 2,Ω q − ∆u 2 2,Ω + ∆q + f 2 2,Ω KdV u t + d i=1 u x i x i x i = f (x) p − ∇u 2 2,[0,T ]×Ω + q − diag(∇p) 2 2,[0,T ]×Ω + u t + ∇ · q − f (x) 2 2,[0,T ]×Ω Neural network architecture ResNet [21] is used to approximate the PDE solution and its high-order derivatives. It consists of m blocks in the following form s k = σ(W 2,k σ(W 1,k s k−1 + b 1,k ) + b 2,k ) + s k−1 , k = 1, 2, · · · , m.(5) Here s k , b 1,k , b 2,k ∈ R n , W 1,k , W 2,k ∈ R n×n . m is the depth of network, n is the width of network, and σ is the (scalar) activation function. Explicit formulas of activation functions used in this work are given in Table 4. The last term on the right-hand side of (5) is called the shortcut connection or residual connection. Each block has two linear transforms, two activation functions, and one shortcut; see Figure 1 for demonstration. Such a structure can automatically solve the notorious problem of vanishing/exploding gradient [22]. Since x is in R d rather than R n , we can pad x by a zero vector to get the network input s 0 . A linear transform can be used as well without much difference. Meanwhile, s m has n outputs which cannot be directly used for the PDE solution and its derivatives employed in the loss function. Therefore, a linear transform T is applied to s m to transform it into a suitable dimension. Let {θ} be the whole set of parameters which include parameters in ResNet Activation function Formula Square Table 4: Activation functions used in numerical tests. ({W 1,k , b 1,k , W 2,k , b 2,k } m k=1 ) and parameters in the linear transform T . Note that the output dimension in MIM depends on both the PDE problem and the mixed residual loss. We illustrate network structures for biharmonic equation as an example in Figure 2. From Figure 2, we see that DGM has only 1 output, MIM 1 a has 2d + 2 outputs, and MIM 1 p has 2 outputs. In Figure 3, we illustrate networks structures of MIM 1 and MIM 2 for Poisson equation. In MIM 2 , two DNNs are used: one to approximate the solution and the other one to approximate its derivatives. It is clear from Figure 2 that network structures in DGM and MIM 1 only differ in the output layer and thus they have comparable numbers of parameters to be optimized. To be precise, we calculate their numbers of parameters in Table 5 a ), respectively. In Section 3, from numerical results, we observe a better performance of MIM 1 for all four equations, not only for derivatives of the PDE solution, but also for the solution itself. x 2 ReLU max{x, 0} ReQU (max{x, 0}) 2 ReCU (max{x, 0}) 3•••••• • !"•#$%••%$• &'%(•) +•",$•"'••&••+•"'• -•••••• • .# "/•$0 +'••#+•"'• !"•#$%••%$• &'%(•) +•",$•"'••&••+•"'••••••••• ••• !•"#•$•%&#" •!•"#' (•')•& +,••"• &"•,'-#".•• ••• •••••• !••• •"# # # # # Method Equation Size of the parameter set DGM Four equations (2m − 1)n 2 + (2m + d + 1)n + 1 MIM 1 Poisson (2m − 1)n 2 + (2m + 2d + 1)n + d + 1 Monge-Ampére Biharmonic (MIM 1 a ) (2m − 1)n 2 + (2m + 3d + 2)n + 2d + 2 Biharmonic (MIM 1 p ) (2m − 1)n 2 + (2m + d + 2)n + 2 KdV (2m − 1)n 2 + (2m + 3d + 1)n + 2d + 1 MIM 2 Poisson (4m − 2)n 2 + (4m + 3d + 1)n + d + 1 Monge-Ampére Biharmonic (MIM 2 a ) (8m − 4)n 2 + (8m + 6d + 2)n + 2d + 2 Biharmonic (MIM 2 p ) (4m − 2)n 2 + (4m + 2d + 2)n + 2 KdV (6m − 3)n 2 + (6m + 5d + 1)n + 2d + 1 Quantity DGM MIM u Ω (u θ −u) 2 dx Ω u 2 dx Ω (u θ −u) 2 dx Ω u 2 dx ∇u Ω (∇u θ −∇u) 2 dx Ω (∇u) 2 dx Ω (p θ −∇u) 2 dx Ω (∇u) 2 dx ∆u Ω (∆u θ −∆u) 2 dx Ω (∆u) 2 dx Ω (q θ −∆u) 2 dx Ω (∆u) 2 dx ∇∆u Ω (∇(∆u θ )−∇(∆u)) 2 dx Ω (∇(∆u)) 2 dx Ω (w θ −∇(∆u)) 2 dx Ω (∇(∆u)) 2 dx diag(∇ 2 u) Ω (diag(∇ 2 u θ )−diag(∇ 2 u)) 2 dx Ω (diag(∇ 2 u)) 2 dx Ω (q θ −diag(∇ 2 u)) 2 dx Ω (diag(∇ 2 u)) 2 dx Stochastic Gradient Descent For completeness, we also briefly introduce stochastic gradient descent method. For the loss function defined in (3), we generate two sets of points uniformly distributed over Ω and ∂Ω: {x i } N i=1 in Ω and {x j } M j=1 on ∂Ω. θ k+1 = θ k − α∇ θ \|Ω\| N N i=1 [Lu θ (x i ; θ k )] 2 + λα∇ θ \|∂Ω\| M M j=1 [u θ (x j ; θ k ) − g(x j )] 2 ,(6) where α is the learning rate chosen to be 1e − 3 here. \|Ω\| and \|∂Ω\| are measures of Ω and ∂Ω, respectively. u θ is the DNN approximation of PDE solution parameterized by {θ}. Sampling points {x i } N i=1 and {x j } M j=1 are updated at each iteration. In implementation, we use ADAM optimizer [26] and automatic differentiation [31] for derivatives in PyTorch. Numerical Result In this section, we show numerical results of MIM for four types of equations. We use relative L 2 errors of u, ∇u, ∆u, and ∇(∆u) defined in Table 6 for comparison. In all figures, relative L 2 errors are in log 10 scale. Poisson Equation Consider the following Neumann problem          − ∆u + π 2 u = 2π 2 d k=1 cos(πx k ) x ∈ Ω = [0, 1] d ∂u ∂n = 0 x ∈ ∂Ω(7) with the exact solution u(x) = d k=1 cos(πx k ). The neural network structure in DGM is the same as that for biharmonic equation shown in Figure 2. Following Table 1 and Table 2, we use the loss function for (7) L(u) = − ∆u + π 2 u − 2π 2 d k=1 cos(πx k ) 2 2,Ω + λ ∂u ∂n 2 2,∂Ω .(8) Since both u and p are explicitly used, one more advantage of MIM is the enforcement of boundary conditions. For (7), we multiply p i , i = 1, · · · , d by x i (1 − x i ) to satisfy the Neumann boundary condition automatically; see Figure 3. DGM only has u as its unknown function, and thus it is unclear that how the exact Neumann boundary condition can be imposed. Therefore, for DNNs in Figure 3, the loss function in MIM can be simplified as L(u, p) = p − ∇u 2 2,Ω + − ∇ · p + π 2 u − 2π 2 d k=1 cos(πx k ) 2 2,Ω . (9) We emphasize that Dirichlet boundary condition can be exactly imposed in DGM [4] and no penalty term is needed. For Neumann boundary condition, mixed boundary condition, and Robin boundary condition, however, it is difficult to build up a DNN representation which satisfies the exact boundary condition. Building up a DNN approximation which satisfies the exact boundary condition can have a couple of advantages [8]: 1) make ease of the training process by avoiding unnecessary divergence; 2) improve the approximation accuracy; 3) save the execution time. In MIM, however, we have the direct access to both u and p. Therefore, all these boundary conditions can be imposed exactly in principle. This will be presented in a subsequent work [29]. For (7), average errors of u and ∇u over the last 100 iterations are recorded in Table 7. The network depth m = 2 and the activation function x 2 is used. Network widths are 5, 10, 15, 20 for 2, 4, 8, 16 dimensional problems, respectively. Time is recorded as the average CPU time per iteration. It is not surprising that MIM 1 costs less time than DGM since the DNN approximation in MIM satisfies the Neumann boundary condition automatically and both methods have similar network structures. It is surprising that MIM 2 costs less time than DGM since the number of parameters in MIM 2 is about twice of that in DGM. In terms of execution time, MIM 1 < MIM 2 < DGM. Figure 4 and Figure 5 plot training processes of DGM and MIM in •••••••• ••• !•"#•$•%&#" '()••"• &"•)+#", ••• (a) MIM 1 : one network to approximate the PDE solution and its derivatives. terms of relative L 2 errors for u and ∇u. Generally speaking, in terms of approximation error, MIM 2 < MIM 1 < DGM as expected. Therefore, MIM provides a better strategy over DGM. MIM provides better approximations in terms of relative L 2 errors for both u and ∇u. For ∇u, the improvement of MIM 1 over DGM is about several times and that of MIM 2 over MIM 1 is about one order of magnitude. For u, the improvement is about several times. Moreover, a dimensional dependence is observed for both u and ∇u. The higher the dimension is, the better the approximation is. Table 8 records approximation errors of MIM and DGM in terms of activation function and network depth when d = 4. MIM provides better approximations for both ∇u and u. It is not surprising that ReLU is not a suitable function for DGM due to high-order derivatives, but is suitable in MIM since only first-order derivatives are present in MIM. •••••••• ••• !•"#•$•%&#" '()••"• &"•)+#", ••• •••••••• ••• !•"#•$•%&#" '()••"• &"•)+#", ••• Monge-Ampére equation Consider the nonlinear Monge-Ampére equation with the exact solution defined as u( Table 1 In this example, we fix the network depth m = 2 and the activation function as σ(x) = ReQU(x). Relative L 2 errors in the last 1000 iterations with respect to the network width in different dimensions are recorded in Table 9. Figure 6 plots errors in terms of network width for different dimensions. The advantage of MIM is obvious from these results. det(∇ 2 u) = f (x) x ∈ Ω = [−1, 1] d u(x) = g(x) x ∈ ∂Ω(x) = e 1/d( d i=1 x 2 i ) . Following Biharmonic equation Consider the biharmonic equation                      ∆ 2 u = π 4 16 d k=1 sin( π 2 x) x ∈ Ω u(x) = d k=1 sin( πx 2 ) x ∈ ∂Ω ∂u ∂n = 0 x ∈ ∂Ω(11) with the exact solution u(x) = Table 9: Relative L 2 errors in the last 1000 iterations with respect to the network width for Monge-Ampére equation defined in (10) for different dimensions. The network depth is fixed to be m = 2 and the activation function is fixed to be σ(x) = ReQU(x). The loss function in MIM a is L(u, p, q, w) = p − ∇u 2 2,Ω + q − ∇ · p 2 2,Ω + w − ∇q 2 2,Ω + ∇ · w − π 4 16 d k=1 sin( π 2 x) 2 2,Ω + λ 1 u − d k=1 sin( πx 2 ) 2 2,∂Ω + λ 2 p 2 2,∂Ω ,(12) and the loss function in MIM p is d = 2. Generally speaking, MIM provides better approximations for u, ∇u, ∆u, and ∇(∆u) than DGM. For MIM a and MIM p , MIM p has a slightly better approximation accuracy comparable to that of MIM a , although MIM a has 2d + 2 more outputs. These results are of interests since they are connected with results of local discontinuous Galerkin method that the formulation with a subset of derivatives has a better numerical performance [37,9]. We point out that MIM a has the advantage that the exact boundary condition can be enforced, although we use penalty terms for this example. (12) and (13), respectively. L(u, q) = q − ∆u 2 2,Ω + ∆q − π 4 16 d k=1 sin( π 2 x) 2 2,Ω + λ 1 u − d k=1 sin( πx 2 ) 2 2,∂Ω + λ 2 ∂u ∂n KdV equation Consider a time-dependent linear KdV-type equation                  u t + d k=1 u x k x k x k = 0 (t, x) ∈ [0, T ] × Ω u(0, x) = u 0 (x) = sin( d k=1 x k ) (t, x) ∈ [0] × Ω u(t, x) is periodic in x(14) defined over Ω = [0, 2π] d , where the exact solution u(t, x) = sin( d k=1 x k +dt). We first rewrite it into the first-order system p = ∇u, q = diag(∇p), u t + ∇ · q = 0. The loss function in DGM is L(u) = u t + d k=1 u x k x k x k 2 2,[0,1]×Ω + λ 1 u − sin( d k=1 x k + dt) 2 2,[0,1]×∂Ω + λ 2 d k=1 u(x, t) − u(x ± 2πe k , t) 2 2,Ω + λ 3 d k=1 ∇u(x, t) − ∇u(x ± 2πe k , t) 2 2,Ω . Here {e k } d k=1 is the standard basis set of R d . The loss function in MIM is L(u, p, q) = p − ∇u 2 2,[0,1]×Ω + q − diag(∇p) 2 2,[0,1]×Ω + u t + ∇ · q 2 2,[0,1]×Ω + λ 1 u − sin( d k=1 x k + dt) 2 2,[0,1]×∂Ω + λ 2 d k=1 u(x, t) − u(x ± 2πe k , t) 2 2,Ω + λ 3 d k=1 p(x, t) − p(x ± 2πe k , t) 2 2,Ω . Relative L 2 errors of u, ∇u, and diag(∇ 2 u) are recorded in Table 11. Again, as shown in previous examples, MIM provides better results compared to DGM, especially for ReQU activation function. No obvious improvement of MIM 2 over MIM 1 is observed. Conclusion and Discussion Motivated by classical numerical methods such as local discontinuous Galerkin method, mixed finite element method, and least-squares finite element method, we develop a deep mixed residual method to solve high-order PDEs in this paper. The deep mixed residual method inherits several advantages of classical numerical methods: • Flexibility for the choice of loss function; • Larger solution space with flexible choice of deep neural networks; • Enforcement of exact boundary conditions; d σ Method Relative L 2 error (×10 −2 ) u ∇u diag(∇ 2 u) • Better approximations of high-order derivations with almost the same cost. Meanwhile, the deep mixed residual method also provides a better approximation for the PDE solution itself. These features make deep mixed residual method suitable for solving high-order PDEs in high dimensions. Boundary condition is another issue which is important for solving PDEs by DNNs. Enforcement of exact boundary conditions not only makes the training process easier, but also improves the approximation accuracy; see [4,8] for examples. The deep mixed residual method has the potential for imposing exact boundary conditions such as Neumann boundary condition, mixed boundary condition, and Robin boundary condition. All these conditions cannot be enforced exactly in deep Galerkin method. This shall be investigated in a subsequent work [29]. So far, in the deep mixed residual method, only experiences from classical numerical methods at the basic level are transferred into deep learning. We have seen its obvious advantages. To further improve the deep mixed residual method, we need to transfer our experiences from classical numerical analysis at a deeper level. For example, the choice of solution space relies heavily on the choice of residual in order to maximize the performance of least-squares finite element method [5]. Many other connections exist in discontinuous Galerkin method [10] and mixed finite element method [6]. For examples, since only first-order derivatives appear in the deep mixed residual method, ReLU works well for all time-independent equations we have tested but does not work well for KdV equation. Therefore, it deserves a theoretical understanding of the proposed method in the language of linear finite element method [20]. Another possible connection is to use the weak formulation of the mixed residual instead of least-squares loss, as done in deep learning by [38] and in discontinuous Galerkin method by [10]. Realizing these connections in the deep mixed residual method will allow for a systematic way to understand and improve deep learning for solving PDEs. Acknowledgments This work was supported by National Key R&D Program of China (No. 2018YFB0204404) and National Natural Science Foundation of China via grant 11971021. We thank Qifeng Liao and Xiang Zhou for helpful discussions. Figure 1 : 1One block of ResNet. A deep neural network contains a sequence of blocks, each of which consists of two fully-connected layers and one shortcut connection. , from which one can see the number of parameters in DGM and MIM 1 is close. The number of parameters in MIM 2 is nearly double for Poisson equation, Monge-Ampére equation and biharmonic equation (MIM 2 p ), tripled for KdV equation, and quadrupled for biharmonic equation (MIM 2 Figure 2 : 2Network structures for biharmonic equation with deep Galerkin method and deep mixed residual method. DGM only approximates solution u. MIM 1 p approximate solution u and ∆u. MIM 1 a approximates solution u and all of its derivatives used in the equation ∇u, ∆u, ∇(∆u). MIM 2 a uses four networks to approximate u, ∇u, ∆u, ∇(∆u) and MIM 2 p uses two networks to approximate u, ∆u. Each network has a similar structure with different output dimensions. (b) MIM 2 2: multiple networks to appriximate the PDE solution and its derivatives. Figure 3 : 3Detailed network structures of MIM 1 and MIM 2 to solve Poisson equation. DNN part is the same as that in Figure 2. x i (1 − x i ) are multipliers which make MIM 1 and MIM 2 satisfy the exact Neumann boundary condition. Figure 4 :Figure 5 : 45Relative L 2 error of u in terms of iteration number for Poisson equation defined in(7). Relative L 2 error of ∇u in terms of iteration number for Poisson equation defined in(7). , 3 and 2, we have the loss function in DGML(u) = det(∇ 2 u) − f 2 2,Ω + λ u − g 2 2,∂Ω , and the loss function in MIML(u, p) = p − ∇u 2 2,Ω + det(∇p) − f 2 2,Ω + λ u − g 2 2,∂Ω , respectively. For (10), the Dirichlet boundary condition can be enforced for both DGM and MIM. For comparison purpose, instead, we have the penalty term in both DGM and MIM. However, imposing exact boundary conditions is always encouraged in practice. we can enforce the exact boundary condition in MIM but cannot enforce it in DGM. For comparison purpose, we use penalty terms in both methods.Set m = 2 and n = 8, 10, 20 when d = 2, 4, 8, respectively.Table 10records averaged errors in the last 1000 iterations. Relative L 2 errors for u , ∇u, ∆u and ∇(∆u) in terms of iteration number are plotted inFigure 7when Figure 6 : 6Relative L 2 errors of u and ∇u for Monge-Ampére equation defined in (7). Figure 7 : 7Relative L 2 errors of u, ∇u, ∆u, ∇(∆u) in terms of iteration number for biharmonic equation. Both the solution and its derivatives are approximated by the same network in MIM 1 , while different networks are used for the solution and its derivatives in MIM 2 . MIM a means all derivatives are approximated and MIM p means only a subsect of derivatives (∆u here) are approximated. Table 1 : 1Loss functions for four types of PDEs in the deep Galerkin method.Condition Explicit form Contribution to the loss function Dirichlet u(x) = g u − g 2 2,[0,T ]×∂Ω Neumann ∂u ∂n = g ∂u ∂n − g 2 2,[0,T ]×∂Ω or p − g 2 2,[0,T ]×∂Ω Table 2 : 2Contributions to the loss function for the initial condition and different types of boundary conditions used in the deep Galerkin method and the deep mixed residual method. Table 3 : 3Loss functions in the deep mixed residual method for four types of equations. Two different loss functions for biharmonic equation are denoted by MIM a and MIM p , in which all high-order derivatives or part of high-order derivatives are included, respectively. Table 5 : 5Number of parameters for different network structures used for different equations and different loss functions. n, m, and d are the network width, the network depth, and the problem dimension, respectively. 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[ "BOX COMPLEXES AND HOMOTOPY THEORY OF GRAPHS", "BOX COMPLEXES AND HOMOTOPY THEORY OF GRAPHS" ]	[ "Takahiro Matsushita " ]	[]	[]	We introduce a model structure on the category of graphs, which is Quillen equivalent to the category of Z 2 -spaces. A weak equivalence is a graph homomorphism which induces a Z 2 -homotopy equivalence between their box complexes. The box complex is a Z 2 -space associated to a graph, considered in the context of the graph coloring problem. In the proof, we discuss the universality problem of the Hom complex.	10.4310/hha.2017.v19.n2.a10	[ "https://arxiv.org/pdf/1605.06222v2.pdf" ]	119,575,978	1605.06222	83357fecfe6ee0c228e663ebbbe8303a0ee0bd50	BOX COMPLEXES AND HOMOTOPY THEORY OF GRAPHS 29 Jul 2017 Takahiro Matsushita BOX COMPLEXES AND HOMOTOPY THEORY OF GRAPHS 29 Jul 2017arXiv:1605.06222v2 [math.AT] We introduce a model structure on the category of graphs, which is Quillen equivalent to the category of Z 2 -spaces. A weak equivalence is a graph homomorphism which induces a Z 2 -homotopy equivalence between their box complexes. The box complex is a Z 2 -space associated to a graph, considered in the context of the graph coloring problem. In the proof, we discuss the universality problem of the Hom complex. Introduction We consider the category of graphs from the viewpoint of homotopical algebra. As a result, we construct a model structure on the category of graphs which is Quillen equivalent to the category of Z 2 -spaces. A weak equivalence is a graph homomorphism which induces a Z 2 -homotopy equivalence between their box complexes. The box complex was introduced in the context of the graph coloring problem. An n-coloring of a graph G is a map from the vertex set of G to the n-point set {1, · · · , n} so that adjacent vertices have different values. The chromatic number χ(G) of G is the smallest integer n such that G has an n-coloring. The graph coloring problem is to determine the chromatic number, and this is one of the most classical problems in graph theory. The first application of algebraic topology to this subject is Lovász's proof of the Kneser conjecture [16]. Lovász introduced the neighborhood complex N (G) of a graph G, and showed that if the neighborhood complex of G is n-connected, then the chromatic number of G is greater than n + 2. The box complex B(G) is a Z 2 -space which is homotopy equivalent to the neighborhood complex N (G). The precise definition will be found in Section 2. The Hom complex Hom(T, G) is a generalization of the box complex. If T is a right Γ-graph, then the Hom complex Hom(T, G) becomes a left Γ-space and a graph homomorphism f : G 1 → G 2 induces a Γ-map f * : Hom(T, G 1 ) → Hom(T, G 2 ). Since an n-coloring of a graph G is identified with a graph homomorphism from G to K n , we have that if there is no Γ-map from Hom(T, G) to Hom(T, K n ) then we have χ(G) ≥ n. The equivariant homotopy of the Hom complexes has been extensively researched (see [1], [2], [10], [17], and [20] for example). Therefore it is important to compare the category of graphs with the category of Γ-spaces of some group Γ. This is the motivation of this research. Our main result (Theorem 3.13) asserts that the category of graphs has the model structure whose homotopy category is equivalent to the homotopy category of Z 2 -spaces. 1.1. Singular complex functor and its adjoint. Let Γ be a finite group and T a finite right Γ-graph. The functor G → Hom T (G) = Hom(T, G) has neither a left nor a right adjoint, and hence it is not a Quillen functor. So we use the singular complex functor Sing T (G) = Sing(T, G) introduced in [18]. It is known that the singular complexes and the Hom complexes are homotopy equivalent (see Theorem 3.1 and Corollary 3.2), and we will show that the functor Sing T : G −→ SSet Γ , G → Sing(T, G) is a right adjoint functor (Proposition 3.3). Here G denotes the category of graphs and SSet Γ is the category of Γ-simplicial sets. Let A T denote the left adjoint of Sing T . In the case of simplicial complexes, a similar construction A T (K) was obtained in [10] (see the end of Section 3.1). Consider the unit of the adjoint pair (A T • Sd k , Ex k • Sing T ) : SSet Γ −→ G, where Ex is Kan's extension functor (see Section 2.2). We take k to be sufficiently large. If this adjoint pair is a Quillen equivalence between SSet Γ and G, then the unit Id −→ Ex k • Sing T • A T • Sd k must be a natural weak equivalence. The main task of Section 3 is to characterize the condition of T that the unit is a natural Γ-weak equivalence. For a right Γ-graph T and for an element γ of Γ, let α γ denote the graph homomorphism G → G, v → vγ. Consider the following two conditions concerning a finite right Γ-graph T : (A) For each subgroup Γ ′ of Γ, the map Γ/Γ ′ → Hom(T, T /Γ ′ ), γΓ ′ → p • α γ is a Γ-homotopy equivalence. The Γ-action on Hom(T, T /Γ ′ ) is described in Section 2.1. (B) The map Γ → Hom(T, T ), γ → α γ is a Γ-homotopy equivalence. Here we consider Γ/Γ ′ as a discrete space and p denotes the projection T → T /Γ ′ . Clearly, (A) implies (B). The main structural results of Section 3 are the following two theorems: Theorem 1.1 (Theorem 3.4). Let Γ be a finite group and T a finite connected right Γ-graph with at least one edge. Let k be an integer such that 2 k−2 is greater than the diameter of T . Then the following are equivalent: (1) The unit of the adjoint pair (A T • Sd k , Ex k • Sing T ) is a natural Γ-weak equivalence. (2) The right Γ-graph T satisfies the condition (A). Theorem 1.2 (Theorem 3.5). Let Γ be a finite group and T a finite connected right Γ-graph with at least one edge. Let k be an integer such that 2 k−2 is greater than the diameter of T . Then the following are equivalent: (1) For a free Γ-simplicial set K, the unit map K → Ex k • Sing T • A T • Sd k (K) is a Γ-weak equivalence. (2) The right Γ-graph T satisfies the condition (B). If the graph T is stiff (see Section 2.1 for the definition), then the condition (B) has the following combinatorial characterization: Every endomorphism of T is an automorphism and the group Γ is isomorphic to the automorphism group of T (Lemma 3.6). Such examples are given by complete graphs, odd cycles, and stable Kneser graphs (see Example 3.7). Theorem 1.2 concerns the universality problem of the Hom complexes. Csorba [7] showed that for every finite free Z 2 -complex X, there is a graph G such that Hom(K 2 , G) and X are Z 2 -homotopy equivalent. Dochterman and Schultz [10] showed that for every free S n -complex X, there is a graph G such that Hom(K n , G) and X are S n -homotopy equivalent. Here S n is the symmetric group of the nelement set {1, · · · , n}. Thus Theorem 1.2 is a generalization of their results. On the other hand, the condition (A) is rarely satisifed. I think that the following are (essntially) only examples naturally arising: (1) Γ is trivial and T is the graph 1 consisting of one looped vertex. This corresponds to the clique complex of the maximal reflexive subgraph. (2) Γ is the cyclic group Z 2 of order 2 and T is K 2 with the flipping involution. This corresponds to the box complex. This is a specific difference between the box complex and the other Hom complexes. 1.2. Organization of the paper. In Section 2, we introduce the notation and the terminology, and review some relevant facts. In Section 3, we state Theorem 1.1 and Theorem 1.2 and construct the model structures on the category of graphs (Theorem 3.13 and Theorem 3.14). Here we give their proofs based on some structural results (Lemma 3.10 and Proposition 3.11). These structural results are obtained in Section 4 by comparing the (equivariant) strong homotopy theory of simplicial complexes [3] with the ×-homotopy theory established by Dochtermann [8]. Preliminaries In this section we shall introduce the notation and terminology, and review relevant facts which we will use in later sections. Throughout the paper, Γ denotes a finite group unless otherwise stated. We consider not only left group actions but also right group actions. However, we shall use the term "a Γ-action" to mean "a left Γ-action". For a category C, we write C Γ to indicate the category of objects in C equipped with a Γ-action. For a poset P , the classifying space of P is denoted by \|P \|. We often regard a poset as a topological space by its classifying space, and assign topological terminology by the classifying space functor. For example, two order-preserving maps f, g : P → Q are homotopic if and only if \|f \| and \|g\| are homotopic. 2.1. Box complexes and Hom complexes of graphs. For a concrete introduction to this subject, we refer to [15]. A graph is a pair G = (V (G), E(G)) consisting of a set V (G) together with a symmetric subset E(G) of V (G)×V (G), i.e. (x, y) ∈ E(G) implies (y, x) ∈ E(G). A graph homomorphism is a map f : V (G) → V (H) such that (f ×f )(E(G)) ⊂ E(H). Let G denote the category of graphs whose morphisms are graph homomorphisms. For a vertex v ∈ V (G), the neighborhood N G (v) of v is the set of vertices of G adjacent to v. We sometimes abbreviate N G (v) to N (v). Define the complete graph K n with n-vertices by V (K n ) = {1, · · · , n} and E(K n ) = {(x, y) \| x = y}. Then an n-coloring of G is identified with a graph homomorphism from G to K n , and the chromatic number is χ(G) = inf{n ≥ 0 \| There is a graph homomorphism from G to K n .}. The box complex of a graph G is the Z 2 -poset A multi-homomorphism from G to H is a map η : B(G) = {(σ, τ ) \| σ, τ ∈ 2 V (G) \ {∅}, σ × τ ⊂ E(G)}V (G) → 2 V (H) \ {∅} such that (v, w) ∈ E(G) implies η(v) × η(w) ⊂ E(H) . For a pair of multi-homomorphisms η and η ′ , we write η ≤ η ′ to mean that η(v) ⊂ η ′ (v) for every vertex v of G. The Hom complex from G to H is the poset of the multi-homomorphisms from G to H, and denoted by Hom(G, H). This definition of the Hom complex is slightly different from the one of Babson and Kozlov [1]. They define the Hom complex Hom(G, H) as a certain subcomplex of a direct product of simplices when G and H are finite. Our Hom complex is the face poset of theirs, and thus the topological types of these two definitions coincide. Graph homomorphisms f, g : G → H are ×-homotopic (see [8]) if they belong to the same connected component of Hom(G, H). We write f ≃ × g to mean that f and g are ×-homotopic. A graph homomorphism f : G → H is a ×-homotopy equivalence if there is a graph homomorphism h : H → G such that hf ≃ × id G and f h ≃ × id H . Let G, H, and K be graphs. Define the composition map * : Hom(H, K) × Hom(G, H) → Hom(G, K), (τ, η) → τ * η by (τ * η)(v) = w∈η(v) τ (w). If f : G → H and g : H → K are graph homomorphisms, we write g * (η) (or f * (τ )) instead of g * η (or τ * f , respectively). Let T be a right Γ-graph. For γ ∈ Γ, we write α γ to indicate the graph homomorphism T → T , v → vγ. Then we have a Γ-action on Hom(T, G) defined by γη = α * γ (η), and a graph homomorphism f : G 1 → G 2 induces a Γ-poset map f * : Hom(T, G 1 ) → Hom(T, G 2 ). Consider K 2 as a Z 2 -graph by the flipping involution. Then it is clear that Hom(K 2 , G) and B(G) are isomorphic as Z 2 -posets. [8]). Let f and g be graph homomorphisms from G to H and suppose f ≃ × g. For a right Γ-graph T , the following hold: (1) f * , g * : Hom(T, G) → Hom(T, H) are Γ-homotopic. (2) f * , g * : Hom(H, T ) → Hom(G, T ) are Γ-homotopic. Proof. We only prove (1) since (2) is similarly proved. Let ϕ : [0, 1] → \|Hom(G, H)\| be a path joining f to g. Then the composition of [0, 1] × \|Hom(T, G)\| ϕ×id − −−− → \|Hom(G, H)\| × \|Hom(T, G)\| \| * \| − −−− → \|Hom(T, H)\| is a Γ-homotopy from f * to g * . A vertex v of G is dismantlable if there is another vertex w of G with N (v) ⊂ N (w). If v is dismantlable, then the inclusion G \ v ֒→ G is a ×-homotopy equiva- lence. Here G \ v denotes the induced subgraph of G whose vertex set is V (G) \ {v}. In particular, we have the following: [14]). Let G be a graph and v a dismantlable vertex of G. For every right Γ-graph T , the inclusion Hom(T, G \ v) ֒→ Hom(T, G) is a Γ-homotopy equivalence. Corollary 2.2 (Kozlov A graph G is stiff if G has no dismantlable vertices. Lemma 2.3 (Lemma 6.5 of [8]). If G is a stiff graph, then every automorphism of G is an isolated point of Hom(G, G). Γ-simplicial sets. For a concrete explanation of simplicial sets, we refer to [11]. We write ∆ to indicate the cosimplicial indexing category. Let SSet denote the category of simplicial sets. The geometric realization of a simplicial set K is denoted by \|K\|. A simplicial map f : K → L is called a weak equivalence if the map \|f \| : \|K\| → \|L\| induced by f is a homotopy equivalence. For a Γ-simplicial set K and a subgroup of Γ ′ , let K Γ ′ denote the subcomplex of K consisting of the simplices fixed by Γ ′ . A Γ-simplicial map f : K → L is a Γ-weak equivalence if f Γ ′ : K Γ ′ → L Γ ′ is a weak equivalence for every subgroup Γ ′ of Γ. Then SSet Γ has the model structure described as follows (see [4] or [21]) whose generating cofibrations is I Γ = {(Γ/Γ ′ ) × ∂∆[n] ֒→ (Γ/Γ ′ ) × ∆[n] \| n ≥ 0 and Γ ′ is a subgroup of Γ.} and whose generating trivial cofibrations is J Γ = {(Γ/Γ ′ )×Λ r [n] ֒→ (Γ/Γ ′ )×∆[n] \| n ≥ 1, 0 ≤ r ≤ n, and Γ ′ is a subgroup of Γ.}. Here ∆[n] is the Yoneda functor [m] → ∆([m], [n] ) and Λ r [n] is the r-horn. The class of weak equivalences is the class of Γ-weak equivalences. Let Ex denote Kan's extension functor, and Sd the barycentric subdivision functor. There is a natural weak equivalence Sd(K) → K whose adjoint K → Ex(K) is also a natural weak equivalence (Theorem 4.6 of [11]). The adjoint pair (Sd, Ex) : SSet → SSet gives rise to the adjoint pair (Sd, Ex) : SSet Γ → SSet Γ . It is easy to see that if K is a Γ-simplicial sets, both of the above maps Sd(K) → K and K → Ex(K) are also Γ-weak equivalences. 2.3. Bredon's theorem. We will use the following proposition several times. Proposition 2.4 (Bredon [6]). Let Γ be a finite group and f : X → Y a Γmap between Γ-CW-complexes. Then f is a Γ-homotopy equivalence if and only if f Γ ′ : X Γ ′ → Y Γ ′ is a homotopy equivalence for every subgroup Γ ′ of Γ. 2.4. Some homotopy colimits. The following proposition is sometimes called the gluing lemma or cube lemma (Lemma 8.8 of [11] or Proposition 15.10.10 of [12]). Proposition 2.5. Let C be a model category and let A i ← −−− − B − −−− → C fA     fB   fC A ′ i ′ ← −−− − B ′ − −−− → C ′ be a commutative diagram in C. Suppose that the all vertical arrows are weak equivalences, and the all objects appearing in the above diagram are cofibrant. If i and i ′ are cofibrations in C, then the natural map A ∪ B C → A ′ ∪ B ′ C ′ is a weak equivalence. For an ordinal λ, the minimum of λ is denoted by 0. Proposition 2.6. Let C be a model category, X • : λ → C, Y • : λ → C functors from an ordinal λ, and f • : X • → Y • a natural transformation. Suppose the following conditions are satisfied: (1) f α : X α → Y α is a weak equivalence for every α < λ. (2) For each α < λ, the map colim β<α X β −→ X α is a cofibration of C. In particular, X 0 is cofibrant. Then the colimit f λ : X λ → Y λ of f • is a weak equivalence. Proof. The proof is similar to Proposition 15.10.12 of [12]. Simplicial methods Let Γ be a finite group and T a right Γ-graph. Let P denote the category of posets. As was mentioned in Section 1, the functor Hom T : G −→ P Γ , G → Hom(T, G) is neither a left nor right adjoint functor. So we use the singular complex functor Sing T : G → SSet Γ , which is reviewed in Section 3.1. This is a right adjoint functor (Proposition 3.3) and let A T denote the left adjoint. Then we have an adjoint pair (A T • Sd k , Ex k • Sing T ) : SSet Γ −→ G for k > 0. In Section 3.2, we characterize the condition that the unit Id −→ Ex k • Sing T • A T • Sd k of the adjunction is a natural Γ-weak equivalences for sufficiently large k (Theorem 3.4). In this section we give the proof of Theorem 3.4 based on some structural results proved in Section 4. In Section 3.3, we construct a model structure on G which is Quillen equivalent to SSet Z2 (Theorem 3.13). Singular complexes. For a non-negative integer n, define the graph Σ n by V (Σ n ) = [n] and E(Σ n ) = V (Σ n ) × V (Σ n ). For a pair of graphs T and G, the singular complex (see [18]) is the simplicial set Sing(T, G) whose n-simplices are the graph homomorphisms from T × Σ n to G, i.e. Sing(T, G) n = G(T × Σ n , G). The face and degeneracy maps are defined in an obvious way. A 0-simplex of Sing(T, G) is identified with a graph homomorphism from T to G. The fundamental result of the singular complex is the following: [18]). There is a homotopy equivalence Theorem 3.1 (MatsushitaΦ : \|Sing(T, G)\| −→ \|Hom(T, G)\|, which is natural with respect to both T and G. Moreover, for a graph homomorphism Proof. The naturality with respect to T implies that Φ : f : T → G, we have Φ(f ) = f .\|Sing(T, G)\| → \|Hom(T, G)\| is Γ-equivariant. It suffices to show that, for every subgroup Γ ′ of Γ, the map Φ Γ ′ : \|Sing(T, G)\| Γ ′ −→ \|Hom(T, G)\| Γ ′ is a homotopy equivalence (see Proposition 2.4). Since the geometric realization preserves equalizers, we have \|Sing(T, G)\| Γ ′ ∼ = \|Sing(T, G) Γ ′ \|, \|Hom(T, G)\| Γ ′ ∼ = \|Hom(T, G) Γ ′ \|. Let p : T → T /Γ ′ be the quotient map. Clearly, the maps Sing T : G −→ SSet Γ , G −→ Sing(T, G) has a left adjoint. I am pleased to mention that Shouta Tounai provides me the following sophisticated proof. Proof of Proposition 3.3. Let Γ be a group and consider Γ as a small category in the usual way. Namely, the object set is the one point set { * }, the set of endomorphisms of * is Γ, and the composition is the multiplication of Γ. A right Γ-graph is identified with a functor from Γ op to G. Thus we have a functor from Γ op × ∆ → G, [n] → T × Σ n . The associated functor Let T be a right Γ-graph. Let A T denote the left adjoint of the functor Sing T : G −→ SSet Γ . We shall precisely describe the graph A T (K) for a Γ-simplicial set K. First we construct a Γ-graph A(K). The vertex set of A(K) is the set K 0 of 0-simplices, and two 0-simplices v and w of K are adjacent in A(K) if and only if there is a 1-simplex connecting them. The group Γ acts on T × A(K) by γ(x, v) = (xγ −1 , γv). Then the graph A T (K) is the quotient graph Γ\(T × A(K)). G(T × Σ • , −) : G −→ Set Γ op ×∆ = Set Γ×∆ op ∼ = SSet Γ In the case of simplicial complexes, the following similar construction A T (K) was known: We first construct a Γ-graph A(K) for a Γ-simplicial complex K. The vertex set of A(K) is the vertex set V (K) of K. Two vertices v and w of K are adjacent in A(K) if and only if the set {v, w} is a simplex of K. Then the graph A T (K) is the quotient A T (K) = Γ\(T × A(K) ). This construction was first considered by Csorba [7] in case T is K 2 , and was later generalized in Dochtermann and Schultz [10]. Let K be a simplicial set whose cell structure is isomorphic to some simplicial complex K. Then it is clear that the graphs A T (K) and A T (K) are isomorphic. 3.2. Unit of the adjoint pair. Throughout this section, T is a finite connected right Γ-graph with at least one edge. In this section, we characterize the condition that the unit Id −→ Ex k • Sing T • A T • Sd k (1) is a natural Γ-weak equivalence for sufficiently large k (Theorem 3.4). First we consider the following two conditions concerning a finite right Γ-graph T . Here we consider a Γ-set X as the simplicial set whose 0-simplices are the elements of X and which has no other non-degenerate simplices. Recall that for an element γ of Γ, the graph homomorphism α γ : T → T is defined by x → xγ. (A) For each subgroup Γ ′ of Γ, the map Γ/Γ ′ −→ Sing(T, T /Γ ′ ), γΓ ′ −→ p • α γ is a Γ-weak equivalence. (B) The map Γ −→ Sing(T, T ), γ → α γ is a Γ-weak equivalence. The condition (A) implies the condition (B). One can show that the map Γ/Γ ′ −→ Sing(T, T /Γ ′ ) = Sing T • A T (Γ/Γ ′ ) in the condition (A) is the unit map of (A T , Sing T ). Thus if the unit (1) is a natural Γ-weak equivalence for every Γ-simplicial set K, then the condition (A) holds (see also (4) of Section 2.3). On the other hand, the following result asserts that the converse also holds: Theorem 3.4. Let Γ be a finite group and T a finite connected right Γ-graph having at least one edge and diameter r. Let k be a positive integer such that 2 k−2 > r. If T satisfies the condition (A), then the unit map u K : K −→ Ex k • Sing T • A T • Sd k (K) is a Γ-weak equivalence for every Γ-simplicial set K. If we restrict our attention to free Γ-simplicial sets, the following holds: Theorem 3.5. Let Γ be a finite group and T a finite connected right Γ-graph having at least one edge and diameter r. Let k be an integer with 2 k−2 > r. If T satisfies the condition (B), then the unit map u K : K −→ Ex k • Sing T • A T • Sd k (K) is a Γ-weak equivalence for every free Γ-simplicial set K. Before giving the proofs, we consider when the right Γ-graph T satisfies the above conditions. In fact the condition (A) is a quite strong requirement. I think that the following are the only examples naturally arising. (1) Γ is trivial and T is the graph 1 consisting of one looped vertex. (2) Γ is Z 2 and T is K 2 with the Z 2 -action which flips the edge of K 2 . Of course, if T is ×-homotopy equivalent to 1 or Z 2 -×-homotopy equivalent to K 2 , then T satisfies the condition (A). Note that Hom(1, G) is the face poset of the clique complex of the maximal reflexive subgraph of G, i.e. the induced subgraph of G whose vertices are looped vertices of G. On the other hand, the case (2) corresponds to the box complex B(G) = Hom(K 2 , G). On the other hand, there are several graphs such that the condition (B) is satisfied. In case the graph T is stiff (see Section 2.1), the condition (B) has the following combinatorial characterization. Here Aut(T ) denotes the automorphism group of T . (1) Γ is the symmetric group S n of the n-element set {1, · · · , n}, and T is the complete graph K n for n ≥ 2. (2) Γ is the dihedral group D 2n+1 of order (4n + 2), and T is the odd cycle C 2n+1 with length 2n + 1 for n ≥ 1. (3) A subset σ of the cyclic group Z n of order n is stable if x ∈ σ implies x + 1 ∈ σ. The stable Kneser graph SG n,k is the graph whose vertex set consists of the stable k-subsets of Z n and two stable k-subsets are adjacent if and only if they are disjoint. The stable Kneser graphs were introduced by Schrijver [19], and he showed that stable Kneser graphs are vertex critical, i.e. every subgraph G of SG n,k such that V (G) V (SG n,k ) has a chromatic number smaller than χ(SG n,k ). It is easy to see that the vertex critical finite graph satisfies the condition of Lemma 3.6. Braun [5] showed that the group of automorphisms of SG n,k is isomorphic to the dihedral group D n of order 2n. Csorba [7] showed that for every Z 2 -CW-complex X, there is a simple graph G such that Hom(K 2 , G) and X are Z 2 -homotopy equivalent. Theorem 3.4 implies that the free assumption is redundant: Corollary 3.8. For every Z 2 -CW-complex X, there is a graph G such that X and Hom(K 2 , G) are Z 2 -homotopy equivalent. Let S n be the symmetric group of the n-element set {1, · · · , n}. Then S n acts on K n in an obvious way. Dochtermann and Schultz [10] proved that for every free S n -CW-complex X, there exists a graph G such that Hom(K n , G) and X are S n -homotopy equivalent [10]. The following is a generalization of [10]. Corollary 3.9. Suppose that a finite connected right Γ-graph T with at least one edge satisfies the condition (B). Then for every free Γ-complex X, there is a graph G such that Hom(T, G) and X are Γ-homotopy equivalent. Here we need the assumption that the group action is free. In fact, Hom(K n , G) is free if and only if G has no looped vertices. On the other hand, if G has looped vertices, then Hom(K n , G) has fixed points. The converse of Corollary 3.9 is false. In fact Dochtermann [9] showed that for every finite connected graph T with at least one edge and for every CW-complex X, there is a graph G such that Hom(T, G) and X are homotopy equivalent. Now we turn to the proofs of Theorem 3.4 and Theorem 3.5. Let ∆ n to indicate the simplicial complex ([n], 2 [n] ). For n > 0 and 0 ≤ r ≤ n, define Λ n r to be the simplicial complex whose vertex set is [n] and whose simplex is a subset σ with σ ∪ {r} = [n]. Clearly, we have A T (∆[n]) = A T (∆ n ), A T (Λ r [n]) = A T (Λ n r ) . Let k be an integer such that 2 k−2 is greater than the diameter of T . In the rest of this section, we writeÂ T instead of A T • Sd k andŜ T instead of Ex k • Sing T . For a subgroup Γ ′ of Γ and a simplicial set K, we write K Γ ′ instead of (Γ/Γ ′ ) × K. Now we turn to the proof of Theorem 3.4. The proof is based on the following two assertions which will be proved in Section 4. Lemma 3.10. Let Γ ′ be a subgroup of the finite group Γ, and suppose that T satisfies the condition (A). Then the following hold: (1) The inclusionÂ T (Γ/Γ ′ ) ֒→Â T ((Γ/Γ ′ ) × Λ r [n]) induced by the inclusion Γ/Γ ′ ֒→ (Γ/Γ ′ ) × Λ r [n], γ → (γ, r) is a ×-homotopy equivalence. (2) The inclusionÂ T (Γ/Γ ′ ) ֒→Â T ((Γ/Γ ′ )×∆[n]) is a ×-homotopy equivalence. (3) The unit map Λ r [n] Γ ′ −→Ŝ T •Â T (Λ r [n]) Γ ′ is a Γ-weak equivalence. (4) The unit map ∆[n] Γ ′ −→Ŝ T •Â T (∆[n] Γ ′ ) is a Γ-weak equivalence. For a Γ-simplicial set K, we write u K to indicate the unit K →Ŝ T •Â T (K). Proposition 3.11. Let (K, L) be a pair of Γ-simplicial sets, f : L → L ′ a Γsimplicial map, and let K ′ = K ∪ L L ′ . If the unit maps u L , u K , u L ′ are Γ-weak equivalences, then the unit map u K ′ is a Γ-weak equivalence. Proof of Theorem 3.4. Let K be a Γ-simplicial set. Recall that a generating cofibrations I Γ of SSet Γ (Section 2.2) is described as follows: I Γ = {(Γ/Γ ′ ) × ∂∆[n] ֒→ (Γ/Γ ′ ) × ∆[n] \| n ≥ 0 and Γ ′ is a subgroup of Γ.} Since every Γ-simplicial set is an I Γ -cell complex, there exist an ordinal λ and a colimit preserving functor X • : λ → SSet Γ such that X 0 = ∅, the colimit X λ of X • is isomorphic to K, and X α → X α+1 is a pushout of an element of I Γ for every α < λ. We want to show that the unit map X λ →Ŝ T •Â T (X λ ) is a Γweak equivalence by the transfinite induction on α 0 < λ. Suppose that for every α < α 0 , the unit map X α →Ŝ T •Â T (X α ) is a Γ-weak equivalence. If α 0 − 1 exists, then it follows from Lemma 3.10, Proposition 3.11, and Corollary 3.12 that the unit X α0 →Ŝ T •Â T (X α0 ) is a Γ-weak equivalence. If α 0 is a limit ordinal, then Proposition 2.6 implies that X α0 →Ŝ T •Â T (X α0 ) is a Γ-weak equivalence since Sing T and Ex preserve sequential colimits. This completes the proof of Theorem 3.4. We can prove Theorem 3.5 in a similar way. Define the small family I ′ Γ of Γ-simplicial maps as follows: I ′ Γ = {Γ × ∂∆[n] ֒→ Γ × ∆[n] \| n ≥ 0} . Then a Γ-simplicial set K is free if and only if K is an I ′ Γ -cell complex. Thus, in the proof of Theorem 3.4, replacing "condition (A)" to "condition (B)" and considering only the trivial subgroup 1, we have the proof of Theorem 3.5. 3.3. Model structure. In this section, we introduce two model structures on the category of graphs. In the case of Sing 1 , we have the following theorem. Note that the Hom complex Hom(1, G) is the face poset of the clique complex of the maximal reflexive subgraph of G. We write I and J for I Γ and J Γ if the group Γ is the trivial group 1. Theorem 3.14. The category G of graphs has the cofibrantly generated model structure with generating cofibrations A 1 • Sd 2 (I) and with generating trivial cofibrations We only give the proof of Theorem 3.13 since the other is similar. First we show that G has the model structure described in Theorem 3.13. It is clear that every object of G is a small object in the sense of Definition 10.4.1 of [12]. Thus by Theorem 11.3.2 of [12], it suffices to show that Ex 3 • Sing K2 takes a pushout of an element of A K2 • Sd 3 (J Z2 ) to a Γ-weak equivalence. But this follows from (3) of Lemma 3.10 and Proposition 3.11. Thus G has the model structure described in Theorem 3.13. Next we show that the adjoint pair (Â K2 ,Ŝ K2 ) = (A K2 • Sd 3 , Ex 3 • Sing K2 ) is a Quillen equivalence. By Corollary 1.3.16 of [13], it suffices to verify the following: (1) Let f : X → Y be a graph homomorphism between fibrant objects in G. If S K2 (f ) is a Z 2 -weak equivalence, then f is a weak equivalence. (2) For every Z 2 -simplicial set K, the composition of K →Ŝ K2ÂK2 (K) →Ŝ K2 RÂ K2 (K) is a Γ-weak equivalence. Here R denotes a fibrant replacement functor of SSet Γ . The definition of weak equivalences of G follows (1) and that the right arrow in (2) is a Γ-weak equivalence. Thus (2) follows from Theorem 3.4 since K 2 satisfies the condition (A) in Section 3.2. This completes the proof. Strong homotopy theory The purpose of this section is to show Lemma 3.10 and Proposition 3.11. Lemma 3.10 is proved in Section 4.4 and Proposition 3.11 is proved in Section 4.6. The difficulty of the proofs seems to lie in the following fact: Let K be a Γsimplicial set and L a Γ-subcomplex of it. In general, \|Sing T • A T (L)\| is not a deformation retract of \|Sing T •A T (K)\| even if \|L\| is a deformation retract of \|K\|. On the other hand, the strong collapses of Γ-complexes and ×-homotopy deformation retracts, which will be introduced later, have the following important properties: • If a graph H is a ×-deformation retract of G, then \|Sing T (H)\| is a deformation retract of \|Sing T (G)\| (Lemma 2.1). • If a Γ-simplicial complex K strongly Γ-collapses to its Γ-subomplex L, then A T (L) is a ×-deformation retract of A T (K) (Proposition 4.12). • For any pair of (K, L) of finite Γ-simplicial complexes, Sd k (L) has a large neighborhood in Sd k (K) which strongly Γ-collapses to Sd k (L) if we take k to be sufficiently large (Corollary 4.18). 4.1. Simplicial complexes. An (abstract) simplicial complex consists of a family K of finite subsets of a set S such that σ ∈ K and τ ⊂ σ imply τ ∈ K. The vertex set V (K) of K is the union of the all simplices of K. Let K and L be simplicial complexes. A simplicial map from K to L is a map f : V (K) → V (L) such that σ ∈ K implies f (σ) ∈ L. The geometric realization of K (see [15]) is denoted by \|K\|. We assign the topological terminology to simplicial complexes by the geometric realization functor. For example, a simplicial map f : K → L is a homotopy equivalence if and only if the continuous map \|f \| : \|K\| → \|L\| induced by f is a homotopy equivalence. Let K and L be simplicial complexes. A map η : V (K) → 2 V (L) \{∅} is a simplicial multi-map if, for every simplex σ of K, the subset v∈σ η(v) of V (L) is a simplex of L. For a pair of simplicial multi-maps η and η ′ , we write η ≤ η ′ to mean that η(v) ⊂ η ′ (v) for every v ∈ V (K). The poset of simplicial multi-maps from K to L is denoted by Map(K, L). A simplicial map is identified with a minimal point of Map(K, L). Two simplicial maps f and g are strongly homotopic if they belong to the same connected component of Map(K, L). Recall that two simplicial maps f, g : K → L are contiguous if σ ∈ K implies f (σ) ∪ g(σ) ∈ L. Hence f and g are contiguous if and only if there is an element η of Map(K, L) such that f ≤ η and g ≤ η. Thus our definition of strong homotopy coincides with the original one of [3]. Let K and L be Γ-simplicial complexes. A simplicial multi-map η ∈ Map(K, L) is Γ-equivariant if γ(η(v)) = η(γv) for each v ∈ V (K) and γ ∈ Γ. The induced subposet of Map(K, L) consisting of the Γ-equivariant multi-maps is denoted by Map Γ (K, L). Two Γ-simplicial maps are strongly Γ-homotopic if they belong to the same connected component of Map Γ (K, L). 4.2. Posets. For a poset P , the order complex ∆(P ) of P is the abstract simplicial complex whose vertex set is the underlying set of P and whose simplices are the finite chains of P . The classifying space of P is the geometric realization of ∆(P ), and is denoted by \|P \|. It is easy to see that this definition coincides with the usual definition of the classifying space, i.e. the geometric realization of the nerve of P . Let f and g be order-preserving maps from P to Q. We write f ≤ g to mean that f (x) ≤ g(x) for every element x of P . The poset of order-preserving maps from P to Q is denoted by Poset(P, Q). The order-preserving maps f and g are strongly homotopic if they belong to the same connected component of Poset(P, Q). If f and g are strongly homotopic, then they are homotopic, i.e. \|f \|, \|g\| : \|P \| → \|Q\| are homotopic. In fact f and g induce simplicially homotopic maps between the nerves (see Proposition 14.2.10 of [12]), and many results concerning strong homotopy theory in this section and Lemma 2.1 and Corollary 2.2 hold for simplicial homotopy. We call a Γ-equivariant order-preserving map between Γ-posets a Γ-poset map, for short. If P and Q are Γ-posets, then we denote by Poset Γ (P, Q) the induced subposet of Poset(P, Q) consisting of Γ-poset maps. Two Γ-poset maps are strongly Γ-homotopic if they belong to the same connected component of Poset Γ (P, Q). It is easy to see that if f and g are strongly Γ-homotopic, then they are also Γ-homotopic. The face poset of a simplicial complex K is the poset of non-empty simplices of K ordered by inclusion, and is denoted by F K. For a simplicial map f : K → L, define the order-preserving map F f : F K → F L by the correspondence σ → f (σ). [3]). Let K and L be Γ-simplicial complexes, f and g Γ-simplicial maps from K to L, and suppose that f and g are strongly Γ-homotopic. Then the order-preserving maps F f and F g are strongly Γ-homotopic. Lemma 4.1 (Barmak-Minian Proof. This lemma clearly follows from the fact that the map F : Map Γ (K, L) −→ Poset Γ (F K, F L), η → σ → v∈σ η(v) is an order-preserving map. Thus if two simplicial maps f, g : K → L are strongly Γ-homotopic, then they are actually Γ-homotopic, i.e. \|f \| and \|g\| are Γ-homotopic. Strong collapse. In this section, we consider the notion of the deformation retracts in the sense of the strong homotopy theory of Γ-posets and Γ-simplicial complexes. The goal of this section is Corollary 4.11. Although the precise statements of many of the results given here did not appear in Barmak and Minian [3], the ideas of the proofs already appearing there. However, for the reader's convenience, we shall give precise proofs. Remark 4.5. Let P be a poset and Q is an induced subposet of P . Then P strongly collapses to Q if and only if there is a sequence (f 0 , · · · , f n ) with finite length which satisfies the following conditions: (1) f 0 = id P and f n (P ) ⊂ Q. (2) f i (y) = y for every y ∈ Q and i = 0, 1, · · · , n. (3) f i and f i−1 are comparable for i = 1, · · · , n. Replacing "comparable" to "contiguous", we have a similar formulation for simplicial complexes. Recall that the group Γ is assumed to be finite. Here we note the following obvious lemma, whose proof is omitted. Lemma 4.6. Let Γ be a finite group, and x an element of a Γ-poset P . If γx and x are comparable, then we have γx = x. Let P be a Γ-poset. Recall that a point x ∈ P is an upper beat point if P >x = {y ∈ P \| y > x} has the minimum, and x is a lower beat point of P if P <x = {y ∈ P \| y < x} has the maximum. A point x is a beat point if x is either an upper or lower beat point of P . If x is a beat point, then P strongly collapses to P \ Γx. In fact, if x is an upper beat point and y is the minimum of P >x , then y does not belong the orbit Γx of x (Lemma 4.6). Since the map f : P → P, f (z) = z (z ∈ Γx) γy (z = γx, γ ∈ Γ) is a closure operator and f (P ) = P \ Γx. Thus P strongly Γ-collapses to P \ Γx. Lemma 4.7. Let Q ⊂ P ′ ⊂ P be a sequence of Γ-posets. Suppose that P strongly Γ-collapses to P ′ . Then P strongly Γ-collapses to Q if and only if P ′ strongly Γ-collapses to Q. Proof. Let r : P → P ′ be a retract of the inclusion i : P ′ ֒→ P . Then the map Def Γ (P, Q) → Def Γ (P ′ , Q), f → r • f • i is an order-preserving map. This implies that P ′ strongly Γ-collapses to Q if P strongly Γ-collapses to Q. It is easy to show the converse by Remark 4.5. Proposition 4.8. Let Q be an induced Γ-subposet of a finite Γ-poset P . Then P strongly Γ-collapses to Q if and only if there is a linear order {α 1 , · · · , α n } on Γ\(P \ Q) which satisfies the following: For each i = 1, · · · , n, α i is a family of beat points of P \ (α 1 ∪ · · · ∪ α i−1 ) of P (This order of Γ\(P \ Q) is independent from the order of P ). Proof. By Lemma 4.7, it suffices to show the following claim: Suppose that a finite poset P strongly Γ-collapses to an induced subposet Q of P and P = Q. Then there is a beat point of P not belonging to Q. By the hypothesis, there is a map f ∈ Def Γ (P, Q) such that f < id P or f > id P . If f > id P , a maximal element of {x ∈ P \| f (x) > x} is an upper beat point of P not belonging to Q. The case f < id P is similarly proved. Let K be a finite simplicial complex. A vertex v of K is dominated (see Definition 2.1 of [3]) if there exists another vertex w which satisfies the following condition: If a simplex σ of K contains v, then σ ∪ {w} is a simplex of K. It is easy to see that if v is dominated in K, then K strongly collapses to K \ Γv. Here K \ Γv is the subcomplex of K whose simplex is a simplex of K does not contain an element of Γv. On the other hand, we have the following: Proposition 4.9. Let P be a finite Γ-poset and Q an induced Γ-subposet of P . If P strongly Γ-collapses to Q, then ∆(P ) strongly Γ-collapses to ∆(Q). Proof. If x is a beat point of P , then x is dominated in ∆(P ). Thus this proposition follows from Proposition 4.8. Proposition 4.10. Let K be a Γ-simplicial complex and L an induced Γ-subcomplex of K. Suppose that K strongly Γ-collapses to L. Then F K strongly collapses to F L. Proof. It suffices to note that the order-preserving map F : Map Γ (K, L) −→ Poset Γ (F K, F L) in the proof of Lemma 4.1 maps Def Γ (K, L) to Def Γ (F K, F L). Let K be a simplicial complex. The simplicial complex ∆(F K) is called the barycentric subdivision of K, and is denoted by Sd(K). Proposition 4.12. Let (K, L) be a pair of Γ-simplicial complexes such that L is K strongly Γ-collapses to L, and let T be a right Γ-graph. Then A T (L) is a ×-homotopy deformation retract of A T (K). Proof. Consider the map Φ : Map Γ (K, K) −→ Hom Γ (A(K), A(K)), η −→ (v → η(v)). It is easy to see that Φ is well-defined and order-preserving. Next define the map T × (−) : Hom Γ (A(K), A(K)) → Hom Γ (T × A(K), T × A(K)) by (T × η)(x, v) = {x} × η(v). Thus we have a sequence Map Γ (K, K) Φ − −−− → Hom Γ (A(K), A(K)) T ×(−) −−−−→ Hom Γ (T × A(K), T × A(K)) Q − −−− → Hom(A T (K), A T (K)) of order-preserving maps, where the last map Q is described in the previous paragraph of this proposition. The composition of the above sequence maps Def Γ (K, L) to Def(A T (K), A T (L)). Thus the proposition follows. Proof of Lemma 3.10. Note that A T ((Γ/Γ ′ ) × Λ r [n]) = A T • Sd k ((Γ/Γ ′ ) × Λ r [n]) ∼ = A T • Sd k ((Γ/Γ ′ ) × Λ n r ) . Since any vertex of (Γ/Γ ′ ) × Λ r n other than elements of (Γ/Γ ′ ) × {r} is dominated, (Γ/Γ ′ ) × Λ n r strongly Γ-collapses to Γ/Γ ′ . Thus A T • Sd k (Γ/Γ ′ ) is a ×-deformation retract of A T • Sd k ((Γ/Γ ′ ) × Λ n r ) (Proposition 4.12). The proof of (2) is similar. We now show (3). Consider the diagram (Γ/Γ ′ ) ≃Γ − −−− → (Γ/Γ ′ ) × Λ r [n]     Ŝ T •Â T (Γ/Γ ′ ) − −−− →Ŝ T •Â T ((Γ/Γ ′ ) × Λ r [n]), where the vertical arrows are the unit maps. Since T satisfies the condition (A), the left vertical arrow is a Γ-weak equivalence. By Lemma 2.1 and (1) of this proposition, the lower horizontal arrow is a Γ-weak equivalence. The proof of (4) is similar. We need the following assertion later: Lemma 4.13. Let H be a ×-deformation retract of a graph G, and f : H → Y a graph homomorphism. Let X be the pushout G ∪ f Y . If H is a ×-homotopy deformation retract of G, then Y is a ×-deformation retract of X. Proof. Let u : G → X be the natural map. Define Φ : Def(G, H) → Def(X, Y ) by Φ(η)(v) = {v} (v ∈ V (Y )) u(η(v)) (v ∈ V (G)). It is easy to see that this map Φ is well-defined and order-preserving. 4.5. r-NDR of Γ-simplicial complex. In this section, we introduce the notion of r-NDR's of Γ-simplicial complexes. A pair of Γ-simplicial complexes is a pair (K, L) consisting of a Γ-simplicial complex K together with a Γ-subcomplex L of K. Let K be a simplicial complex. Recall that the star of v ∈ V (K) is the subcomplex of K defined by st K (v) = {σ ∈ K \| σ ∪ {v} ∈ K}. Definition 4.14. Let K be a finite Γ-simplicial complex and L a Γ-subcomplex of K. The neighborhood of L in K is the Γ-subcomplex ν K (L) = ν(L) = v∈V (L) st K (v) ⊂ K. For a positive integer r, define the r-neighborhood ν r (L) inductively by ν 1 (L) = ν(L), and ν s+1 (L) = ν(ν s (L)). A pair of Γ-simplicial complexes (K, L) is an r-NDR pair if there exists a Γsubcomplex A of K containing ν r (L) such that A strongly Γ-collapses to L. To prove Proposition 4.15, we use the following lemma: Proof. Recall that a simplex of Sd(K) = ∆(F K) is a chain of the face poset of K. Let c be a simplex of ν 2 (Sd(L)) ⊂ Sd(K). By the definition of ν 2 , there is a vertex σ of ν(Sd(L)) such that c ∈ st Sd(K) (σ), namely, c ∪ {σ} is a chain of F K. Since σ ∈ V (ν(Sd(L))), there is τ ∈ V (Sd(L)) such that {σ} ∈ st Sd(K) (τ ), namely, {σ, τ } is a chain in F K. Then the maximum σ ′ of c ∪ {σ} contains some element v of τ , and hence we have σ ′ ∈ st K (v) ⊂ ν(L). Therefore every element of c ∪ {σ} belongs to ν(L), and hence c ⊂ c ∪ {σ} ∈ Sd(ν(L)). Proof of Proposirion 4.15. Suppose that (K, L) is an r-NDR pair of Γ-simplicial complexes, and let A be a Γ-subcomplex of K containing ν r (L) such that A strongly Γ-collapses to L. By Lemma 4.16, we have ν 2r (Sd(L)) ⊂ Sd(ν r (L)) ⊂ Sd(A). Corollary 4.11 implies that Sd(A) strongly Γ-collapses to Sd(L). Therefore the pair (Sd(K), Sd(L)) is a (2r)-NDR pair. Theorem 4.17. Let K be a finite Γ-simplicial complex and L a Γ-subcomplex of K. Then the pair (Sd 2 (K), Sd 2 (L)) is a 1-NDR pair. Proof. Note that Sd(K) is the Γ-simplicial complex whose simplex is a finite chain of K \ {∅} with respect to the inclusion ordering. Set X = K \ L = {σ ∈ K \| σ ∈ L}. Then we have F Sd(L) = {c ∈ F Sd(K) \| c ∩ X = ∅}. Set P = {c ∈ F Sd(K) \| There exists σ ∈ L with σ ∈ c.} = {c ∈ F Sd(K) \| c ⊂ X}. Note that ∆(P ) is the 1-neighborhood of Sd 2 (L) in Sd 2 (K). Thus it suffices to show that P strongly Γ-collapses to F Sd(L) (Proposition 4.9). Define the closure operator (see 4.6. r-NDR for graphs. In this section, we introduce the r-NDR pair of graphs and prove Proposition 3.11. ). Proof. The case r = 1 is deduced from the construction of A T (Section 3.1) and is omitted. Thus we have ν r AT (K) (A T (K)) ⊂ ν r−1 AT (K) (A T (ν K (L))) ⊂ · · · ⊂ A T (ν r K (L)). Corollary 4.21. If (K, L) is an r-NDR pair of Γ-simplicial complexes, then the pair of graphs (A T (K), A T (L)) is an r-NDR pair of graphs. Proof. Let L ′ be a subcomplex of K containing ν r K (L) such that L ′ strongly Γcollapses to L. By Proposition 4.20, we have ν r AT (K) (A T (L)) ⊂ A T (ν r K (L)) ⊂ A T (L ′ ). Proposition 4.12 implies that A T (L) is a ×-deformation retract of A T (L ′ ). Corollary 4.22. Let (K, L) be a pair of finite Γ-simplicial complexes. Then for r ≥ 2, the pair (A T (Sd r (K)), A T (Sd r (L))) is a 2 r−2 -NDR pair of graphs. Proof. This follows from Corollary 4.11, Proposition 4.12, and Corollary 4.21. Let T be a finite connected right Γ-graph. The following theorem asserts that if r is sufficiently large, then the class of r-NDR's satisfies the gluing lemma with respect to Sing T -complexes. is a pushout diagram in the category Set Γ of Γ-sets. Let ϕ : T × Σ n → X be a graph homomorphism. If the image of ϕ does not intersect Y , then ϕ factors through G. Suppose that there is a vertex v of T × Σ n with ϕ(v) ∈ V (Y ). Since the diameter of T × Σ n is smaller than or equal to r and Y ′ is the r-neighborhood of Y in X, we have that ϕ factors through Y ′ . Thus we have shown that the map f * ⊔ j * : G(T × Σ n , G) ⊔ G(T × Σ n , Y ′ ) −→ G(T × Σ n , X) is surjective. Next let ψ 0 : T × Σ n → G and ψ 1 : T × Σ n → Y ′ be graph homomorphisms withf ψ 0 = jψ 1 . Since the image off ψ 0 is contained in Y ′ , we have that ψ 0 factors through H ′ , and let ψ : T × Σ n → H ′ with iψ = ψ 0 . Since jf ′ ψ =f iψ =f ψ 0 = jψ 1 and j is a monomorphism, we have that f ′ ψ = ψ 1 . Thus the diagram (3) is a pushout diagram. ordered by the product of the inclusion orderings. The Z 2 -action of B(G) is the exchange of the first and the second entries, i.e. (σ, τ ) ↔ (τ, σ). Corollary 3 . 2 . 32Let Γ be a finite group and T a right Γ-graph. Then the map Φ : \|Sing(T, G)\| −→ \|Hom(T, G)\| in Theorem 3.1 is a Γ-homotopy equivalence. p * : Sing(T /Γ ′ , G) −→ Sing(T, G), p * : Hom(T /Γ ′ , G) −→ Hom(T, G)are monomorphisms and their images coincide with Sing(T, G) Γ ′ and Hom(T, G) Γ ′ , respectively. Since Φ : \|Sing(T /Γ ′ , G)\| → \|Hom(T /Γ ′ , G)\| is a homotopy equivalence (Theorem 3.1), this completes the proof. Proposition 3. 3 . 3Let Γ be a group and T a right Γ-graph. Then the functor coincides with Sing T . Thus the left Kan extension of T × Σ • along the Yoneda functor Γ op × ∆ → Set is the left adjoint of Sing T . Lemma 3. 6 . 6Suppose that T is stiff. Then T satisfies the condition (B) if and only if the homomorphism α : Γ → Aut(T ), γ → α γ is an isomorphism and every endomorphism of T is an automorphism.Proof. Recall that the composition ofΓ −→ Aut(T ) −→ Sing(T, T )(2)coincides with the unit map. Suppose that the homomorphism Γ → Aut(T ) is an isomorphism and every endomorphism of T is an automorphism. Lemma 2.3 implies Aut(T ) ∼ = Hom(T, T ) and hence the unit Γ → Sing(T, T ) ≃ Hom(T, T ) is a Γ-weak equivalence. The proof of the converse is similar and is omitted. Example 3. 7 . 7There are several examples which satisfy the condition (B): Corollary 3 . 12 . 312Suppose that T satisfies the condition (A). For n ≥ 0, the unit map u ∂∆[n] : (Γ/Γ ′ ) × ∂∆[n] →Ŝ T •Â T ((Γ/Γ ′ ) × ∂∆[n]) is a Γ-weak equivalence for every subgroup Γ ′ . Proof. We show this by the induction on n. The case n = 0 is obvious. Suppose n > 0. Regard ∆[n−1] as a subcomplex of ∆[n] by the map induced by the inclusion [n − 1] ֒→ [n]. Apply Proposition 3.11 to the case of K = Λ n [n], L = ∂∆[n − 1], and L ′ = ∆[n − 1]. Theorem 3 . 13 . 313The category G of graphs has the cofibrantly generated model structure with generating cofibrations A K2 • Sd 3 (I Z2 ) and with generating trivial cofibrations A K2 • Sd 3 (J Z2 ) (see Section 2.2). A graph homomorphism f : G → H is a weak equivalence if and only if the map f * : B(G) → B(H) induced by f is a Z 2 -homotopy equivalence. Moreover, the adjoint pair(A K2 • Sd 3 , Ex 3 • Sing K2 ) : SSet Z2 − −−− → Gis a Quillen equivalence. A 1 • 1Sd 2 (J ). A graph homomorphism f : G → H is a weak equivalence if and only if f * : Hom(1, G) → Hom(1, H) induced by f is a homotopy equivalence. Moreover, the adjoint pair (A 1 • Sd 2 , Ex 2 • Sing 1 ) : SSet − −−− → G is a Quillen equivalence. Definition 4. 2 . 2Let P be a Γ-poset and Q an induced Γ-subposet of P . Define Def Γ (P, Q) to be the induced subposet of Poset Γ (P, P ) consisting of the Γ-poset maps which fix each point of Q. P strongly Γ-collapses to Q if there is an element f belonging to the identity component of Def Γ (P, Q) with image contained in Q. Example 4 . 3 . 43Let c : P → P be a Γ-equivariant closure operator, i.e. c 2 = c and either c ≥ id or c ≤ id holds. Then P strongly Γ-collapses to c(P ).The associated notion of simplicial complexes is similarly defined:Definition 4.4. Let L be an induced Γ-subcomplex of a Γ-simplicial complex K. Let Def Γ (K, L) be the induced subposet of Map Γ (K, K) consisting of the Γequivariant simplicial multi-maps η such that η(v) = {v} for every v ∈ V (L). K strongly collapses to L if there is a simplicial map f belonging to the identity component of Def(K, L) with image contained in L. Corollary 4. 11 . 11Let K be a finite Γ-simplicial complex and L a Γ-subcomplex of K. If K strongly Γ-collapses to L, then Sd(K) strongly Γ-collapses to Sd(L). 4. 4 . 4×-homotopy deformation retract. In this section, we introduce the ×homotopy deformation retract of graphs and prove Lemma 3.10.Let G be a graph and H an induced subgraph of G.Define the poset Def(G, H) to be the induced subposet of Hom(G, G) consisting of the multi-homomorphisms η such that η(w) = {w} for every vertex w of H. H is a ×-homotopy deformation retract of G if there is a graph homomorphism f belonging to the identity component of Def(G, H) such that f (v) ∈ V (H) for every v ∈ V (G). Let G and H be left Γ-graphs. A multi-homomorphism η from G to H is Γequivariant if γ(η(v)) = η(γv) for every v ∈ V (G) and γ ∈ Γ. We let Hom Γ (G, H) be the induced subposet of Hom(G, H) consisting of Γ-equivariant multi-homomorphisms. Then we have a poset map Q : Hom Γ (G, H) −→ Hom(Γ\G, Γ\H) defined as follows: For a vertex v ∈ V (G), Q(η)(Γv) = {Γw \| w ∈ η(v)}. Then we have the following proposition. For the definitions of A(K) and A T (K), see the end of Section 3.1. Proposition 4 . 15 . 415If (K, L) is an r-NDR pair of finite Γ-simplicial complexes, then the pair (Sd(K), Sd(L)) is a (2r)-NDR pair. Lemma 4 . 16 . 416Let (K, L) be a pair of simplicial complexes. Then we have ν 2 (Sd(L)) ⊂ Sd(ν(L)). Example 4.3) f : P → P by f (c) = c ∩ L. Then f (P ) = F Sd(L). Thus the theorem follows. Combining Proposition 4.15 and Theorem 4.17, we have the following: Corollary 4 . 18 . 418Let (K, L) be a pair of finite Γ-simplicial complexes. Then for r ≥ 2, the pair (Sd r (K), Sd r (L)) is a 2 r−2 -NDR. Definition 4 . 19 . 419Let G be a graph and H a subgraph of G. Let ν G (H) = ν(H) be the subgraph of G defined byV (ν(H)) = {v ∈ V (G) \| There is w ∈ V (H) such that (v, w) ∈ E(G).}, E(ν(H)) = {(v, w) ∈ E(G) \| One of v and w is a vertex of H.}.For r ≥ 1, define the r-neighborhood ν r G (H) = ν r (H) of H inductively by ν 1 (H) = ν(H) and ν s+1 (H) = ν(ν s (H)). Proposition 4 . 20 . 420Let (K, L) be a pair of Γ-simplicial complexes. Then ν r AT (K) (A T (L)) ⊂ A T (ν r K (L) Theorem 4 . 23 . 423Let r be a positive integer, (G, H) an r-NDR pair of graphs, f : H → Y a graph homomorphism, and X the pushout Y ∪ H G. Suppose that the finite right Γ-graph T has at least one edge and the diameter of T is smaller than r. Then the diagramSing T (H) − −−− → Sing T (G) Sing T (Y ) − −−− → Sing T (X)is a homotopy pushout square in the category SSet Γ of Γ-simplicial sets. In other words, the natural map\|Sing T (Y )\| ∪ \|Sing T (H)\| \|Sing T (G)\| −→ \|Sing T (X)\|is a Γ-homotopy equivalence. A similar assertion holds for Hom T -complexes.Proof. By Proposition 2.5 and Corollary 3.2, the case of Hom T -complexes follows from the case of Sing T -complexes. Thus we only give the proof of the case of Sing T .Since (G, H) is an r-NDR pair, there is a subgraph H ′ of G containing ν r (H) such that H is a ×-deformation retract of H ′ . Let Y ′ be the pushout Y ∪ H H ′ . Then Y is a ×-deformation retract of Y ′ .Consider the commutative squareSing T (H ′ ) Sing T (X),wheref : G → X = Y ∪ H G and f ′ : H ′ → Y ′ = Y ∪ H H ′ are the natural maps, i : H ′ ֒→ G and j : Y ′ ֒→ X are inclusions. We claim that the above square is a pushout square. To see this, we want to show that the diagram G(T × Σ n , H ′ ) Next we consider the commutative diagramSing T (Y ) ← −−− − Sing T (H) − −−− → Sing T (G) j * Acknowledgements. I would like to express my gratitude to Toshitake Kohno for his indispensable advice and support. I would like to thank Dai Tamaki for his kind support during my stay at Shinshu University. I would like to thank Shouta Tounai for careful reading and helpful suggestions. Moreover, he provided me the proof of Proposition 3.3. I thank Anton Dochtermann for valuable comments and a detailed answer to my question concerning his result. I also thank Jun Yoshida and an anonymous referee for valuable comments. 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[ "Measurement of the b-jet cross-section with associated vector boson production with the ATLAS experiment at the LHC", "Measurement of the b-jet cross-section with associated vector boson production with the ATLAS experiment at the LHC" ]	[ "Heather M Gray " ]	[]	[]	on behalf of the ATLAS Collaboration CERN Abstract. A measurement of the cross-section for vector boson production in association with jets containing b-hadrons is presented using 35 pb −1 of data from the LHC collected by the ATLAS experiment in 2010. Such processes are not only important tests of pQCD but also large, irreducible backgrounds to searches such as a low mass Higgs boson decaying to pairs of b-quarks when the Higgs is produced in association with a vector boson. Theoretical predictions of the V+b production rate have large uncertainties and previous measurements have reported discrepancies. Cross-sections measured using in the electron and muon channels will be shown. Comparisons will be made to recent theoretical predictions at the next-to-leading order in α S .	10.1051/epjconf/20122812050	[ "https://www.epj-conferences.org/articles/epjconf/pdf/2012/10/epjconf_hcp2012_12050.pdf" ]	118,600,352	1201.4976	bc4fa57073731b9fb8ee124bc470d0d41ea0465a	Measurement of the b-jet cross-section with associated vector boson production with the ATLAS experiment at the LHC Heather M Gray Measurement of the b-jet cross-section with associated vector boson production with the ATLAS experiment at the LHC 10.1051/epjconf/20122812050 on behalf of the ATLAS Collaboration CERN Abstract. A measurement of the cross-section for vector boson production in association with jets containing b-hadrons is presented using 35 pb −1 of data from the LHC collected by the ATLAS experiment in 2010. Such processes are not only important tests of pQCD but also large, irreducible backgrounds to searches such as a low mass Higgs boson decaying to pairs of b-quarks when the Higgs is produced in association with a vector boson. Theoretical predictions of the V+b production rate have large uncertainties and previous measurements have reported discrepancies. Cross-sections measured using in the electron and muon channels will be shown. Comparisons will be made to recent theoretical predictions at the next-to-leading order in α S . Vector boson production in association with jets (V+jets) is often used as a testing ground for perturbative QCD calculations. Despite substantial progress in the understanding and modelling of inclusive jet production in vector boson events, less study has been made of heavy-flavour jet production. V+jets is also an important background to many searches at the LHC. These include searches for the Higgs boson produced in association with a vector boson and with the Higgs decaying to a pair of b-quarks [1] or supersymmetric models with b-quarks in the final state[2]. Significant progress has been made recently in increasing the accuracy of theoretical calculations [3,4]. Challenges to theoretical calculations stem from the non-necessarily negligible b-quark mass and the interplay between contributions from b-quark production in initial and final states. We discuss results of a measurement of the W and Z + bjet cross-sections made by the ATLAS experiment using 35 pb −1 of LHC data collected in 2010 [5,6]. Both measurements are challenging, but in different respects: the Z+b-jet is statistically limited by its small cross-section, while the W+b-jet process has a larger cross-section, but also larger backgrounds resulting in a low signal to background ratio. The V+b-jet cross-sections are measured in a fiducial phase space, by correcting the results back to the particle level while accounting for all detector effects. The cuts applied to define the fiducial cross-section, closely matching the experimental acceptance, are listed in Table 1. The lepton momentum includes the energy from photons radiated within a cone of radius 0.1. The neutrino p T is used as the transverse missing energy. Particle level jets are reconstructed using the anti-k T algorithm using all stable particles (τ > 10 ps). A b-jet is defined as a jet with a b-hadron with p T > 5 GeV with an angular separation from the jet ∆R < 0.3. The Z+b-jet cross-section is measured inclusively for events containing at least one b-jet. Due to the larger W+bjet cross-section a differential measurement is made for events containing either one or two b-jets, but events cona e-mail: [email protected] taining additional jets are vetoed to control backgrounds. The Z+b-jet cross-section is measured at the jet level, such that events with two jets enter the distributions twice. The W+b-jet cross-section, is measured at the event level, with each event entering each distribution once. The ATLAS detector consists of an inner tracking system (\|η\| < 2.5) surrounded by a 2 T super-conducting solenoid, electromagnetic and hadronic calorimeters (\|η\| < 4.9), and a muon spectrometer (\|η\| < 2.7) [7]. Events are collected using single electron or muon high p T triggers and required to contain at least one reconstructed primary vertex. Leptons are required to have p T > 20 GeV with electrons within \|η\| < 2.47 and muons within \|η\| < 2.4. Events containing a Z boson are selected by requiring two isolated leptons of the same flavour but with opposite charge and with the invariant mass of the leptons to be consistent with the Z boson mass: 76 < m ll < 106 GeV. Events containing a W boson are selected by requiring exactly one isolated lepton with missing transverse energy (E miss T > 25 GeV) and transverse mass, m W T = 2p T p ν T (1 − cos(φ ell − φ ν )), consistent with that of a W boson (m W T > 40 GeV). Jets are reconstructed using the anti-k T algorithm with a radius parameter R = 0.4 and required to have p T > 25 GeV. Jets with an angular separation ∆R < 0.5 from a selected lepton are removed. Jets are required to have \|η\| < 2.1 to ensure good b-tagging performance with the full jet contained within the Inner Detector acceptance. Jets containing b-quarks are identified using the decay length significance between the primary and a secondary vertex [8]. Jets with a significance greater than 5.85 are tagged as b-jets. In a simulated tt sample such a cut has a 50% bjet identification efficiency and a mis-tag rate of 10% and 0.5% for c and light jets respectively [8]. The b-tagging efficiency was measured in data using a b-jet enriched sample obtained by selecting jets containing muons, which are produced predominantly from b-hadron decay. The distribution of the muon transverse momentum relative to the jet axis, p rel T , is then used to discriminate between b, c-and light jets, before and after b-tagging. This is an Open Access article distributed under the terms of the Creative Commons Attribution License 2.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (2012) Article available at http://www.epj-conferences.org or http://dx.doi.org/10.1051/epjconf/20122812050 Table 1. Definition of the phase space for the fiducial cross section for the W+b-jet and Z+b-jet measurements [5,6]. Table 2. Number of signal and background events in 35 pb −1 of data [5,6]. The W+b numbers show the ALPGEN predictions normalised to the inclusive NNLO W cross-section, while the Z+jet yield has been normalised to the fit result. The QCD background in both cases has been estimated from data. The top background in the W+b-jet analysis was estimated using a partially data-driven method, while for Z+b it was estimated from simulation. The smaller single top and dibosons backgrounds were estimated from simulation. Uncertainties are not indicated. The estimated number of signal and background events in 35 pb −1 of data is shown in Table. 2. The dilepton invariant mass in the muon channel for the Z+b-jet analysis is shown in Fig. 1. The dominant background is Z+jet events with a light or c-jet has been mistagged as a b-jet. Other backgrounds, including top and single-top, are small and therefore estimated from simulation. The small background from multi-jet production, referred to as QCD, was estimated from data, by fitting an exponential distribution in the dilepton mass in a QCD enriched sample obtained with a relaxed lepton selection criteria. Requirement W + b Z + b Lepton p T p T > 20 GeV Lepton η \|η \| < 2.5 Dilepton mass - 76 − 106 GeV Neutrino p T p ν T > 25 GeV - W transverse mass m T > 40 GeV - Jet p T p j T > 25 GeV Jet rapidity \|y j \| < 2.1 Jet multiplicity n ≤ 2 - b-jet multiplicity n b = 1 or n b = 2 n b ≥ 1 Jet-lepton separation ∆R( , jet) > 0.5 -Z + ≥ 1 b-jet W + 1 b-jet W+ 2-jet V + b 64 43 45 V + c 60 192 81 V + l 0. In the W+b-jet measurement, particularly in the 2-jet selection, the contribution from non W+jet backgrounds is large with a S /B 0.1-0.2. Therefore both the QCD multijet and the top background were estimated from data. The top background was estimated in a control region with more than 4 jets and extrapolated into the signal region using Monte Carlo (MC) simulation. The uncertainty on the btagging efficiency cancels in this ratio therefore obtaining an estimate largely independent of the b-tagging uncertainty. The veto on additional jets was used to reduce the top background. Heavy-flavour jet production is the dominant process contributing to the QCD multi-jet background. As the QCD background is large and has a large uncertainty a very tight requirement on lepton isolation was introduced. In the muon channel the QCD background largely results from nonprompt muons and was extracted by exploiting the differ- Fig. 1. Di-lepton mass distribution for events with at least one btagged jet with p T > 25GeV and \|y\| < 2.1 for the muon channel in the Z+b-jet analysis [5]. The contribution estimated from the simulated MC samples of the signal and various backgrounds is shown. The multi-jet background, estimated with a data-driven method, is not shown. [6]. All backgrounds, but QCD multi-jet, are normalised to their MC prediction. ence in efficiency between prompt and non-prompt muons to pass the standard selection criteria. In the electron channel the QCD background is estimated from a fit to the missing energy distribution with a template obtained by reversing certain electron identification criteria (see Fig. 2). The 50% uncertainty on the QCD background estimate translates into a 7% uncertainty on the W+b-jet cross-section. A maximum likelihood fit to the secondary vertex mass distribution was used to discriminate between b−, c− and light-jets and to extract the flavour fraction on a statistical basis. The other non-V+jets backgrounds were allowed to float within uncertainties. The template shapes for each jet flavour were estimated from simulation and the systematic uncertainties on the shape were derived by comparing data and MC in multi-jet samples enriched in either light or band c-jets. Examples of the fits are shown in Fig. 3 and 4. For the Z+b-jet measurement, the fit is performed for all b-jets in the electron and muon channels combined, while for the W+b-jet measurement it is performed separately by lepton flavour and jet multiplicity. Figure 3 shows the secondary vertex mass distribution for W+1-jet events. The distribution is shown both with the W+jet contribution normalised to the NNLO W crosssection (left) and after the W+jet contribution has been scaled by the fit results (right). The secondary vertex distribution for the Z+b-jet analysis is shown with the flavour Fig. 3. The secondary vertex mass distribution for b-tagged jets in W+b-jet events containing an electron and a single jet [6]. The distribution is shown with the b, c and light contributions normalised to the inclusive NNLO W+jet cross-section (left) and to the fit results (right). contributions normalised to the fit results. There is no visible contribution from Z+light jets because the best fit value was consistent with zero within uncertainties. The fiducial V+jet cross-sections were obtained by unfolding the measured b-fraction using MC simulation samples. Alpgen was used for the W+b-jet measurement and Sherpa for the Z+b-jet measurement. The measured crosssections have uncertainties from the backgrounds, template shape and the unfolding factors. Uncertainties on the b-tagging efficiency result in a systematic uncertainty of 10% (12%) for the Z(W)+b-jet measurement. The unfolding factor is sensitive to the accuracy of the Monte Carlo modelling such as the b-jet p T spectrum and uncertainty on the opening angle between the pairs of b-quarks. The b-jet p T spectrum is important because the b-tagging efficiency depends on p T and the opening angle distribution determines the probability that the pair of b-quarks will be reconstructed as a single jet. Uncertainties on the jet energy scale results in a 4% (7%) uncertainty in the Z(W)+b-jet measurement. The results for the Z+b-jet measurement are summarised in Table 3. The Z+b-jet cross-section agrees well with the NLO MCFM prediction of 3.88 ± 0.58 pb [9,10]. The results for the W+b-jet measurement are summarised in Fig. 5. The measurements are compared to NLO predictions obtained using the 5 flavour number scheme [3] after including a non-perturbative correction to the crosssection of 0.93 ± 0.07 [6]. The correction includes a contribution from a bb pair in the final state with massive b- Fig. 5. Comparison of the measured and predicted fiducial crosssections of b-jet production in association with a W boson [6]. The cross-section is shown in exclusive bins for 1, 2 and 1+2 jets and separately for lepton flavours. The measurement is compared to NLO predictions obtained using the 5 flavour number scheme and to those from ALPGEN and Pythia. Table 3. Measured and predicted fiducial cross-section of inclusive b-jet production in association with a Z-boson [5] quarks and contribution with a massless b quark in the initial state treated using a scheme based on b quarks PDFs. The W+b-jet cross-section is systematically found to have a small excess over theoretical predictions, both in the 1 and 2-jet exclusive bins and the 1+2-jet exclusive bin. However the excess is small and the measurement is consistent at the 1.5 σ level. In conclusion, ATLAS measurements of vector boson in association with b-jets using the first 35 pb −1 of data have been presented. Despite large uncertainties, the Z+bjet cross-section is found to be consistent with NLO predictions, while a small excess at the 1.5σ level is observed in the W+b-jet measurement. Fig. 2 . 2Missing ET distribution in the W+b-jet electron channel in the combined 1-and 2-jet bin after applying the b-tagging requirement Fig. 4 . 4The secondary vertex mass distribution for b-tagged jets in selected Z+b events. The b, c and light templates have been normalised to the fit results. The small contribution from other backgrounds is also indicated[5]. Cross-section [pb] Sherpa 3.29 ± 0.04 (stat only) ALPGEN 2.23 ± 0.01 (stat only) MCFM 3.88 ± 0.58 ATLAS 3.55 +0.82 −0.74 (stat.) ±0.12 (syst) . J M Campbell, Phys. Rev. D. 7934023Campbell, J.M. et al, Phys. Rev. D. 79 (2009) 034023. . F Caola, arXiv:1107.3714Caola, F. et al, arXiv:1107.3714, (2011) . arXiv:1109.1403Phys. Lett. B. 706ATLAS Collaboration, Phys. Lett. B 706 (2012) 295- 313, arXiv:1109.1403 . arXiv:1109:1470Phys. Lett. B. 707ATLAS Collaboration, Phys. Lett. B 707 (2012) 418- 437, arXiv:1109:1470 ATLAS-CONF-2011-089S08003 8. ATLAS Collaboration. 01ATLAS Collaboration, JINST 01 (2008), S08003 8. ATLAS Collaboration, ATLAS-CONF-2011-089 (2011), https://cdsweb.cern.ch/record/1356198 . J M Campbell, arXiv:1007.3492Nucl.Phys.Proc.Suppl. Campbell, J.M. et al, Nucl.Phys.Proc.Suppl. 205-206 (2010) 10-15, arXiv:1007.3492 . John M Campbell, hep-ph/0312024Phys.Rev. 6974021Campbell, John M. et al. Phys.Rev. D69 (2004) 074021, hep-ph/0312024	[]
[ "Light tetraquark state candidates", "Light tetraquark state candidates" ]	[ "Zhi-Gang Wang \nDepartment of Physics\nNorth China Electric Power University\n071003BaodingP. R. China\n" ]	[ "Department of Physics\nNorth China Electric Power University\n071003BaodingP. R. China" ]	[]	In this article, we study the axialvector-diquark-axialvector-antidiquark type scalar, axialvector, tensor and vector ssss tetraquark states with the QCD sum rules. The predicted mass mX = 2.08 ± 0.12 GeV for the axialvector tetraquark state is in excellent agreement with the experimental value (2062.8 ± 13.1 ± 4.2) MeV from the BESIII collaboration and supports assigning the new X state to be a ssss tetraquark state with J P C = 1 +− . The predicted mass mX = 3.08 ± 0.11 GeV disfavors assigning the φ(2170) or Y (2175) to be the vector partner of the new X state. As a byproduct, we obtain the masses of the corresponding qqqq tetraquark states. The light tetraquark states lie in the region about 2 GeV rather than 1 GeV.	10.1155/2020/6438730	[ "https://arxiv.org/pdf/1901.04815v2.pdf" ]	119,454,576	1901.04815	783d8073319b020b28a8241ddc1d17c79d7f7d92	Light tetraquark state candidates 15 Jan 2019 Zhi-Gang Wang Department of Physics North China Electric Power University 071003BaodingP. R. China Light tetraquark state candidates 15 Jan 2019number: 1239Mk1238Lg Key words: Tetraquark stateQCD sum rules In this article, we study the axialvector-diquark-axialvector-antidiquark type scalar, axialvector, tensor and vector ssss tetraquark states with the QCD sum rules. The predicted mass mX = 2.08 ± 0.12 GeV for the axialvector tetraquark state is in excellent agreement with the experimental value (2062.8 ± 13.1 ± 4.2) MeV from the BESIII collaboration and supports assigning the new X state to be a ssss tetraquark state with J P C = 1 +− . The predicted mass mX = 3.08 ± 0.11 GeV disfavors assigning the φ(2170) or Y (2175) to be the vector partner of the new X state. As a byproduct, we obtain the masses of the corresponding qqqq tetraquark states. The light tetraquark states lie in the region about 2 GeV rather than 1 GeV. Introduction Recently, the BESIII collaboration studied the process J/ψ → φηη ′ and observed a structure X in the φη ′ mass spectrum [1]. The fitted mass and width are m X = (2002.1 ± 27.5 ± 15.0) MeV and Γ X = (129 ± 17 ± 7) MeV respectively with assumption of the spin-parity J P = 1 − , the corresponding significance is 5.3σ; while the fitted mass and width are m X = ((2062.8 ± 13.1 ± 4.2) MeV and Γ X = (177 ± 36 ± 20) MeV respectively with assumption of the spin-parity J P = 1 + , the corresponding significance is 4.9σ. The X state was observed in the φη ′ decay model rather than in the φη decay model, they maybe contain a large ssss component, in other words, it maybe have a large tetraquark component. In Ref. [2], Wang, Luo and Liu assign the X state to be the second radial excitation of the h 1 (1380). In Ref. [3], Cui et al assign the X to be the partner of the tetraquark state Y (2175) with the J P C = 1 +− . We usually assign the lowest scalar nonet mesons {f 0 (500), a 0 (980), κ 0 (800), f 0 (980)} to be tetraquark states, and assign the higher scalar nonet mesons {f 0 (1370), a 0 (1450), K * 0 (1430), f 0 (1500)} to be the conventional 3 P 0 quark-antiquark states [4,5,6]. In Ref. [7], we take the nonet scalar mesons below 1 GeV as the two-quark-tetraquark mixed states and study their masses and pole residues with the QCD sum rules in details, and observe that the dominant components of the nonet scalar mesons below 1 GeV are conventional two-quark states. The light tetraquark states maybe lie in the region about 2 GeV rather than lie in the region about 1 GeV. In this article, we take the axialvector diquark operators as the basic constituents to construct the tetraquark current operators to study the scalar (S), axialvector (A), tensor (T ) and vector (V ) tetraquark states with the QCD sum rules, explore the possible assignments of the new X state. We take the axialvector diquark operators as the basic constituents because the favored configurations from the QCD sum rules are the scalar and axialvector diquark states [8,9], the current operators or quark structures chosen in the present work differ from that in Ref. [3] completely. The article is arranged as follows: we derive the QCD sum rules for the masses and pole residues of the ssss tetraquark states in section 2; in section 3, we present the numerical results and discussions; section 4 is reserved for our conclusion. QCD sum rules for the ssss tetraquark states We write down the two-point correlation functions Π µναβ (p) and Π(p) firstly, Π µναβ (p) = i d 4 xe ip·x 0\|T J µν (x)J † αβ (0) \|0 ,(1)Π(p) = i d 4 xe ip·x 0\|T J 0 (x)J † 0 (0) \|0 ,(2) where J µν (x) = J 2,µν (x), J 1,µν (x), J 2,µν (x) = ε ijk ε imn √ 2 s T j (x)Cγ µ s k (x)s m (x)γ ν Cs T n (x) + s T j (x)Cγ ν s k (x)s m (x)γ µ Cs T n (x) , J 1,µν (x) = ε ijk ε imn √ 2 s T j (x)Cγ µ s k (x)s m (x)γ ν Cs T n (x) − s T j (x)Cγ ν s k (x)s m (x)γ µ Cs T n (x) , J 0 (x) = ε ijk ε imn s T j (x)Cγ µ s k (x)s m (x)γ µ Cs T n (x) ,(3) where the i, j, k, m, n are color indexes, the C is the charge conjugation matrix. Under charge conjugation transform C, the currents J µν (x) and J 0 (x) have the properties, C J 2,µν (x) C −1 = + J 2,µν (x) , C J 1,µν (x) C −1 = − J 1,µν (x) , C J 0 (x) C −1 = +J 0 (x) .(4) At the hadronic side, we can insert a complete set of intermediate hadronic states with the same quantum numbers as the current operators J µν (x) and J 0 (x) into the correlation functions Π µναβ (p) and Π(p) to obtain the hadronic representation [10,11]. After isolating the ground state contributions of the scalar, axialvector, vector and tensor tetraquark states, we get the results, Π 2,µναβ (p) = λ 2 XT m 2 XT − p 2 g µα g νβ + g µβ g να 2 − g µν g αβ 3 + · · · = Π 2 + (p) g µα g νβ + g µβ g να 2 − g µν g αβ 3 + · · · ,(5)Π 1,µναβ (p) = λ 2 XA m 2 XA − p 2 p 2 g µα g νβ − p 2 g µβ g να − g µα p ν p β − g νβ p µ p α + g µβ p ν p α + g να p µ p β + λ 2 XV m 2 XV − p 2 (−g µα p ν p β − g νβ p µ p α + g µβ p ν p α + g να p µ p β ) + · · · = Π 1 + (p 2 ) p 2 g µα g νβ − p 2 g µβ g να − g µα p ν p β − g νβ p µ p α + g µβ p ν p α + g να p µ p β +Π 1 − (p 2 ) (−g µα p ν p β − g νβ p µ p α + g µβ p ν p α + g να p µ p β ) ,(6)Π(p) = Π 0 + (p 2 ) = λ 2 XS m 2 XS − p 2 + · · · ,(7) where g µν = g µν − pµpν p 2 , the subscripts 2 + , 1 + , 1 − and 0 + denote the spin-parity J P of the corresponding tetraquark states. The pole residues λ X and λ X are defined by 0\|J 2,µν (0)\|X T (p) = λ XT ε µν , 0\|J 1,µν (0)\|X A (p) = λ XA ε µναβ ε α p β , 0\|J 1,µν (0)\|X V (p) = λ XV (ε µ p ν − ε ν p µ ) , 0\|J 0 (0)\|X S (p) = λ XS ,(8) where the ε µν and ε µ are the polarization vectors of the tetraquark states. Now we contract the s quarks in the correlation functions with Wick theorem, there are four s-quark propagators, if two s-quark lines emit a gluon by itself and the other two s-quark lines contribute a quark pair by itself, we obtain a operator GGssss, which is of order O(α k s ) with k = 1 and of dimension 10. In this article, we take into account the vacuum condensates up to dimension 10 and k ≤ 1 in a consistent way. For the technical details, one can consult Refs. [7,12]. Once the analytical expressions of the QCD spectral densities are obtained, we take the quarkhadron duality below the continuum thresholds s 0 and perform Borel transform with respect to the variable P 2 = −p 2 to obtain the QCD sum rules: λ 2 X exp − m 2 X T 2 = s0 0 ds ρ(s) exp − s T 2 ,(9) where ρ(s) = ρ S (s), ρ A (s), ρ V (s) and ρ T (s), ρ S (s) = s 4 3840π 6 − 13s m s sg s σGs 384π 4 + 2s ss 2 3π 2 − 17 ss sg s σGs 48π 2 + s 2 192π 4 α s GG π + 19m s ss 96π 2 α s GG π − 16m s ss 3 3 δ(s) + sg s σGs 2 192π 2 δ(s) − ss 2 24 α s GG π δ(s) , (10) ρ A (s) = s 4 11520π 6 − s 2 m s ss 12π 4 + s m s sg s σGs 9π 4 + 4s ss 2 9π 2 − 5 ss sg s σGs 18π 2 − s 2 2304π 4 α s GG π + 3m s ss 64π 2 α s GG π − 32m s ss 3 9 δ(s) − 2 ss 2 27 α s GG π δ(s) ,(11)+ s 2 768π 4 α s GG π − 79m s ss 1728π 2 α s GG π + 16m s ss 3 9 δ(s) − 2 ss 2 81 α s GG π δ(s) − sg s σGs 2 18π 2 δ(s) ,(12)α s GG π + 43m s ss 864π 2 α s GG π − 64m s ss 3 9 δ(s) − 4 ss 2 27 α s GG π δ(s) ,(13) and λ X A/V = m X A/V λ X A/V . We derive Eq.(9) with respect to τ = 1 T 2 , then obtain the QCD sum rules for the masses of the tetraquark states through a fraction, m 2 X = − s0 0 ds d dτ ρ(s) exp (−τ s) s0 0 dsρ(s) exp (−τ s) .(14) 3 Numerical results and discussions at the energy scale µ = 1 GeV [10,11,13], and choose the M S mass m s (µ = 2 GeV) = 0.095 ± 0.005 GeV from the Particle Data Group [14], and evolve the s-quark mass to the energy scale µ = 1 GeV with the renormalization group equation, furthermore, we neglect the small u and d quark masses. We choose suitable Borel parameters and continuum threshold parameters to warrant the pole contributions (PC) are larger than 40%, i.e. PC = s0 0 ds ρ (s) exp − s T 2 ∞ 0 ds ρ (s) exp − s T 2 ≥ 40% ,(15) and convergence of the operator product expansion. The contributions of the vacuum condensates D(n) in the operator product expansion are defined by, D(n) = s0 0 ds ρ n (s) exp − s T 2 s0 0 ds ρ (s) exp − s T 2 ,(16) where the subscript n in the QCD spectral density ρ n (s) denotes the dimension of the vacuum condensates. We choose the values \|D(10)\| ∼ 1% to warrant the convergence of the operator product expansion. In Table 1, we present the ideal Borel parameters, continuum threshold parameters, pole contributions and contributions of the vacuum condensates of dimension 10. From the Table, we can see that the pole dominance is well satisfied and the operator product expansion is well convergent, we expect to make reliable predictions. We take into account all uncertainties of the input parameters, and obtain the values of the masses and pole residues of the ssss tetraquark states, which are shown explicitly in Fig.1 and Table 1. In this article, we have assumed that the energy gaps between the ground state and the first radial state is about 0.6 GeV [15]. In Fig.1, we plot the masses of the scalar, axialvector, tensor and vector ssss tetraquark states with variations of the Borel parameters at larger regions than the Borel windows shown in Table 1. From the figure, we can see that there appear platforms in the Borel windows. The predicted mass m X = 2.08 ± 0.12 GeV for the axialvector tetraquark state is in excellent agreement with the experimental value (2062.8 ± 13.1 ± 4.2) MeV from the BESIII collaboration [1], which supports assigning the new X state to be an axialvector-diquark-axialvector-antidiquark type ssss tetraquark state. The predicted mass m X = 3.08 ± 0.11 GeV for the vector tetraquark state lies above the experimental value of the mass of the φ(2170) or Y (2175), m φ = 2188±10 MeV, from the Particle Data Group, and disfavors assigning the φ(2170) or Y (2175) to be vector partner of the new X state. If the φ(2170) have tetraquark component, it maybe have color octet-octet component [16]. As a byproduct, we obtain the masses and pole residues of the corresponding qqqq tetraquark states, which are shown in Table 1. The present predictions can be confronted to the experimental data in the future. Now we perform Fierz rearrangement to the currents both in the color and Dirac-spinor spaces, J 0 = 2ssss + 2siγ 5 ssiγ 5 s +sγ α ssγ α s −sγ α γ 5 ssγ α γ 5 s , J 1,µν = √ 2 isssσ µν s −sσ µν γ 5 ssiγ 5 s + iε µναβs γ α γ 5 ssγ β s , J 2,µν = 1 √ 2 2sγ µ γ 5 ssγ ν γ 5 s − 2sγ µ ssγ ν s + 2g αβs σ µα ssσ νβ s + g µν ssss +siγ 5 ssiγ 5 s +sγ α ssγ α s −sγ α γ 5 ssγ α γ 5 s − 1 2s σ αβ ssσ αβ s . The diquark-antidiquark type currents can be re-arranged into currents as special superpositions of color singlet-singlet type currents, which couple potentially to the meson-meson pairs or molecular states, the diquark-antidiquark type tetraquark states can be taken as special superpositions of meson-meson pairs, and embodies the net effects. The decays to their components are Okubo-Zweig-Iizuka supper-allowed, we can search for those tetraquark states in the decays, X S → η ′ η ′ , f 0 (980)f 0 (980) , φ(1020)φ(1020) , X A/V → f 0 (980)h 1 (1380) , φ(1020)η ′ , φ(1020)φ(1020) , X T → η ′ η ′ , f 0 (980)f 0 (980) , φ(1020)φ(1020) .(18) Conclusion In this article, we construct the axialvector-diquark-axialvector-antidiquark type currents to interpolate the scalar, axialvector, tensor and vector ssss tetraquark states, then calculate the contributions of the vacuum condensates up to dimension-10 in the operator product expansion, and obtain the QCD sum rules for the masses and pole residues of those tetraquark states. The predicted mass m X = 2.08 ± 0.12 GeV for the axialvector tetraquark state is in excellent agreement with the experimental value, m X = (2062.8 ± 13.1 ± 4.2) MeV, from the BESIII collaboration and supports assigning the new X state to be an axialvector-diquark-axialvector-antidiquark type ssss tetraquark state. The predicted mass m X = 3.08 ± 0.11 GeV for the vector tetraquark state lies above the experimental value of the mass of the φ(2170), m φ = 2188 ± 10 MeV, from the Particle Data Group, and disfavors assigning the φ(2170) to be the vector partner of the new X state. As a byproduct, we also obtain the masses and pole residues of the corresponding qqqq tetraquark states. The present predictions can be confronted to the experimental data in the future. take the standard values of the vacuum condensates qq = −(0.24 ± 0.01 GeV) 3 , qg s σGq = m 2 0 qq , m 2 0 = (0.8 ± 0.1) GeV 2 , ss = (0.8 ± 0.1) qq , sg s σGs = m \| m X (GeV) λ X (10 −2 Figure 1 : 1The masses with variations of the Borel parameters T 2 , where the A, B, C and D denote the scalar, axialvector, tensor and vector tetraquark states, respectively. AcknowledgementsThis work is supported by National Natural Science Foundation, Grant Number 11775079. . M Ablikim, arXiv:1901.00085M. Ablikim et al, arXiv:1901.00085. . L M Wang, S Q Luo, X Liu, arXiv:1901.00636L. M. Wang, S. Q. Luo and X. Liu, arXiv:1901.00636. . E L Cui, H M Yang, H X Chen, W Chen, C P Shen, arXiv:1901.01724E. L. Cui, H. M. Yang, H. X. Chen, W. Chen and C. P. Shen, arXiv:1901.01724. . F E Close, N A Tornqvist, J. Phys. 28249F. E. Close and N. A. Tornqvist, J. Phys. G28 (2002) R249. . C Amsler, N A Tornqvist, Phys. Rept. 38961C. 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[ "Dual approach for object tracking based on optical flow and swarm intelligence", "Dual approach for object tracking based on optical flow and swarm intelligence" ]	[ "Mr Rajesh Misra \nS.A.Jaipuria College\n10 Raja Naba Krishna StreetShobhabazarKolkataIndia\n", "DrS Kumar \nIndian Statistical Institute\n203 B.T.Road, Kolkata-108India\n", "Ray \nIndian Statistical Institute\n203 B.T.Road, Kolkata-108India\n" ]	[ "S.A.Jaipuria College\n10 Raja Naba Krishna StreetShobhabazarKolkataIndia", "Indian Statistical Institute\n203 B.T.Road, Kolkata-108India", "Indian Statistical Institute\n203 B.T.Road, Kolkata-108India" ]	[]	Though object tracking is a very old problem still there are several challenges to be solved;for instance, variation of illumination of light, noise, occlusion, sudden start and stop of moving object, shading etc.In this paper we propose a dual approach for object tracking based on optical flow and swarm Intelligence.The optical flow based KLT tracker, tracks the dominant points of the target object from first frame to last frame of a video sequence;whereas swarm Intelligence based PSO tracker simultaneously tracks the boundary information of the target object from second frame to last frame of the same video sequence.The boundary information of the target object is captured by the polygonal approximation of the same.The dual approach to object tracking is inherently robust with respect to the above stated problems.We compare the performance of the proposed dual tracking algorithm with several benchmark datasets and in most of the cases we obtain superior results. of tracking-by-detection with kernels[CSK] [21] Henriques et.al use theory of circulant matrices with Fast Fourier Transformation to detect and track the moving object. Real-time compressive tracking[CT] [22] zhang et.al create an appearance model based on feature extracted from multi-scale image space and compute a sparse measurement matrix. Later using that sparse matrix they compress foreground and background targets, and perform tracking by using naive-Bayes classifier. In [23] Moudgil et.al provide a benchmark dataset for long duration video sequence which they name as 'Track Long and Prosper(TLP)'. This dataset is important because most tracking algorithms work well in short sequences but drastically failed on long challenging video sequence. This dataset contains 50 long time running video nearly 400 minutes. This [23] includes some of the recent tracking algorithms: Learning multi-domain convolutional neural networks for visual tracking [24] Nam et.al use convolutional Neural Network which is composed of different domain specific layers which are trained to capture different parts of moving object to track. Fully-convolutional siamese networks for object tracking [25] Bertinetto et.al create a fully-convolutional Siamese Network which is trained with ILSVRC15 dataset for video tracking. Crest: Convolutional residual learning for visual tracking [26] Song et.al reformulate Discriminative correlation filters as a one-layer convolutional neural network and apply residual learning to take appearance changes into consideration. Actiondecision networks for visual tracking with deep reinforcement learning [27] Yun et.al propose a tracking algorithm which sequentially pursue actions learned by deep reinforcement learning. MEEM: robust tracking via multiple experts using entropy minimization [28] Zhang et.al propose multi-expert restoration method for problem of drifting of model in on line tracking by creating an expert ensemble where best expert is selected based on minimum entropy criteria to correct undesirable model updates. Bio-inspired based methods are effective tools for object tracking and are given extensive attention in past few decades [29] Among other Bio-inspired methods like Genetic Algorithm (GA),Ant Colony optimization(ACO), Particle	null	[ "https://arxiv.org/pdf/1808.08186v2.pdf" ]	52,090,118	1808.08186	48950f0859e96adcfa676eb332b9fa7c8c877391	Dual approach for object tracking based on optical flow and swarm intelligence 1 Aug 2021 Mr Rajesh Misra S.A.Jaipuria College 10 Raja Naba Krishna StreetShobhabazarKolkataIndia DrS Kumar Indian Statistical Institute 203 B.T.Road, Kolkata-108India Ray Indian Statistical Institute 203 B.T.Road, Kolkata-108India Dual approach for object tracking based on optical flow and swarm intelligence 1 Aug 2021Preprint submitted to Engineering Application of Artificial Intillegence August 3, 2021(Mr.Rajesh Misra), [email protected] (Dr.Kumar S Ray)Dual trackingDominant PointKLT(Kanade-Lucas-Tomasi) trackerPSO trackerParticle Swarm Optimization (PSO)Polygonal approximation Though object tracking is a very old problem still there are several challenges to be solved;for instance, variation of illumination of light, noise, occlusion, sudden start and stop of moving object, shading etc.In this paper we propose a dual approach for object tracking based on optical flow and swarm Intelligence.The optical flow based KLT tracker, tracks the dominant points of the target object from first frame to last frame of a video sequence;whereas swarm Intelligence based PSO tracker simultaneously tracks the boundary information of the target object from second frame to last frame of the same video sequence.The boundary information of the target object is captured by the polygonal approximation of the same.The dual approach to object tracking is inherently robust with respect to the above stated problems.We compare the performance of the proposed dual tracking algorithm with several benchmark datasets and in most of the cases we obtain superior results. of tracking-by-detection with kernels[CSK] [21] Henriques et.al use theory of circulant matrices with Fast Fourier Transformation to detect and track the moving object. Real-time compressive tracking[CT] [22] zhang et.al create an appearance model based on feature extracted from multi-scale image space and compute a sparse measurement matrix. Later using that sparse matrix they compress foreground and background targets, and perform tracking by using naive-Bayes classifier. In [23] Moudgil et.al provide a benchmark dataset for long duration video sequence which they name as 'Track Long and Prosper(TLP)'. This dataset is important because most tracking algorithms work well in short sequences but drastically failed on long challenging video sequence. This dataset contains 50 long time running video nearly 400 minutes. This [23] includes some of the recent tracking algorithms: Learning multi-domain convolutional neural networks for visual tracking [24] Nam et.al use convolutional Neural Network which is composed of different domain specific layers which are trained to capture different parts of moving object to track. Fully-convolutional siamese networks for object tracking [25] Bertinetto et.al create a fully-convolutional Siamese Network which is trained with ILSVRC15 dataset for video tracking. Crest: Convolutional residual learning for visual tracking [26] Song et.al reformulate Discriminative correlation filters as a one-layer convolutional neural network and apply residual learning to take appearance changes into consideration. Actiondecision networks for visual tracking with deep reinforcement learning [27] Yun et.al propose a tracking algorithm which sequentially pursue actions learned by deep reinforcement learning. MEEM: robust tracking via multiple experts using entropy minimization [28] Zhang et.al propose multi-expert restoration method for problem of drifting of model in on line tracking by creating an expert ensemble where best expert is selected based on minimum entropy criteria to correct undesirable model updates. Bio-inspired based methods are effective tools for object tracking and are given extensive attention in past few decades [29] Among other Bio-inspired methods like Genetic Algorithm (GA),Ant Colony optimization(ACO), Particle Introduction Object Tracking employs the idea of following an object as long as its move- Optical flow is the pattern of motion of images between two consecutive frames generated by movement of object or camera. The resultant vector of the optical flow is the displacement vector containing position of pixels from first frame to second frame. This optical flow provides a good amount of motion information of moving object, and thus encourage researchers to apply that information in moving object detection as well as tracking. There exist several optical flow methods, like Lucas--Kanade method [1], Horn--Schunck method [2], Buxton-Buxton method [3], Black-Jepson method [4] etc. Among all these methods on optical flow [5], Horn-Schunck and Lucas--Kanade are more popular than others. Both these methods have their own merits and demerits. Shin et.al [6] proposed optical flow based object tracking under non-prior training active feature model by localizing of an object-of-interest then applying correction based model using spatio-temporal information and then applying NPT-AFM framework. In object tracking KLT method has been applied by Sundaram et.al [7] for multiple point tracking in a parallel environment. Chen et.al [8] perform segmentation of video object and apply optical flow method to track each segments. Cui .et .al [9]proposed a probabilistic fusion approach between low and high dimensional tracking approaches. Schwarz et.al [10] use optical flow in subsequent intensity image frames to get the motion information about the moving object body and apply graph based representation to track the entire object. Tsutsui [11] proposed an optical flow based object tracking method where multiple camera has been used instead of single camera. If the object is occluded by some other object different camera images helps the method to estimate the position of the object.In 2012 Liu et.al [12] propose a fusion approach for object tracking of low and high dimensional tracking sampling information. In 2006, Brox et.al show a way of tracking 3D pose by optical method utilizing contour and flow based constraints in [13]. Aslani et.al [14] used optical flow together with some image processing method to estimate the position of the object in consecutive frames, and using that positional pixel values they track the whole object. Kale et.al [15] use optical flow to compute motion vector which provides an estimation of object position in consecutive frames. Though optical flow method has been applied extensively in object detection and tracking still there is no method which can extract perfect flow of data. Thus, the use of optical flow in object tracking is still a widely open problem [16]. In [17] Carlo tracking method with particle filter. Exploiting the circulant structure Swarm Optimization (PSO) emerges real fast because of its efficient, robust and quick convergence. Some of the earlier works successfully considered tracking problems using PSO. Particle Swarm Optimization is applied by Zheng et.al [30], [31] on high dimensional feature space for searching optimal matching in Haar-Like features detected by a pre-defined classifier set. Xiaoqin et.al [32] calculate temporal continuity between two frames and use that information for swarm particle to fly and track that information. Vijay et.al [33] construct Human Body Model as a collection of truncated cones and numbering those cones and PSO cost function checks how well a pose matches with data taken from multiple cameras. Multiple people tracking is considered by Chen et.al [34] where target object is modeled by feature vector and then PSO particles search the search space for optimal matching. Fakheredine shows [35] the use of multiple swarms for multiple parts of object during object tracking. Those swarms share information with each other to make tracking of object as a whole. Multiple object tracking is also considered by Chen-Chien et.al [36] using PSO, construct a feature model using gray-level histogram and apply PSO particles to track the difference between gray level histogram information of consecutive frames in a video sequence. Bogdan [37] represents an approach where object is represented by image template and a covariance matrix is formed on that. Using similarity measure PSO tracks the difference between the movement of object and target template [38]. We Compare our proposed approach with some other PSO based tracking algorithm: Multiple object tracking using particle swarm optimization [36] Hsu et.al first create a grey-level histogram feature model and then distribute PSO particles where target object used as fitness function. Real-Time Multiview Human Body Tracking using GPU-Accelerated PSO [? ] Boguslaw et. al show that movement is tracked by a 3D human model in the pose described by each particle and then rasterizing it in each particle's 2D plane. Hierarchical Annealed Particle Swarm Optimization for Articulated Object Tracking [39] Xuan et.al show articulate object tracking by decomposing the search space into subspaces and then using particle swarms to optimize over these subspaces hierarchically. Monocular Video Human Motion Tracking based on Hybrid PSO [40] Ben shows tracking human pose in monocular video human motion by using hybrid PSO method. Object tracking using Particle Swarm Optimization and Earth mover's distance [41] shading etc. we observe that the deep learning method for object tracking don't produce satisfactory results and in many cases deep neural network(DNN) requires further training under the new environment and unknown objects [42]. In [42] Simpson states that "It is an embarrassing fact that while deep neural networks(DNN) are frequently compared to the brain, and even their performance found to be similar in specific static tasks, there remains a critical difference; DNN do not exhibit the fluid and dynamic learning of the brain but are static once trained. For example, to add a new class of data to a trained DNN it is necessary to add the respective new training data to the preexisting training data and re-train (probably from scratch) to account for the new class. By contrast, learning is essentially additive in the brain -if we want to learn a new thing, we do". Based on such observation [42] and a critical appraisal [43] and also based on our recent experimental studies on VGGNet we categories DNN approach as a representation of crystallized intelligence [44], [45] of the network under learned or accumulated knowledge and has low capability of handling unknown environment specially under unknown objects. We suggest that for new environment with unknown objects DNN should have an added feature of fluid intelligence with some working memory which can handle novel or abstract problem solving environment [46], [47], [44], [48], [49], [45], [50], [51], [52]. However all such challenging issues and several other proposals [53], [54], [55], [56], [57], [58], [59], [60], [61] should be throughly reevaluated before we come to any conclusion. Such issues should be separately considered elsewhere as an independent work. In this paper we essentially try to extract the merit of fusion between KLT tracker and PSO based tracker. Such fusion is considered to supplement each other in an intelligent fashion so that dual trackers become very simple, very robust and cost effective under variable background and static background and very much capable of handling the challenges of object tracking which cannot be tackled by DNN based tracking algorithms as stated above. The proposed dual tracking algorithm can successfully tracks object for short video sequence as well as long challenging video sequence. In case of unknown object , dual tracking approach simply needs to recalculate the dominant points on the contour of the unknown target object(objects) and no need to spend huge time to learn/train the unknown environment with unknown object form the beginning of tracking of the target object as we have seen in case of DNN based tracking algorithms. The KLT trackers based on optical flow concepts tracks the dominant points of the target objects from the first frame to last frame of the video sequence. Tracking of dominants points by KLT tracker is supplemented by swarm intelligence based PSO tracker from frame 2 to last frame. Swarm Intelligence based PSO (Particle Swarm Optimization) tracker basically tracks the boundary information of the target object from frame 2 to last frame. The flexibility of our approach is that it can be successfully applicable to variable background as well as static background. The basic tracking sequences of the propose dual tracking approach is as follows; In the first frame of the video sequence we obtain the dominant points of the target object and start tracking it by KLT tracker till the last frame. In frame 2 of the same video sequence the boundary of the target object is polygonally approximated for the first time. An environment of multiswarms is generated and an annular ring(strip) of swarms is formed within which the approximated ploygon is embedded.At frame -3 onwards the shape of the annular ring (strip) of the multiswarms changes simply because the shape of the dynamically generated polygon which, at each frame, continuously captures the boundary information of the target object, changes due to the movement of the target object which is a non-rigid body in general. The newly generated polygon embedded over the newly generated annular ring(strip) of the multiswarms is tracked from frame -3 by PSO tracker along with KLT tracker. The above process of dual tracking continues till the last frame with dynamic change of shape of the approximated polygon and the change of shape of the annular ring(strip) at each frame of the video sequence. In course of tracking if there is any loss of dominant points due to some sort of unpredictable disturbances then the tracking procedures by KLT and PSO are disturbed. In that case instead of recomputation of dominant points, we reinitialize the missing dominant points by some heuristic approach which essentially exploits the intelligence level of swarms. Similar to the reinitialization of the missing dominant points sometimes positions of the particles of the individual swarm may need to be reinitialized due to its distraction from the individual swarm by some process of disturbances. When all the swarms around the boundary of the target object reach the optimal solution a bounding box is generated around the target object based on particles final positions. This entire method of tracking uses only one feature which is basically dominant points on the contour of the target object and other information of the target object boundary is captured by the annular ring(strip) of multiswarms within which polygonally approximated target object is embedded. Note that the notion of using dominant points on the contour of a target object as good features for object tracking is basically derived from the concept of interest points as proposed by Shi et.al [62] and Tomasi et.al [1]. In our approach, instead of searching for interest points of an object for tracking we directly compute dominant points on the contour of the target object to be tracked and thereby reduces the search complexity of KLT algorithm for object tracking. In sec 2.3.3 we experimentally demonstrate that the set of dominant points on the contour of the target object is basically a subset of interest points.Further note that the use of dominant points as good features for object tracking is an important and unique concept which is not used by classical KLT algorithm for object tracking. The robustness of the proposed dual tracking algorithm, under several existing challenges of object tracking as stated earlier, is verified and established through several experimental studies on benchmark datasets [17], [23], [63]. Another specialty of the proposed dual tracking algorithm is its robustness under short video sequence as well as long challenging video sequence where most of the existing classical approaches fail [23]. Note that in the proposed dual tracking algorithm the KLT tracker for tracking the dominant points of the target object is continuously supplemented by PSO tracker from frame-2 to last frame. And due to embedding of the target object approximated by polygon in the annular ring(strip) of multiswarms,the target object is tightly captured throughout the tracking sequence by the multiswarms environment and there is no loss of meaningful information about the target object during tracking. Hence the proposed dual tracking algorithm is inherently robust. There are several striking features of the proposed dual tracking algorithm.The overall performance of the proposed dual tracking algorithm, with respect to several benchmark datasets, are very much competitive and in most of the cases superior than the others. The paper is organized as follows: Section 2 discusses the basic concepts and tools and techniques required for dual tracking algorithm. Section 3 essentially deals with salient features of the proposed dual tracking algorithm.Section 4 pictorially describes the proposed dual tracking algorithm. Section 5 pro-vides the pseducode and complexity analysis of the proposed dual tracking algorithm.Section 6 provides detail experimental studies on 3 benchmark datasets and also provides some analysis and performance measure of the proposed dual tracking algorithm. Section 7 provides Conclusion and future work. This dual function of tracking makes the trackers very much robust with respect to the above stated problems. In our approach, in the first frame of the video sequence we calculate the dominant points of the target object and start tracking it till the last frame. From frame 2 of the same video sequence the boundary information of the target object is captured by a dynamically generated polygon of the target object. The polygonal approximation of the target object at each frame is achieved by joining two consecutive dominant points on the target object by a straight line segment. In frame -2 of the same video sequence a group of particles is distributed randomly over the image search space. This particles form swarm over each line segment of the dynamically generated polygon of the target object. [64], [65]. If the population of the particles at frame-2 is very large the computational complexity of the entire algorithm may increase. Keeping this in mind we have to select the population of particles at frame 2.The flexibility of our approach for dual tracking is that it can be successfully applicable to variable background as well as static background. The tools and techniques used for implementing the basic concepts of dual tracking are discussed in the following - Dominant Point Detection For the detection of the dominant point on the contour of the target object we use the methods [66], [67] and [68] and [17]. We first perform contour tracking of the target object to find the Chain Code based on Freeman's Chain Code [69]. Freeman Chain code gives us list of pixels around object body. Among those pixels we eliminate linear points(pixels), as those points(pixels) do not provide us any significant curvature information. For elimination of linear points(pixels) we consider the following rule - if C i−1 = C i then point P i is a linear point,(1) where C i−1 is the previous chain code value and C i is the current one, on the point P i . For each group of breakpoints we calculate k-Cosine values for each of them and apply the following rule -Let us start with k =1 to form a group. Increase the value of k by 1 until we reach all breakpoints on that group. k i = k if cos ik = max{cos ij \|j = K min ...K max }, for j = 1, 2, 3...n.(2) We chose dominant point as those points which are max k−Cosine values, i.e. D i = max(cos(P i )).(3) Dominant Point Calculation : . Thus the entire procedure for calculating dominant points can be summarized as follows; • Use Freeman Chain Code for performing contour tracking.Get those pixels and store them in a file. • Form the stored pixels eliminate linear points using equation (1). Save them in a file and call breakpoints. • Perform K -Cosine for each of the breakpoints using Eq-(2). • Select those points as dominant points which has max k-cosine values and collect a set of Dominant points as per Eq-(3). M in(λ 1 , λ 2 ) = λ(4) where, λ 1 , λ 2 are two eigenvalues and λ is a predefined threshold. Rather applying a separate algorithm for finding good interest points which satisfy the above equation-(4), we consider dominant point as our interest point. Let us define the image gradient as follows - G =   G x G y   .(5) We, consider the product of gradient and its transpose as follows - GG T =   G 2 x G x G y G x G y G 2 y   .(6) If we integrate the matrix defined above over the area W(selected window),we get. Z = W   G 2 x G x G y G x G y G 2 y   W dx.(7) Z is a 2x2 matrix containing texture information along X and Y axis. Analyzing the eigenvalues of the matrix Z we get the W, which is window of pixels that Dominant points as subset of interest points In section 2.2 we state that dominant point holds maximum curvature information on the contour of a target object and provides enough texture for tracking. In this subsection we further clarify this concept through a simple experiment, as example, that dominant points are the subset of interest points which are the key elements of KLT tracking algorithm. In Figure- This amount can be defined as the displacement d = (∆x, ∆y) and the main goal of tracking is to calculate d. I(x, y, t + τ) = I(x − ∆x, y − ∆y, t)(8) Calculation Feature displacement Now we have basic information to solve the displacement d mentioned above. The solution is explained in [70]. According to [70] -we can calculate displacement d from from image frame I to image frame J.Thus we obtain- = W J(x + d 2 ) − I(x − d 2 ) 2 W (x)dx (9) where x = [x y] T , the displacement d = [dx dy] T ,J( ) ≈ J(p) + ( x − a x ) δj δx (a) + ( y − a y ) δj δy (a)(10) where, = [δx δy] T . Following the derivation, we let (x + d 2 ) = .To get the final derivation, (11) In continuation of equation-(11) we calculate, δ δd = 2 J(x + d 2 ) − I(x − d 2 ) δJ(x + d 2 ) δd − δI(x − d 2 ) δd W (x)dxδ δd ≈ W J(x) − I(x) + g T d g(x)W (x)dx (12) where, g = δ δx ( I+J 2 ) δ δy ( I+J 2 ) T . To calculate the displacement d , we need to set the derivative 0. δ δd = 0. Solving further, we get a simplified equation - Zd = e.(14) where, Z is the 2x2 matrix : Z = W g(x)g T (x)W (x)dx and e is the 2x1 vector: e = W I(x) − J(x) g(x)w(x)dx. So the displacement d is the solution of equation-(14). KLT algorithm We summaries the KLT algorithm as follows - Step 1: Find the dominant points which satisfy min(λ 1 , λ 2 ) > λ(see equation -(4). Step 2: For each dominant point compute displacement to next frame using the Lucas-Kanade method (see equation - (14)). Step 3: Store displacement of each dominant point, update the position of the dominant point. Step 4: Go to step 2 until all dominant points are exhausted. Particles Swarm Optimization (PSO) method for tracking. In 1995 James Kennedy and Russell Eberhart proposed an evolutionary algorithm that creates a ripple among Bio-inspired algorithms. This particular algorithm is called Particle Swarm Optimization (PSO) [71]. In a simple term it is a method of optimization for continuous non-linear function. This method is influenced by swarming theory form biological world like fish schooling, bird swarming etc [72]. PSO is effectively applied to the problems in which each solution of that problem can be considered as a point in a solution space. Each point in the solution space is called one particle. Analogically suppose there is a food source and a swarm of birds tries to reach that food source. Every bird, tries by its own choice to reach there. Whoever is reached. or nearly reached to that food source share that information with other birds who are close neighbor.As a ripple in water information flows among entire swarm of birds and every bird synchronously update their velocity and position. If it gets better position in terms of nearest position to the food source. As a result after certain period of time entire swarm eventually gathers to the food source. Every solution considered as particle computes its value based on some cost function, until it satisfies certain criterion known as stopping condition. It keeps updating its velocity and position, provided its neighbor has better solution. Position and Velocity are two associated terms in Particle Swarm Optimization. Position of every particle is calculated by particle's own velocity. Let X i (t) denote position of particle i in the search space at time t. Position updation formula is as follows - X i (t + 1) = X i (t) + V i (t + 1)(15) where, V i (t + 1) is the velocity of particle i at time (t+1), which is computed based on this following formula- V i (t) = V i (t − 1) + C 1 .R 1 (P LB (t) − X i (t − 1)) + C 2 .R 2 (P GB (t) − X i (t − 1))(16) where, C 1 ,C 2 represent the relative influence on social and cognitive components respectively. They are also known as learning rates and are often set to same constants value, to give each component equal weight. called "inertia weight" or "w". After addition of inertia weight the Eq (14) becomes as follows - V i (t) = w * V i (t − 1) + C 1 .R 1 (P LB (t) − X i (t − 1)) + C 2 .R 2 (P GB (t) − X i (t − 1))(17) This inertia weight helps to balance local and global search abilities. Small weight means local search and larger weight means global search [73]. Pseudo code of the basic PSO algorithm is given in appendix. In this paper the PSO based tracker tracks the dynamically approximated polygon of the target object and continuously supplements the tracking of the dominant points of the target object by KLT. Setting PSO parameters and Initialization Because of dynamic nature, setting PSO parameters to right value is a crucial task. Below we discuss some of the major parameters. we increase the population size the computational complexity of the PSO tracker will be increased. It is also obvious that more the length of the line segment more the particles will converge on that to form a swarm as per the PSO algorithm. • Position and Velocity initialization -According to PSO methodology we need to initialize the position and velocity of every particle of the swarm. This position of particle for each swarm will be inside the search space and randomly defined. In our case we first consider the range of the image space within which the target object are lying. If the image space is represented by [x,y] range for each frame then we first select some values for V x V y as velocity component in x-direction and y-direction to be [74]; 1 ≤ V x , V y ≤3 where V x and V y denotes velocity towards X and Y direction. Velocity signifies how far a single particle will jump, as we are working on image pixels. It cannot be negative value or fractional value and also setting high value is not a practical approach as the particles work in close vicinity on the dominant points. • Local best value of Particle i (P lbesti ) -local best value of an individual particle in a swarm indicates its current best position it achieved to converge on the target line segment between two consecutive dominant points. We initialize each particle's Plbest value with its initial position inside the search space [x,y] which is randomly defined at the very beginning as stated above. Latter it will be modified according to the Plbest updating rule. • Global best value of Particle i (P gbesti ) -In a particular swarm, the particle which holds the best position such as close to the line segment between two consecutive dominant points is considered as global particle and its position is P gbesti . Each particle first compute the perpendicular distance from line segment connected by dominant points. The particle which hold minimum distance considered as P gbesti . P gbesti = min      (X D1 − X i ) 2 + (Y D1 − Y i ) 2 (X D2 − X i ) 2 + (Y D2 − Y i ) 2(18) Polygonal approximation of the target object For polygonal approximation of the target object we draw small line segments between two consecutive dominant points of the target object. Let us consider two dominant points D 1 and D 2 which are calculated using equation- (3). The path between this two points is a small line segments joining the said two points. There could be infinitely many curves( not straight line segments) that may pass through the said two dominant points, but in this paper we consider Euclidean Distances between the two said points. In Cartesian coordinate, D 1 (X 1 , Y 1 ) and D 2 (X 2 , Y 2 ) are the two points in Euclidean space. The distance between this two points is calculated as follows - D 1 D 2 = (X 2 − X 1 ) 2 + (Y 2 − Y 1 ) 2 .(19) In the following we illustrate this phenomenon using an arbitrary curve as shown Fitness Function for PSO tracker Every PSO model is based on some cost function. Each particle of the swarm computes that fitness function in each iteration to confirm whether it converges to the final solution or not. In this paper, our cost function is the distance of the particle i to the small line segment which is a part of the approximated polygon of the arbitrary curve C. Based on figure-(6), we draw figure-(7). Here we have a particle P 1 and the small line is L 1 . We compute the distance from the point P 1 to the line L 1 . As L 1 passes through two dominant points D 1 (X 1 , Y 1 ) and D 2 (X 2 , Y 2 ) then the distance from the point P 1 (X 0 , Y 0 ) is - Dist(D 1 , D 2 , P 1 ) = \|(Y 2 − Y 1 ) * X 0 − (X 2 − X 1 ) * Y 0 + X 2 * Y 1 − Y 2 * X 1 \| (Y 2 − Y 1 ) 2 + (X 2 − X 1 ) 2 .(20) The denominator is the length between D 1 and D 2 . Numerator is the twice the area of triangle with its vertices's at 3 points D 1 , D 2 , P 1 . For Every particle we compute the distance from the particle to the small line joining two dominant points as stated above and if the distance is in the acceptable range iteration stops else this procedure continues. Formation of Multiswarms Once the task of constructing the polygon of the target object is completed, for the first time,at frame 2 of the video sequence, we distribute particles over the entire image space. Note that the the population of the particles is a heuristic parameter which depends on the need of the problem and which has several options as stated in [75], [76], [77]. P gbest and P lbest are updated as -N ew P gbest (P i ) = min(P erDist(D 1 , D 2 , P i ))∀i (21) N ew P lbest (P i ) =            P erDist(D 1 , D 2 , P i ), PerDist(D 1 , D 2 , P i ) < Previous PerDist(D 1 , D 2 , P i ). P reviousP erDist(D 1 , D 2 , P i ), Otherwise.(22) Reinitialization of the particle of the individual swarm At the time of updating the position the particles of the individual swarms, instead of converging over the small line segments of the changed polygon, the particle(particles) may be distracted from the said line segments to a far away distance even after several iteration of updation. In that case we need to reinitialize the particle(particles). Reinitializing particles over entire image space certainly feasible but not a practical idea. For further illustration see section-3. Bounding Box formulation To identify tracked target object usually a rectangular bounding box is utilized. There are some pre-defined algorithms exist for this purpose, but here we design our own bounding box based on PSO particle position which will best suite our object tracking algorithm. The main idea is whenever all particles in all swarms successfully converge for a particular image frame we find p number of particles which have smallest X -direction and smallest Y-Direction. These particle are close to (0,0) in our image space. These p values can be the first 10 particles with minimum values in X-direction and Y-directions. This choice of number of particles entirely depends on application. As per our experimental experience this number of particles should lie from 10 to 20 particles with smallest X,Y direction. We take an average of these p-points, which is the starting point for bounding box formation. Let us consider a particle q, which is calculated as follows - Length is the euclidean distance from point q(q x , q y ) to point len(l x , l y ) as shown below. Length(L) = (q x − l x ) 2 + (q y − l y ) 2 .(25) Similarly, for breadth calculation, first select b number of particle which have minimum X-direction values and maximum Y-direction values. bre = b i=1 b i b ∀i = 1, 2, 3...b.(26) Breadth is the euclidean distance from point q(q x , q y ) to point bre(b x , b y ) as follows. Breadth(B) = (q x − b x ) 2 + (q y − b y ) 2 .(27) Once we get the 3 parameters; length(L), breadth(B) and starting point q, using equation- (28) we construct the bounding box as shown in Fig-9. q = (q x , q y ) ql = (q x , q y+L ) qb = (q x+B , q y ) qlb = (q x+B , q y+L )                  four boundary position formula. 3. Salient features of the proposed algorithm. Re-initialization of missing dominant points Pictorial Illustration of Reinitialization of missing dominant points We explain reinitialization process of missing Dominant points using an example. Let's consider a curvature C whose start (s) and end (e) points are dominant points. We track these dominant point using KLT method.In the fol- We can see that, in figure-(11) KLT missed tracking of dominant point s. We also assume that some of the swarms may be lost during tracking because they are distributed over the path which originates from point s. This phenomenon is shown by yellow dots near to the lost dominant point. In figure-(12) we show how re-initialization happens. The PSO algorithm tracks the curvature of the object which is approximated by a straight line between s and e. After few frames when we observe that s is not moving as its pixel position is not changing, we then consider that KLT has lost point s. This heuristic approach to design a tracker is basically an attempt to extract the element of intelligence of a swarm. Reinitialization of the particles of the swarms Reinitialization of particle(s) of individual swarm is sometime required as it is inherent in nature of PSO that few particles are too diverged from their desired position and even after several updation may not bring them towards their goal. In our case it is also possible that some particles are too far away from curvature boundary and after a finite number of iteration they still unable to converge. Then we need to reinitialize those particle. Reinitializing particles over entire image space certainly feasible but not a practical idea, because it again may diverge. So we have a better possibility to converge by assigning the position of the diverged particle on the current position of dominant point. Diagrammatically we can represent this phenomenon in figure- (14) Let us consider two swarm particles start diverging at frame number f and we detect this after few more frames are processed. Let say at frame (f + t), we find two particles are diverged. In frame (f+t+1) we took action about its repositioning. Lots of research works have been done about repositioning of diverged particle [64], [78], [75]. In [78], At frame number (f+t+1), we simply consider the X and Y direction of those diverged particles and update their position according to the following formula. p new x ik = dompts x i (29) p new y ik = dompts y i(30) where, p new x ik -is the latest X directional positional value of kth diverged Figure 13: Newly selected point is marked as s and curvature tracking is resumed as earlier. particle from i th swarm. p new y ik -is the latest Y directional positional value of kth diverged particle from i th swarm. Basic intuition behind the above equations is that if some particles diverge, their main objective is to reach as near as possible to the straight line joining two Let us consider the following issue; Thus random sampling of the line segment between two dominant points itself provides us the same effect as PSO ? The answer is many fold as stated below : • • Now the approximate curvature between two dominant points is not exactly a line segment. It could be any curve. That is why random sampling is not giving any better results. • As the two dominant points are not so close to each other, the curvature between two dominant points is captured by the particles of the swarm; that is the main reason we need PSO. • From frame to frame, we may loose dominant points during tracking, then we don't need to calculate dominant points again and again. The particles close to the dominant points become next dominant points. Thats why using PSO is beneficial rather than random sampling between two dominant points. 4. Some further illustration on the proposed dual tracking algorithm. Step 1: First we extract the first frame from input video and convert it into binary image.By trial and error we find a pixel point on the boundary of target object as shown in figure- (16). Here P i (x i , y i ) is the boundary point. Note that for simplicity of illustration we consider the front view of an object. In practice it can be any given aspect of an object to be tracked. Step 2: Apply Freeman Chain code to find the breakpoints started with P i as shown in figure- (17). Step 3 (20). Again for simplicity of illustration we show front view of an object in Frame-2. In practice it can be any given aspect of an object as frame-2. Step 5: On frame -2 as stated above and as shown in figure Step 6: In figure- (22) we have shown that from frame -2 to frame -3 op- Step 8: From frame 2 upto the last frame of the video sequence a bounding box, as shown in figure- (24), is designed based on position of PSO particles. Step 7: Note that by process of dual tracking when the target object, which is polygonally approximated , reaches the 3rd frame of the video sequence the shape of the annular ring(strip) of the multi-swarm changes due to the change of the shape of the dynamically generated polygon for the movement of the non-rigid target object. The newly generated polygon of the target object is automatically embedded over the changed annular ring(strip) of the multiswarm and the process of dual tracking proceed from frame-3 to last frame. Algorithmic summary and Complexity analysis The Dual Tracking Algorithm(DTA) is represented in pseuducode as follows for particles ← 1 to ss do 8: for pi ← 1 to ss do 9: Initialize particles pi velocity and position. 10: Initialize plbest and pgbest. 11: Compute Procedure FitnessComputePSO(pi). [Initialization of each particle's position and velocity for the swarm. ] for swarm ← 2 to nSwarm do 24: for pi ← 1 to ss do 25: AcceptedParticles ← FitnessComputePSO(pi) 26: if pi NOT in AcceptedParticles then 27: Update velocity of pi using Eq-(2) 28: Update position of pi using Eq-(1) 29: end if O(br) where br is b number of particle in breadth computation .so over all in BoundingBox() algorithm time complexity will be O(b+h). Complexity of Algorithm DominantPointReInitialization(AcceptedParticles, dompts(dmpt)) -If total number of accepted particle is n and for sorting that vector containing x particle will at best take O(nlogn). After sorting we will took first particle as next new dominant point, so overall time complexity will ne O(nlogn). Complexity of Algorithm DominantPointReInitialization(AcceptedParticles, dompts(dmpt)) -If total number of accepted particle is n and for sorting that vector containing x particle will at best take O(nlogn). After sorting we will took first particle as next new dominant point, so overall time complexity will ne O(nlogn). Based on these video streams we demonstrate the tracking results of the proposed dual tracking algorithm. Static background. The first experiment is on static background. We consider - (25) to figure- (34). [18],GREEN box CPF [20], BLUE box CT [22]. YELLOW box ASLA [18],GREEN box CPF [20], BLUE box CT [22]. Variable Background Now we perform our experiment on a video where background is moving with object. Video is taken by a moving camera. Here we consider 3 video frames from TB-100 sequence namely - [18],GREEN box CPF [20], BLUE box CT [22]. DTA, YELLOW box ASLA [18],GREEN box CPF [20], BLUE box CT [22] 5.Football1[IPR,OPR,BC]. Both tracking results and Bounding Box representations are shown below from figure- (35) to (44). DTA, YELLOW box ASLA [18],GREEN box CPF [20], BLUE box CT [22] We have tested the overall performance of our proposed Dual Tracking Algorithm(DTA) not only for the above data sets but also for other datasets such as; 1.Dog[SV,DEF,OPR] Tracking results have been shown in figure-(45). Analysis and Evaluation We evaluate the proposed dual tracking algorithm(DTA) using three parameters: True Detection(TD), False Detection(FD), Missed Detection(MD). We consider the parameter Frames per Seconds(FPS) to denote the number of frames per second. There is substantial amount of impacts in tracking due to high speed (high FPS) video sequence. TD is evaluated as the percentage of frames that successfully detect object and track in each video sequence. Following is the mathematical formulation of TD : T D = n td N × 100.(31) where N = Total number of frames in a video sequence and n td = number of frames that qualify as Truly Detected objects. Successful object detection and tracking can be computed as per the following rule -we consider a frame is successful in marking with the target object if our proposed bounding box both of them do not overlap i.e. \|Ct t g − Ct g t\| > B t /B g ; then the detection is false and is represented as: F D = n f d n td + n f d × 100.(32) If the algorithm is unable to detect target object in a frame, but ground truth value exists; then the situation is considered as Missed Detection and is represented as: M D = n md n td + n md × 100.(33) We have compared the proposed Dual Tracking Algorithm(DTA) with other state of the art algorithms: Visual tracking via adaptive structural local sparse appearance model(ASLA) [18], Beyond semi-supervised tracking: Tracking should be as simple as detection, but not simpler than recognition(BSBT) [19], Colorbased probabilistic tracking(CPF) [20], Exploiting the circulant structure of tracking-by-detection with kernels(CSK) [21], Real-time compressive tracking(CT) [22]. [23]. In all cases We achieve superior results in comparison with other algorithms. Analysis and Evaluation Methodology We further compare the proposed Dual Tracking Algorithm(DTA) in terms of three evaluation method: Precision plot, Success plot and Longest Subsequence Measure(LSM) [23]. Precision plot -It is the most common and widely used method in object tracking [17] [79]. It shows the percentages of frames whose calculated pixel position(location) of the image is within the given threshold distance of the ground truth value. We use threshold distance value as 20 [80]. Success plot -Another evaluation metric is success plot [17]. It provides the result of computing Intersection over Union(IoU) between computed and ground-truth bounding box position and also computes the number of successful frames whose IoU values is larger than given threshold values. If computed bounding box position of the target object is -BT c and given groundtruth bounding box position value is -GT c , then the overlap score [81] is - OS = \|BT c ∩ GT c \| \|BT c ∪ GT c \|(34) where ∩ and ∪ represent intersection and union operators respectively and \|.\| represents number of pixels in that region. We take Average Overlap Score(AOS) as the performance metric. AOS value decides weather a frame is successfully tracked or not. Experimental dataset-3(Performance analysis with other PSO algorithms) We extend our experiment with other state-of-the-art Particle Swarm Optimization algorithms. We choose several published PSO algorithms and compare their tracking performance with the proposed dual tracking algorithm(DTA). We consider the Object Tracking Evaluation 2012 from The KITTI Vision Benchmark Suite [63]. Web link is as followshttp://www.cvlibs.net/datasets/ kitti/eval_tracking.php. Proceedings of the 21st international conference on pattern recognition ment can be captured by a camera in various environments under Variable Background and Static Background. Moving object detection and tracking pose a challenge in real world scenarios like automatic surveillance system, traffic monitoring, vehicle navigation etc. In many scenarios where background changes dynamically due to motion of camera, abrupt changes in speed of the tracked object, change in illumination of light, noise, occlusion etc., tracking becomes very complex and challenging. Therefore tracking algorithm under such situation should be robust, flexible and adaptive. It should be capable of real time execution. Moving object tracking is existing for past several decades. Many methods have been proposed with a certain degree of accuracy and effectiveness. Still there remains several challenging problems in tracking due to the reasons stated earlier. In this paper we adopt a dual tracking approach based on optical flow and swarm intelligence so that the tracking becomes very robust under the challenges as stated. Wu.et.al provide object tracking benchmark. In [17] Wu.et.al provide tracking results of some of the top performing object tracking algorithms: Visual Tracking via Adaptive Structural Local Sparse Appearance Model(ASLA) [18] Jia et.al use sparse representation to find possible match with target template with minimum reconstruction error, Beyond semi-supervised tracking: Tracking should be as simple as detection, but not simpler than recognition(BSBT) [19] Stalder et.al use multiple supervised and semi supervised classifier to perform the task of detection, recognition and tracking. Color-based probabilistic tracking[CPF] [20] Perez' et.al use Monte Xia et.al use Particle Swarm Optimization (PSO) as the object localization method based on the Bayesian tracking framework. Though several approaches for object tracking are existing for past several decades but none of them consider dual tracking approach which is the major novelty of the present work.The proposed dual tracking approach for object tracking is very much robust for short video sequence under static background and variable background as well as long challenging video sequence under static background and variable background. Though an exhaustive survey on existing object tracking algorithms and a through review on deep learning approaches to object tracking are not within the scope of the present work; but in this particular context we like to make a critical appreciation on deep neural network(DNN) learning for object tracking. Through several experimental studies we reveal that deep learning approaches(based on Microsoft's ResNet, Google's Inception and Oxford's VGGNet etc.) for object tracking are good for known classes of object under fixed environments; but under variable background(in a time critical situation) with several uncertainties like unknown object,variation of illumination of light,noise, occlusion, sudden start and stop of moving object, paper we propose a dual approach for object tracking based on optical flow and swarm Intelligence. The optical flow based tracker i.e. KLT, tracks the dominant points of the target object from frame 1 to last frame; whereas swarm Intelligence based PSO (Particle Swarm Optimization) tracker simultaneously tracks the boundary information of the target object from frame 2 to last frame. Formation of swarm on each line segment is based on the smallest distance of each particle from the individual line segment.Thus, a multiswarms environment is formed and an annular ring(strip) of swarms is generated over which the dynamically generated polygon of the target object is embedded. If the target object is a closed digital curve then the annular ring of swarms is formed as shown infig-(1); otherwise a strip of swarms is formed as shown infig-(2). Figure 1 : 1Annular ring of multiswarms. Figure 2 : 2Strip of multiswarms. The vertices's (dominant points) of the polygon are tracked by KLT tracker and the boundary information of the target object, which is approximated by dynamically generated polygon and which is embedded over the annular ring(strip) formed by multiswarms, is tracked by the pso tracker from frame -2 to last frame. At frame -3 the shape of the annular ring(strip) of the multiswarms changes simply because the shape of the dynamically generated polygon, which continuously captures the boundary information of the target object, changes due to the movement of the target object which is a non-rigid body in general. During the said process of shape change of the annular ring(strip) of multiswarms, the in-dividual swarm of each small line segment further rearranges the position of the particles of each swarm to converge on the individual line segment of the newly generated polygon. During the said process of convergence, until all the particles of individual swarm over individual line segment successfully converges over all the line segments of the newly generated polygon, they (particles) update their velocity and position based on previous local best and global best position. Thus local best and global best positions are further updated. Again the newly generated polygon embedded over the newly generated annular ring(strip) of the multiswarms is tracked from frame -3 by PSO tracker along with KLT tracker. The above process of dual tracking continues till the last frame with dynamic change of shape of the polygon and the change of shape of the annular ring(strip) at each frame of the video sequence. Thus the dual tracking approach for object tracking tracks the dominant points on the contour of the target object and simultaneously tracks the tightly captured and embedded approximated polygon of the target object. The basic purpose of this dual tracking approach is that during tracking the multiswarms environment within which the approximated polygon is embedded continuously supplement the KLT tracker for tracking the dominant points from frame-2 to last frame. As the polygonally approximated target object is embedded and tightly captured within the frame of multiswarms ring(strips) so under any kind of environmental disturbances as stated earlier the tracking of the target object is not lost in the midway of any video sequence of tracking. Another specialty and uniqueness of this dual tracking approach is that it very successfully tracks the long challenging video sequences where many classical approaches for tracking drastically fails. This achievement of successful tracking of long challenging video sequence is mainly due to the fact that the approximated polygonal version of the target object is embedded and tightly captured in a multiswarms environment. And there is a very little possibility that the target object is lost during tracking in a long challenging video sequence. In course of tracking if there is any loss of dominant point (points) due to some environmental disturbances then the tracking procedures by KLT and PSO are disturbed. In that case instead of recomputation of dominant point (points), we reinitialize the missing dominant point (points) by some heuristic approach which essentially exploits the intelligence level of swarms. Similar to the reinitialization of the missing dominant point (points), particles of the individual swarm over individual line segment may require reinitialization during convergence process, which starts from frame -3 till the end of the last frame. If it is detected, during the said convergence process, a particular particle(particles) of an individual swarm over individual line segment diverges ( instead of converges) from its global best position ( even after several iteration of convergence) then the position(positions) of that particular particle(particles) is (are) reinitialized to a position(positions) for convergence over the line segment of the corresponding swarm from where the particle(particles) is (are) displaced to an undesirable position. After successful convergence of all particles over individual swarm of each line segment of the polygon a bounding box around the target object is formed based on a new concept of PSO-based bounding box generation algorithm.Note that, as stated earlier,at frame-2 a group of particles are randomly distributed over the image search space. These particles essentially take part in formation of swarms on individual line segments of the dynamically generated polygon of the target object. The population of particles is not fixed. It depends upon the need of the problem and basically a heuristic parameter in nature 2. 3 . 3Tracking of Dominant point(points) by KLT 2.3.1. Feature selection Before any tracking of moving object the most fundamental step is the selection of "trackable" features. For the present problem we consider sparse optical flow method based on KLT algorithm. First we have to determine the parameters to find out good features. According to Tomasi and Kanade[1] 'a single pixel cannot be tracked until it has s a very distinctive brightness with respect to all of its neighbors'. Hence, they prefer a "Window" of pixels which should contain sufficient texture. By "texture" we mean a group of neighboring pixels (window of pixels) which shows significant variation or changes of intensity or brightness between consecutive frames. Areas with a varying texture pattern are mostly unique in an image, while uniform or linear intensity areas are often common and not unique. Based on these guideline we proceed as follows; 2.3.2. Selecting Dominant point(points) as good feature The main reason for choosing dominant point as a trackable feature is that by definition [67] dominant point itself holds maximum curvature information on the contour of a target object. So quite obviously a window centered at dominant point should always give us enough texture for tracking from one frame to another. The area of such window can vary, depending on the number of features. This dominant point act as "interest point" which captures maximal local intensity information. Every basic KLT algorithm starts with finding corners or interest points satisfying the equation. [62] - are trackable.The equation for Z forms an intricate part of the Kanade-Lucas-Tomasi tracking algorithm. It is necessary to establish a minimum threshold for the value of the eigenvalues. If the two eigen values of Z are λ 1 and λ 2 , we accept a window which satisfies equation-(4). Figure 3 : 3original image; no dominant point is selected so far. (a) image gradient according X-axis. (b) image gradient according Y-axis. Figure 4 : 4image gradient according X and Y-axis from left to right. ( 5 ) 5the set of RED dots are the interest points as per equation -(4) and our chosen dominant points for target object are taken from this set of RED dots as a subset. In figure-(3) we show original image and in figure-(4) we show image gradient in x axis and y axis. Figure-(5) is the results of the feature points which satisfy the above equation-(4).Experimentally we obtained that the calculated Figure 5 : 5RED dots indicate the pixels which qualify the equation-(4). dominant points using equation -(3) [66] is a subset of the interest points of a selected window of feature points as stated above. So we can move to the next step of the KLT tracking algorithm by considering dominant points as our interest points which we do not have to search for [62] 2.3.4. Concepts of tracking dominant points by KLT The basic notion of tracking by KLT can be explained by looking at two images in an image sequence. Let us assume that the first image is captured at time t and the second image is captured at time t + τ . It is important to keep in mind that the incremental time τ depends on the frame rate of the video camera and should be as small as possible. An image can be represented as function of variables x and y. If we define a window in an image taken at time t+τ as I(x,y,t+τ). The basic assumption of the KLT tracking algorithm is; From equation -( 8 ) 8it is clear that every point in the second window can by obtained by shifting every point in the first window by an amount (∆x, ∆y). and the weighting function w(x) is usually set to constant 1. Now according to Taylors series expansion of J about a point p(x, y)T , truncated to the linear term is − R 1 , 1R 2 = random values associated with learning rate components to give more robustness. P LB = Particle Local Best position − it is the historically best position of the i th particle achieved so far. P GB = Particle Global Best position − it is the historically best position of the entire swarm. Which is basically the position of a particle which achieves closest solution. Equation (16) is Kennedy and Eberhart's original idea. After that lot of different researches have been going on.Based on those researches a remarkable idea comes up [? ]. In [? ] Shi and Eberhart add a a new factor • Multiswarms -In the proposed dual tracking algorithm one tracker is PSO based approach. In the basic concept of section 2.1 we have clearly explain a key feature of dual tracking algorithm is ring(strip) of multi swarms within which the approximated polygonal representation is embedded during tracking. Number of swarms are decided by number of dominant points of the target object. If we have D number of dominant points of a target object the number of line segments which polygonally approximate the said target object is equal to (D -1). Thus there will be (D -1) number of swarms. During tracking, due to several disturbances as stated earlier the dominant point(points) of the target object may be lost at the midway of tracking and thereby some of the particle of the swarm which is based on that dominant point will be also distracted. In such cases as mentioned in section 3.1 and section 3.2 the algorithm will automatically reinitialize the lost dominant points and the lost particle of the swarms. During experiment of tracking we have seen in worst case approximately 10% percent of the particle including dominant points need to be reinitialized. • Population of particles -We initialize the population of particles needed for construction of swarm around individual line segment of the approximated polygon of the target object. It is chosen heuristically depending upon the need of the application. In case of object tracking under static background the particle population is 25 and object tracking under variable background the particle population is 33. It is obvious that if where (X D1 , Y D1 ) is the position of the 1st dominant point and (X D2 , Y D2 ) position of the 2nd dominant point and X i , Y i is the coordinate of the i th particle.• Initialization of w, C 1 , C 2 , R 1 , R 2 values. -Earlier we define the meaning of these terms in equation -(16) and equation-(17). Initialization of these variables based entirely on application . In this paper after some experiment we choose w = 0.3 , C 1 = 0.1, C 2 = 0.1. R 1 and R 2 are set to some integer between 1 to 3. The convergence of the PSO algorithm is based on these parameters and we are basically guided by the information provided in[74]. in figure-(6):. In the following figure-(6) , we have an arbitrary curve C as stated above, which contains dominant points like D 1 , D 2 , D 3 , D 4 . The line segments L 1 , L 2 and L 3 passing through D 1 D 2 , D 2 D 3 and D 3 D 4 respectively. Though it is not exactly the curve connecting dominant points D 1 D 2 , D 2 D 3 or D 3 D 4 but as shown in the figure it serves the purpose of approximately representing the contour(boundary) of the curve as shown in fig-1. Thus we obtain polygonal approximation of the arbitrary chosen curve C as stated above. As we are not detecting or tracking the exact contour of the target object, we focus only on moving area of the target object approximated by polygon, so polygonal approximation of the target object does not produce any serious threat for tracking. It is always possible to construct the exact curvature between two consecutive dominant points of the target object. But such construction of the curvature is always time consuming and does not really improve the tracking result. This can be a scope for future work. Figure 6 : 6Diagram shows approximate Curvature calculation using Pythagorean formula. Figure 7 : 7Generation of Fitness function. These said particles form swarm over each small line segment of the polygon according to the smallest value of the fitness function. For the first time each particle of frame 2 measures its perpendicular distance from each small line segment and chooses the particular line segment as the line over which it will lie to form a swarm. Thus all particles of the image space of frame 2 are distributed over the small line segments of the approximated polygon of the target object and form a multiswarms scenario at frame-2 of the video sequence. These multiswarms scenario is nothing but an annular ring(strip) of swarms within which the approximated polygon of the target object is embedded(see figure-1 and 2 of section 2.1). Dual tracking of the target object starts from frame -2 . The vertices's of the polygon which are essentially the dominant points of the target object and which are computed at frame 1 of the video sequence are tracked at frame -2 where these vertices's are embedded in the annular ring(strip) of the multiswarms as dominant points of the approximated polygon of the target object. These ver-tices's (dominant points) are tracked by KLT. KLT tracks the dominant point of the target object from frame 1 to the last frame of the video sequence. Whereas the entire target object which is approximated by polygon and embedded in the multiswarms environment is tracked by the PSO tracker. PSO tracks the approximated polygon of the target object embedded in the multiswarms from frame 2 to last frame. 2.4.5. p lbest , p gbest Updation and Reshaping of the annular ring of the multiswarms When the dual trackers arrive at frame-3 of the video sequence, the shape of the polygon is automatically changed due to the movement of the target object which is in general non rigid in nature. In case of rigid object the shape of the polygon of the target object remains same. Once the shape of the polygon changes at frame 3 of video sequence the particles inside a swarm are redistributed on the small line segments of the changed polygon as per the built in function p lbest , p gbest function. Until all the particles inside a swarm successfully converge on the small line segment of the changed polygon, they (particles) keep updating their velocity and position using formula-(16) and (15) respectively. Figure 8 : 81 + p 2 + ... + p p ) p (23) Figure-8 represent the point q. Now we compute the Length and Breadth of the bounding box. Length is the vertical line and they are calculated as follows -First select l number of particles which have maximum X-direction values and minimum Y-direction values. These l values can be the first 10 particles which have values maximum in X-direction and minimum in Y-direction. This number of points depends on designer choice and application. According to our experience it is effective if we take first 10 to 20 particles which are maximum in X direction and minimum in Y direction. Thus we get as followslen = l i=1 l i l ∀i = 1, 2, 3...l. (24) Convergence of particles towards object boundary for construction of bounding box. Figure 9 : 9Representation of the bounding Box. Due to background clutter, occlusion, illumination Variation, low resolution and scale variation of various video sequences, change of image background occurs frequently. So the optical flow method based Kanade-Lucas-Tomasi(KLT) tracker which is basically a point tracker is unable to track a single point throughout the video duration. Hence the proposed algorithm is developed based on the fusion between optical flow and swarm intelligence. After the first frame of tracking using KLT, the PSO provides a continuous support to capture the overall information including the dominant points of the object by automatic generation of polygon of the object to be tracked where the vertices's of the polygon is basically the dominant points of the object being tracked.This polygon is automatically updated with the moving object from frame to frame.With the movements of the object being tracked the shape of the object(usually non rigid) changes which is continuously updated by the newly generated polygon of the object at each frame.Thus a total information, in an approximated sense, is provided to the tracking algorithm by the dual function of optical flow and swarm intelligence.We track dominant points using KLT. As the video sequence changes a lot there is a very high probability that KLT tracker may loose some of these dominant points in course of its tracking. During tracking, if the tracking of a dominant point is disturbed then the particle(particles) of the corresponding swarm is(are) also distracted and PSO tracker may failed to track. To avoid this situation we propose reinitialization of missing dominant points. lowing figure-(10) there are two dominant points (s and e) which are marked as RED. These two points are tracked by KLT tracking algorithm and all yellow color points are swarm particles which spread over the line joining between two dominant points. If we consider that the curved object which is being tracked is moving from left to right then due to various reasons stated above KLT may loose tracking of one of the dominant points as shown in the figure-(11). Figure 10 : 10Initial curvature C(s,e) with two dominant points (RED dots) tracked by KLT and yellow dots represent the swarm particles. When we detect that loss, we need another dominant point for continuation of our curvature tracking. But we do not compute another dominant point using formula-3 [66] as in real time tracking re-computation of lost dominant point is not a feasible solution. Instead of computing of another dominant point we assign a moving PSO particle which is nearest to the lost dominant point s. As we already keep tracking of all particles which follow the path joining between two dominant points,it is much more feasible and viable approach to follow. It does not require any computation and also we do not need to find where the closest particle will be. In figure-(13) we have shown, after selection of a new dominant point, the entire curvature tracking is resumed. By this approach, neither we have lost our tracking nor we have made any delay/break in tracking due to loss of dominant point. But the question is, weather the approach to replace lost dominant points by a new one rather then actual computation of dominant point is feasible? We need to remember we are not tracking exact contour of tracked object. Hence we do not need to follow exact curvature of the object body. Rather PSO Particles are tracking the approximated curvature of the object by simply a straight line joining two dominant points in our case. Though the newly selected point is not exact dominant point but it can easily solve our purpose to follow the object boundary. Figure 11 : 11Lost dominant point which is shifted left is shown by RED color. Blue arrows show the direction of movement of the object. One of the swarm particles shown by yellow dot is lost;but other particles successfully track the curvature. Figure 12 : 12Nearest neighboring point selected as new dominant point. Other particles track as usual without considerable amount of delay. Richards et.al use generators from centroidal Voronoi tessellations as the starting points for the swarm. In [75], de Melo et.al consider the algorithm named Smart Sampling (SS) finds regions with high possibility of containing a global optimum. A meta-heuristic can be used to initialize inside each region to find that optimum. Smart Sampling(SS)and Differential Evaluation (DE) are combined to establish SSDE algorithm to evaluate the approximate position of the diverged particles. So we can choose and apply any of these methods which works successfully. But in the present context, instead of doing this we consider another approach we think to be more effective in our case. dompts x i -X directional value of the dominant point of the i th swarm. dompts y i -Y directional value of the dominant point of the i th swarm. We need to keep in mind that, dompts is a set of dominant point which continuously updated frame by frame as per our algorithm. So whenever we mention dompts we always refer latest updated dominant points. Another point worth referring here is that swarm particles try to converge over the straight line joining two dominant points. So whenever any swarm particle is diverged from its desired location over a particular line segment, if we directly place the said diverged particle on any dominant points of the line segment, according to the above formula then the question is which dominant point we should choose among two dominant points joining which we get the line segment?. Actually it does not produce any serious impact if we choose arbitrarily any dominant point among these two. Figure 14 : 14At frame number-f we detect two swarm particles marked as RED points are diverged.consecutive dominant points. So rather computing any complex mathematical function and performing extensive number of iteration we consider the most simple approach by placing the diverged particles positions directly on any of the two dominant points around which a swarm was already formed and from where particle(particles) were diverged.Figure-(15) explains the newly updated position of the diverged particles. Figure 15 : 15Convergence of the diverged swarm particle at frame (f+t+1). Figure 16 : 16A human body whereP i (x i , y i ) is mapped using trial and error method. Figure 17 : 17Boundary points are detected using freeman chain code(see equation-(1)). : Find out dominant points using max cosine values(Ray,1992). Initially it eliminates all linear points and subsequently find out those points which have maximum cosine values.We denote those dominant point set as D. ={D 1 , D 2 , D 3 ...D 15 }. The resultant figure is shown in figure -(18). Figure 18 : 18Selected Dominant points calculated from breakpoints are shown here (see equation(2) and (3)) Step 4: These dominant points as shown in figure-(18) by RED dots are tracked by KLT tracker from f rame 1 to f rame n . In figure-(19) blue arrows show how KLT tracks dominant point independently from frame to frame.On f rame 2 we distribute PSO particles over the image space randomly. The distribution of PSO particles shown in figure - Figure 19 : 19-(21) we draw the lines joining two consecutive dominant points. All green dots are PSO particles and RED dots are dominant points. The straight line joining two consecutive dominant points has been shown by black straight line and green PSO particles spread on those straight lines. The yellow arrows show the movement of PSO particles. The right hand object of figure-(21) is basically a polygonal approx-Tracking of dominant points as shown by Red Dots,by KLT tracker from one frame to another. imation of the left hand object of figure -(21). The vertices's of the polygonal approximation(i.e the right hand object) represent the dominant points of the object at frame -2. Figure 21 : 21Polygonal approximation of the Object Boundary at frame 2. tical flow and swarm intelligence simultaneously performing the task of dual tracking. KLT(based on the concept of optical flow) tracks the dominant points of the object(i.e.the vertices's of the polygon)and PSO(based in the concept of Figure 20 : 20PSO particles distributed over the image space randomly Swarm Intelligence)tracks the boundary (approximated by the straight line of the polygon) of the object. The green points are PSO particles and they are distributed over a straight line between two consecutive dominant points. The green points of PSO particles which are distributed over each small line segments of the dynamically generated polygon of the target object form a swarm. Thus around the polygonally approximated target object a multi-swarm scenario is generated and the approximated polygon of the target object is embedded over the annular ring (strip) of the multi-swarms (see figure (1)and(2)). Dominant points are marked as RED. Blue arrows show that the tracking of dominant points is performed by KLT.Yellow arrows show the tracking of the boundary(approximated by a straight line)of the curved object by PSO particles. Figure 22 : 22Simultaneous tracking of moving object by both KLT and PSO. Figure 23 : 23Object at frame-4 shows the polygonal approximation dynamically created and tracked by PSO particles. Figure 24 : 24Bounding box is designed based on PSO particles position(see equation-(26)). Figure - ( -23) shows the polygon created by PSO particles at frame no.4. Thus, in addition to the tracking of the dominant points of the object, which are basically vertices's of the polygon dynamically created due to the change in shape of the nonrigid object(see the discussion on 3rd paragraph of section 2.1 and frame -4 of figure-23), we simultaneously track the boundary of the dynamically changed curved object which is approximated by straight line segments of the newly generated polygon(see frame-4 of figure-23) by PSO particles. calculate time complexity of Procedure DualTrackingAlgorithm(DTA)() we need to compute complexity of all the sub algorithms it called and summing up all those complexity will give us approximated time complexity of this algorithm. Complexity of Algorithm Frame Extraction( Video input)− reading a video file and extracting each frame and storing them in a separate file requires O(f). Where f is the number of frames. Complexity of Algorithm BrPtCal(Frames) -if the targeted object contains p number of pixels for the entire boundary, then freeman chain code at maximum will check 8 direction for boundary condition for each pixels. So at maximum the time required to find all the breakpoints for the object is -O(7p), which we can consider as linear time O(q), where q approximately 7p. Complexity of Algorithm DominantPt(brpts) -let's consider number of breakpoints are -b, then no region = b/[5-10], lets that is b 1 . Now computing k−cosine values required constant time. So final time required is O(b 1 * b * k), where k is a constant time, b 1 -no region, b -no of breakpoints. Complexity of Algorithm klt(dmpt)− Assume that the number of warp parameters is n and the number of pixels in T is N. The total computational cost of each iteration of Lucas-Kanade algorithm is O(n 2 * N + n 3 ), detail discussion explained in Lucas-Kanade 20 Years On: A Unifying Framework: Part 1 (Sundaram,2010).Complexity of Algorithm BoundingBox( AcceptedParticles)-In order to get time complexity of this algorithm we need only to calculate time complexity of height and breadth procedure. According to algorithm it will be O(h) where is h is number of particle we need to check whether they lie on the object boundary range. Similarly for breadth it will be Final complexity of DualTrackingAlgorithm(DTA)-Frame Extraction(),BrPtCal(), and DominantPt() will be called only one time so time required to compute will be -=O(f ) + O(q)+ O(b 1 * b * k) =O(f) + O(q) +O(b 1 b) [ k is constant] =O(f) + O(q) + O(b 2 ) [ b 1 is very less than b] =O(f+q) + O(b 2 ) = O(b 2 )Initialization of PSO particles will give us time O(n) where n is no. of particles. Now KLT will be called for every frame, so if there is f number of frame then this will give -f O(n 2 N + n 3 ).and DominantPointReInitialization()will called very few times so it is approximately O(nlogn),where n is the number of particles.And finially PSO will run for each frames required O(f), f is frame number. Total time complexity = [O(b 2 ) + f O(n 2 *N +n 3 ) + O(f )] where bno. of breakpoints, f -no of frames and n -number of pso particle, and Nno. of pixels in KLT tracking. Figure 25 : 25It shows a sequence of frames of a single person moving towards a camera where background is static. Proposed dual tracking algorithm successfully tracks the target object as it moves in Walking dataset. Green dots show the dominant points and RED dots show the swarm particles. Figure 26 : 26In Walking dataset, a bounding box based on PSO based algorithm is shown. For representational clarity the dominant points and swarm particles are not shown explicitly inside the bounding box.Comparison With other State-of-art algorithms shown here. RED box indicate DTA, YELLOW box ASLA Figure 27 : 27The Dual tracking approach tracks the movement of the face in GIRLS dataset. Green dots show the dominant points and RED dots show the swarm particles. Figure 28 : 28The Bounding Box is shown around the face of GIRLS dataset.For representational clarity the dominant points and swarm particles are not shown explicitly inside the bounding box.Comparison With other State-of-art algorithms shown here. RED box indicate DTA, Figure 29 : 29The Dual tracking approach tracks the movement of the target object in Walking2 dataset.Green dots show the dominant points and RED dots show the swarm particles. Figure 30 : 30The Bounding Box is shown around the target object in Walking2 dataset.For representational clarity the dominant points and swarm particles are not shown explicitly inside the bounding box.Comparison With other State-of-art algorithms shown here. RED box indicate DTA, YELLOW box ASLA Figure 31 : 31The Dual tracking approach tracks the movement of the target object in FaceOcc1 dataset.Green dots show the dominant points and RED dots show the swarm particles. Figure 32 :Figure 33 : 3233The Bounding Box is shown around the target object in FaceOcc1 dataset.For representational clarity the dominant points and swarm particles are not shown explicitly inside the bounding box.Comparison With other State-of-art algorithms shown here. RED box indicate DTA, YELLOW box ASLA [18],GREEN box CPF [20], BLUE box CT [22] The Dual tracking approach tracks the movement of the target object in Boy dataset.Green dots show the dominant points and RED dots show the swarm particles. Figure 34 : 34The Bounding Box is shown around the target object in Boy dataset.For representational clarity the dominant points and swarm particles are not shown explicitly inside the bounding box.Comparison With other State-of-art algorithms shown here. RED box indicate Figure 35 : 35The tracking result obtain by the dual tracking approach for jogging(1)(2)dataset.Green dots show the dominant points and RED dots show the swarm particles. Figure 36 :Figure 37 : 3637The bounding box representation is shown for jogging (1)(2) dataset. For representational clarity the dominant points and swarm particles are not shown explicitly inside the bounding box.Comparison With other State-of-art algorithms shown here. RED box indicate DTA, YELLOW box ASLA[18],GREEN box CPF[20], BLUE box CT[22] The tracking result obtained by the dual tracking approach for SUV dataset.Green dots show the dominant points and RED dots show the swarm particles. Figure 38 :Figure 39 : 3839The bounding box representation is shown for SUV dataset.For representational clarity the dominant points and swarm particles are not shown explicitly inside the bounding box.Comparison With other State-of-art algorithms shown here. RED box indicate DTA, YELLOW box ASLA[18],GREEN box CPF[20], BLUE box CT[22] The tracking result is obtained by the dual tracking approach for Walking dataset.Green dots show the dominant points and RED dots show the swarm particles. Figure 40 : 40The bounding box representation is shown for Walking dataset. For representational clarity the dominant points and swarm particles are not shown explicitly inside the bounding box.Comparison With other State-of-art algorithms shown here. RED box indicate Figure 41 : 41The tracking result is obtained by the dual tracking approach for Skater2 dataset.Green dots show the dominant points and RED dots show the swarm particles. Figure 42 :Figure 43 : 4243The bounding box representation is shown for Skater2 dataset. For representational clarity the dominant points and swarm particles are not shown explicitly inside the bounding box.Comparison With other State-of-art algorithms shown here. RED box indicate DTA, YELLOW box ASLA[18],GREEN box CPF[20], BLUE box CT[22] The tracking result is obtained by the dual tracking approach for Football1 dataset.Green dots show the dominant points and RED dots show the swarm particles. Figure 45 : 45The tracking result obtained by the dual tracking approach for above mentioned datasets are shown.Green dots show the dominant points and RED dots show the swarm particles. around a detected and tracked object overlaps with the bounding box of the given ground truth. Mathematically, \|Ct to − Ct gt \| ≤ B t /B g , where Ct's are the centroids and B's are the bounding boxes, Ct tg centroid of the ground truth, Ct to centroid of the target object. In this ratio B t /B g , B t represents area of the proposed bounding box of the target object and B g represents area of the bounding box of the ground truth. If the detected object position of our algorithm does not match with the position indicated by ground truth value or (SV,MB,OCC,OV). This 6 datasets contains other attributes as well. We mention those attributes which we consider for comparison with the proposed dual tracking algorithm. Figure - ( -46) shows tracking results on 6 challenging video streams consecutively. LSM plot -It shows[23] which tracking algorithm successfully tracks the length of longest tracked subsequence per sequence. If F percentage of frames in a long video sequence is successfully tracked then we call it Longest Subsequence(LS), where F is an appropriate large value.We pick up 5 state-of-art algorithm form the TLP dataset[23] : Learning Multi-Domain Convolutional Neural Networks for Visual Tracking(MDNet)[24], Fully-convolutional siamese networks for object tracking(SiamFC) [25], CREST: Convolutional Residual Learning for Visual Tracking(CREST) [26],Action-Decision Networks for Visual Tracking with Deep Reinforcement Learning (ADNet) [27],MEEM: Robust Tracking via Multiple Experts using Entropy Minimization(MEEM) [28] We perform the task of comparison with the proposed Dual Tracking Algorithm(DTA) in terms of Success plot , Precision plot and LSM plot. The proposed Dual Tracking Algorithm(DTA) achieves superior result in all three categories. In figure-(47), we show the success plot and the precision plot and in figure-(48) we show the LSM plot. a (Hsu et al.,2012), b (Boguslaw,2014), c (Xuan et al.,2013), d (Zhang et al.,2014), e (Xia et al.,2017) a (Hsu et al.,2012), b (Boguslaw,2014), c (Xuan et al.,2013), d (Zhang et al.,2014), e (Xia et al.,2017) Crowd PETS09 S2 L3 Time 14-41 View 01 dataset from KITTI Vision Benchmark Suite.In figure -(49) we show how successfully the proposed dual tracking algorithm(DTA) performs the tracking in comparison with other PSO algorithms.7. Conclusion and Future WorkIn this paper we propose a dual tracking algorithm based on optical flow and swarm intelligence. KLT tracker which tracks the dominant points of the target object is based on optical flow method whereas PSO tracker tracks the boundary information of the target object which is approximated by polygon.The proposed dual tracking algorithm (DTA) is inherently robust mainly because of two reasons; i)each tracker continuously supplement the performance of the other and thus acts as a corrective measure for each other under several disturbances during tracking as stated earlier in this paper. ii) the multiswarms annular rings(strips) where is approximated polygon of the target object is embedded captures the target object very tightly so that during tracking under several undesirable disturbances as stated earlier there is no chance for loss of tracking the target object.Hence the proposed dual tracking algorithm is robust for short video sequences and long challenging video sequence. In both the cases DTA is equally effective under static background as well as variable background.We consider dominant point as a primary feature of the target object. It is considered as a good feature to track[62]. Another major advantage of choosing dominant points as good features to track is that it help constructing the approximated polygon of the target object just by joining by two consecutive dominant points. Thus from frame-2 the PSO tracker which is an important part of the dual tracking algorithm is readily supplied with approximated polygon of the target object and a multiswarms environment is generated which provides the automatic mechanism for robustness of the dual tracking algorithm.Also the fitness function of the PSO algorithm is based on the coordinates of the dominant points. Thus dominant points of the target object have many important roles to play in dual tracking algorithm(DTA). Also it is very easy to calculate the dominant point of a new object which may arrive at any instance during tracking. If a new object appear there is no need to start the tracking from very beginning The dual tracking algorithm(DTA) will calculate the dominant point and automatically approximate the contour of the new object in the form of a polygon which will be embedded further in multiswarms annular ring(strip).Construction of the bounding box around the target object is unique and is based on the concept of PSO algorithm We test the performance of the dual tracking algorithm under several benchmark datasets andshow that the performance of the proposed dual tracking algorithm(DTA) is superior then the existing algorithms as shown in section -6.The proposed dual tracking algorithm can be further improved by some finer tuning of the parameters like w, C 1 , C 2 , R 1 , R 2 of the PSO tracker. The basic concepts of the multiswarms environment,with certain modification, can be further extended to object recognition and action recognition problems. Figure 46 : 46The tracking result obtain by the proposed Dual Tracking Algorithm(DTA) for above mentioned datasets.Green dot shows the dominant points and RED dots shows the swarm particles and Bounding box also present in every dataset.Comparison With other Stateof-art algorithms shown here. RED box indicate DTA, YELLOW box ASLA [18],GREEN box CPF [20], BLUE box CT [22] Figure 47: precision and Success plot evaluated on TLP dataset with Five other state-of-art algorithms. Figure 48 : 48Longest Subsequence Measure(LSM) plots evaluated on TLP dataset with Five other state-of-art algorithms. Figure 49 : 49successful tracking images on Crowd PETS09 S2 L3 Time 14-41 View 01. From Top to Bottom and left to right frames number 101,134,160,170,184,197,209,216,228,232,237 and 239 are tracked successfully. more stable and we can use much longer curvature. Therefore less number of breakpoints suffices for dominant point calculation.After excluding those linear points(pixels) rest of the points(pixels) are called breakpoints, which are candidates for dominant points. We have to consider the region of support of only for those breakpoints. We calculate the length of support of each breakpoint. Rather considering all breakpoints at once we collect them as a group of 10 for variable background and group of 5 for static background. The number of breakpoints in a group is decided based on which background we perform the tracking. Normally on variable background object shape changes fast. Hence we need the curvature of the object body smaller so that large number of breakpoints are close to each other. That's why we chose large number of breakpoints, compare to static background where the object is In our code , in each iteration we keep on running unless all the particles achieve desired threshold values, that means all the particles are as much as close to the line segment joining two dominant points. So it is not just a single run of PSO algorithm rather an continuous loop until all the particles achieve threshold value.One obvious question is why we choose line curve as a fitness function? We use a fitness function that is most suitable for tracking a curve. Though, we can use other fitness function like Beizer Curve or Hermite Curve but we think that when we are tracking a line segment joining two dominant points, using a line function as fitness function is most appropriate and easy to implement. We need approximation of the curvature not exact contour, so we stay away from complex fitness function and chose line function. We need to remember that, dominant points are basically those pointswhich hold high amount of curvature information. Now for a moving objects, especially fast car or running human body etc, there are so many dominant points that if we consider most of them then entire process of tracking become cumbersome and clumsy. So we need to be careful during choosing dominant points. Now if we minimize the distance between two dominant points, then for a whole moving object we need huge number of dominant points. That is why we keep number of dominant points minimum and distance between two dominant points are not very close. 6 . 6Experimental Result and Analysis.6.1. Experimental Setting. The proposed dual tracking approach for variable background and static background under different challenges as stated earlier, is tested by MatLab 2015a on a 64 bit PC with Intel i5 processor with 3 GHz speed. The image size of the frame 180 X 144. Static video is 20 sec duration whereas variable background is 13 sec duration. 6.2. Experimental Dataset-1 (Wu et.all).[17] All the experimental dataset has been taken from benchmark library cre- ated by Wu, Yi Wu, Jongwoo Lim and Ming-Hsuan Yang[17] and [68] which is available on http://cvlab.hanyang.ac.kr/tracker_benchmark/. 6.2.1. Tracking Results of the proposed method. Form the experimental test data set we pick up 5 video streams which have static background and point of interest is moving high to moderate rate. From TB-50 sequence - 1.Girl[SV,OCC,IPR,OPR], 2.Walking2[SV,OCC,LR], 3.Walking[LR,IV], 4.FaceOcc1[OCC], 5.Boy[SV,FM,IPR,OPR] and video stream from TB-100 sequence for dynamic background; 1.jogging(1)(2)[OCC,DEF,OPR], 2.Suv[OCC,IPR,OV], 3.Walking[OCC], 4.Skater2[SV,DEF,FM,IPR,OPR], 5.Football1[IPR,OPR]. 2 . 2Football[OCC,IPR,OPR,BC] 3.Human2[IV,SB,OPR]Figure 44: The bounding box representation is shown for Football1 dataset. For representational clarity the dominant points and swarm particles are not shown explicitly inside the bounding box.Comparison With other State-of-art algorithms shown here. RED box indicate DTA, YELLOW box ASLA [18],GREEN box CPF [20], BLUE box CT [22] 4.Human3[SV,OCC,DEF] 5.Girl[SV,OCC,IPR,OPR] 6.Singer1[IV,SV,OCC, OPR] 7.Skater2[SV,DEF,FM, IPR, OPR] 8.Women[IV, SV, OCC, DEF]. Table - 1 -shows the comparative result of execution time with various tracking algorithm stated above based on FPS with respect to all six attributes stated above in dataset-1[17] and[68]. In all cases We achieve superior results in comparison with other algorithms.For each of the above mentioned tracking algorithm , based on six attribute , True Detection(TD), False Detection(FD), Missed Detection(MD) are evaluated and presented in Table-2. Recently, Moudgil et.al developed a new benchmark dataset which contain long duration video sequence which they name as "Track Long and Prosper"(TLP). This dataset contain 50 real world videos which is approximately 400 minutes nearly 676K frames. This dataset is important because most tracking algorithms work best in short sequences but drastically fail on long challenging6.3. Experimental dataset-2(TLP dataset)[23] BC 59 78 34 78 56 87 a (Jia et al.,2012), b (Stalder,2009), c (Perez et al.,2012), d (henriques et al.,2012), e (Zhang et al.,2012) Table 1 : 1Attribute wise Execution time based on Frames per Second (FPS) on the benchmark datasets [68] and [17]. red: rank1, blue: rank2 ASLA BSBT CPF CSK CT DTA ASLA BSBT CPF CSK CT DTA ASLA BSBT CPF CSK CT DTAAttribute Mean TD(%) Mean FD(%) Mean MD(%) OCC 77 73.77 72.36 72.9 77.5 80.1 4.72 4.2 4.81 4.12 3.92 3.63 2.0 3.41 3.67 3.12 3.41 1.97 SV 85.23 80 78.1 71.1 79.23 88.19 4.20 2.9 2.2 2.7 2.9 2.0 3.17 1.2 1.43 2.8 2.91 1.2 DEF 70.63 67.21 67.92 60.51 69.86 75.13 4.25 4.21 3.16 3.84 4.3 3.27 3.17 2.92 2.86 3.62 3.19 2.57 OPR 60.45 68 57.20 61.4 62.4 69.23 5.67 5.91 5.57 5.9 5.7 5.28 3.13 2.92 3.96 2.7 3.7 2.4 IPR 60.24 61 52.1 59.1 59.4 65.23 5.7 5.81 5.77 6.3 6.7 5.21 3.19 3.32 3.95 3.12 3.17 3.0 BC 51.45 68 57.1 60.1 62.4 69.73 5.93 5.81 5.77 5.99 5.87 5.28 3.13 3.92 3.83 2.6 3.9 2.1 Table 2 : 2Attribute-wise Experimental Results on Benchmark Datasets[17] red: rank1, blue: rank2 video sequence. We perform experiment with our proposed dual tracking approach on 6 different dataset from this benchmark suite. This benchmark suite is available here: https://amoudgl.github.io/tlp/From the TLP dataset we pick up 6 video streams which have 5 attributes: Scale Variation(SV), Motion Blur(MB),Occlusion(OCC), Multiple Instances(MI),Out of view(OV). This 5 attributes are very challenging attributes as: OV indicates a situation where target fully out of the viewing window momentarily, similarly, MI indicates more than one objects with similar appearance as the target exist in the sequence and interact with it. This six video sequence with their at-a (Jia et al.,2012), b (Stalder,2009), c (Perez et al.,2012), d (henriques et al.,2012), e (Zhang et al.,2012)Attributes ASLA a BSBT b CPF c CSK d CT e DTA SV 67 78 54 35 77 91 MB 47 45 62 70 69 77 OCC 79 66 77 53 95 71 MI 65 70 51 85 65 71 OV 68 54 46 51 56 69 Table 3 : 3Attribute wise Execution time based on Frames per Second (FPS) on the benchmark datasets-2(TLP dataset)[23]. Table - 3 -shows the comparative results of execution time (based on Frames per Second(FPS))with various tracking algorithms with respect to all Five attributes stated above in dataset-2(TLP dataset) Table - 4 -shows the comparative results of execution time(based on Frames per Second(FPS)) with various tracking algorithms with respect to all Five attributes stated above in dataset-3(KITTI Vision Benchmark Suite)[63]. In all cases We achieve superior results in comparison with other algorithms.Attributes A a B b C c D d E e DTA SV 77 71 42 61 67 80 MB 77 78 61 87 63 83 OCC 76 63 47 73 90 73 MI 75 72 61 77 73 91 Table 4 : 4Attribute wise Execution time based on Frames per Second (FPS) with other PSO algorithms on the benchmark datasets-3 [63].red: rank1, blue: rank2The DTA algorithm is tested with CLEAR matrix[82]. We consider fewparameters: TheMulti Objective Tracking Accuracy [MOTA], which counts all missed target, false positive and identity mismatches, the Multiobjective Tracking Precision[MOTP] which considers the normalized distance between ground truth location and actual location. Another two parameters are, Mostly Tracked [MT] and Mostly Lost[ML].Table-5gives comparative performance of all this parameters with 5 state-of-the-art algorithms.We also perform experiment on video streamParameters A a B b C c D d E e DTA MOTA 81% 87.6% 89.5% 93.6% 77.5% 98.2% MOTP 79% 74.1% 81.1% 88.6% 92.8% 90.3% MT - 83.4% 78.2% 72.8% 79.3% 84.1% ML - 2.3% 3.5% 2.9% 3.9% 2.6% Table 5 : 5Quantitative Comparison with our proposed Dual Tracking Algorithm (DTA) and other state-of-the-art algorithm. Red, green and blue represent First, Second and Third top performance values respectively. Detection and tracking of point feature. T K , Carlo Tomasi, School of Computer Science, Carnegie Mellon Univ. Pittsburgh.T. K. Carlo Tomasi, Detection and tracking of point feature, School of Computer Science, Carnegie Mellon Univ. Pittsburgh. (1991). Determining optical flow: a retrospective. B K Horn, B Schunck, Artificial Intelligence. ElsevierB. K. Horn, B. 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[ "Electrical Detection of Individual Skyrmions in Graphene Devices", "Electrical Detection of Individual Skyrmions in Graphene Devices" ]	[ "F Finocchiaro \nIMDEA Nanociencia\nCalle de Faraday 928049Cantoblanco, MadridSpain\n", "J L Lado \nQuantaLab\nInternational Iberian Nanotechnology Laboratory (INL)\nAv. Mestre Jose Veiga4715-310BragaPortugal\n", "J Fernandez-Rossier \nQuantaLab\nInternational Iberian Nanotechnology Laboratory (INL)\nAv. Mestre Jose Veiga4715-310BragaPortugal\n\nDepartamento de Fisica Aplicada\nUniversidad de Alicante\nSan Vicente del Raspeig03690Spain\n" ]	[ "IMDEA Nanociencia\nCalle de Faraday 928049Cantoblanco, MadridSpain", "QuantaLab\nInternational Iberian Nanotechnology Laboratory (INL)\nAv. Mestre Jose Veiga4715-310BragaPortugal", "QuantaLab\nInternational Iberian Nanotechnology Laboratory (INL)\nAv. Mestre Jose Veiga4715-310BragaPortugal", "Departamento de Fisica Aplicada\nUniversidad de Alicante\nSan Vicente del Raspeig03690Spain" ]	[]	We study a graphene Hall probe located on top of a magnetic surface as a detector of skyrmions, using as working principle the anomalous Hall effect produced by the exchange interaction of the graphene electrons with the non-coplanar magnetization of the skyrmion. We study the magnitude of the effect as a function of the exchange interaction, skyrmion size and device dimensions. Our calculations for multiterminal graphene nanodevices, working in the ballistic regime, indicate that for realistic exchange interactions a single skyrmion would give Hall voltages well within reach of the experimental state of the art. The proposed device could act as an electrical transducer that marks the presence of a single skyrmion in a nanoscale region, paving the way towards the integration of skyrmion-based spintronics and graphene electronics.	10.1103/physrevb.96.155422	[ "https://arxiv.org/pdf/1702.06889v1.pdf" ]	119,102,301	1702.06889	cbb6bc8e071e6e2c29aca9eec011455a54c931d2	Electrical Detection of Individual Skyrmions in Graphene Devices (Dated: September 13, 2018) F Finocchiaro IMDEA Nanociencia Calle de Faraday 928049Cantoblanco, MadridSpain J L Lado QuantaLab International Iberian Nanotechnology Laboratory (INL) Av. Mestre Jose Veiga4715-310BragaPortugal J Fernandez-Rossier QuantaLab International Iberian Nanotechnology Laboratory (INL) Av. Mestre Jose Veiga4715-310BragaPortugal Departamento de Fisica Aplicada Universidad de Alicante San Vicente del Raspeig03690Spain Electrical Detection of Individual Skyrmions in Graphene Devices (Dated: September 13, 2018) We study a graphene Hall probe located on top of a magnetic surface as a detector of skyrmions, using as working principle the anomalous Hall effect produced by the exchange interaction of the graphene electrons with the non-coplanar magnetization of the skyrmion. We study the magnitude of the effect as a function of the exchange interaction, skyrmion size and device dimensions. Our calculations for multiterminal graphene nanodevices, working in the ballistic regime, indicate that for realistic exchange interactions a single skyrmion would give Hall voltages well within reach of the experimental state of the art. The proposed device could act as an electrical transducer that marks the presence of a single skyrmion in a nanoscale region, paving the way towards the integration of skyrmion-based spintronics and graphene electronics. I. INTRODUCTION Skyrmions are magnetic non-coplanar spin textures that are attracting a great deal of attention for both their appealing physical properties 1 and their potential use in spintronics [2][3][4][5] . They have been observed forming lattices in a variety of non-centrosymmetric magnetic crystals [6][7][8][9] , including insulating materials such as the chiral-lattice magnet Cu 2 OSeO 3 [10][11][12] . They also form two dimensional arrays in atomically thin layers of Fe deposited on Ir(111) 13,14 . In these systems the spins typically feel a competition between aligning with their neighbors and being perpendicular to them, what favors chiral ordering. A variety of interactions can assist non-collinear arrangements, including Dzyaloshinskii-Moryia interactions, dipolar interactions and frustrated exchange interactions and the size of an individual skyrmion can range from 1 nm to 1 µm depending on which specific mechanism is involved. To date, these magnetic structures are detected by means of neutron scattering 6 , electron microscopy 15 and even individually, with atomic scale resolution, by means of spin polarized scanning tunneling microscopy 13,16 and atomic size sensors 17 . The particle-like nature of skyrmions has motivated proposals to use them as elementary units to store classical digital information, inspired by the magnetic domainwall racetrack memories 18 . Such a perspective has become increasingly attractive since it has been experimentally proved 14 the possibility of manipulating twodimensional magnetic lattices by creating and destroying individual skyrmions by means of spin-polarized currents in STM devices. This, along with the experimental finding 19 of skyrmion motion driven by ultralow current densities of the order of 10 −6 A m −2 , considerably smaller than those needed for domain wall motion in ferromagnets, makes skyrmions potentially optimal candidates for the next generation of magnetoelectronic readout devices. Mathematically, skyrmions are topologically nontrivial objects whose topology content is embedded in an index, the winding number N , defined as N = 1 4π A n(x, y) · ∂n(x, y) ∂x × ∂n(x, y) ∂y dx dy (1) where n(x, y) : R 2 → R 3 is a classical magnetization field and the two-dimensional integral is performed over the overall area occupied by the skyrmion. The winding number N can only acquire integer values, and a skyrmion is distinguished from other topologically trivial magnetic textures for exhibiting a non-zero value of the integer N . The magnetization field n(x, y) of a skyrmion can be expressed as a mapping from the polar plane coordinates r = (r, φ) to the unit sphere coordinates (Φ, Θ) n(r) = (cos Φ(φ) sin Θ(r), sin Φ(φ) sin Θ(r), cos Θ(r)) (2) provided the spin configuration at r = ∞ is φindependent so that it can be mapped to a single point on the sphere. The mapping is specified by the two functions 20 : Φ(φ) = N φ + γ(3) and Θ(r) varies from 0 for large r to π as we approach r = 0, the core of the skyrmion. Here we adopt the following model: Θ(r) =    π for r = 0 f (r) = π (1 − r/R) for 0 < r ≤ R 0 for r > R(4) where N is the skyrmion winding number introduced in (1), γ is a phase termed helicity that can be gauged away by rotation around the z-axis, and f (r) = π (1 − r/R) is a function of the radial coordinate that describes a smooth radial profile inside of the skyrmion radius R. Such a texture describes a magnetic configuration where the spins are all aligned perpendicular to the film plane with the exception of those comprised within the radius R where they all progressively align along the anti-parallel arXiv:1702.06889v1 [cond-mat.mes-hall] 22 Feb 2017 direction, that is picked up exactly at r = 0. The condition that the spins at r = 0 and r = ∞ are oppositely oriented is crucial in order to ensure a non-trivial topology of the magnetic texture. e - Figure 1. A graphene triangular quantum dot (the transmission region) proximized with a skyrmion and connected to three leads. Due to the anomalous Hall effect, a net transverse voltage is generated by the skew scattering of Dirac electrons traveling though the central region. Several recent theoretical works 21-23 point out that two-dimensional systems coupled either weakly or strongly to individual skyrmions or skyrmionic lattices can develop an Anomalous Hall (AH) or Quantum Anomalous Hall (QAH) phase owing to the non-trivial topology of these structures in real space. This effect refers to the onset of a transverse Hall response arising in magnetic systems driven by anomalous velocities, associated to Berry curvature, without the need of an applied magnetic field 24 . This anomalous Hall response can be either of extrinsic or intrinsic nature. In the case of proximizing a pristine 2D system with magnetic skyrmions, the generation of a transverse voltage is of extrinsic nature and ascribable to the imprinting of the skyrmions real space topology onto the (trivial) reciprocal space topology of the non-magnetic system 23 . Based on these findings, along with a recent work demonstrating the possibility of growing a graphene flake on top of a single atomic layer of Fe on a Ir(111) substrate 21,25 , here we consider graphene flakes weakly coupled to magnetic films as skyrmion detectors. To this aim, we compute the skewness of the scattering and the associated Hall signal induced in a graphene island coupled to a single skyrmion within a multi-terminal geometry. Graphene unique properties are ideal to implement the proposed device. As a fact, being atomically thin maximizes proximity effects, making it an optimal material to grow on top of magnetic materials. Furthermore, the fabrication of high quality graphene electronic devices both at the micron and nanometer scale is absolutely well demonstrated [26][27][28] and its use as a magnetic sensor for magnetic adsorbates has been already tested experimentally 29,30 and studied theoretically 31 . The paper is organized as follows. In section II we discuss a 2D Dirac system in the continuum coupled to a non-uniform spin texture and performing a standard rotation in spin space we unveil two types of influence on the Dirac electrons. In section III we introduce Landauer's formalism for quantum transport on the lattice and describe the setup of the proposed Hall experiment. Finally, in section IV, we discuss the results obtained by applying Landauer's formula to a graphene flake coupled to a single skyrmion, characterizing the Hall conductance as a function of several parameters and comparing the effectiveness of graphene with that of a standard twodimensional electron gas (2DEG). II. ANALYTIC APPROACH IN THE CONTINUUM In this section we describe graphene electrons interacting with a non-coplanar magnetization field n, as given by equation (2), using a 2D Dirac Hamiltonian: H = H 0 + H ex = −i τ v F (∂ x σ x + τ ∂ y σ y ) + Jn · s (5) with s = (s x , s y , s z ) the vector of Pauli matrices acting in spin space and σ = (σ x , σ y , σ z ) the vector of Pauli matrices acting in pseudo-spin space. We perform a rotation of the Hamiltonian so that in every point of space the spin quantization axis is chosen along the direction of the spin texture n. As a result, the representation of the exchange term is diagonal in the rotated frame, but the Dirac Hamiltonian acquires new terms that encode the influence of the exchange interaction of the Dirac electrons with the non-coplanar field. The unitary matrix R that performs such a transformation in the basis ψ = (A ↑, B ↑, A ↓, B ↓) T is R =    u 0 v 0 0 u 0 v −v * 0 u * 0 0 −v * 0 u *    = u v −v * u * ⊗ σ 0 (6) where u = cos Θ(r) 2 e iΦ(φ)/2 v = sin Θ(r) 2 e −iΦ(φ)/2 (7) The transformed Hamiltonian H → H = RHR −1 reads H = τ v F [σ x (p x + A x ) + τ σ y (p y + A y )] + + τ v F 2 −σ x N r s x n y + ∂ r θs y cos φ + + τ σ y N r s x n x − ∂ r θs y sin φ + Js z(8) with Figure 2. Mapping of a system characterized by real hopping and with a double exchange interaction with a non-coplanar magnetic texture to a system with spatially uniform magnetization field and with a complex hopping function mimicking the coexistence of spin-orbit with a vector gauge field. A x = N 2r cos θ sin φ ⊗ s z A y = − N 2r cos θ cos φ ⊗ s z (9) t R i ( ⇥ r) z + t e i ẽ R A·dr and n x = cos Φ sin Θ, n y = sin Φ sin Θ. In the rotated reference frame, the exchange term is manifestly diagonal. Besides, the Hamiltonian has acquired additional kinetic terms. The A = (A x , A y ) field acts as a spindependent gauge vector potential that couples with the momenta of the Dirac electrons, whereas the remaining two terms closely resemble a spin-orbit (SO) interaction of the Rashba type. On the lattice, this corresponds to mapping a system characterized by a non-collinear exchange field and real hopping to a ferromagnetic system with a purely imaginary hopping mimicking the effect of SO coupling plus a complex hopping supported by a gauge field entering as a Peierls phase. This is schematized in figure 2. From the gauge field, one can compute the effective magnetic field acting on the system as B = ∇ × A = N 2r s z ∂ r θ sin θ − 1 r cos θ ẑ(10) that reads B z = − N 2r s z π sin θ/R + r −1 cos θ for r ≤ R r −1 for r > R(11) This transformation of the Hamiltonian therefore allows to interpret the topological content embedded in the skyrmion texture as a superposition of two effects: (i) The generation of an effective emergent electromagnetic field (EEMF) described by the gauge potential A; (ii) The coexistence of ferromagnetic exchange with a Rashba-like SO interaction, what has been predicted to give rise to a QAH phase 32 . Both ingredients are endowed with a topological character that the skyrmion texture is able to imprint onto the Dirac electrons and are therefore responsible for generating a Hall response in the system. An analog result has been derived ? for Schrodinger electrons, with the remarkable difference that in the strong coupling limit (J t) the spin-mixing terms vanish and the problem is exactly mapped to a spinless one-band system where the electrons momenta are coupled to a vector potential describing an emergent magnetic field. In the case of Dirac electrons, the spin-mixing term survives at all coupling regimes and the mapping to a pure EEMF is an incomplete description of the physics taking place in the system. Whereas this picture provides some physical insight of what happens to graphene Dirac electrons surfing a skyrmions, it does not provide a straightforward method to compute the Hall response. III. TIGHT-BINDING QUANTUM TRANSPORT APPROACH In this section we overview the quantum transport methodology that we will employ to compute the Hall response induced by an individual magnetic skyrmion in a graphene device. Importantly, we are implicitly assuming that the substrate material is an insulating skyrmion crystal such as CuGeO 3 33 and Cu 2 OSeO 3 10-12 in such a way that the current only flows through graphene. The graphene electrons are described with the standard tight-binding Hammiltonian for the honeycomb lattice with one p z orbital per atom 34 , plus their exchange interaction with the classical magnetization of the skyrmion n: H = −t <i,j>,σ c † iσ c jσ + J i S i · n i(12) Here n i is the classical continuous magnetization texture (2) discretized over the graphene lattice and taken at site i and S i = σσ c † iσ s σσ c iσ is the vector whose components are the Pauli matrices acting in spin space associated with the i-th lattice site. The < i, j > symbol implies summation over all nearest neighboring pairs of atoms, and we are assuming that the magnitude of the magnetization is uniform over the whole graphene lattice. This Hamiltonian has been considered before 23 for the case of 2D graphene interacting with a skyrmion crystal. In contrast, here we consider a graphene device that hosts an individual skyrmion. The mathematical framework that we use to study quantum transport is based on Landauer's formalism for conductance 35 . Given an experimental setup where a device is attached to N metallic contacts, Landauer's multiterminal technique allows to compute the transmission amplitude between the m-th and the n-th contact from the relation T mn = Tr G + d Γ n G d Γ m(13) where G d and G + d are respectively the retarded and advanced Green's functions of the device, that is the Green's function of the isolated device corrected by the self-energies Σ m of the N leads G d ( ) = ( + iδ) I − H d − N −1 m=0 Σ m −1(14) where H d is the Hamiltonian of the isolated device. The Γ m 's are quantities associated to the leads' selfenergies as Γ m = i (Σ m − Σ + m ). The leads' self-energies incorporate the coupling between the device and the leads as Σ m = t + m g m t m , with g m the surface Green's function 36 of the m-th lead, and t m the hopping matrix between the device and the m-th lead. From the knowledge of the transmission amplitudes, the expression for the total current flowing from the lead m follows straightforwardly: I m = e h n =m +∞ −∞ d [f ( − µ m ) − f ( − µ n )] T mn ( )(15) with f ( − µ) the Fermi distribution function, so that at zero temperature the previous expression reduces to I m = e h n =m µm µn d T mn ( F ) and for a sufficiently small energy interval µ m − µ n one can expand the transmission coefficient T mn ( ) around the Fermi energy F and stick to zeroth order. By doing so, one finally finds that the formula for the current flowing from the lead m becomes: I m = e h n =m (µ m − µ n )T mn ( F )(16) This equation can be used to derive the Hall response in a given multiterminal device in two different ways. In both cases, the first step of the calculation is the numerical determination of the transmission coefficients T mn ( ). Then we can either impose (i) the voltage drops eV , defined as the difference between the chemical potentials of the different electrodes, and compute the resulting current (inverse Hall effect), or (ii) impose a longitudinal current flow and a null transverse current, find the resulting chemical potentials and determine the Hall response (direct Hall effect). When the methods just described are implemented in an ordinary four-terminal geometry 22 , the resulting relation between the Hall conductance and the transmission coefficients is far from intuitive. In this paper, for the sake of simplicity, we consider a three terminal device (TTD) of the kind of the one shown in figure 3a. We choose to fix the chemical potentials of the three electrodes, labeled as 0, 1 and 2, and compute the resulting current. Specifically, we impose that V 0 = V and V 1 = V 2 = −V . In this way, the voltage difference between leads 1 and 2 is automatically set to zero whereas the voltage difference between lead 0 and leads 1,2 is V y = V 0 −V 1,2 = 2V . The expression for the current flowing from leads 1 and 2 is I i = 2V T 0i for i = 1, 2. From this expressions it is straightforward to deduce the current imbalance δI, that reflects the presence of a transverse force, δI = I 1 − I 2 = 2V (T 01 − T 02 ), whence our definition of Hall conductance in this geometry G H = δI V 0 − V 1,2 = e 2 h (T 01 − T 02 ) ≡ e 2 h δT(17) In the following we present the numerical results for the normalized transmission imbalance, that is T = δT T ≡ (T 01 − T 02 )/(T 01 + T 02 )(18) in order to work with quantities that do not depend on the number of conduction channels in the device. This 3terminal setup simplifies considerably the analysis of the numerical results, and also matches the C 3 symmetry of the graphene lattice. However, in a real device, disorder and contact asymmetries might result in additional transmission imbalances that might obscure the detection of skyrmions. Thus, in real devices a standard 4 terminal geometry should be used, given that the principles and magnitude of the physical effect are expected to be the same. IV. RESULTS AND DISCUSSION We now present the results obtained by calculating the imbalance in the transmission coefficients T eq. (18) for a graphene quantum dot coupled to a skyrmion. For a better physical insight, we provide an estimate for the equivalent magnetic field B eq that would give rise to a conventional Hall response of the same magnitude of that induced by the skyrmion. Details on the determination of such a field are given in the Appendix. In the following we consider flakes sizes of the order of ∼ 50 nm 2 , and skyrmions with radius of the order of 2-3 nm and winding number N = 1. Also, we are solely interested in realistic 37,38 weak exchange proximity effects, that do not alter the graphene spectrum substantially, so we explore coupling constants up to J ∼ 100 meV 39 . In order to simulate standard metallic contacts in some of the calculations square leads have been used instead of hexagonal leads. Results obtained with different leads geometries are consistent, so we chose to present curves associated to one or the other geometry in order to minimize resonance effects due to confinement inside of the central island. A. Anomalous Hall effect We first investigate the magnitude and behavior of the transmission asymmetry T as a function of the coupling constant J, comparing the results for Dirac electrons (half filled honeycomb lattice), and Schrodinger electrons (heavily doped honeycomb lattice). The result is shown in fig. 3(b) in both linear and logarithmic scale, for a skyrmion with radius R = 2.3 nm and a device of linear dimension L = 10.6 nm. The first thing to notice is that, even for small J 1 meV, the equivalent field B eq is of the order of 1 Tesla, which shows that the anomalous Hall effect is very large. For J < 100 meV the transmission imbalance T of Dirac electrons shows an approximately linear behavior with J in contrast with the case of Schrodinger electrons (Fermi energy away from the Dirac point) for which T ∝ J 3 . For all the values of J, the Hall response for Dirac electrons is much larger than for Schrodinger electrons, most notably for the experimentally relevant case of small J, for which T is up to 4 orders of magnitude larger. This difference is reduced and eventually canceled at higher and unrealistic couplings larger than 100 meV. We now characterize the Hall conductance of a graphene TTD by investigating its dependence on the system parameters, such as the Fermi energy of the leads, the skyrmion size R and the size of the graphene island coupled to the skyrmion. The results are shown in fig. 4. The anomalous Hall response as a function of the chemical potential of graphene ( fig. 4(a,b)), shows a local maxima at charge neutrality, and other two local maxima of opposite sign at symmetric electron/hole doping, a behavior resembling graphene coupled to a skyrmion crystal. 23 Such phenomenology can be understood in terms of the modification of the Dirac cone due to the non-coplanar magnetization field. As we have seen in section II, the problem can be mapped to one where spatially uniform exchange field and Rashba-like spin-mixing terms coexist. The first contribution has the effect of lifting spin degeneracy, whereas the latter opens small gaps at both the Fermi energy and at crossing points forming at higher energies of the order of ±J. Within these gaps, the absolute value of the Berry curvature reaches local maxima and this is reflected in the behavior of T as a function of the transmission energy ε shown in fig. 4. In fig. 4(c) we show the behavior of T as a function of the skyrmion radius R, keeping the dimension of the device constant and equal to L = 10.6 nm, and J = 80 meV. We consider the case of small skyrmions with nanometric radius such as those found in systems with frustrated exchange interactions 40 . Two competing effects are at play as the radius of the skyrmion increases: on the one side the change in magnetization as a function of the distance from the skyrmion center becomes smoother, so that the effective skew scattering is weaker, and on the other the surface where the skew scattering is non zero increases. The normalized scattering asymmetry resulting from our calculations behaves as R 4 indicating that the second mechanism is dominant, and therefore that larger skyrmions yield a stronger Hall signal. The dependence of the Hall response on the size of the graphene flake is shown in Fig. 4(d), for a fixed radius of R = 1.4 nm and an exchange of J = 80 meV. We see that by increasing the flake size while keeping the skyrmion radius fixed, the Hall signal decreases as L −1 , where L is the linear size of the triangular transmission region. From these results we infer that the Hall conductance behaves as T (R, L) ∼ R 4 /L as a function of the radius and of the linear size of the central island. This scaling reflects the fact that the Hall response is proportional to the probability that the electrons surf over the skyrmion, which is manifestly an increasing function of R and a decreasing function of L. By changing both the radius and the device size by a common factor α, T scales as T (αR, αL) ∼ α 3 T (R, L) indicating that the Hall conductance is not scale invariant under simultaneous rescaling of R and L. Now, since we are considering flakes of the minimum experimentally achievable dimensions proximized with the smallest skyrmions experimentally detected so far (of the order of the nm, whereas observation of skyrmions with radius of up to 100 nm has been reported 15,41 ), the presented scaling argument evidences that our estimates of Hall conductances of the order of 10 −5 -10 −4 G 0 merely set a lower bound for the range of values that this parameter can undertake in actual laboratory measurements. A general example of this non-linear scaling trend is shown in fig. 4(e) where a comparison of two systems with L and R scaled by a common factor is presented. We note that most systems in the brink of hosting skyrmion lattices need a non-zero external magnetic flux to drive them into the skyrmionic phase, as they typically exhibit spiral spin phases at zero magnetic field. This implies that an additional non-zero Hall contribution is to be expected from the external field that sums up to the one driven by the skyrmion alone. An effective way to discriminate between the two effects relies on their different symmetry properties. In fact, while the skyrmionic contribution is electron-hole symmetric (as made clear by fig. 4(b)) and changes sign only by switching the sign of either J or N , the Hall effect induced by the magnetic field is electron-hole asymmetric as holes have opposite charge with respect to electrons and thus respond with an opposite velocity to an applied external magnetic field. It is thus the ε → −ε asymmetry of the overall scattering cross-section that allows to subtract the spurious external contribution and determine the intrinsic skyrmionic one. B. Effects of disorder So far we have dealt with a graphene flake perfectly clean. However, some current degradation brought about by defects or impurities in the sample is to be expected. In order to provide a more realistic estimate of the extent to which the Hall responses that our results anticipate are robust with respect to this loss of conductance, we now consider the effect of introducing an amount of scalar disorder in the samples. We do so by averaging over N = 50 Anderson disorder configurations in each of which we assign a random scalar on-site potential W i ∈ [−W/2 : W/2] to each atom in the quantum dot and tune the parameter controlling the disorder degree W from 0 to a maximum of ∼ 400 meV, an upper limit for the energy scale associated with disorder that is consistent with the assumption of Coulomb long-range scattering 42,43 . The clean limit is recovered for W = 0. We employ square leads and compare two disorder configurations with different symmetry: one where the disorder distribution preserves mirror symmetry with respect to the y axis and one where the distribution is completely random in the whole sample. A realization of each of these different disorder profiles is shown in fig. 5(a,b). Error bars associated with the standard deviation of the data are shown for completeness. From the resulting T Symmetric disorder Random disorder curves shown in fig. 5(c,d) we see that symmetric disorder barely affects the Hall response of the problem, as it provokes changes in the normalized transmission imbalance of the order of ∆T /T ≈ 10 −2 . On the other side, a randomly distributed disorder that does not respect y → −y symmetry affects the conductance more sizeably, yielding variations ∆T of the order of T . The difference could be explained by noting that in the symmetric case the defects simply act as a fluctuating potential that does not contribute to the asymmetry of the scattering, whereas in the random case an additional transverse conductance driven by the disorder asymmetry rather than by the skyrmion-induced AHE is generated. However, significant alterations of the Hall response only take place at relatively high values of the disorder potential of the order of ∼ 400 meV, whereas for weaker and more reasonable disorder strengths the change in the conductance is smaller and comparable to the one obtained in the symmetric configuration. We can therefore safely rely on the results obtained so far for pristine graphene, as the unavoidable presence of a low concentration of defects and noise in the actual samples is not able to turn down the figure of merit of the problem. W/2 W/2 (a) (b) (c) (d) V. CONCLUSIONS Our results strongly indicate that graphene would be an excellent skyrmion detector at realistic exchange couplings of the order of ∼ 1-10 meV, exhibiting minimum Hall conductances G H of the order of 10 −5 -10 −4 G 0 , several orders of magnitude larger than the minimum experimentally detectable conductance of the order of 10 −10 G 0 44,45 . The equivalent magnetic field B eq can easily reach one Tesla for J ≈ 1 meV , R ≈ 2 nm and L ≈ 10 nm. Besides, these values merely set a lower bound estimate for the conductances that are detectable in actual experimental devices where sample dimensions, skyrmion radius and even skyrmion number can be consistently larger than those considered in this work. Our results also show that at weak coupling Schrodinger electrons are less sensitive to the non-trivial magnetic ordering and respond with a conductance that is some orders of magnitude smaller than that displayed by Dirac electrons. Finally, we proved that scalar disorder does not affect the transverse conductance in a dramatic manner. In conclusion, we suggest that graphene might be exploited as a non-invasive probe to readout the presence of an individual skyrmion in a material underneath. The underlying physical principle is the enhanced anomalous Hall effect due to the interaction of Dirac graphene fermions with non-coplanar spin textures. Our work establishes the principles of hybrid devices combining graphene Hall probes and insulating skyrmionic materials 10-12 . ACKNOWLEDGMENTS The authors acknowledge financial support by Marie-Curie-ITN 607904-SPINOGRAPH. JFR acknowledges fi-nancial supported by MEC-Spain (FIS2013-47328-C2-2-P and MAT2016-78625-C2) and Generalitat Valenciana (ACOMP/2010/070), Prometeo, by ERDF funds through the Portuguese Operational Program for Competitiveness and Internationalization COMPETE 2020, and National Funds through FCT-The Portuguese Foundation for Science and Technology, under the project PTDC/FIS-NAN/4662/2014 (016656). This work has been financially supported in part by FEDER funds. JLL and FF thank the hospitality of the Departamento de Fisica Aplicada at the Universidad de Alicante. We are grateful to F. Guinea and P. San-Jose for useful discussions. APPENDIX: DETERMINATION OF Beq In order to determine the equivalent magnetic field B eq , we have performed a calculation of the transmission imbalance T of a three-terminal triangular device where a perpendicular magnetic field B ⊥ is applied to the transmission region. To include such field, we retain only the hopping term of eq. 12 where we perform the standard Peierls substitution t → t exp −i e rj ri A · dr such that H = −t <i,j>,σ c † iσ c jσ e −i e r j r i A·dr By calculating the transmission imbalance between left and right lead, one gets a linear relation T ≈ 20 B ⊥ as shown in fig. 6. The linear relation between B ⊥ and T , in the absence of a skyrmion, permit to assign an equivalent field B eq to characterize the transmission imbalance calculated in the presence of a skyrmion at B ⊥ = 0. Figure 3 . 3(a) Three terminal device setup for the inverse Hall measurement with C3 rotational symmetry. (b) Normalized transmission imbalance T (eq. 18) and equivalent magnetic field Beq as a function of the coupling constant J, comparison of a Dirac-like (undoped graphene) and a Schrodinger-like (heavily doped graphene) material for an island with side of 10.6 nm and a skyrmion radius of 2.3 nm. Inset: log-log representation of T (J) and Beq(J). Figure 4 . 4(a) Schematics of the effect on the local electronic structure of graphene of being proximized to a skyrmion. (b) Left-right normalized transmission imbalance T of a graphene TTD as a function of the transmission energy of the leads ε for an island of 15.5 nm, skyrmion radius of 3.4 nm and coupling constant J = 80 meV. Energies characterized by maximum absolute Berry curvature in the infinite system are evidenced. (c) and (d) Transmission imbalance of a graphene TTD as a function of skyrmion radius (with fixed flake size of L = 10.6 nm) and flake size (with fixed skyrmion radius of R = 1.4 nm), respectively. 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[]	[ "\nJIANQI LIU\n\n" ]	[ "JIANQI LIU\n" ]	[]	The filtrations on Zhu's algebra A(V ) and bimodules A(M ) are studied. As an application, we prove that A(V ) is noetherian when V is strongly finitely generated. By using the associated graded module grA(M ), we find some connections between different tensor products of A(V ) bimodules A(N ) and A(M ).	null	[ "https://arxiv.org/pdf/2103.08090v2.pdf" ]	232,233,706	2103.08090	50f427cc0828239927c0b0066a50897d64b3aa1d	6 Mar 2022 JIANQI LIU 6 Mar 2022arXiv:2103.08090v2 [math.QA] ON FILTRATIONS OF A(V ) The filtrations on Zhu's algebra A(V ) and bimodules A(M ) are studied. As an application, we prove that A(V ) is noetherian when V is strongly finitely generated. By using the associated graded module grA(M ), we find some connections between different tensor products of A(V ) bimodules A(N ) and A(M ). Introduction Let V be a vertex operator algebra(VOA) over the ground field C. If V is a rational VOA, then by the main result in [3] the Zhu's algebra A(V ) is finite dimensional semisimple over C. In particular, A(V ) is (left) noetherian as a ring. In fact, the noetherian property of A(V ) actually holds for some irrational VOAs as well. For example, if V = Mĥ(1, 0) be the level one Heisenberg VOA, it is well-know that its Zhu's algebra A(Mĥ(1, 0)) is isomorphic to S(h) the polynomial ring over the finite dimensional vector space h, hence A(Mĥ(1, 0)) is noetherian. More generally, if V = Vĝ(k, 0) the level k ∈ Z >0 vacuum module VOA associated to the finite dimensional Lie algebra g, then it is also well-know that A(Vĝ(k, 0)) ∼ = U(g) which is noetherian as well. And if V =V (c, 0) is the Virasoro VOA of central charge c, then by [18], A(V (c, 0)) ∼ = C[x] the polynomial ring with one variable, which is also noetherian. These VOAs have one common property, which is that they are all strongly finitely generated (see [12] for the definition), so it is natural to expect A(V ) to be noetherian for every strongly finitely generated VOA V . We prove this fact in section 2 by using the level filtration on A(V ) given in [19]. The strongly generating condition of a VOA V was systematically studied by Li (see [14], [15] and [13]), it is proved that V is strongly generated by a subspace U ⊆ V if and only if the C 2 -algebra V /C 2 (V ) defined in [19] is generated by (U + C 2 (V ))/C 2 (V ) as an algebra, if and only if V = U + C 1 (V ), where C 1 (V ) is defined and studied in [15]. In particular, V is finitely strongly generated if and only if V is C 1 -cofinite [13]. In section 3, we define a similar strongly generating property for an admissible Vmodule M, and we show that M is strongly generated by a subspace W if and only if M = W + C 1 (M), where C 1 (M) is the one defined in [9]. In particular, our definition of a finitely strongly generated admissible module is the same as a C 1 -cofinite module defined in [9]. On the other hand, the A(V )-bimodule A(M) also has a natural filtration given by its gradings: A(M) n = n i=0 M(i) for all n ≥ 0. Under this filtration A(M) becomes a filtered A(V )-bimodule, and by using the associated graded module grA(M) over grA(V ) we prove that A(M) is generated as a filtered A(V )-bimodule by W when M is strongly generated by W . The filtration on A(M) is also useful in the study of the tensor product and fusion rules. In fact, A(M) was defined in the first place with the purpose of studying the fusion rules of VOA [8], and the connections between the fusion tensor product of VOAs defined by Huang and Lepowsky [11] and bimodules A(M) was noticed in [5]. In particular, when the VOA V is rational and C 2 -cofinite, the associativity of the fusion tensor [10] is equivalent to the isomorphism between A(M) ⊗ A(V ) A(N) and A(N) ⊗ A(V ) A(M) as A(V )-bimodules. But one would also expect to find an isomorphic map directly from the construction of A(M). However this is not an easy task, since the left and right module structures of A(M) are not interchangeable. In this paper, we use the filtration on the tensor product A(M) ⊗ A(V ) A(N) to present such an isomorphic map with an additional assumption. This paper is organized as follows: In section 2, we first recall some basic definitions and properties of filtered rings and Zhu's algebra A(V ), then prove the noetherianess of A(V ) for strongly finitely generated VOA V . We also studied the relations between the C 2 -algebra V /C 2 (V ) and the graded algebra grA(V ) for various cases of VOAs. In section 3, we first define the concept of strongly generated and quasi-strongly generated modules over VOAs and explore the connections between these concepts with the Poisson module M/C 2 (M) and grA(M). We use the filtrations on A(V )-bimodules to study A(M) ⊗ A(V ) A(N). 2. Noetherianess of A(V ) when V is strongly finitely generated. We will first recall some basic facts about filtered rings, some of them will be used in the proof of noetherianess of A(V ) and section 3. Most of these notions can be found in [16] and [17]. Definition 2.1. A ring R is called a filtered if there exists subgroups F n R ≤ R for n = 0, 1, 2... such that: (a) 1 ∈ F 0 R ⊆ F 1 R ⊆ F 2 R ⊆ ..., (b) F i R·F j R ⊆ F i+j R for all i, j ≥ 0, (c) R = ∞ n=0 F n R. The family {F n R} ∞ n=0 is called a filtration of R. A filtered ring R has the following related constructions: (a) Given a filtration {F n R} ∞ n=0 of R, there is an associated graded ring grR = F n R/F n−1 R = ∞ n=0 (grR) n , with product given byx ·ȳ := xy for any x ∈ F n R, y ∈ F m R. We adopt the convention that F −1 R = 0. The well-definedness of the product follows immediately from the definition of a filtration. (b) Let I ⊳ R be a left ideal. Set (grI) n := (I + F n−1 R) ∩ F n R/F n−1 R ≤ (grR) n for each n ≥ 0, then let grI = ∞ n=0 (grI) n ≤ grR. Clearly, grI is a graded left ideal of grR withā.x := ax forā ∈ (grR) m ,x ∈ (grI) n . (c) A R-module M is called filtered if there exists a sequence of subgroups (grM) n is a graded module over grR, with action given byx.w = x.w for any x ∈ F n R, w ∈ F m M. 0 = F −1 M ⊆ F 0 M ⊆ F 1 M ⊆ ... such that M = ∞ n=0 F n M and (F m R).F n M ⊆ F m+n M for all m, n ∈ N. One can find the following well known fact in [16]. Proposition 2.2. Let R be a filtered ring such that grR is left Noetherian, then R is left Noetherian. The definition of a vertex operator algebra can be found in [6]. In this paper, we assume that a VOA V is always N-gradable: V = ∞ n=0 V n , and is of CFT-type: V 0 = C1. Then we can write V = V 0 ⊕ V + , where V + is the sum of all positive levels. Now we recall some basic facts about the Zhu's Algebra A(V ) defined in [19]. Let a, b ∈ V be homogeneous elements. (a) A(V ) = V /O(V ) is an associated algebra with 1 + O(V ) as identity and the product: a * b = Res z Y (a, z)b (1 + z) wta z = wta j=0 wta j a j−1 b (2.1) (b) For positive integers m ≥ n ≥ 0: Res z Y (a, z)b (1 + z) wta+n z 2+m ≡ 0 mod O(V ) (2.2) (c) There is a commutative formula: a * b − b * a ≡ Res z Y (a, z)b(1 + z) wta−1 = j≥0 wta − 1 j a j b (mod O(V )) (2.3) (d) A(V ) has a filtration A(V ) 0 ⊆ A(V ) 1 ⊆ A(V ) 2 ⊆ ..., where A(V ) n is the image of ⊕ n i=0 V i in A(V ). i.e. for a ∈ A(V ) m and b ∈ A(V ) n we have a * b ∈ A(V ) m+n , and 1 + O(V ) ∈ A(V ) 0 . We call the filtration in (d) the level filtration of A(V ). These properties indicate that A(V ) is a filtered ring, so we have the associated graded ring as in Definition 2.1: grA(V ) = ∞ n=0 A(V ) n /A(V ) n−1 = ∞ n=0 (grA(V )) n ,(2.4) where (grA(V )) n = A(V ) n /A(V ) n−1 for all n ≥ 0 and A(V ) −1 = 0. By Definition 2.1, for a ∈ A(V ) m /A(V ) m−1 andb ∈ A(V ) n /A(V ) n−1 , their product is given bȳ a * b = a * b ∈ A(V ) m+n /A(V ) m+n−1 (2.5) The graded ring grA(V ) satisfies the following property: Lemma 2.3. grA(V ) is a commutative Poisson algebra with the product and the Lie bracket given by: ā * b = a −1 b + A(V ) m+n−1 ∈ (grA(V )) m+n , (2.6) {ā,b} = a 0 b + A(V ) m+n−2 ∈ (grA(V )) m+n−1 , (2.7) for allā ∈ A(V ) m /A(V ) m−1 ,b ∈ A(V ) n /A(V ) n−1 , and all m, n ∈ N. Proof. By (2.1) and (2.5) we have: a * b = a −1 b + wta j=1 wta j a j−1 b = a −1 b, since for any j ≥ 1 we have wt(a j−1 b) = wta + wtb − j ≤ m + n − 1, so a j−1 b ∈ A(V ) m+n−1 and a j−1 b = 0 in A(V ) m+n /A(V ) m+n−1 . Moreover, by (2.3) and (2.5) we have: a * b −b * ā = wta−1 j=0 wta − 1 j a j b = 0, since wt(a j b) = wta + wtb − j − 1 ≤ m + n − 1 for all j ≥ 0. This shows that grA(V ) is a commutative algebra over C. Since grA(V ) is commutative, it follows from a standard fact of filtered rings (cf. [17]) that grA(V ) is a Poisson algebra with respect to the bracket {ā,b} = a * b − b * a + A(V ) m+n−2 ∈ (grA(V )) m+n−1 . Since we have a * b−b * a ≡ a 0 b (mod A(V ) m+n−2 ) , it follows that grA(V ) is a commutative Poisson algebra with respect to the bracket given in (2.7). The notion of a strongly generated vertex operator algebra is defined by Kac[12]: Definition 2.4. Let V be a VOA, and let U ⊆ V be a subset. V is said to be strongly generated by U if V is spanned by elements of the form: a 1 −n 1 . ..a r −nr u, where a 1 , ..., a r , u ∈ U, and n i ≥ 1 for all i. If V is strongly generated by a finite dimensional subspace, then V is called strongly finitely generated. Recall the following result of Li (cf. Theorem 4.11 [14]): Proposition 2.5. Let V be a VOA, and let U ⊆ V + be a graded subspace. The following conditions are equivalent: (a) V is strongly generated by U (b) V + = U + C 1 (V ), where C 1 (V ) = span({u −1 v : u, v ∈ V + } ∪ {L(−1)u : u ∈ V }). (c) (U + C 2 (V ))/C 2 (V ) generates V /C 2 (V ) as commutative algebra. Theorem 2.6. Let V be a VOA. If V is strongly finitely generated, or equivalently, C 1cofinite, then A(V ) is (left) noetherian as an algebra. Proof. First, we show that there is a well-defined epimorphism of commutative Poission algebras: φ : V /C 2 (V ) → grA(V ) = ∞ n=0 A(V ) n /A(V ) n−1 , x + C 2 (V ) →x ∈ A(V ) n /A(V ) n−1 for x ∈ V n . (2.8) Define φ : V = ⊕ ∞ n=0 V n → grA(V ) : φ(x 1 + ... + x r ) = x 1 + ...x r , where x i ∈ V n i and x i ∈ A(V ) n i /A(V ) n i −1 for all i. Clearly φ is linear, and we claim that φ(C 2 (V )) = 0. Indeed, let a −2 b be a spanning element in C 2 (V ), with a ∈ V m and b ∈ V n , where m ≥ 1 and n ≥ 0. Then a −2 b ∈ V m+n+1 and φ(a −2 b) = a −2 b ∈ A(V ) m+n+1 /A(V ) m+n . Recall that in A(V ) we have: L(−1)a + L(0)a = 0. Thus, a −2 b = (L(−1)a) −1 b = (−L(0)a) −1 b = −ma −1 b in A(V ), with wt(a −1 b) = m + n. This implies that a −2 b = −ma −1 b =0 ∈ A(V ) m+n+1 /A(V ) m+n . Thus, φ(C 2 (V )) = 0 and φ gives rise to the map in (2.8). Since A(V ) n is the image of ⊕ n i=0 V n in A(V ), it is clear that φ is surjective. Moreover, by Lemma 2.3 we have: φ((a + C 2 (V )) · (b + C 2 (V ))) = φ(a −1 b + C 2 (V )) = a −1 b + A(V ) wta+wtb = φ(a + C 2 (V )) * φ(b + C 2 (V )), and similarly φ({a + C 2 (V ), b + C 2 (V )}) = φ(a 0 b + C 2 (V )) = a 0 b + A(V ) wta+wtb−1 = {φ(a + C 2 (V )), φ(b + C 2 (V ))} for all homogeneous a, b ∈ V . Therefore, φ given in (2.8) is an epimorphism of commutative Poisson algebras. Now let U = span{x 1 , ..., x n } be a subspace that strongly generates V . By proposition 2.5, V /C 2 (V ) is generated by {x 1 + C 2 (V ), ..., x n + C 2 (V )} as an algebra, in particular V /C 2 (V ) is finitely generated, hence its image grA(V ) under the epimorphism φ is also finitely generated. Thus grA(V ) is noetherian, since it is quotient of a polynomial ring with finitely many variables. Then by Proposition2.2, A(V ) is also (left) noetherian. It is natural to ask whether or not the epimorphism (2.8) is an isomorphism between the commutative Poisson algebras V /C 2 (V ) and grA(V ). In general, this is not true, we will give a counterexample later. Nevertheless, we have the following result regarding this question: Proposition 2.7. The epimosphism φ in (2.8) is an isomorphism, if and only if for all a = a 1 + ... + a r ∈ O(V ), with a i ∈ V n i for each i and n 1 < n 2 < ... < n r , the highest weight summand a r of a belongs to C 2 (V ). Proof. By the proof of Theorem 2.6 we already have C 2 (V ) ⊆ ker φ, and so φ is an isomorphism if and only if C 2 (V ) = ker φ. Also note that φ in (2.8) is clearly gradation preserving. Assume the condition for O(V ) in the proposition is satisfied, let x + C 2 (V ) ∈ ker φ with x ∈ V n , then we have x + O(V ) ∈ A(V ) n−1 , and so there exists y ∈ n−1 i=0 V i s.t. x − y = a = a 1 + ... + a r ∈ O(V ), with a i ∈ V n i for each i and n 1 < n 2 < ... < n r . By comparing the highest weight elements on both sides of this equation, we have x = a r ∈ C 2 (V ). Hence C 2 (V ) = ker φ and φ is an isomorphism. Conversely, if C 2 (V ) = ker φ, let a = a 1 + ... + a r ∈ O(V ) , with a i ∈ V n i for each i and n 1 < n 2 < ... < n r , we have a r + O(V ) = −a 1 − a 2 − ... − a r−1 + O(V ) in A(V ) nr . But the right hand side lies in A(V ) nr−1 since n 1 < n 2 ... < n r−1 ≤ n r − 1, so φ(a r ) = a r =0 ∈ A(V ) nr /A(V ) nr−1 ⊆ grA(V ), this implies a r ∈ ker φ = C 2 (V ).u • v = u −2 v + j≥1 wtu j u j−2 v, it is not true for a general element r i=1 u i • v i in O(V ) , because the highest weight components u i −2 v i may cancel with each other. But for certain examples of VOAs, especially the VOAs that are also universal highest weight modules over infinite dimensional Lie algebras, the C 2 -algebra V /C 2 (V ) is indeed isomorphic to grA(V ), and we can prove it in a direct way. We denote the C 2 -algebra V /C 2 (V ) by P 2 (V ) as in [4]. Proposition 2.9. Let g be a finite dimensional Lie algebra, for the vacuum module VOA [8] V = Vĝ(k, 0) of level k, we have: P 2 (Vĝ(k, 0)) ∼ = grA(Vĝ(k, 0)) as commutative Poisson algebras. Proof. It is well-known (cf. [4] Proposition 5.16) that in this case P 2 (Vĝ(k, 0)) ∼ = S(g), with a 1 (−1)...a r (−1)1 + C 2 (V ) → a 1 ...a r , for a 1 , ..., a r ∈ g. On the other hand, we have the following identification of the Zhu's algebra [8]: A(Vĝ(k, 0)) ∼ = U(g), with [a 1 (−1)...a r (−1)1] → a r ...a 1 . Note that under this isomorphism we have A(Vĝ(k, 0)) n = span{[a 1 (−1)...a r (−1)1] : a i ∈ g, 0 ≤ r ≤ n} ∼ = span{a r ...a 1 : a i ∈ g, 0 ≤ r ≤ n} = U(g) n , the standard filtration of U(g) [1]. It follows that grA(Vĝ(k, 0))( ∼ = grU(g)) ∼ = S(g), [a 1 (−1)...a r (−1)1] + A(V ) r−1 → a r ...a 1 = a 1 ...a r , thus we have an isomorphism: P 2 (Vĝ(k, 0)) ∼ = grA(Vĝ(k, 0)), a 1 (−1)...a r (−1)1 + C 2 (V ) → [a 1 (−1)...a r (−1)1] + A(V ) r−1 , (2.9) which is exactly the map φ in (2.8). It is easy to see that by adopting a similar method, we can also show that for V = Mĥ(k, 0), the Heisenberg VOA, P 2 (Mĥ(k, 0)) ∼ = S(h) ∼ = grA(Mĥ(k, 0)) under the same identification map (2.9). Proposition 2.10. Let V =V (c, 0) = V (c, 0)/ L −1 1 be the Virasoro VOA associated to the (universal) highest weight moduleV (c, 0) [8]. We also have: P 2 (V (c, 0)) ∼ = grA(V (c, 0)) as commutative Poisson algebras. Proof. Recall that V (c, 0) = span{L −n 1 ...L −n k 1 : k ≥ 0, n 1 ≥ n 2 ≥ ... ≥ n k ≥ 2}, and the spanning elements are linearly independent. Thus, we have a linear isomorphism: P 2 (V (c, 0)) = span{(L −2 ) n 1 + C 2 (V ) : n ≥ 0} ∼ = C[y], (L −2 ) n 1 + C 2 (V ) → y n , for all n ∈ N. On the other hand, it is proved in [18] that the Zhu's algebra ofV (c, 0) is isomorphic to C[x] via A(V (c, 0)) ∼ = C[x], [ω] n → x n , for all n ∈ N. Moreover, it is also noticed in [18] that for every n ≥ 1, one has the following relation: L −n ≡ (−1) n ((n − 1)(L −2 + L −1 ) + L 0 ) mod O(V (c, 0)), and [b] * [ω] = [(L −2 + L −1 )b] for any b ∈V (c, 0). Thus, in A(V (c, 0)) we have: [L −n 1 ...L −n k 1] = P ([ω]), for some P (x) ∈ C[x] with deg P ≤ k. So the level filtration of A(V ) satisfies: A(V (c, 0)) n = span{[L −n 1 ...L −n k 1] : k ≥ 0, n 1 + ... + n k = n, n i ≥ 2, ∀i} = span{P ([ω]) : deg P ≤ k ≤ ⌊n/2⌋} = span{[1], [ω], [ω] 2 , ..., [ω] r : r ≤ ⌊n/2⌋}, for all n ∈ N. In particular, we have A(V (c, 0)) 2p = A(V (c, 0)) 2p+1 = span{[1], [ω], ..., [ω] p }, for all p ∈ N. This resluts in a filtration {F p C[x]} p∈N of C[x], where F 2p C[x] = F 2p+1 C[x] = span{1, x, x 2 , ..., x p }. It is obvious that under this filtration we have an isomorphism: gr F C[x] = ∞ p=0 F 2p C[x]/F 2p−1 C[x] ∼ = C[y], x p + F 2p−1 C[x] → y p , for all p ∈ N. Moreover, note that in A(V (c, 0)) 2p /A(V (c, 0)) 2p−1 [ω] p + A(V (c, 0)) 2p−1 = [(L −2 + L −1 ) p 1] + A(V (c, 0)) 2p−1 = [(L −2 ) p 1] + A(V (c, 0)) 2p−1 . Hence we have an isomorphism: P 2 (V (c, 0))( ∼ = C[y]) ∼ = grA(V (c, 0)), (L −2 ) p 1 + C 2 (V ) → [ω] p + A(V (c, 0)) 2p−1 = [(L −2 ) p 1] + A(V (c, 0)), (2.10) which is the same map φ in (2.8). Let L be a positive definite even lattice. For the lattice VOA V L , P 2 (V L ) is not isomorphic to grA(V L ) in general. Here is a counterexample: Example 2.11. Let L = E 8 be the root lattice of type E 8 . It is well-known that this lattice is unimodular. By the main reslut in [2], V E 8 is rational, and it has only one irreducible module, namely itself, and the bottom level of this module is C1. It follows from [19] (cf. Theorem 2.2.1) that dim A(V E 8 ) = 1 = dim grA(V E 8 ). On the other hand, note that for any VOA V of CFT-type, we have V 1 ∩ C 2 (V ) = 0 because wta −2 b = wta + wtb + 1 ≥ 2 when a −2 b is nonzero. Hence dim P 2 (V E 8 ) ≥ dim(V E 8 ) 1 ≥ rank E 8 = 8, and so dim P 2 (V E 8 ) > dim grA(V E 8 ). In fact, a similar argument also shows for any unimodular lattice L, P 2 (V L ) ≇ grA(V L ). However, for certain positive definite even lattices, we do have the isomorphism: Proposition 2.12. Let L = Zα be the positive definite lattice of rank 1 with (α\|α) = 2k, where k ∈ Z >0 . Then for the lattice VOA V = V Zα , we have: P 2 (V Zα ) ∼ = grA(V Zα ) as commutative Poisson algebras. Proof. Since φ : P 2 (V Zα ) → grA(V Zα ) in (2.8) is already an epimorphism, we only have to show that dim P 2 (V Zα ) = dim grA(V Zα ). Note that in this case the dual lattice L • = k n=−k+1 L + n 2k α, and by the main result in [2] , the irreducible V Zα modules are V L+ n 2k α , for −k + 1 ≤ n ≤ k. When n = k, the bottom level of V L+ 1 2 α is Ce α/2 ⊕ Ce −α/2 . When \|n\| < k, we have (mα + n 2k α\|mα + n 2k α) = (m + n 2k ) 2 (α\|α) > n 2k 2 (α\|α), for all m ∈ Z\{0}, since m 2 + n k m > 0. So the bottom level of V L+ n 2k α is one-dimenional for any −k + 1 ≤ n < k. Thus, by Theorem 2.2.1 in [19], dim grA(V Zα ) = dim A(V Zα ) = 2 2 + (2k − 1) · 1 2 = 2k + 3. On the other hand, it is proved in [4] (cf. Proposition 5.19) that P 2 (V Zα ) is a quotient of the polynomial algebra C[X, Y, Z], moudulo the relations: X 2 = Y 2 = XZ = Y Z = 0, XY = 1 (2k)! Z 2k . In particular, P 2 (V L ) has a basis1,X,Ȳ ,Z, , ...,Z 2k−1 ,Z 2k = (2k)!XY . So dim P 2 (V Zα ) = 2k + 3 = dim grA(V Zα ). Remark 2.13. Example (2.11) and Proposition (2.12) also indicate that for an affine VOA Lĝ(k, 0) with positive integer level k, the C 2 algebra P 2 (Lĝ(k, 0)) may or may not be isomorphic to grA(Lĝ(k, 0)). Indeed, since L = E 8 is a simple laced root lattice, the lattice VOA V E 8 is isomorphic to the affine VOA L g E 8 (1, 0) (see [7]), where g E 8 is the simple Lie algebra whose root system is of the type E 8 . So P 2 (L g E 8 (1, 0)) ≇ grA(L g E 8 (1, 0)). On the other hand, for L = Zα with (α\|α) = 2, L is the root lattice of type A 1 . Hence V L ∼ = Lŝ l 2 (1, 0) as VOAs (see [7]), and by the conclusion of Proposition (2.12), we have P 2 (Lŝ l 2 (1, 0)) ∼ = grA(Lŝ l 2 (1, 0)). We conclude this section by giving an application of the notherianess of A(V ) in theorem (2.6). Recall that a V -module M is called admissible (or N-gradable) [19], [3] if M = ∞ n=0 M(n), and a m M(n) ⊆ M(wta − m − 1 + n), for all a ∈ V , m ∈ Z, and n ∈ N. Note that each graded subspace M(n) of an admissible module M needs not to be finite dimensional. For instance, let U be an infinite-dimensional module over a simple Lie algebra g, then the induced module Vĝ(k, U) = U(ĝ) ⊗ U (ĝ ≥0 ) U is an admissible module over the VOA Vĝ(k, 0), with bottom level Vĝ(k, U)(0) = U that is not finite dimensional [8]. Also recall that the bottom level M(0) of any admissible module M is an A(V )module [19], with the action given by [a].w = a wta−1 w, for all [a] ∈ A(V ) and w ∈ M(0). Proposition 2.14. Let V be a VOA that is C 1 -cofinite. Assume M is an admissible V -module s.t. M Graded module grA(M) and the tensor product We first give a definition of strongly generated module over the VOA V . Recall that we assume V to be of CFT-type: x = a 1 −n 1 ...a k −n k w, V = V 0 ⊕ V + , with V 0 = C1 and V + = ∞ n=1 V n .(3.1) where k ≥ 0, a i ∈ V homogeneous, n i ≥ 0 for all i, and w ∈ W . (b) We say that M is strongly generated by W , if M is spanned by elements of the form: x = a 1 −n 1 ...a k −n k w, (3.2) where k ≥ 0, a i ∈ V homogeneous, n i ≥ 1 for all i, and w ∈ W . Note that under this definition, if W strongly generates M, it must quasi-strongly generates M. Moreover, since 1 0 = 0 and wta i −n i = wta i + n i − 1 ≥ n i for a i ∈ V + , it follows that W ∩ M(0) = 0 when W quasi-strongly generates M, and M(0) ⊆ W when W strongly generate M. Example 3.2. The module Vĝ(k, λ) = U(ĝ) ⊗ U (ĝ ≥0 ) L(λ) and its irreducible quotient Lĝ(k, λ) over the vacuum module VOA Vĝ(k, 0) (or over the affine VOA Lĝ(k, 0) with k ∈ Z + and λ ∈ P k + ) are both strongly generated by their bottom level L(λ), and are quasi-strongly generated by the highest-weight vector v λ ∈ L(λ). The irreducible module M = Mĥ(k, λ) over the Heisenberg VOA Mĥ(k, 0) is strongly generated its bottom level M(0) = Ce λ . The irreducible module M = L(c, h) over the Virasoro VOA L(c, 0) is quasi-strongly generated by its bottom level M(0) = Cv c,h , and it is strongly generated by the set W = {L(−1) n v c,h : n ≥ 0}. Let L = Zα be a rank 1 lattice with (α\|α) = 2k, for some k ∈ N. Consider the irreducible module V L+ n 2k α for some 0 ≤ n ≤ 2k − 1 over the lattice VOA V L . It is strongly generated by two elements e It is proved in [13] (cf. Proposition 3.2.) that a CFT-type VOA V is strongly generated by a subspace U ≤ V + if and only if V = U + C 1 (V ). By a slight modification of their proof, we can derive a similar result for the modules. For the sake of correctness, we will write out the details of the proof in the next proposition. Let M = ∞ n=0 M(n) be an admissible V module. We follow [9] and set C 1 (M) to be the subspace of W spanned by a −1 M, for all homogeneous a ∈ V + (Note that this definition of C 1 (M) is slightly different from the one given in [15]). It follows immediately from the definition that C 1 (M) ⊆ n≥1 M(n). We follow [9] again and call M C 1 -cofinite if dim M/C 1 (M) < ∞. u 1 −n 1 ...u r −nr .w (3.3) for r ≥ 0, n i ≥ 1, u i ∈ U and w ∈ W homogeneous. In particular, M is strongly generated by W . Proof. Since W ⊆ M is homogeneous, it is clear that M(0) ⊆ W . Denote the subspace spanned by elements in (3.3) by P M . We follow [13] and give a filtration P 0 ⊆ P 1 ⊆ ... on V by letting P s be the subspace of V spanned by u 1 −n 1 ...u k −nr 1 for 0 ≤ r ≤ s, u i ∈ V + , n i ≥ 1. Claim 1. If M(0) + M(1) + ... + M(n) ⊆ P M for some n ≥ 0, and if a ∈ V + , v ∈ M homogeneous such that a −r v ∈ M(n + 1) for some r ≥ 1, then a −r v ∈ P M . Indeed, since n + 1 = deg(a −r v) = wta + deg v + r − 1 ≥ wta + deg v and wta ≥ 1, we have deg v ≤ n, and so v ∈ P M by assumption of the claim. By Proposition 3.2 in [13], V = ∞ s=0 P s . So a ∈ P s for some s ≥ 0, and we may use induction on s to show a −r v ∈ P M . If a ∈ P 1 , say a = u −m 1 for some m ≥ 1, then a −r v = (u −m 1) −r v = j≥0 −r j (−1) j u −m−j 1 −r+j v − (−1) −m−j −r j 1 −m−r−j (u j v) = −r r − 1 (−1) r−1 u −m−r+1 v ∈ P M . Now assume that a ∈ P s , and the conclusion holds for smaller s. Write a = u 1 −k b for some u 1 ∈ U, k ≥ 1 and b = u 2 −n 2 ...u r −nr 1 ∈ P s−1 . Then a −r v = (u 1 −k b) −r v = j≥0 −k j (−1) j u 1 −k−j (b −r+j v) − −k j (−1) −k+j b −k−r−j (u 1 j v), (3.4) with each summand on the right hand side has the same degree n + 1. Since wtu 1 ≥ 1, we have: Proof. If M is strongly generated by W then every spanning element of M as in (3.2), except for x = w ∈ W , is contained in C 1 (W ), since n + 1 = deg(u 1 −k−j (b −r+j v)) = wtu 1 + k + j − 1 + deg(b −r+j v) > deg(b −r+j v), thus, deg(b −r+j v) ≤ n and by assumption of the claim, b −r+j v ∈ P M . Hence u 1 −k−j (b −r+j v) ∈ P M for all j ≥ 0. On the other hand, since k + r + j ≥ 1, b ∈ P s−1 and b −k−r−j (u 1 j v) ∈ M(n + 1), by induction hypothesis, b −k−r−j (u 1 j v) ∈ P M for all j ≥ 0. Therefore,u −n w = 1 (n − 1)! (L(−1) n−1 u) −1 w ∈ C 1 (W ) for all u ∈ V + homogeneous and n ≥ 1. Thus, M = W +C 1 (W ). Conversely, assume M = W + C 1 (M), we may choose a homogeneous subspace U ⊆ V + such that V + = C 1 (V ) ⊕ U. Then by Proposition 3.2 in [13], V is strongly generated by U, and by Propostion 3.3, M is strongly generated by W . Note that for an admissible module M, M/C 2 (M) is a module over the commutative algebra V /C 2 (V ), with (a + C 2 (V )).(v + C 2 (M)) = a −1 v + C 2 (M) (3.5) for all a ∈ V and v ∈ M. In fact, M/C 2 (M) is also a Poisson module over V /C 2 (V ), because we can define a bilinear map {·, ·} : V /C 2 (V )×M/C 2 (M) → M/C 2 (M) by letting {a + C 2 (V ), w + C 2 (M)} := a 0 w + C 2 (M), (3.6) for all a ∈ V and w ∈ M, then it is easy to check that {{a + C 2 (V ), b + C 2 (V )}, w + C 2 (M)} = {(a + C 2 (V )), {b + C 2 (V ), w + C 2 (M)}} − {b + C 2 (V ), {a + C 2 (V, w + C 2 (M))}}, {a + C 2 (V ), b + C 2 (V )}.(w + C 2 (M)) = (a + C 2 (V )).{b + C 2 (V ), w + C 2 (M)} − {b + C 2 (V ), {a + C 2 (V ), w + C 2 (M)}}.... ⊆ 0 ⊆ 0 ⊆ A(M) 0 ⊆ A(M) 1 ⊆ A(M) 2 ⊆ ... with n∈Z A(M) n = 0 and n∈Z A(M) n = A(M). i.e. the filtration {A(M) n } n∈Z is exhaustive, separated and discrete (see [17] for the definitions). Moreover, for a ∈ A(V ) n and v ∈ A(M) m with homogeneous representatives a ∈ V and v ∈ M, by the formulas in [8] we have: a * v = wta j=0 wta j a j−1 v ∈ A(M) m+n , (3.7) v * a = wta−1 j=0 wta − 1 j a j−1 v ∈ A(M) m+n , (3.8) a * v − v * a = wta j=0 wta − 1 j a j v ∈ A(M) m+n−1 . (3.9) So it is easy to see that A(M) together with the filtration {A(M) n } n∈Z forms a filtered A(V )-bimodule, and by (3.7)-(3.9), the associated graded space grA(M) = ∞ n=0 A(M) n /A(M) n−1 (3.10) is a graded Poisson module over the commutative graded Poisson algebra grA(V ), with the module actions given by: a * v =v * ā := a −1 v + (grA(M)) m+n−1 ∈ (grA(M)) m+n , (3.11) {ā,v} := a * v − v * a = a 0 v + A(M) m+n−2 ∈ (grA(M)) m+n−1 , (3.12) for allā ∈ (grA(V )) n = A(V ) n /A(V ) n−1 andv ∈ (grA(M)) m = A(M) m /A(M) m−1 . Note that grA(M) is naturally a module over V /C 2 (V ) via the epimorphism φ : (2.8), and similar as Theorem 2.6, we have the following relation between M/C 2 (M) and grA(M): V /C 2 (V ) → grA(V ) in Lemma 3.6. There exists an epimophism of Poisson modules over the commutative Poisson algebra V /C 2 (V ): ψ : M/C 2 (M) → grA(M) : v + C 2 (M) →v ∈ A(M) m /A(M) m−1 for v ∈ M(m) (3.13) Proof. The proof is the same as the proof of Theorem (2.6), we omit it. Proof. This is a direct consequence of Lemma (3.5) and Lemma (3.6). Next, we will use our graded module grA(M) to prove a similar but slightly refined result regarding the generators of A(M), and we will use it in our later discussion of the tensor product. The following lemma is a variation of Proposition 5.3. in [17]. It can be applied nicely to our case of the A(V ) bimodules A(M). Since our assumptions here are different from the ones in [17], we write out the proof of it. Lemma 3.9. Let R be a filtered ring with filtration {F p R} p∈N , and let M be a filtered R-module with filtration {F p M} p∈N . If there exists w 1 , ..., w r ∈ M such that w i ∈ F n i M for each i, and grM = grR.w 1 + ... + grR.w r , then F p M = (F p−n 1 R).w 1 + ... + (F p−nr R).w r for all p ≥ 0. Proof. Recall that by definition, grR = ∞ p=0 (grR) p = ∞ p=0 F p R/F p−1 R, grM = ∞ p=0 (grM) p = ∞ p=0 F p M/F p−1 M. It follows that for all p ≥ 0 we have (grR) p−n 1 .w 1 + ... + (grR) p−nr .w r ⊆ (grM) p . On the other hand, by assumption grM = grR.w 1 + ... + grR.w r , we can write any y ∈ (grM) p as y = x 1 .w 1 + ... + x r .w r with x j = x j,k , where x j,k ∈ (grR) k for all j, k. Since y is homogeneous of degree p, we must have: y = x 1,p−n 1 .w 1 + ... + x r,p−nr .w r , hence (grM) p ⊆ (grR) p−n 1 .w 1 + ... + (grR) p−nr .w r . for all p ≥ 0. So we have: (grM) p = (grR) p−n 1 .w 1 + ... + (grR) p−nr .w r (3.14) for all p ≥ 0. We now use induction on p to show F p M = (F p−n 1 R).w 1 + ... + (F p−nr R).w r . The base case p = 0 follows from (3.14) and the facts that (grM) 0 = F 0 M and (grR) −n i = 0 if n i > 0 F 0 R if n i = 0. Let p > 0 and assume that the conclusion holds for smaller p. By definition we have (F p R).w 1 + ... + (F p R).w r ⊆ F p M. Let y ∈ F p M\F p−1 M, by (3.14), y ∈ F p M/F p−1 M = (grM) p = (grR) p−n 1 .w 1 + ... + (grR) p−nr .w r , so there exists x i ∈ F p−n i R for i = 1, 2, ..., r such that y − (x 1 .w 1 + ... + x r .w r ) ∈ F p−1 M. By induction hypothesis, we have F p−1 M = F p−1−n 1 .w 1 + ... + F p−1−nr .w r , then we can find z i ∈ F p−1−n i R ⊆ F p−n i R for i = 1, 2, ..., r such that y − (x 1 .w 1 + ... + x r .w r ) = z 1 .w 1 + ... + z r .w r . Therefore, y = (x 1 + z 1 ).w 1 + ... + (x r + z r ).w r ∈ (F p−n 1 R).w 1 + ... + (F p−nr R).w r . Remark 3.10. By duality, the conclusion in the previous lemma holds for right filtered modules M over R and right graded modules as well. The rest of this paper is dedicated to exploring the possibility of finding an isomorphism between the following two A(V )-bimodules: (3.15) under the conditions that V is rational, and M, N are simple V -modules. Note that in this case the Zhu's algebra A(V ) is semisimple [3], and its envelpoing algebra A(V ) e = A(V ) ⊗ C A(V ) op is also semisimple. In fact, it is not different to show that they are isomorphic as vector spaces: Let A(M) ⊗ A(V ) A(N) and A(N) ⊗ A(V ) A(M),φ M : A(M) → A(M), u → e L(1) (−1) L(0) u, φ N : A(N) → A(N), v → e L(1) (−1) L(0) v, for all u ∈ A(M) and v ∈ A(N), be the anti-involutions of A(M) and A(N), and let φ : A(V ) → A(V ), a → e L(1) (−1) L(0) a, for all a ∈ A(V ), be the anti-involution of A(V ) [19]. By the computation in [8] (see also [4]), it is easy to deduce that φ M (a * u) = φ M (u) * φ(a), φ M (u * a) = φ(a) * φ M (u), φ N (a * v) = φ N (v) * φ(a), φ N (v * a) = φ(a) * φ N (v), for all u ∈ A(M), v ∈ A(N), and a ∈ A(V ). Then we can definẽ φ : A(M) ⊗ A(V ) A(N) → A(N) ⊗ A(V ) A(M), u ⊗ v → φ N (v) ⊗ φ M (u). It is well-defined sinceφ (u * a ⊗ v) = φ N (v) ⊗ φ M (u * a) = φ N (v) ⊗ φ(a) * φ M (u) = φ N (v) * φ(a) ⊗ φ M (u) = φ N (a * v) ⊗ φ M (u) =φ(u ⊗ a * v), and similarlyφ(u ⊗ a * v) =φ(u * a ⊗ v). Clearlyφ satisfiesφ 2 = Id, henceφ is a linear isomorphism. However,φ is in general not a homomorphism of A(V )-bimodules. Indeed, φ(a.(u ⊗ v)) = φ N (v) ⊗ φ M (a * u) = φ N (v) ⊗ φ M (u) * φ(a) =φ(u ⊗ v).φ(a), andφ(u⊗v).φ(a) = a.φ(u⊗v) in general, because on the irreducible direct-sum component M i (0) ⊗ C M j (0) of A(N) ⊗ A(V ) A(M), where M 1 , ..., M p are all irreducible modules over V , the left action of a is not the same as the right φ(a) action. So we need another way to study the possible isomorphism between these two bimodules. By using our graded module grA(M), we can find such an isomorphism under an additional assumption. Since both A(M) and A(N) are N-filtered, following [17] I.8. we can give the tensor product A(M) ⊗ A(V ) A(N) a filtration {(A(M) ⊗ A(V ) A(N))} n∈N by letting (A(M) ⊗ A(V ) A(N)) n := span{u ⊗ v : u ∈ A(M) r , v ∈ A(N) s , r + s ≤ n}. (3.16) By this definition, clearly we have A(V ) m * (A(M) ⊗ A(V ) A(N)) n ⊆ (A(M) ⊗ A(V ) A(N)) m+n , (A(M) ⊗ A(V ) A(N)) n * A(V ) m ⊆ (A(M) ⊗ A(V ) A(N)) m+n . Let us give the semisimple algebra A(V ) ⊗ C A(V ) op a similar filtration: (A(V ) ⊗ C A(V ) op ) m := span{a ⊗ b : a ∈ A(V ) k , b ∈ A(V ) l , k + l ≤ m}, (3.17) then for a ⊗ b ∈ (A(V ) ⊗ C A(V ) op ) m and u ⊗ v ∈ (A(M) ⊗ A(V ) A(N)) n , we have (a ⊗ b).(u ⊗ v) = a * u ⊗ v * b ∈ (A(M) ⊗ A(V ) A(N)) m+n . Thus, A(M) ⊗ A(V ) A(N) becomes a filtered left A(V ) ⊗ C A(V ) op -module. Corollary 3.12. Assume that M is strongly generated u i ∈ M(m i ) for i = 1, 2, ..., p, and N is strongly generated by v j ∈ N(n j ) for j = 1, 2, ..., q, then the filtered left A(V ) ⊗ C A(V ) op -module A(M) ⊗ A(V ) A(N) satisfies: (A(M) ⊗ A(V ) A(N)) n = p i=1 q j=1 (A(V ) ⊗ C A(V ) op ) n−m i −n j .(u i ⊗ v j ). (3.18) for all n ≥ 0. Proof. By the definition of filtration on the tensor product (3.16) and (3.17), it is clear (N)) n for all i, j and n ≥ 0. On the other hand, let u ⊗ v be a spanning element of A(M) ⊗ A(V ) A(N), where u ∈ A(M) s and v ∈ A(N) t with s + t ≤ n. By Proposition 3.11, we can write u = a 1 * u 1 + ... + a p * u p Proof. Define a map that (A(V ) ⊗ C A(V ) op ) n−m i −n j .u i ⊗ v j ⊆ (A(M) ⊗ A(V ) Aand v = v 1 * b 1 + ... + v q * b q for some a i ∈ A(V ) s−m i and b j ∈ A(V ) t−n j for all i, j, then we have a i ⊗ b j ∈ (A(V ) ⊗ C A(V ) op ) n−m i −n j for all i, j, and u ⊗ v = (a 1 * u 1 + ... + a m * u m ) ⊗ (v 1 * b 1 + ... + v n * b n ) = m i=1 n j=1 (a i * u i ⊗ v j * b j ) = m i=1 n j=1 (a i ⊗ b j ).(u i ⊗ v j ) ∈ m i=1 n j=1 (A(V ) ⊗ C A(V ) op ) n−m i −n j .(u i ⊗ v j ).φ : gr(A(M) ⊗ A(V ) A(N)) → gr(A(N) ⊗ A(V ) A(M)) : u ⊗ v → v ⊗ u,(3.19) where u ∈ A(M) r , v ∈ A(N) s with r + s ≤ n and u ⊗ v ∈ (A(M) ⊗ A(N)) n /(A(M) ⊗ A(N)) n−1 . To show this map is well defined, first we note that if r + s ≤ n − 1 then v ⊗ u ∈ (A(N) ⊗ A(V ) A(M)) n−1 , hence φ(0) =0. Moreover, for any a ∈ A(V ) m since a * u − u * a = wta j=0 wta − 1 j a j u ∈ A(M) r+m−1 , and similarly v * a − a * v ∈ A(N) s+m−1 , then we have by the definition of filtration (3.16): a * u ⊗ v = u * a ⊗ v = u ⊗ a * v = u ⊗ v * a. (3.20) Apply (3.20) we get: φ(u * a ⊗ v) = v ⊗ u * a = a * u ⊗ v = φ(u ⊗ a * v), and similarly, φ(u ⊗ a * v) = φ(u * a ⊗ v). Therefore, φ is well-defined. Moreover, we can apply (3.20) again and get: φ(ā * u ⊗ v * b) = φ(a * u ⊗ v * b) = v * b ⊗ a * u =ā * v ⊗ u * b =ā * φ(u ⊗ v) * b. i.e. φ is a homomorphism of left gr(A(V ) ⊗ C A(V ) op )-modules. Since φ is clearly an involution, it follows that φ is an isomorphism of left gr(A(V ) ⊗ C A(V ) op )-modules. Our idea is to lift the isomorphism φ : gr(A(M) ⊗ A(V ) A(N)) → gr(A(N) ⊗ A(V ) A(M)) up to the level of filtered modules A(M) ⊗ A(V ) A(N) → A(N) ⊗ A(V ) A(M) . We shall need the following concepts in [17] for our further discussions Then HOM A (U, V ) ≤ Hom A (U, V ) is an abelian subgroup. The following result (Lemma 6.4. in [17]) gives us a connection between the Hom sets defined above. We also have the following characterization (Proposition 5.14. in [17]) of filt-projective objects. Lemma 3.16. Let R be a filtered ring, P ∈ R-filt such that F P is exhaustive. i.e. P = n∈Z F n P , then P is filt-projective if and only if P is a direct summand in R-filt of a filt-free object. Now assume that the admissible modules M and N over V are strongly generated by u i ∈ M(m i ) for i = 1, 2, ..., p and v j ∈ N(n j ) for j = 1, 2, ..., q, respectively. Note that this is not a strong condition imposed on irreducible modules; see Example (3.2). By Corollary 3.12, we can construct a linear map: In particular, the isomorphism φ in (3.19) corresponds to a filtration preserving isomor- is generated by finite elements from its bottom level. Then M must have a maximal submodule.Proof. By assumption, there exists a finite set S ⊂ M(0) s.t.M = span{a 1 n 1 ...a k n k w : a i ∈ V, n i ∈ Z, w ∈ S}. Given a spanning element x = a 1 n 1 ...a k n k w of M, if wta i − n i − 1 < 0 for some i,then a i n i w = 0, and x can be written as a sum of elements of shorter length. So it follows from an easy induction that the bottom level M(0) of M is span by elements of the form: a 1 wta 1 −1 ...a k wta k −1 w, for a 1 , ..., a k ∈ V homogeneous, and w ∈ S. i.e. M(0) is a finitely generated A(V )module. Since A(V ) is noetherian by Theorem 2.6, M(0) is a noetherian module, and so M(0) has a maximal submodule U. Let W ≤ M be the V -submodule generated by U. Then the bottom level of the quotient module M/W is an irreducible A(V )-module M(0)/U, and M/W is generated by its bottom level. Hence M/W is a quotient of the generalized Verma moduleM(M(0)/U) constructed in [4]. By Theorem 6.3 in [4], M/W has a maximal submoduleW with the property thatW ∩ (M(0)/U) = 0. But then (M/W )/W ∼ = L(M(0)/U), which is an irreducible V -module since M(0)/U is an irreducible A(V )-module. Thus, π −1 (W ) + W ≤ M is a maximal submodule, where π : M → M/W is the quotient map. Definition 3 . 1 . 31Let M = ∞ n=0 M(n) be an admissible(or N-gradable) V module, and let W ⊆ M be a subset. (a) We say that M is quasi-strongly generated by W , if M is spanned by elements of the form: α , because for any m ≥ 0, we have e mα+ n 2k α = ±(e mα ) −(mα\| n 2k α)−1 e n 2k α = ±(e mα ) mn − 1 ≤ −1, and m(n − 2k) − 1 ≤ −1. Proposition 3 . 3 . 33Let V be strongly generated by a homogeneous subspace U ⊆ V + . Let M = ∞ n=0 M(n) be an admissible V -module such that M = W + C 1 (M) for some homogeneous subspace W ⊆ M. Then M is spanned by elements of the form: each summand on the right hand side of (3.3) lies in P M , and so a −r v ∈ P M . claim 2. M = P M . i.e. M is spanned by elements of the form (3.3). Indeed, we may use induction on n to show M(0) + M(1)... + M(n) ⊆ P M . The base case is clear since M(0) ⊆ W ⊆ P M . Now assume that M(0) + M(1)... + M(n) ⊆ P M , then for any x ∈ M(n + 1) = M(n + 1) ∩ W + M(n + 1) ∩ C 1 (M), we may write x = w + a 1 −1 v 1 + ... + a k −1 v k some w ∈ W , a i ∈ V + and v i ∈ M homogeneous, with a i −1 v i ∈ M(n + 1) for all i. By claim 1, we have a i −1 v i ∈ P M for i = 1, 2, ..., k. Thus, x ∈ P M . Corollary 3.4. An admissible V module M is strongly generated by a homogeneous subspace W ⊆ M if and only if M = W + C 1 (W ). In particular, M is strongly finitely generated if and only if M is C 1 -cofinite. Lemma 3. 5 . 5Let M be an admissible V -module, and let W ⊆ M be a homogeneous subspace. (a) If M is strongly generated byW , then M/C 2 (M) is generated by W/C 2 (M) = {w + C 2 (M) : w ∈ W } as a V /C 2 (V )-module. (b) If M is quasi-strongly generated by W , then M/C 2 (M) is generated by W/C 2 (M) as a Poisson module over V /C 2 (V ) Proof. Assume W strongly generates M. Since a −1 C 2 (M) ⊆ C 2 (M) for all a ∈ V ,it follows that for any spanning element of M: x = a 1 −n 1 ...a r −nr w as in (3.2), we have x ∈ C 2 (M) unless n 1 = ...n r = 1, in which case we have x + C 2 (M) = a 1 −1 ...a r −1 w + C 2 (M) ∈ V /C 2 (V ).(W + C 2 (M)). Similarly, if W quasi-strongly generates M, then because a 0 C 2 (M) ⊆ C 2 (M), any x = a 1 −n 1 ...a r −nr w as in (3.1) is contained in C 2 (M) unless n i = 0 or 1 for i = 1, 2, ..., r, in which case by (3.5) and (3.6) x + C 2 (M) is in the Poisson submodule of M/C 2 (M) generated by W/C 2 (M). Let M be an admissible V module. For the A(V )-bimodule A(M) defined in [8], we construct a filtration as follows: For any positive integer n ≥ 0 set A(M) n := n i=0 (M(n) + O(M))/O(M), and set A(M) −n−1 = 0. Then we have: Proposition 3 . 7 . 37Let M be an admissible V -module, and let W ⊆ M be a homogeneous subspace. (a) If W strongly generates M, then grA(M) is generated by ψ(W/C 2 (M)) as grA(V ) module. In particular, if M is finitely strongly generated then grA(M) is finitely generated. (b) If W quasi-strongly generates M, then grA(M) is generated by ψ(W/C 2 (M)) as a Poisson grA(V )-module. Remark 3 . 8 . 38It is proved in [15] (Proposition 3.6) that the A(M) is generated by (M 0 + O(M))/O(M) as A(V )-bimodule if M = M 0 + B(M), where B(M) = C 1 (M) + span{a 0 M : wta ≥ 2}. In particular, if M/B(M) is finite dimensional then A(M) is a finitely generated A(V )-bimodule. Proposition 3 . 11 . 311Let M = ∞ n=0 M(n) be an admissible module over the VOA V . If M is stronly finitely generated by w 1 , ..., w r with w i ∈ M(n i ) for all i, thenA(M) n = A(V ) n−n 1 * w 1 + ... + A(V ) n−nr * w r = w 1 * A(V ) n−n 1 + ... + w r * A(V ) n−nras left or right A(V ) module for all n ≥ 0, where we use the same symbol w i for its image in A(M).Proof. By proposition3.7, grA(M) = grA(V ).w 1 + ... + grA(V ).w r , and by the definition of filtration on A(M) we have w i ∈ A(M) n i for all i. Thus, the filtered ring A(V ) and its filtered module A(M) satisfies the conditions in Lemma 3.9, and so the conclusion follows from Lemma 3.9. Proposition 3 . 13 . 313There is an isomorphism: gr(A(M) ⊗ A(V ) A(N)) ∼ = gr(A(N) ⊗ A(V ) A(M)), as graded grA(V )-bimodules or as graded left gr(A(V ) ⊗ C A(V ) op )-modules. F p HOM R (M, N) := {f ∈ Hom R (M, N) : f (F i M) ⊆ F i+p N, ∀i ∈ Z},then we have F p HOM R (M, N) ⊆ F q HOM R (M, N) for p ≤ q. Let HOM R (M, N) = p∈Z F p HOM R (M, N) ≤ Hom R (M, N), then HOM R (M, N) ≤ Hom R (M, N) is an abelian Z-filtered subgroup.(2) (The category R-filt). Objects in R-filt is defined to be the filtered left R-modules, and the morphism set between two objects M and N is defined to be F 0 Hom R (M, N). i.e. a map f : M → N is called a morphism if f ∈ Hom R (M, N) and f (F p M) ⊆ F p N for all p ∈ Z. Note that R-filt is NOT an abelian category.(3) Let A = p∈Z A p be a graded ring, and let U, V be two graded left A-modules. SetHOM A (U, V ) p := {f ∈ Hom A (U, V ) : f (U i ) ⊆ V i+p , ∀i ∈ Z}.Then we have p∈Z HOM A (U, V ) p = p∈Z HOM A (U, V ) p . LetHOM A (U, V ) := p∈Z HOM A (U, V ) p ≤ Hom A (U, V ). Lemma 3. 15 . 15If M, N ∈ R-filt, the natural mapϕ : gr(HOM R (M, N)) = p∈Z F p HOM R (M, N) F p−1 HOM R (M, N) → HOM gr(R) (gr(M), gr(N)) defined by ϕ(f )(x) := f (x) = f (x) + F p+q N for f ∈ F p HOM R (M, N) and x ∈ F q M is an monomorphism. Moreover, ϕ is an isomorphism if M is filt-projective.i.e. a projective object in R-filt. f : (A(V ) ⊗ C A(V ) op ) ⊕pq → A(M) ⊗ A(V ) A(N) : (x 11 , ..., x mn ) ij .(u i ⊗ v j ). f is a homomorphism between left A(V ) ⊗ C A(V ) op modules. Since A(V ) is semisimple, the universal algebra A(V )⊗ C A(V ) op of A(V ) is also semisimple, and so A(M) ⊗ A(V ) A(N) is isomorphic to a direct sum of simple left A(V ) ⊗ C A(V ) op modules. In particular, A(M) ⊗ A(V ) A(N) is projective in the category A(V ) ⊗ C A(V ) op -Mod,since it is a direct sum of projective modules. Thus, the map f has a section g in view of the following diagram:A(M) ⊗ A(V ) A(N) (A(V ) ⊗ C A(V ) op ) pq A(M) ⊗ A(V ) e. f g = Id. Theorem 3 . 17 . 317If the section g in(3.22) of f is filtration preserving, that is, g(u i ⊗v j ) ∈ (A(V ) ⊗ C A(V ) op ) ⊕pq m i +n j for all i, j, then the isomorphism in Proposition 3.13 can be lift up to an isomorphism:A(M) ⊗ A(V ) A(N) ∼ = A(N) ⊗ A(V ) A(M) of A(V )-bimodules.Proof. By assumption and Corollary 3.12, we haveg((A(M) ⊗ A(V ) A(N)) r ) ⊆ (A(V ) ⊗ C A(V ) op ) ⊕mn rfor all r ≥ 0. Therefore, for any r ≥ 0 we have(A(V ) ⊗ C A(V ) op ) ⊕mn r = g((A(M) ⊗ A(V ) A(N)) r ) ⊕ (ker f ∩ (A(V ) ⊗ C A(V ) op ) ⊕mn r ). By Lemma 3.16, A(M) ⊗ A(V ) A(N) ∼ = g(A(M) ⊗ A(V ) A(N)) is filt-projective, then by Lemma 3.15, there is an natural isomorphism between the degree zero hom set F 0 0HOM A(V )⊗ C A(V ) op (A(M) ⊗ A(V ) A(N), A(N) ⊗ A(V ) A(M))and the hom setHOM gr(A(V )⊗ C A(V ) op ) (gr(A(M) ⊗ A(V ) A(N)), gr(A(N) ⊗ A(V ) A(M))) 0 . phism θ : A(M) ⊗ A(V ) A(N) → A(N) ⊗ A(V ) A(M) Definition 3.14. (1) Let R be a filtered algebra with filtration {F p R} p∈Z , and let M, N be two filtered (left) R-modules with filtrations {F p M} p∈Z and {F p N} p∈Z respectively. Set Printing of the 1977 English Translation. J Dixmier, Graduate Studies in Mathematics. IIAmerican Mathematical SocietyEnveloping AlgebrasJ. Dixmier, Enveloping Algebras, (The 1996 Printing of the 1977 English Translation) Graduate Studies in Mathematics, Vol. II, American Mathematical Society, 1996. Vertex algebras associated with even lattices. C Dong, J. Algebra. 1611C. Dong, Vertex algebras associated with even lattices, J. 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[ "A micro-CT of a human skull Data descriptor", "A micro-CT of a human skull Data descriptor" ]	[ "Thomas Kirchner [email protected] \nInstitut für Physik Martin-Luther-Universität Halle-Wittenberg\nHalle (Saale)Germany\n" ]	[ "Institut für Physik Martin-Luther-Universität Halle-Wittenberg\nHalle (Saale)Germany" ]	[]	This is a data descriptor for an X-ray microtomography of a human skull.	10.5281/zenodo.6108435	[ "https://arxiv.org/pdf/2202.11519v1.pdf" ]	247,058,756	2202.11519	90aed5459d1ba771e85885a823e3eada21b6b86e	A micro-CT of a human skull Data descriptor 21 Feb 2022 Thomas Kirchner [email protected] Institut für Physik Martin-Luther-Universität Halle-Wittenberg Halle (Saale)Germany A micro-CT of a human skull Data descriptor 21 Feb 2022 This is a data descriptor for an X-ray microtomography of a human skull. Data set The data set includes the X-ray projection images (the measured raw data), a 3D reconstruction in approximate Hounsfield units (HU) and a semi-manual bone segmentation. Raw data 'halle_skull_2D_projections.zip' contains 2501 projection TIFF images, recorded by a cone beam micro-CT as described in section 3. Reconstructed 3D volume 'halle_skull.nrrd' is a nearly raw raster data (NRRD) [2] file containing the reconstructed 3D micro-CT. The file was produced by 1. reconstructing a volume with the proprietary software CT Pro (Metris, Tring, UK) using the settings defined in the file 'etc/halle_skull_3D_reconstruction.xtekVolume', then 2. converting the CT Pro output to NRRD using ImageJ [3], and then 3. scaling the reconstruction to approximate HU. HU in a voxel are defined by HU = 1000 × (µ voxel − µ water )/(µ water − µ air ). The absorption values µ air for air and µ water for water were manually segmented from three embedded water samples and surrounding air. Note that these HU are only approximated -as this is an ad hoc calibrated cone beam CT, the resulting HU values should not be interpreted as quantitative [4]. The finished 3D volume has a isotropic spacing of 0.125 mm with 1150 × 1700 × 1400 16-bit signed integer voxels. Bone segmentation 'halle_skull_segmented.nrrd' is a semi-manual segmentation of the skull bone based on 'etc/threshold.py' which combines a fixed threshold and a Sauvola binarization [5]. Incorrectly segmented voxels, such as reconstruction artifacts, were cleaned up from the segmentation by hand using MITK [6]. Subject The subject is an ex vivo skull of a body donor, a 70 year old male. He signed a declaration of consent at lifetime for the use of his body for scientific purposes. The skull was provided by the Institute of Anatomy and Cell Biology, Martin-Luther-University Halle-Wittenberg, Halle (Saale), Germany. Other than the calcifications around a small trepanation in the right parietal bone, this is a typical human skull. There is one coronal cut, which was necessary during preparation of the skull. The skull was imaged without jaw bone. 3 Measurement X-ray tomography was performed with an industrial cone beam micro-CT (XT H 225, Nikon Metrology) with 150 kV, 400 µA and using a 1 mm copper filter. The detector had 1750 × 2000 pixels with 200 µm isotropic spacing. This and other metadata of the acquisition can be found in the files 'etc/halle_skull.xtekct' and 'etc/halle_skull.ctprofile.xml'. 2501 angles were measured, each by averaging 8 exposures with 708 ms exposure times. This covered a 360.349°rotation -exact angles are listed in the file 'halle_skull_2D_projections/_ctdata.txt'. The skull was placed on a low-density polyethylene packing foam together with three water filled 1.5 mL polypropylene micro-centrifuge tubes. The cut-off top of the skull was held in position by 48 µm thick polypropylene tape. Figure 1 : 1Raw data example -one of the 2501 2D cone beam X-ray projection images. arXiv:2202.11519v1 [physics.med-ph] 21 Feb 2022 The FAIR Guiding Principles for scientific data management and stewardship. Scientific data. M D Wilkinson, 10.1038/sdata.2016.18M. D. Wilkinson et al. The FAIR Guiding Principles for scientific data management and stewardship. Scientific data, 2016, doi:10.1038/sdata.2016.18 Nearly raw raster data (NRRD) filetype. Nearly raw raster data (NRRD) filetype, http://teem.sourceforge.net/nrrd/ . ImageJ image processing software. ImageJ image processing software, https://imagej.net/software/imagej/ Deriving Hounsfield units using grey levels in cone beam computed tomography. Dentomaxillofacial Radiology. P Mah, T E Reeves, W D Mcdavid, 10.1259/dmfr/19603304P. Mah, T. E. Reeves, and W. D. McDavid. Deriving Hounsfield units using grey levels in cone beam computed tomography. Dentomaxillofacial Radiology, 2010. doi:10.1259/dmfr/19603304. Adaptive document image binarization. J Sauvola, M Pietikainen, 10.1016/S0031-3203(99)00055-2Pattern Recognition. J. Sauvola and M. Pietikainen. Adaptive document image binarization. Pattern Recognition, 2000. doi:10.1016/S0031-3203(99)00055-2 . The Medical Imaging Interaction Toolkit. The Medical Imaging Interaction Toolkit (MITK), https://mitk.org	[]
[ "An interaction-driven topological insulator in fermionic cold atoms on an optical lattice: A design with a density functional formalism", "An interaction-driven topological insulator in fermionic cold atoms on an optical lattice: A design with a density functional formalism" ]	[ "Sota Kitamura \nDepartment of Physics\nUniversity of Tokyo\n113-0033HongoTokyoJapan\n", "Naoto Tsuji \nDepartment of Physics\nUniversity of Tokyo\n113-0033HongoTokyoJapan\n", "Hideo Aoki \nDepartment of Physics\nUniversity of Tokyo\n113-0033HongoTokyoJapan\n" ]	[ "Department of Physics\nUniversity of Tokyo\n113-0033HongoTokyoJapan", "Department of Physics\nUniversity of Tokyo\n113-0033HongoTokyoJapan", "Department of Physics\nUniversity of Tokyo\n113-0033HongoTokyoJapan" ]	[]	We design an interaction-driven topological insulator for fermionic cold atoms in an optical lattice, i.e., we pose a question whether we can realize in a continuous space a spontaneous symmetry breaking induced by the inter-atom interaction into a topological Chern insulator. Such a state, sometimes called topological Mott insulator (TMI), has yet to be realized in solid-state systems, since this requires, in the tight-binding model, large off-site interactions on top of a small on-site interaction. Here we overcome the difficulty by introducing a spin-dependent potential, where a spin-selective occupation of fermions in A and B sublattices makes the on-site interaction absent, while a sizeable inter-site interaction is achieved by a shallow optical potential with a large overlap between neighboring Wannier orbitals. This puts the system away from the tight-binding model, so that we adopt the density functional theory for cold-atoms, here extended to accommodate noncollinear spin structures emerging in the topological regime, to quantitatively demonstrate the phase transition to TMI.	10.1103/physrevlett.115.045304	[ "https://arxiv.org/pdf/1411.3345v3.pdf" ]	33,747,898	1411.3345	00bf3a71ab1844c674a4dd7b768e1955f8560a98	An interaction-driven topological insulator in fermionic cold atoms on an optical lattice: A design with a density functional formalism Sota Kitamura Department of Physics University of Tokyo 113-0033HongoTokyoJapan Naoto Tsuji Department of Physics University of Tokyo 113-0033HongoTokyoJapan Hideo Aoki Department of Physics University of Tokyo 113-0033HongoTokyoJapan An interaction-driven topological insulator in fermionic cold atoms on an optical lattice: A design with a density functional formalism (Dated: November 14, 2014) We design an interaction-driven topological insulator for fermionic cold atoms in an optical lattice, i.e., we pose a question whether we can realize in a continuous space a spontaneous symmetry breaking induced by the inter-atom interaction into a topological Chern insulator. Such a state, sometimes called topological Mott insulator (TMI), has yet to be realized in solid-state systems, since this requires, in the tight-binding model, large off-site interactions on top of a small on-site interaction. Here we overcome the difficulty by introducing a spin-dependent potential, where a spin-selective occupation of fermions in A and B sublattices makes the on-site interaction absent, while a sizeable inter-site interaction is achieved by a shallow optical potential with a large overlap between neighboring Wannier orbitals. This puts the system away from the tight-binding model, so that we adopt the density functional theory for cold-atoms, here extended to accommodate noncollinear spin structures emerging in the topological regime, to quantitatively demonstrate the phase transition to TMI. Introduction.-There are growing fascinations with topological phases in condensed-matter physics 1,2 . The topological systems are characterized by various topological invariants 3,4 , e.g., Chern numbers 5,6 , as exemplified by the quantum Hall system, and Z 2 numbers 7,8 for the topological insulator. Such topologically non-trivial phases emerge from one-body physics: A non-zero Chern number arises when the time-reversal symmetry is broken, e.g., by a strong external magnetic field. A non-zero Z 2 number can be realized without breaking the timereversal symmetry, while typically a large spin-orbit coupling is required. Recently, a class of topological phases that do not have such one-body terms but that can still be made topological has been proposed, in terms of spontaneous symmetry breaking due to many-body interactions 9,10 . Such systems, so-called the "topological Mott insulator" (TMI), accompany interaction-driven loop currents, which act as an effective magnetic field or spin-orbit coupling. These have been suggested to arise from repulsive inter-site interactions, mainly from Fock-term contributions, and existence of such anomalous topological phases has been proposed for the tight-binding extended Hubbard model on various lattices [9][10][11][12][13] . Although some experimental schemes or candidate materials are suggested 11,[13][14][15][16][17] , condensed-matter realization of such topological phases has yet to be achieved. The difficulty here is that the TMI often requires large inter-site interactions to trigger the desired symmetry breaking, while other interactions with closer sites should be suppressed to avoid competing instabilities. On the other hand, ultracold atom systems, now attracting much interests as quantum simulators for various physical phenomena 18,19 , provide clean and tunable systems as a platform for exploring exotic topological phenomena [20][21][22][23] . Hence there is a chance of realizing the TMI in cold-atom systems, although some trick to encode large inter-site interaction is required: Typically we manipulate interactions by changing an s-wave scat-tering length via the Feshbach resonance 24 , while it is short-ranged and thus usually provides too small intersite interactions. Several studies 9,11,14 propose schemes with other interactions, e.g., molecular dipole-dipole interactions, which are experimentally challenging and still lack quantitative estimates. Thus we propose in the present paper an optical lattice system that is tuned to exhibit, in spite of employing an s-wave scattering, the interaction-driven topological phase transition, and demonstrate quantitatively the existence of a significant topological gap. In the proposal, a spin-dependent optical lattice potential, which constitutes a square optical lattice, resolves the primary difficulty: spin-dependent potential minima yield a spin-selective occupation of fermions in A and B sublattices, which changes the leading interaction derived from s-wave scattering from on-site interactions to nearest-neighbor (NN) interactions 25 . First we discuss this trick in detail in terms of a tight-binding limit, and shall show that the limit corresponds to a checkerboard tight-binding model studied by Sun et al. 10 . On the other hand, sufficient breadth of Wannier orbitals are required for the strong inter-site interaction, where the tight-binding picture is inadequate: Thus we employ the density-functional theory (DFT) for coldatom systems 26 , here extended to accommodate noncollinear spin-density functionals to describe topological spin structures. We confirm quantitatively that the proposed cold-atom system in a continuous space does indeed exhibit a topological phase transition from a semimetallic phase to a Chern insulator, as the repulsive interaction is increased. Basic idea.-In the present work, we consider ultracold fermions of spin-1/2 in a continuous space in the presence of an optical lattice potential, with a Hamilto-arXiv:1411.3345v1 [cond-mat.quant-gas] 12 Nov 2014 nian, H OL = σ,σ drψ † σ (r) − 2 2M δ σσ ∇ 2 + V σσ (r) ψ σ (r) + drdr ψ † ↑ (r)ψ † ↓ (r )U (r − r )ψ ↓ (r )ψ ↑ (r).(1) Hereψ σ (r) is the fermion field operator, V σσ (r) = W (r)δ σσ + B(r) · s σσ the spin-dependent optical lattice potential, consisting of a periodic potential W (r) and a periodic Zeeman field B(r), with the Pauli matrix s σσ , and U (r − r ) the hard-core fermion-fermion interaction with an s-wave scattering length a s . In tight-binding models 9-13 , realization of the TMI is shown to require repulsive intetactions extending to neighboring sites and a semimetallic band structure in the non-interacting case. However, even in cold atoms with a short-range interaction, we can still generate offsite interactions by employing a spin-dependent optical potential as discussed below. It turns out that a form of the potential, V ↑↑ (r) = −V A cos π(y − z) d − V B cos π(y + z) d + V T (x),(2)V ↓↓ (r) = V B cos π(y − z) d + V A cos π(y + z) d + V T (x),(3)V ↑↓ (r) = V ↓↑ (r) = V C sin πy d + sin πz d ,(4) where d is a lattice constant, acomplishes the desired natures, semimetallic band structure and sizeable NN interactions. Here V T (x) is a trapping potential along x, taken to have a form V T (x) = V x sin 2 (πx/2d) with a cutoff of x to [−d, d] for simplicity. Spatial patterns at x = 0 are shown in Fig.1(a-c). Although we work in a continuous space for cold atoms, in order to explain why we adopt the form of the potential Eqs.(2-4), we can first have an intuitive look at what the system would look like in the tight-binding limit. Let us first focus on the spin-diagonal components, where the optical lattice sites (the minima of V ↑↑ (r) and V ↓↓ (r)) constitute square lattices (see Fig.1(a,b)). Because the positions of the minima are here spin-dependent unlike ordinary lattice models, fermions with spin-up occupy A sublattice sites while spin-down B sites. We can thus regard the system as that of spinless fermions, if we translate the spin into the sublattice index. In this situation on-site interactions are absent, so that the leading interaction is the NN (density-density) one, which is caused by overlapping tails of neighboring Wannier orbitals. While this idea of encoding NN interactions is adopted from Ref. 25, where a kagomé lattice with NN interactions is described in a tripod scheme of resonant transitions between atomic levels (atoms in three levels correspond to those in the three sublattices in kagomé), the scheme we adopt is fairly different as it employs off-resonant lasers 27 , although we finally get a spin-dependent potential 28,29 . -2 -1 0 1 2 y/d -2 -1 0 1 2 z/d -2 -1 0 1 2 -2 -1 0 1 2 y/d -2 -1 0 1 2 z/d -2 -1 0 1 2 −2 −2 −1 −1 0 0 1 1 2 2 z/d y/d -2 -1 0 1 2 y/d -2 -1 0 1 2 z/d -2 -1 0 1 2 -2 -1 0 1 2 y/d -2 -1 0 1 2 z/d -2 -1 0 1 2 −2 −2 −1 −1 0 0 1 1 2 2 z/d y/d -2 -1 0 1 2 y/d -2 -1 0 1 2 z/d -0.5 -0.25 0 0.25 0.5 -2 -1 0 1 2 y/d -2 -1 0 1 2 z/d -0.5 -0.25 0 0.25 0.5 −2 −2 −1 −1 0 0 1 1 2 2 z/d y/d −2 −2 −1 −1 0 0 1 1 2 2 −0.5 −0.25 0 0.25 0.5 (a) V ↑↑ (r) (b) V ↓↓ (r) (c) V ↑↓ (r) • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •(d) (e) t −t t A t B −t A In the situation so far, NN hopping is absent in the model. We can induce it by adding the spin-offdiagonal part, Eq.(4), as a perturbation (V C V A , V B ), which indeed induces transitions between spin-up (A site) and spin-down (B) orbitals. Since Eq.(4) is a Zeeman field along x-axis in a staggered form, the hopping amplitude takes a real value with alternating signs (see Fig.1(c)). Then a corresponding tight-binding model with NN interactions can be depicted as shown in Fig.1(d), where t A (t B ) denotes hopping amplitude from tunneling through potential barriers in Eqs. (2,3), while t denotes those from the off-diagonal part, Eq.(4) (all parameters are taken as positive). We can make the tight-binding model simpler by performing a unitary transformation 27 which puts the model into the one depicted in Fig.1(e). The transformed model is equivalent to an extended Hubbard model for spinless fermions on a checkerboard lattice, with alternating signs for the second-neighbor hoppings. The tightbinding model is semimetallic at half-filling in the noninteracting case, and theoretically reported to have a nonzero Chern number for infinitesimal NN repulsions due to a spontaneous breaking of the time-reversal symmetry 10 . Hence we can expect that the Hamiltonian in the original continuous spaceĤ OL may realize a Chern insulating state driven by repulsive interactions. In passing, we remark on the symmetry of the present system: The time reversal symmetry, defined as a physical symmetry (which inverts spin directions), is explicitly broken by the Zeeman fields (B z and B x ). However, B ycomponent is absent so that the Hamiltonian is real, and the system has a symmtery against the complex conjugation. The complex conjugation corresponds to the timereversal symmetry in the spinless system (which does not invert sublattice indice), where the breaking of that symmetry signifies the topological phase transition 35 . Hence the ordered phase of the present system should accompany a (staggered) magnetization along y-axis inducing a magnetic field along y-axis, i.e., the imaginary spinoffdiagonal part of the mean-field potential 27 . In terms of the spinless tight-binding model, that translates into a complex NN hopping amplitude and is consistent with spontaneous loop currents in the topological phase 10 . In short, the present idea is summerized as follows: (i) By employing the spin-dependent potential minima, the s-wave scattering translates into the NN interaction of a spinless square lattice system, which is a key ingredient of the TMI. (ii) By the staggered Zeeman field along x, a checkerboard pattern of the second-neighbor hopping with alternating signs is realized (after a unitary transformation), which acomplish another important feature, a semimetallic behavior for the non-interacting case. Formulation.-Now we go back to the original problem in the continuous space. The whole point is that, although the reduced tight-binding model has a desired form, there is no guarantee that this reduction is adequate. In fact this becomes especially serious, since we have to employ, for a realization of a TMI, a shallow lattice potential to enhance the NN interaction from a large overlap between neighboring Wannier orbitals. The shallow potential will then also enhance the longer-distance hoppings and effects of excited bands, which may well degrade the desired situation. This is precisely why we have to employ the DFT to directly solveĤ OL . The exchangecorrelation functional for ultracold fermions within the local spin-density approximation (LSDA) has been formulated in a pioneering work by Ma et al. 26 , where it is reported that LSDA quantitatively reproduces the total energy of the shallow optical lattice system as estimated from the diffusion Monte Carlo method. If we apply the DFT to the present system, however, the potential V σσ (r) is not spin-diagonal, so that we have to deal with non-collinear spin DFT 30,31 , where the spatial pattern of spin directions are allowed to have general structures. We adopt here the local approximation, i.e., we employ the Hartree exchangecorrelation functional of the form E HXC [n(r), \|m(r)\|], where n(r) is the atomic density and m(r) the magnetization. It can be obtained from the collinear functional given in Ref. 26, E HXC [n ↑ (r), n ↓ (r)], by replacing the collinear spin density n ↑,↓ (r) with n(r) ± \|m(r)\|, as is done in electron systems 31 . The resulting Kohn-Sham potential then reads V σσ (r)+[δE HXC /δn(r)]δ σσ + [δE HXC /δ\|m(r)\|]\|m(r)\| −1 m(r) · s σσ . We can mention that the non-collinear LSDA formalism should be particularly appropriate to the TMI in cold-atom systems: While for the long-ranged Coulom- bic interaction, the Fock term, which is essential for the topological transition, is non-local and may be difficult to capture with LDA, the term in the present system can be expressed explicitly as the non-collinear LSDA functional 36 . Further, the topological phase suggested in the checkerboard lattice is estimated to occur in a weakly-correlated regime, where the DFT should be adequate. Results.-Now we present the density-functional results. We consider the periodic boundary condition to employ Bloch wavefunctions. The number of k-points is taken as 1 × 32 × 32, and each Bloch function is represented by 9 × 21 × 21 plane waves. We set the parameters of the lattice potential as V A = 0.8, V B = 1.2, V C = 0.25 and V x = 10, all in units of E R = 2 π 2 /(4M d 2 ). In the non-interacting case, a s = 0, we have a band structure as depicted in Fig.2(a). Due to the staggered fields, a unit cell contains four lattice sites (two A and two B sites), so that the bottom four bands correspond to those in the tight-binding model. Two dispersive bands (second and third from bottom) touch with each other at the corner of the Brillouin zone, which is called a quadratic band-crossing point (QBCP), associated with the symmetry against complex conjugation. At halffilling, the lowest two bands are fully-occupied, and the system is a semimetal. When the interaction is switched on in Fig.2(b), however, we can see that a gap opens at QBCP, where a spontaneously broken symmetry makes the system an insulator. The size of the gap as a function of a s in Fig.2(d) shows the gap grows with the interaction with a threshold behavior. The gap is indeed a topological gap, which is verified from the Chern number 32 . Figure 2(g) shows the Chern density for the lowest two bands 37 , where we can see conspicuous magnitudes around the QBCPs. The Chern number as an integrated value turns out to be −1, and we can conclude that the system is a Chern insulator driven by spontaneous symmetry breaking from a semimetallic phase. The spin structure of the system in continuous space is depicted in Fig.3(a-c). As we have noted above in the discussion on the symmetry, the order parameter for the present system is the staggered magnetization along yaxis. Fig.3(a) shows the non-interacting case, where the spatial spin structure comes from the Zeeman field in the x-z plane, so that the y-component is trivially absent. In the interacting case, by sharp constrast, y-component magnetization spontaneously emerges, as most clearly seen in the Bloch sphere inset, and we can identify the insulator to be topological. Accordingly, the spatial behavior of the spins in the periodic system change from two-dimensional vortices to three-dimensional ones ( Fig.3(b,c)). Hence we conclude that the designed system does indeed realize, in the continuous space, the mechanism for the emergence of the topological phase conceived for tight-binding models. Figure 3(c) indicates that the order parameter has large amplitudes around (y, z) = (∓d/2, ±d/2), where the atomic density gives the upper limit for m y . Hence we can enlarge the topological gap by enhancing the density around these positions. This can be achieved by controlling the anisotropy of the potential barrier separating adjacent A sites (or B sites) (i.e., reducing V A − V B ), as shown in Fig.2(c-e), where V A − V B has an effect of increasing the gap. Figure 3(d) shows that the density around (y, z) = (∓d/2, ±d/2) is sizeable 27 . We can notice a round-off of the gap at stronger interactions in Fig.2(e), which we identify to come from another phase transition into a site-nematic order (i.e., a spontaneous imbalance of the filling between the two sublattices), which is reported for a tight-binding model in Ref. 10. Coexistence of the topological and nematic orders occurs above a s ∼ 0.23d in the present setup (see Fig.3(e)). Discussions.-We should mention we have neglected some factors that may work against the topological transition: (i) thermal fluctuations, (ii) a Zeeman splitting accompanying the Feshbach resonance, (iii) three-body scattering processes, which induce an instability toward a dimerized phase 33,34 , and (iv) non-local/dynamical correlations. Let us discuss (i)(ii) in detail. The critical temperature should have an order of magnitude of the topological gap, which is scaled by the bandwidth. The bandwidth is larger in shallower optical lattices, so that we can expect that the critical temperature can be made accessible. If we introduce a uniform Zeeman field, V M s x σσ /2, we can estimate the upper bound of the magnetic field to be V M ∼ 2C, at which the Zeeman splitting makes V ↑↓ (r) non-staggered. As shown in Fig.2(f), the topological order, although reduced, still remains in a magnetic field of V M = 0.8C. As for experimental detections, the phase transition can be detected from the behavior of the magnetization. There is also a theoretical suggestion to detect the Chern number 23 . Also, the chiral edge states, the inherent topological feature, should emerge along the boundaries of the phase domains and edges of the trapped system, where the latter may be difficult to distinguish from the metallic states arising from the depleted region in the trapped system 14 . To summarize, the present design is the first example of a realistic model in a continuous space that exhibits an interaction-induced spontaneous symmetry breaking toward the Chern insulator. Compared to the other proposals on the TMI in cold-atom systems 9,11,14 , our proposal has some advantages: it employs only a simple and established scheme for cold atoms, i.e., the swave Feshbach resonance and the electric dipole transition between hyperfine states induced by off-resonant lasers, along with shallow lattice potentials, which tend to enhance the transition temperature. The authors would like to thank P. N. Ma, M. Troyer, and T. Oka for illuminating discussions. This work is supported by a Grant-in-Aid for Scientific Research (Grant No. 26247064) from MEXT, and SK by the Advanced leading graduate course for photon science (ALPS). FIG. 1 : 1(Color online). (a-c) Spatial patterns of the optical lattice potentials V ↑↑ (r)(a), V ↓↓ (r)(b), and V ↑↓ (r)(c), given respectively in Eqs.(2,3,4), here shown on the x = 0 plane for (VA, VB) = (0.8, 1.2) in units of ER = 2 π 2 /(4M d 2 ). •(+) indicate positions of minima in V ↑↑ (V ↓↓ ). (d) Spatial pattern of the hopping in the tight-binding limit. Red (blue) circles represent A (B) sites, while red (blue) lines the hopping with positive (negative) amplitudes. (e) A transformed model. FIG. 2 : 2(Color online). Structure of the lowest four bands, in the non-interacting case(a) and for an interacting system(b) with as = 0.25d. (c-f) The topological gap of the system against as. The parameters, all in units of ER, are VA − VB = 0.4(c), −0.4(d), −0.8(e) with (VA + VB, VC ) = (2, 0.25) for all the cases. A round-off of the gap for stronger as for (e)signifies an emergence of a site-nematic order (see text). (f) displays the case of (e) with a magnetic field applied along x-axis with VM = 0.2. (g) The Chern density for the lowest two bands in the system depicted inFig.2(b). ) m y (r)/n(r) (d)n 2D (r)d 2 , a s =0.2d (e)n 2D (r)d 2 , a s =0.27dFIG. 3: (Color online) (a,b) Spatial patterns of the magnetization (arrows) for the non-interacting system(a) and for an interacting one with as = 0.25d(b). The color represents the z-component magnetization up (red) or down (blue). The direction of the arrows are rotated by π/2 around z-axis(a) and y-axis(b) for convenience of viewing. Direction of spins as we sweep the unit cell is depicted in the top insets. (c) The y-component of the magnetization (order parameter for the toplogical phase) for the state depicted in (b). (d) Spatial pattern of the atomic area density for the case Fig.2(e) with as = 0.2d. (e) The same as (d) for a larger as = 0.27d, for which a site-nematic order coexists with a topological gap. FIG. 5 : 5(a) A phase diagram against A − B and as. SM indicates a semimetallic phase, and TMI a insulating phase with a spontaneous topological gap. (b-d) The atomic area density of the non-interacting system for various value of A − B. Namely, the present many-body Hamiltonian has an antiunitary symmetry K with K 2 = 1 and belongs to class AI, where topological phases are absent in two-dimensional noninteracting systems 3 . Nevertheless, many-body effects allow the system to be a Chern insulator, because the Kohn-Sham Hamiltonian of the system after the spontaneous symmetry breaking can be class A.36 The Hartree-Fock energy, (π 2 as/M ) dr(n 2 − m 2 ), for contact interactions indeed coincides with that for hardcore interactions in the dilute limit.37 We have calculated the Chern number as an integral of the Chern density for the lowest two bands, −2Im n=1,2 BZ dk dr[∂u nk (r)/∂kx][∂u nk (r)/∂ky], since they cross with each other. I. Deutsch and P. Jessen, Phys. Rev. A 57, 1972 (1998). 2 A. Dudarev, R. Diener, I. Carusotto, and Q. Niu, Phys. Rev. Lett. 92,153005 (2004). 3 K. Sun, H. Yao, E. Fradkin, and S. Kivelson, Phys. Rev. Lett. 103,046811 (2009).4 In place of ω/ √ 2 we can employ ω 2 + p 2 /2, for which the laser field is obtained by substituting y → y + px in Eq.(A4). It has the desired form on the system plane, x = 0. Appendix A: Implementation of the lattice potentialHere we indicate how we can realize the lattice potential given in Eqs.(2)(3)(4). We consider a situation where the laser electric fields ω E(ω, r)e iωt + c.c. are imposed to atoms with a hyperfine structure. If all the laser frequencies (ω's) are off-resonant with the hyperfine splitting, the effect of the laser field is represented as additional terms in the Hamiltonian 1,2 , a potential ∝ \|E(ω, r)\| 2 and a Zeeman term ∝ −iE (ω, r)×E(ω, r)·F with ω-dependent coefficients, whereF is the total angular momentum operator. For the present system we focus on F = 1/2 multiplets (then F = s σσ ).First, we consider a pair of confronting circularly-polarized lasers along z-axis, and one linearly-polarized laser along y-axis, all with a frequency ω, which can be shown to realize the W -and B z -components of V σσ (r):We further superpose four linearly-polarized lasers of a frequency ω = ω/ √ 2 with a spatial part 4 ,which realizes the W -and B x -components:If we combine laser fields in Eqs.(A1,A4), we end up with the desired Hamiltonian,Ĥ OL . Schematic pictures of these configurations are given inFig.4(a,b). While we can change the strengths of the fields Eq.(A1) and (A4), we have three parameters, V A , V B , and V C , to adjust. If the tuning of the two strengths does not attain the desired parameters, we can introduce additional, linearly-polarized lasers, e.g.,Appendix B: Equivalence of the tight-binding limit to the checkerboard lattice modelHere we describe how the present HamiltonianĤ OL is related to the checkerboard lattice model 3 in the tight-binding limit. As we have discussed in the main text, the tight-binding limit is depicted inFig.1(d), whose Hamiltonian would readĤ7 whereâ i,j annihilates a spin-up (A site) fermion at (y, z) = (i, j) with d = 1, whileb i,j is for a spin-down (B) fermion. In order to relate Eq.(B1) to the checkerboard lattice, we can perform a unitary transformation,Then the transformed model readŝwhich preciselly coincides with the spinless checkerboard lattice model,Fig.1(e). In the derivation we have used thatAppendix C: Order parameter in the mean-field desctiptionHere we discuss the order parameter of the optical lattice system from the corresponding mean-field description of the tight-binding checkerboard lattice model. The order parameter of the Chern insulating phase in the chekcerboard lattice 3 is given as φ, whereWe can readily go back to the tight-binding description of the optical lattice with Eqs.(B2,B3), and the order parameter emerges asWhile this is the order parameter for the optical lattice system expressed in terms of the tight-binding picture, we can introduce an altanative, basis-independent observable appropriate to continuous problems. We can start with an observation that the Fock term corresponding to Eq.(C2) in the mean-field decoupling isas depicted inFig.4(c). As explained in the main text, NN hopping with real amplitudes is obtained by the B xcomponent, while that with imaginary amplitudes is given by the B y -component, since it realizes spin-offdiagonal potential V ↑↓ = −V ↓↑ = iB y . The staggered pattern in Eq.(C3) can be realized by a sinusoidalwhich enables us to define the order parameter as the staggered magnetization along y-axis, proportional to Eq.(C4):Appendix D: Enhancement of the order parameter by a lattice anisotropyAs discussed in the main text, the atomic density around (y, z) = (∓d/2, ±d/2) gives the upper limit for the order parameter, and can be controlled by lattice anisotropy. It is clearly described inFig.5, where A + B and C are fixed to 2E R and 0.25E R , respectively. 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[ "Seasonal Floquet states in a game-driven evolutionary dynamics", "Seasonal Floquet states in a game-driven evolutionary dynamics" ]	[ "Olena Tkachenko \nSumy State University\nRimsky-Korsakov Street 240007SumyUkraine\n", "Juzar Thingna \nInstitut für Physik\nUniversität Augsburg\nUniversitätsstraße 186159AugsburgGermany\n", "Sergey Denisov \nSumy State University\nRimsky-Korsakov Street 240007SumyUkraine\n\nNanosystems Initiative Munich\nSchellingstr, 4D-80799MünchenGermany\n", "Vasily Zaburdaev \nMax Planck Institute for the Physics of Complex Systems\nNöthnitzer Str. 38D-01187DresdenGermany\n", "Peter Hänggi \nInstitut für Physik\nUniversität Augsburg\nUniversitätsstraße 186159AugsburgGermany\n\nNanosystems Initiative Munich\nSchellingstr, 4D-80799MünchenGermany\n" ]	[ "Sumy State University\nRimsky-Korsakov Street 240007SumyUkraine", "Institut für Physik\nUniversität Augsburg\nUniversitätsstraße 186159AugsburgGermany", "Sumy State University\nRimsky-Korsakov Street 240007SumyUkraine", "Nanosystems Initiative Munich\nSchellingstr, 4D-80799MünchenGermany", "Max Planck Institute for the Physics of Complex Systems\nNöthnitzer Str. 38D-01187DresdenGermany", "Institut für Physik\nUniversität Augsburg\nUniversitätsstraße 186159AugsburgGermany", "Nanosystems Initiative Munich\nSchellingstr, 4D-80799MünchenGermany" ]	[]	Mating preferences of many biological species are not constant but season-dependent. Within the framework of evolutionary game theory this can be modeled with two finite opposite-sex populations playing against each other following the rules that are periodically changing. By combining Floquet theory and the concept of quasi-stationary distributions, we reveal existence of metastable timeperiodic states in the evolution of finite game-driven populations. The evolutionary Floquet states correspond to time-periodic probability flows in the strategy space which cannot be resolved within the mean-field framework. The lifetime of metastable Floquet states increases with the size N of populations so that they become attractors in the limit N → ∞.	null	[ "https://arxiv.org/pdf/1505.06726v3.pdf" ]	18,376,086	1505.06726	3498e8ad8c2632241dd5af25125a110c520d55b9	Seasonal Floquet states in a game-driven evolutionary dynamics Olena Tkachenko Sumy State University Rimsky-Korsakov Street 240007SumyUkraine Juzar Thingna Institut für Physik Universität Augsburg Universitätsstraße 186159AugsburgGermany Sergey Denisov Sumy State University Rimsky-Korsakov Street 240007SumyUkraine Nanosystems Initiative Munich Schellingstr, 4D-80799MünchenGermany Vasily Zaburdaev Max Planck Institute for the Physics of Complex Systems Nöthnitzer Str. 38D-01187DresdenGermany Peter Hänggi Institut für Physik Universität Augsburg Universitätsstraße 186159AugsburgGermany Nanosystems Initiative Munich Schellingstr, 4D-80799MünchenGermany Seasonal Floquet states in a game-driven evolutionary dynamics (Dated: August 13, 2015)numbers: 0250Le8723Kg0545-a Mating preferences of many biological species are not constant but season-dependent. Within the framework of evolutionary game theory this can be modeled with two finite opposite-sex populations playing against each other following the rules that are periodically changing. By combining Floquet theory and the concept of quasi-stationary distributions, we reveal existence of metastable timeperiodic states in the evolution of finite game-driven populations. The evolutionary Floquet states correspond to time-periodic probability flows in the strategy space which cannot be resolved within the mean-field framework. The lifetime of metastable Floquet states increases with the size N of populations so that they become attractors in the limit N → ∞. Introduction. The evolutionary dynamics of an animal group is tied to the reproductive activity of its members, a complex process which involves courtship rituals and sharing of parental care [1]. Within the game theory framework, the sex conflict over parental investment was formalized by Dawkins in his famous "Battle of Sexes" (BoS) [2], illustrated in Fig. 1. In this game two opposite-sex members of the group play against each other. Each player can use two behavioral strategies. Entries in the payoff matrix, b ss , quantify the reward received by a female which used a strategy s ∈ {1, 2} after she has played against a male which used a strategy s ∈ {1, 2}. Entries a s s define the reward of the male. A number of observations have shown that mating strategies and preferences of many species are not constant in time but season-dependent [3]. For example, courtship srituals of the males of Carolina anole lizards (Anolis carolinensis), as well as mate selection criteria of the females of the species, are periodically changing during the year [4]. Even the amount of different types of muscle fibers that control the vibrations of a red throat fan (dewlap) -which males employ during the courtshipis a season dependent characteristic [5]. Currently, there is no agreement between the ecologists on the role this seasonal plasticity plays in determining the evolution direction of the species [6]. We address this problem within the BoS framework by allowing the payoffs to periodically vary in time, see Fig. 1. Our goal is to investigate how these modulations influence the game-driven evolutionary dynamics. Here, we first apply the concept of quasi-stationary distributions in absorbing Markov chains [7] to a stochastic evolutionary dynamics of finite populations and define the notion of evolutionary metastable states. Then, by employing the Floquet theory [8,9], we generalize the notion of metastable states [10][11][12] to periodically modulated FIG. 1: (color online) "Battle of Sexes" with seasonal variations. A female of Carolina anole lizards can be either coy and prefer an arduous courtship, to be sure that a mate is ready to contribute to a parental care, or fast, and thus not being much concerned about parental care of offspring. A male can be either faithful and ready to assure the female partner, by performing a long courtship, that he is a faithful potential husband, or philanderer and prefer to shorten the courtship stage. Depending on the strategies, s (s ), played by the female (male), the female (male) gets payoff b ss (a s s ). Both females and males are season-constrained in their strategies and preferences, which is modeled via time-periodic modulations of the payoffs. game-driven evolutionary dynamics. We show that in big but finite populations, the metastable Floquet states survive over extremely long (as compared to the period of modulations) timescales. We argue that, in the limit of infinitely big populations, these states become attractors while still evading the mean-field description. Model. Finite size of animal populations favors a stochastic approach to evolutionary dynamics. Although the convergence to the deterministic mean-field dynamics is typically guaranteed in the limit N → ∞ [8,9], the stochastic dynamics of large but finite populations can still be very different from the mean-field picture [6,[16][17][18]. Here we adapt the game-oriented version of the Moran process [19], introduced in Ref. [5] and generalized to two-player games in Ref. [6]. Players A (males) and B (females) form two populations, each one of a fixed size N and with two available strategies, s = {1, 2}, see Fig. 1. Game payoffs are time-periodic functions, c ss (t) = c ss (t + T ), c = {a, b}, and can be represented as sums of stationary and zeromean time-periodic components, c ss (t) =c ss +c ss (t), c ss (t) T = 0. The time starting from t = 0 is incremented by t = T /M after each round. After M rounds the payoffs return to their initial values. The state of the populations after the m-th round is fully specified by the number of players playing the first strategy, i (males) and j (females), 0 ≤ i, j ≤ N . A detailed description of the corresponding stochastic process is given in Supplemental Material. It can also be shown that, in the limit N → ∞ [6], the dynamics of the variables x = i/N and y = j/N is defined by the adjusted replicator equations [3]; see Refs. [8,11]. For a finite N , the state of the system can be expressed as a N × N matrix p with elements p(i, j), which are the probabilities to find two populations in the states i and j, respectively. Round-to-round dynamics can be evaluated by multiplying the state p with the transition fourth-order tensor S, with elements S(i, j, i , j ) [3]. By using the bijection k = (N − 1)j + i, we can unfold the probability matrix p(i, j) into the vectorp(k), k = 0, ..., N 2 , and the tensor S(i, j, i , j ) into the matrix S(k, l). This reduces the problem to a Markov chain [22], p m+1 =S mp m , where m is the number of the round to be played. The four states (i = {0, N }, j = {0, N }) are absorbing states because the transition rates leading out of them equal zero [3]. The absorbing states are attractors of the dynamics for any finite N , and the finite-size fluctuations will eventually drive a population to one of them [18,23]. This would imply a fixation, so that only one strategy survived in each of now monomorphic populations [6,18]. We are interested in the dynamics before the fixation, so we merge the four states into a single absorbing state by summing the corresponding incoming rates. The boundary states, (i = {0, N }, j ∈ {1, · · · , N − 1}) and (i ∈ {1, · · · , N − 1}, j = {0, N }), can also be merged into this absorbing super-state: Once the population gets to the boundary, it will only move towards one of the two nearest absorbing states. By labeling the absorbing super-state with index k = 0, we end up with a (L + 1) × (L + 1) matrix S m = 1 m 0 0Q m ,(1) where L = (N − 1) 2 , m 0 is a vector of the incoming transition probabilities of the absorbing super-state, 0 is a L × 1 zero vector, andQ m is a L × L reduced transition matrix. With Eq. (1), we arrive at the setup used by Darroch and Seneta to formulate the concept of quasi-stationary distributions [7]. There is the normalized right eigenvector of the reduced transition matrixQ m with the maximum eigenvalue λ [24]. By using the inverse bijection, we can transform this vector into a two-dimensional probability density function (pdf), i.e., a state, d, with maximal mean absorption time. This state is the most resistant to the wash-out by the finite-size fluctuations and it remains near invariant, up to a uniform rescaling, under the action of the tensor S. This is the metastable state of the evolutionary process. Stationary case. As an example, we consider a game with payoffs a 11 , a 22 , b 12 and b 21 equal 1, and payoffs −1 for the rest of strategies [25]. Figure 2 presents the numerically obtained metastable states of the game. We use two methods, the direct diagonalization of the reduced transition matrix, which is stationary in this case, Q m ≡Q, and preconditioned stochastic sampling [3]. For N = 200 we find an agreement between the results of the two methods. The means of the metastable state, x = N −1 i,j=1 i N · d(i, j);ȳ = N −1 i,j=1 j N · d(i, j),(2) coincide with the Nash equilibrium [26] for any N . However, the actual dynamics is determined by the metastable limit cycle encircling the equilibrium (this could be seen by performing short-run stochastic simulations); see Fig. 2. Within the Langevin-oriented approach to the dynamics of finite populations [6,10], the appearance of the metastable limit cycle can be interpreted as a stochastic Hopf bifurcation [28] (see also Ref. [29] for another interpretation). In the limit N → ∞ the cycle collapses to the Nash equilibrium. Note, however, that the convergence to this limit is slow, as indicated by the width of the pdf for N = 400. Case of modulated payoffs. By adding timemodulations to the model, we find that the mean-field dynamics does not exhibit substantial changes. For the choice (t) =ã 11 (t) =b 22 (t) = f cos(ωt) with ω = 2π/T (all other payoffs held stationary) we observed a period-one limit cycle localized near the Nash equilibrium of the stationary case, see Figs. 3(a,b). It collapses to a set of adiabatic Nash equlibria, x N E ( ) = 2− 4− , y N E ( ) = 2 4+ in the limit ω → 0. In the mean-filed limit N = ∞, a trajectory spirals towards a fixed point 1 2 , 1 2 , the Nash equilibrium of the game. For the finite N , metastable states are specified by their quasi-stationary probability density functions (pdf's) (3d plots). For N = 200 the pdf combines the results of the direct diagonalization of the 39 601 × 39 601 matrixQ (left half of the pdf, this procedure was also used to obtain the function for N = 100) and of the preconditioned stochastic sampling (right part of the pdf, this procedure was also used to obtain the function for N = 400) [3]. The baseline fitness w = 0.3 (other parameters are given in the text). The dynamics of a finite N population is different. The stochastic evolution of a trajectory in the (i, j)-space, initiated away from the absorbing boundary, can be divided into two stages. At first the trajectory relaxes towards a metastable state. The timescale of this process is defined by the mixing time t mix (N ) [30], which in this case has to be calculated now for the quasi -stationary state. Then the trajectory wiggles around the metastable state until the fluctuations drive it to the absorbing boundary. Following the random-walk approximation, the mean absorption time t abs (N ), called "mean fixation time" [2,11] in the evolutionary context, seemingly should also scale as N . However, this estimate neglects the presence of the inner attractive manifold and the fact that the noise strength decreases upon approaching the absorbing boundary. In fact, the absorption time scales superlinearly with N [31]. The lifetime of the metastable state is restricted to the time interval [t mix (N ), t abs (N )], whose length scales as t abs (N )[1 − t mix (N )/t abs (N )] ∼ t abs (N ). For ω = 0.1 [32], the stochastic simulations reveal a metastable state which is distinctively different from the limit cycle produced by the mean-field equations, see Fig. 3b. There is a conflict between the evolution of means, described by the adjusted replicator equations, and the results of the stochastic dynamics. The conflict can be resolved with the concept of the quasistationary distribution. Namely, the transition matrices, Eq. (1), are round-specific now and form a set {S m }, m = 1, · · · , M . The propagator over the time interval [0, t], 0 < t < T , is the productŨ(t) = Mt m =1S m with M t = t/ t. All the propagators, including the periodone propagatorŨ(T ), have the same structure as the super-matrix in Eq. (1). We define the metastable state d(T ) as the the quasi-stationary distribution ofŨ(T ). It is also a Floquet state [9] of the reduced propagator U r (T ), which can be obtained by replacing the transition matricesS m with the matricesQ m or by simply cutting out the first line and column from the ma-trixŨ(T ). The Floquet state is a time-periodic state, d(t + T ) = d(t), which changes during one period of modulations, see Fig. 4. The metastable state d(t) at any instant of time t, 0 < t < T , can be obtained by acting on the state d(0) with the reduced propagatorŨ r (t). The evolution of the means of the pdf d(t) (see Fig. 4a), (x(t),ȳ(t)), is close to the period-one limit cycle, see blue dots on Fig. 3a. However, the Floquet state consists of two peaks produced by the noised period-two limit cycle (compare also the positions of the stroboscopic points in Fig. 3b with the pdf for t = 0 in Fig. 4a). The peak contributions balance each other thus reducing the dynamics of the means to the vicinity of the the point 1 2 , 1 2 . The lifetime of the state d(t) can be estimated with the largest eigenvalue λ T , 0 < λ T < 1, of the ma-trixŨ r (T ). To compare it with lifetimes of stationary metastable states, we introduce the mean single-round exponent,λ T = λ 1/M T and define the mean lifetime as t life = 1/(1 −λ T ) [3]. Aside of the slow decay trend, we found the effect of modulations not being strong. This is in stark contrast to the structure of the metastable states. Namely, while in the stationary limit the pdf d is localized near the Nash equilibrium, at the maximal distance from the absorbing boundaries, the metastable Floquet state is localized near the absorbing boundary, see Fig. 4. We also detect the increase of the boundary localization with the increase of the population size beyond N = 200. This suggests that, in the limit N → ∞, the dynamics of the system is governed by a period-two limit cycle localized near the absorbing boundary. The boundary localization of the metastable attractor can be interpreted as the presence of small fractions of mutants [11], i.e. the players that are using strategies different from that used by the majority of populations. The evolutionary dynamics of the mutant fractions looks like a repeating sequence of population bottlenecks [2,33] yet this only weakly affects fraction lifetimes [34] even in the case of finite N . Conclusions. We presented a concept of metastable Floquet states in game-driven populations when mate selection preferences are periodically changing in time. Here we combined the Floquet formalism with the concept of quasi-stationary distributions to reveal the existence of complex, liquid-like nonequlibrium dynamics in the strategy space which cannot be resolved within the mean-field framework. Metastable Floquet states are not restricted to the field of ecology studies but can emerge in different periodically modulated systems with stochastic event-driven dynamics. They may, for example, underlay a gene expression in a single cell, which is modulated by a circadian rhythm [38] and can provide new interpretations of the Bose-Einstein condensation in ac-driven atomic ensembles [39,40]. Fig. 3a (•). Plots for t = 0 (above the diagonal) and t = T (below the diagonal) present the results of the stochastic sampling. Supplemental Material SEASONAL VARIATIONS IN MATE PREFERENCES Stable time variations were found in the female flycatcher preferences or male forehead patch size that resulted in late-breeding females preferring males with larger patches [1]. It was explained by the fact that in the beginning of the breeding season, large-patched males allocate more resources to courting than to parental care but change their habits to the opposite late in the season. Seasonal variations were also found in fiddle crabs (female preference to male claw size) [2], two-spotted goby (female preference to overall male size) [3], and sailfin mol-lies (male preferences for two different kind of females) [4]. MORAN PROCESS Players A (males) and B (females) form two populations, each one of a fixed size N and with two available strategies, s = {1, 2}. Payoffs are specified by four functions, {a ss (t)} and {b s s (t)}, s, s = {1, 2}. The average payoff of the players using strategy s is π A s (j, t) = a s1 (t) j N + a s2 (t) (N − j) N ,(3)π B s (i, t) = b s1 (t) i N + b s2 (t) (N − i) N .(4) Payoffs determine the probabilities for a player to be chosen for reproduction, e.g. for the male population, P A s (i, j, t) = 1 N · 1 − w + wπ A s (j, t) 1 − w + wπ A (i, j, t) ,(5)whereπ A (i, j, t) = [iπ A 1 (j, t) + (N − i)π A 2 (j, t)] /N is the average payoff of the males. The baseline fitness w ∈ [0, 1] is a tunable baseline fitness parameter determining how the player's chance to be chosen for reproduction is related to player's performance [5,6]. When w = 0, the probability to be chosen for reproduction does not depend on player's performance and is uniform across the population. After the choice has been made, another member of the population is chosen completely randomly and replaced with an offspring of the player chosen for reproduction, i.e. with a player using the same strategy as its parent [7]. This update mechanism is acting simultaneously in both populations, A and B, such that a mating pair produces two offspring, a male and a female, on every round. Therefore, the size of the populations N remains constant. A single round can be considered as a one-step Markov process, with transition rates, e.g. for population A, from a state i to states i + 1 and i − 1, are given by [6,10] T + A (i, j, t) = 1 − w + wπ A 1 (t) 1 − w + wπ A i N N − i N , T − A (i, j, t) = 1 − w + wπ A 2 (t) 1 − w + wπ A N − i N i N .(6) TRANSITION TENSOR Here we describe the transition fourth-order tensor S m (i, j, i , j ) in terms of the rates [T +,− A (i, j, t) and T +,− B (i, j, t)] for populations A and B given by Eq. (4) in the main text. The stochastic Moran process can be expressed as a Markov chain [10] p m+1 (i, j) = 1 − T + A (i, j, m t) − T − A (i, j, m t) 1 − T + B (i, j, m t) − T − B (i, j, m t) p m (i, j) +T − B (i, j + 1, m t) 1 − T − A (i, j + 1, m t) − T + A (i, j + 1, m t) p m (i, j + 1) +T + B (i, j − 1, m t) 1 − T − A (i, j − 1, m t) − T + A (i, j − 1, m t) p m (i, j − 1) +T − A (i + 1, j, m t) 1 − T − B (i + 1, j, m t) − T + B (i + 1, j, m t) p m (i + 1, j) +T + A (i − 1, j, m t) 1 − T − B (i − 1, j, m t) − T + B (i − 1, j, m t) p m (i − 1, j) +T − A (i + 1, j + 1, m t)T − B (i + 1, j + 1, m t)p m (i + 1, j + 1) +T + A (i − 1, j + 1, m t)T − B (i − 1, j + 1, m t)p m (i − 1, j + 1) +T − A (i + 1, j − 1, m t)T + B (i + 1, j − 1, m t)p m (i + 1, j − 1) +T + A (i − 1, j − 1, m t)T + B (i − 1, j − 1, m t)p m (i − 1, j − 1).(7) The above equation can be recast into p m+1 (i, j) = i ,j S m (i, j, i , j )p m (i , j ),(8) where the fourth-order tensor S m (i, j, i , j ) is given by, S m (i, j, i , j ) = 1 − T + A (i , j , m t) − T − A (i , j , m t) 1 − T + B (i , j , m t) − T − B (i , j , m t) δ i ,i δ j ,j +T − B (i , j , m t) 1 − T − A (i , j , m t) − T + A (i , j , m t) δ i ,i δ j ,j+1 +T + B (i , j , m t) 1 − T − A (i , j , m t) − T + A (i , j , m t) δ i ,i δ j ,j−1 +T − A (i , j , m t) 1 − T − B (i , j , m t) − T + B (i , j , m t) δ i ,i+1 δ j ,j +T + A (i , j , m t) 1 − T − B (i , j , m t) − T + B (i , j , m t) δ i ,i−1 δ j ,j +T − A (i , j , m t)T − B (i , j , m t)δ i ,i+1 δ j ,j+1 +T + A (i , j , m t)T − B (i , j , m t)δ i ,i−1 δ j ,j+1 +T − A (i , j , m t)T + B (i , j , m t)δ i ,i+1 δ j ,j−1 +T + A (i , j , m t)T + B (i , j , m t)δ i ,i−1 δ j ,j−1 .(9) Above i = 0, · · · , N , j = 0, · · · , N , i = 0, · · · , N , and j = 0, · · · , N . Using the bijection k = (N − 1)j + i and l = (N − 1)j + i , we obtain the required matrix form, see Eq. (7) in the main text. ADJUSTED REPLICATOR EQUATIONS In the continuous limit N → ∞, the dynamics of the variables x = i/N and y = j/N is defined by the adjusted replicator equations [8,11], x = [1 − x][∆ A (t) − Σ A (t)y] 1 Γ +π A (x, y, t) ,(10)y = [1 − y][∆ B (t) − Σ B (t)x] 1 Γ +π B (x, y, t) ,(11) where ∆ C = c 12 −c 22 , Σ C = c 11 +c 22 −c 12 −c 21 , Γ = 1−w w , and C = {A, B}.π A (x, y, t) [π B (x, y, t)] is the averaged (over the population) payoff of the males [females]. THE LIFETIME OF A METASTABLE STATE The lifetime of the state d(t) can be estimated with the largest eigenvalue λ T , 0 < λ T < 1, of the ma-trixŨ r (T ). To compare it with lifetimes of stationary metastable states, we introduce the mean single-round exponent,λ T = λ 1/M T and define the mean lifetime as t life = 1/(1 −λ T ). Figure 1 shows the dependence of t life on the strength of modulations. SIMULATIONS The preconditioned stochastic sampling was performed by launching trajectories from random initial points, uniformly distributed on the N − 1 × N − 1 grid and then sampling the pdf with only those trajectories which remained unabsorbed after 10 · N 2 rounds. For N = 200 the diagonalization of the 39 601 × 39 601 matrixQ T was performed on the cluster of the MPIPKS (Dresden) and Leibniz-Rechenzentrum (München). The stochastic sampling was performed on a GPU cluster consisting of twelve TESLA K20XM cards. That allowed us to obtain 5 · 10 8 realizations for each set of parameters. FIG. 2 : 2(color online) Metastable states of the stationary BoS game. FIG. 3 : 3(color online) Evolutionary dynamics governed by the BoS game with modulated payoffs. (a) Period-one limit cycles of the mean-field dynamics for ω = 0.1 (blue dash-dotted line) and ω = 0.01 (red solid line) are localized near the Nash equilibrium of the stationary game, the direction of motion). In the limit ω → 0, the mean-field attractor shrinks to the set of adiabatic Nash equlibria (black dashed line). Mean position (•) of a finite-N metastable Floquet state, (x(t),ȳ(t)), Eq.(2), moves along the limit cycle localized near the point 1 2 , 1 2 (the means are plotted at the instants tn = nT /5, n = 0, .., 4); (b) A stochastic trajectory (grey line) reveals the existence of a period-two limit cycle [the period doubling can be resolved with stroboscopic points, plotted at the instants 2nT ( ) and (2n + 1)T (♦)]. The trajectory is initiated at the point marked with the open blue square and ends up at the absorbing state (red cross at the upper left corner). The trajectory of the mean of the finite-N metastable Floquet state (•) is distinctively different from the stochastic trajectory [note the change of scale as compared to panel (a)]. The parameters are f = 0.5, N = 200, and M = T / t =10N(corresponds to the driving frequency ω = 0.1 in the mean-field limit)[32]. 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These two consecutive steps, death and birth, can be reinterpreted as a single step of imitation. i.e. adoption of the strategy of the first player by the second one [8, 9These two consecutive steps, death and birth, can be reinterpreted as a single step of imitation, i.e. adoption of the strategy of the first player by the second one [8, 9]. . J Hofbauer, K H Schlag, J. of Evol. Economics. 10523J. Hofbauer and K. H. Schlag, J. of Evol. Economics 10, 523 (2000). . K H Schlag, J. of Econom. Theory. 78130K. H. Schlag, J. of Econom. Theory 78, 130. . A Trauslen, J C Claussen, C Hauert, Phys. Rev. E. 8541901A. Trauslen, J. C. Claussen, and C. Hauert, Phys. Rev. E 85, 041901 (2012). J M Smith, Evolution and the Theory of Games. CambridgeCambridge University PressJ. M. Smith, Evolution and the Theory of Games(Cambridge University Press, Cambridge, 1982).	[]
[ "An application of the BHV theorem to a new conjecture on exponential diophantine equations", "An application of the BHV theorem to a new conjecture on exponential diophantine equations" ]	[ "Maohua Le " ]	[]	[]	Let A, B be fixed positive integers such that min{A, B} > 1, gcd(A, B) = 1 and AB ≡ 0 mod 2, and let n be a positive integer with n > 1. In this paper, using a deep result on the existence of primitive divisors of Lucas numbers due to Y. Bilu, G. Hanrot and P. M. Voutier [1], we prove that if A > 8B 3 , then the equation ( * ) (A 2 n) x + (B 2 n) y = ((A 2 + B 2 )n) z has no positive integer solutions (x, y, z) with x > z > y. Combining the above conclusion with some existing results, we can deduce that if A > 8B 3 and B ≡ 2 mod 4, then (*) has only the positive integer solution (x, y, z) = (1, 1, 1).	null	[ "https://arxiv.org/pdf/1811.00609v1.pdf" ]	119,699,620	1811.00609	ddd17a1b33867221b0d65f9ec7c675fcb4993b54	An application of the BHV theorem to a new conjecture on exponential diophantine equations Nov 2018 Maohua Le An application of the BHV theorem to a new conjecture on exponential diophantine equations Nov 2018 Let A, B be fixed positive integers such that min{A, B} > 1, gcd(A, B) = 1 and AB ≡ 0 mod 2, and let n be a positive integer with n > 1. In this paper, using a deep result on the existence of primitive divisors of Lucas numbers due to Y. Bilu, G. Hanrot and P. M. Voutier [1], we prove that if A > 8B 3 , then the equation ( * ) (A 2 n) x + (B 2 n) y = ((A 2 + B 2 )n) z has no positive integer solutions (x, y, z) with x > z > y. Combining the above conclusion with some existing results, we can deduce that if A > 8B 3 and B ≡ 2 mod 4, then (*) has only the positive integer solution (x, y, z) = (1, 1, 1). Introduction Let Z, N be the sets of all integers and positive integers, respectively. Let a, b be fixed positive integers such that min{a, b} > 1 and gcd(a, b) = 1, and let n be a positive integer with n > 1. Recently, P.-Z. Yuan and Q. Han [7] proposed the following conjecture: has only the solution (x, y, z) = (1, 1, 1). Since Conjecture 1.1 is much broader than Jeśmanowicz' conjecture concerning Pythagorean triples (see [3]), it is unlikely to be solved in the short term. There are only a few scattered results on Conjecture 1.1 at present (see [5]). In the same paper, P.-Z. Yuan and Q. Han [7] deal with the solutions (x, y, z) of (1.1) for the case that (a, b) = (A 2 , B 2 ), where A, B are fixed positive integers such that min{A, B} > 1, gcd(A, B) = 1 and AB ≡ 0 mod 2. Then (1.1) can be rewritten as (A 2 n) x + (B 2 n) y = ((A 2 + B 2 )n) z , x, y, z ∈ N. (1.2) In this respect, they proved that if B ≡ 2 mod 4, then (1.2) has no solutions (x, y, z) with y > z > x, in particular, if B = 2, then Conjecture 1.1 is true for (a, b) = (A 2 , B 2 ). In 2001, Y. Bilu, G. Hanrot and P. M. Voutier [1] completely solved the existence of primitive divisors of Lucas numbers. This result is usually called the BHV theorem. In this paper, using the BHV theorem, we prove the following result: (1.2) has no solutions (x, y, z) with x > z > y. Theorem 1.1. If A > 8B 3 , then By Theorem 1.1 and some results of [5] and [7], we can deduce the following corollary: This implies that, for any fixed B with B ≡ 2 mod 4, then Conjecture 1.1 is true for (a, b) = (A 2 , B 2 ) except for finitely many values of A. Preliminaries Let α, β be algebraic integers. If α + β and αβ are nonzero coprime integers and α/β is not a root of unity, then (α, β) is called a Lucas pair. Further, let u = α + β and w = αβ. Then we have α = 1 2 (u + λ √ v), β = 1 2 (u − λ √ v), λ ∈ {−1, 1}, where v = u 2 − 4w. We call (u, v) the parameters of the Lucas pair (α, β). Two Lucas pairs (α 1 , β 1 ) and (α 2 , β 2 ) are equivalent if α 1 /α 2 = β 1 /β 2 = ±1. Given a Lucas pair (α, β), one defines the corresponding sequence of Lucas numbers by L n (α, β) = α n − β n α − β , n = 0, 1, . . . . (2.1) Obviously, if n > 0, then the Lucas numbers L n (α, β) are nonzero integers. For equivalent Lucas pairs (α 1 , β 1 ) and (α 2 , β 2 ), we have L n (α 1 , β 1 ) = ±L n (α 2 , β 2 ) for any n ≥ 0. A prime p is called a primitive divisor of L n (α, β) (n > 1) if p \| L n (α, β), p \| vL 1 (α, β) . . . L n−1 (α, β). (2.2) A Lucas pair (α, β) such that L n (α, β) has no primitive divisors will be called an n-defective Lucas pair. Further, a positive integer n is called totally non-defective if no Lucas pair is n-defective. Lemma 2.1 (P. M. Voutier [6]). Let n satisfy 4 < n ≤ 30 and n = 6. Then, up to equivalence, all parameters of n-defective Lucas pairs are given as follows: (i) n = 5, (u, v) = (1, 5), (1, −7), (2, −40), (1, −11), (1, −15), (12, −76), (12, −1364). (ii) n = 7, (u, v) = (1, −7), (1, −19). (iii) n = 8, (u, v) = (2, −24), (1, −7) . (iv) n = 10, (u, v) = (2, −8), (5, −3), (5, −47). (v) n = 12, (u, v) = (1, 5), (1, −7), (1, −11), (2, −56), (1, −15), (1, −19). (vi) n ∈ {13, 18, 30}, (u, v) = (1, −7). Lemma 2.2 (Y. Bilu, G. Hanrot, P. M. Voutier [1]). If n > 30 then n is totally non-defective. Let D, k be fixed positive integers such that min{D, k} > 1 and gcd(2D, k) = 1, and let h(−4D) denote the class number of positive definite binary quadratic primitive forms of discriminant −4D. Lemma 2.3. If the equation X 2 + DY 2 = k Z , X, Y, Z ∈ Z, gcd(X, Y ) = 1, Z > 0 (2.3) has solutions (X, Y, Z), then every solution (X, Y, Z) of (2.3) can be expressed as Z = Z 1 t, t ∈ N, (2.4) X + Y √ −D = λ 1 (X 1 + λ 2 Y 1 √ −D) t , λ 1 , λ 2 ∈ {−1, 1}, (2.5) where X 1 , Y 1 , Z 1 are positive integers satisfying X 2 1 + DY 2 1 = k Z1 , gcd(X 1 , Y 1 ) = 1 and h(−4D) ≡ 0 mod Z 1 . (2.6) Proof. This lemma is a special case of Theorems 1 and 2 of [4] for D 1 = 1 and D 2 < −1. α = X 1 + Y 1 √ −D, β = X 1 − Y 1 √ −D. (2.9) Then we have α + β = 2X 1 , αβ = k Z1 , (2.10) α β = 1 k Z1 ((X 2 1 − DY 2 1 ) + 2X 1 Y 1 √ −D). (2.11) Since gcd(X 1 , Y 1 ) = gcd(2D, k) = 1, we see from (2.10) that α + β and αβ are coprime postive integers. Since k > 1, by (2.11), α/β is not a root of unity. Hence, by (2.9), (α, β) is a Lucas pair with parameters Lemma 2.6 (C.-F. Sun and M. Tang [5]). Let (x, y, z) be a solution of (1.1) with (x, y, z) = (1, 1, 1). If min{a, b} ≥ 4, then either (u, v) = (2X 1 , −4DY 2 1 ). (2.12) Let L n (α, β) (n = 0, 1, . . . ) denote the corresponding Lucas numbers. Since X − Y √ −D = λ 1 (X 1 − λ 2 Y 1 √ −D)x > z > y, b = b 1 b 2 , b y 1 = n z−y , b 1 , b 2 ∈ N, b 1 > 1, gcd(b 1 , b 2 ) = 1 or y > z > x, a = a 1 a 2 , a x 1 = n z−x , a 1 , a 2 ∈ N, a 1 > 1, gcd(a 1 , a 2 ) = 1. Proof of Theorem and Corollary Proof of Theorem 1.1. Let (x, y, z) be a solution of (1.2) with x > z > y. By Lemma 2.6, we have B = B 1 B 2 , B 1 , B 2 ∈ N, B 1 > 1, gcd(B 1 , B 2 ) = 1 (3.1) and B 2y = n z−y . (3.2) Substituting (3.1) and (3.2) into (1.2), we get A 2x n x−z + B 2y 2 = (A 2 + B 2 ) z . (3.3) Let D = R(A 2x n x−z ). Since A 2x n x−z /R(A 2x n x−z ) is the square of a positive integer, we see from (3.3) that the equation Since r(n) = r(B 1 ) and n x−z ∈ S(B 1 ) by ( X 2 + DY 2 = (A 2 + B 2 ) Z , X, Y, Z ∈ Z, gcd(X, Y ) = 1, Z > 0 (3.4) has the solution (X, Y, Z) = B y 2 , A 2x n x−z D , z(3.(A 2 n) x−z < 1 + B 2 A 2 z . (3.10) Since log(1 + θ) < θ for any θ > 0, by (3.10), we get (x − z) log(A 2 n) < xB 2 A 2 . (3.11) Further, since A > 8B 3 , by (3.11), we obtain z > (x − z) A 2 B 2 log(A 2 n) ≥ A 2 B 2 log(A 2 n) > 8AB log(A 2 n). (3.12) The combination of (3.7), (3.9) and (3.12) yields 24 π AB 1 log(2eAB 1 ) > 8AB log(A 2 n). (3.13) But, since 24/π < 8, AB 1 ≤ AB and 2eAB 1 < 8AB 3 < A 2 < A 2 n, (3.13) is false. Thus, if A > 8B 3 , then (1.2) has no solutions (x, y, z) with x > z > y. Proof of Corollary 1.1. Under the assumptions, by Theorem 1.1 and the result of [7], (1.2) has no solutions with x > z > y nor with y > z > x. Therefore, by Lemma 2.6, (1.2) has no solutions (x, y, z) with (x, y, z) = (1, 1, 1). Conjecture 1 . 1 . 11If min{a, b} ≥ 4, then the equation (an) x + (bn) y = ((a + b)n) z , x, y, z ∈ N (1.1) Corollary 1 . 1 . 11If A > 8B 3 and B ≡ 2 mod 4, then (1.2) has only the solution (x, y, z) = (1, 1, 1). For a positive integer m with m > 1, let m = p t1 1 . . . p t l l denote the factorization of m. Further, let r(m) = p 1 . . . p l ,R(m) = p e1 1 . . . p e l l , e i = 2, if t i ≡ 0 mod 2, 1, if t 1 ≡ 1 mod 2, i = 1, . . . , l, r(1) = R(1) = 1 and S(m) = {±p s1 1 . . . p s l l \| s i ∈ Z, s i ≥ 0, i = 1, .. . , l}. Obviously, r(m) and R(m) have the following properties:(i) R(m) > 1 if m > 1. (ii) m/R(m) is the square of a positive integer. (iii) For any positive integer m ′ with m ′ ∈ S(m), R(m ′ ) ≤ (r(m)) 2 . (iv) For any coprime positive integers m 1 and m 2 , r(m 1 m 2 ) = r(m 1 )r(m 2 ) and R(m 1 m 2 ) = R(m 1 )R(m 2 ). Lemma 2.5. If D > 2 and (X, Y, Z) is a solution of (2.3) with Y ∈ S(D), (2.7) then Z ≤ 6h(−4D). (2.8) Proof. By Lemma 2.4, the solution (X, Y, Z) can be expressed as (2.4) and (2.5), where X 1 , Y 1 , Z 1 are positive integers satisfying gcd(X 1 , Y 1 ) = 1 and (2.6). Let gcd(A, B) = 1, by (3.1) and (3.2), we have gcd(A, B 1 ) = gcd(A, n) = 1. Hence, we getD = R(A 2x n x−z ) = R(A 2x )R(n x−z ).(3.6) Further, since R(A 2x ) > 1 and R(n x−z ) > 1, by (3.6), we have have D > 2. Since gcd(A, B) = 1 and AB ≡ 0 mod 2, we have gcd(2D, A 2 + B 2 ) = 1. Therefore, applying Lemma 2.5 to (3.5), we get z ≤ 6h(−4D). (3.7) 3.2), we have R(A 2x ) ≤ A 2 and R(n x−z ) ≤ B 2On the other hand, since x > z, by (3.3), we have1 . Hence, by (3.6), we get D ≤ A 2 B 2 1 . (3.8) Further, by Lemma 2.4, we obtain from (3.8) that h(−4D) < 4 π AB 1 log(2eAB 1 ). (3.9) Acknowledgment: The author would like to thank Prof. R. Scott and Prof. R. Styer for reading the original manuscript carefully and giving valuable advice. Especially thanks to Prof. R. Styer for verifying some details of this paper and providing other technical assistance. Existence of primitive divisors of Lucas and Lehmer numbers, with an appendix by M. Mignotte. Y Bilu, G Hanrot, P M Voutier, J. Reine Angew. Math. 539Y. Bilu, G. Hanrot, and P. M. Voutier. Existence of primitive divisors of Lucas and Lehmer numbers, with an appendix by M. Mignotte. J. Reine Angew. Math., 539:75-122, 2001. Introduction to number theory. L.-K Hua, Springer-VerlagBerlinL.-K. Hua. Introduction to number theory. Berlin: Springer-Verlag, 1982. Several remarks on Pythagorean numbers. L Jeśmanowicz, Wiadom. Math. 12in PolishL. Jeśmanowicz. Several remarks on Pythagorean numbers. Wiadom. Math., 1(2):196-202, 1955/1956. (in Polish). Some exponential diophantine equations I: The equation D 1 x 2 − D 2 y 2 = λk z. M.-H Le, J. Number Theory. 552M.-H. Le. Some exponential diophantine equations I: The equation D 1 x 2 − D 2 y 2 = λk z . J. Number Theory, 55(2):209-221, 1995. On the diophantine equation (an) x + (bn) y = (cn) z. C.-F Sun, M Tang, Chinese Math. Ann., Ser. A. 391in ChineseC.-F. Sun and M. Tang. On the diophantine equation (an) x + (bn) y = (cn) z . Chinese Math. Ann., Ser. A, 39(1):87-94, 2018. (in Chinese). Primitive divisors of Lucas and Lehmer sequences. P M Voutier, Math. Comp. 64P. M. Voutier. Primitive divisors of Lucas and Lehmer sequences. Math. Comp., 64: 869-888, 1995. Jeśmanowicz' conjecture and related equations. P.-Z Yuan, Q Han, Acta Arith. 1841P.-Z. Yuan and Q. Han. Jeśmanowicz' conjecture and related equations. Acta Arith., 184(1):37-49, 2018.	[]
[ "Partial local density of states from scanning gate microscopy", "Partial local density of states from scanning gate microscopy" ]	[ "Ousmane Ly \nInstitut de Physique et de Chimie des Matériaux de Strasbourg\nUMR 7504\nUniversité de Strasbourg\nCNRS\nF-67000StrasbourgFrance\n", "Rodolfo A Jalabert \nInstitut de Physique et de Chimie des Matériaux de Strasbourg\nUMR 7504\nUniversité de Strasbourg\nCNRS\nF-67000StrasbourgFrance\n", "Steven Tomsovic \nDepartment of Physics and Astronomy\nWashington State University\nPO Box 64281499164-2814Pullman, WashingtonUSA\n", "Dietmar Weinmann \nInstitut de Physique et de Chimie des Matériaux de Strasbourg\nUMR 7504\nUniversité de Strasbourg\nCNRS\nF-67000StrasbourgFrance\n" ]	[ "Institut de Physique et de Chimie des Matériaux de Strasbourg\nUMR 7504\nUniversité de Strasbourg\nCNRS\nF-67000StrasbourgFrance", "Institut de Physique et de Chimie des Matériaux de Strasbourg\nUMR 7504\nUniversité de Strasbourg\nCNRS\nF-67000StrasbourgFrance", "Department of Physics and Astronomy\nWashington State University\nPO Box 64281499164-2814Pullman, WashingtonUSA", "Institut de Physique et de Chimie des Matériaux de Strasbourg\nUMR 7504\nUniversité de Strasbourg\nCNRS\nF-67000StrasbourgFrance" ]	[]	Scanning gate microscopy (SGM) images from measurements made in the vicinity of quantum point contacts (QPC) were originally interpreted in terms of current flow. Some recent work has analytically connected the local density of states to conductance changes in cases of perfect transmission, and at least qualitatively for a broader range of circumstances. In the present paper, we show analytically that in any time-reversal invariant system there are important deviations that are highly sensitive to imperfect transmission. Nevertheless, the unperturbed partial local density of states can be extracted from a weakly invasive scanning gate microscopy experiment, provided the quantum point contact is tuned anywhere on a conductance plateau. A perturbative treatment in the reflection coefficient shows just how sensitive this correspondence is to the departure from the quantized conductance value and reveals the necessity of local averaging over the tip position. It is also shown that the quality of the extracted partial local density of states decreases with increasing tip radius. arXiv:1706.02220v2 [cond-mat.mes-hall]	10.1103/physrevb.96.125439	[ "https://arxiv.org/pdf/1706.02220v2.pdf" ]	119,096,783	1706.02220	daeba099107d60958d0d7fe504b0fe993941eb7f	Partial local density of states from scanning gate microscopy Ousmane Ly Institut de Physique et de Chimie des Matériaux de Strasbourg UMR 7504 Université de Strasbourg CNRS F-67000StrasbourgFrance Rodolfo A Jalabert Institut de Physique et de Chimie des Matériaux de Strasbourg UMR 7504 Université de Strasbourg CNRS F-67000StrasbourgFrance Steven Tomsovic Department of Physics and Astronomy Washington State University PO Box 64281499164-2814Pullman, WashingtonUSA Dietmar Weinmann Institut de Physique et de Chimie des Matériaux de Strasbourg UMR 7504 Université de Strasbourg CNRS F-67000StrasbourgFrance Partial local density of states from scanning gate microscopy Scanning gate microscopy (SGM) images from measurements made in the vicinity of quantum point contacts (QPC) were originally interpreted in terms of current flow. Some recent work has analytically connected the local density of states to conductance changes in cases of perfect transmission, and at least qualitatively for a broader range of circumstances. In the present paper, we show analytically that in any time-reversal invariant system there are important deviations that are highly sensitive to imperfect transmission. Nevertheless, the unperturbed partial local density of states can be extracted from a weakly invasive scanning gate microscopy experiment, provided the quantum point contact is tuned anywhere on a conductance plateau. A perturbative treatment in the reflection coefficient shows just how sensitive this correspondence is to the departure from the quantized conductance value and reveals the necessity of local averaging over the tip position. It is also shown that the quality of the extracted partial local density of states decreases with increasing tip radius. arXiv:1706.02220v2 [cond-mat.mes-hall] I. INTRODUCTION Since its development 20 years ago, 1 scanning gate microscopy (SGM) has revealed fascinating phenomena in transport processes and has been considered as a powerful tool to probe local properties. 2, 3 In this technique the conductance of an electronic device is measured while the tip of an atomic force microscope (AFM) is scanned above its surface. The AFM tip acts as a movable gate that scatters the electrons leading to a spatially dependent modulation of the conductance. 4 One of the most investigated nanostructures is the quantum point contact (QPC) 5,6 defined in a two dimensional electron gas (2DEG). When the tip is rasterscanned over the surface of the system, electrons are back-scattered to the QPC giving rise to a conductance map that exhibits a branched pattern. In the case of a QPC opening into an unconstrained 2DEG these patterns have been interpreted as a signature of the electron flow in the disordered potential resulting from the ionized donor atoms. 7,8 Thus, a link is presumed to exist between SGM measurements and local properties (local densities of states [LDOS] and current densities) of the unperturbed devices. Typically, the tip voltages used to study QPC setups operating in the regime of conductance quantization are strong enough to create a large depletion disk (much bigger than the Fermi wavelength) in the 2DEG underneath the tip. The connection with local properties has been argued to concern the classical turning point of the electron trajectories with the Fermi energy that leave the QPC and encounter the tip potential. 9 In order to address this problem, the paradigmatic case of a QPC perturbed by a weakly invasive tip has been considered in the linear 10,11 and non-linear 12 regimes (in source-drain bias voltage). In particular, in the regime of conductance quantization of clean 2DEGs, spatial and time-reversal symmetries have been shown to play a key role in establishing a correspondence of the SGM re-sponse with the LDOS and the current density on both sides of the QPC. The SGM technique has also been used to study systems with a variety of electronic confinements, including open quantum dots [13][14][15][16][17][18] and Aharonov-Bohm rings built in high-mobility semiconductor heterostructures, [19][20][21][22] as well as carbon nanotubes 23 and graphene-based microstructures. 24,25 For systems with sufficient electronic confinement charging effects are relevant, and for very small quantum dots a biased SGM tip mainly acts as a gate that modifies the number of electrons in the dot and affects the conductance via the Coulomb-blockade phenomenon. 23,24,[26][27][28] For relatively large and open quantum dots, the charging effects are not crucial and, as in the case of QPC setups, the connection between the SGM measurements and local properties has been pursued. In these systems, qualitative similarity between conductance changes and LDOS has been noted whenever the LDOS exhibits some localized structure. For instance, minima of the SGM response appear where the LDOS vanishes. 21,22 Furthermore, numerical simulations for rectangular resonant cavities 29 indicated that the conductance terms derived in Ref. 10 are correlated with the LDOS when the Fermi energy is close to a resonance with a cavity state. For one-dimensional systems, a perturbative approach has revealed that the first-order conductance change in the presence of a δ-tip is related to the Hilbert transform of the LDOS. 21,30 It is important to note that electronic confinement is associated with a change in the interpretation of SGM maps with respect to the case of a QPC. Specifically tailored experiments have shown the need of such a change of interpretation when the QPC setup is modified by electronic confinement guiding the electron transport. 18,31 The need of different interpretations for setups with and without electronic confinement can be traced, in the case of weakly invasive probes, to special features of conductance quantization characterizing QPCs in the absence of confinement, where the transmission channels are either completely open or closed. 10 The issue of whether the transmission channels are completely open (and otherwise completely closed), i.e. the perfect transmission case, turns out to play a crucial role in the interpretation of measurements and their relationships to local properties. It has been shown that in the case of perfect transmission, the second order conductance change is the first non-vanishing term in a perturbation series 10 and it is proportional to the square of the LDOS. 11 However, the analytic relationship between conductance changes and local properties becomes more complicated for imperfect transmission. In this paper analytical and numerical approaches are developed to study the connection between SGM measurements and local properties for the case of a QPC in which the tip potentials can be perturbative or nonperturbative, local or extended, etc. In addition, cases where the 2DEG surrounding the QPC can be disordered or clean are treated. First, in the perturbative regime and on a perfect conductance plateau (i.e. at perfect transmission), the SGM on one side of the QPC is unambiguously related to the partial LDOS (PLDOS, defined in the next section) of scattering states impinging from the other side, with no requirement of spatial symmetry. Thus, the PLDOS plays a more fundamental role than the LDOS. Next, it turns out that there are significant deviations from the PLDOS that are highly sensitive to how far one is from a perfect transmission case. Nevertheless, averaging over the tip position allows one to develop a quantitative method for extracting the PLDOS even in this regime. Finally, it is shown that increasing the width of the tip reduces the quality of the PLDOS one can extract. In Sec. III the main results of the existing analytic perturbation theory 10,11 are summarized. The analytical derivation of the relationship between SGM and PLDOS for weak local tips is presented in Sec. IV for the case of conductance steps and in Sec. V for the case of perfect unit conductance. The corrections for non-perfect unit conductance are treated perturbatively in Sec. VI. A method for extracting the PLDOS and effectively disentangling first and second order contributions to the conductance response for imperfect transmission are given. Numerical simulations of the second-order conductance correction dominant in the perfect transmission case are presented in Sec. VII for the case of local tips and the full conductance correction is shown in Sec. VIII. The case of extended tips is discussed in Sec. IX. Some technical aspects related with the scattering states are relegated to Appendix A, and Appendix B establishes the link of a particular contribution to the SGM response with the LDOS. II. PARTIAL LOCAL DENSITY OF STATES IN THE SCATTERING FORMALISM The spinless partial local density of states (PLDOS) for electrons impinging into the scatterer from lead l can be defined by 32,33 ρ lε (r) = 2π N a=1 \|Ψ l,ε,a (r)\| 2 ,(1) using the sub-ensemble of the basis of outgoing scattering states (A3) incoming from lead l. N is the number of propagating modes in the lead at the energy ε. The decomposition of the spinless local density of states (LDOS) ρ ε (r) as ρ ε (r) = ρ 1ε (r) + ρ 2ε (r) ,(2) valid for the two-lead case, naturally appears in scattering problems in which one is concerned with the response of the system to a small perturbation of the confining potential. 30 Such is the case of the SGM response, as well as that of the self-consistent treatment of electrical a.c. transport in mesoscopic systems. 33 The definition (1) corresponds to an injectivity, 30,32,33 where the preselection of carriers is done by the incident lead l from where they impinge into the scatterer. Denoting by M the number of open transmitting eigenchannels, the basis of scattering eigenfunctions (A9) gives the expressions for the PLDOS on the right and left, respectively, of the scatterer as ρ 1ε (r) = 2π M m=1 \|χ 1,ε,m (r)\| 2 , x > 0 , (3a) ρ 2ε (r) = 2π M m=1 \|χ 2,ε,m (r)\| 2 , x < 0 .(3b) Since quite generally, M N , and the transmitted parts of the scattering eigenstates (A9) are proportional to the diagonal elements of the transmission submatrices, the expressions (3) are considerably easier to evaluate than (1). However, it is important to keep in mind that the expressions (3a) and (3b) only describe the region opposite to the lead determining their PLDOS, and are not appropriate for obtaining the LDOS using Eq. (2) since they refer to different regions of space. For instance, for a QPC embedded in a clean 2DEG, ρ 1ε (r) ∝ 1/\|r\| far away from the QPC 34 , while the LDOS is independent of r. III. PERTURBATIVE RESULTS An analytical description of SGM in the presence of a strong tip is a challenging theoretical task. However, a perturbative approach 10 is tractable in the weakly invasive case, where the tip-induced potential constitutes a small perturbation of the electrostatic potential seen by the electrons. To begin, consider a weak tip potential V T (r) = v T f (r − r T ), where f (r) is a normalized function with drf (r) = 1, which perturbs the system. The change in the dimensionless (in units of 2e 2 /h) tipposition dependent conductance can be written as g(r T ) = g (0) + δg(r T ) ,(4) with δg(r) = v T g (1) (r) + v 2 T g (2) (r) + O[v 3 T ] .(5) The unperturbed conductance g (0) is given by the Landauer-Büttiker formula as the total transmission probability. It can be expressed as a trace over the propagating modes g (0) = Tr[t † t] = M m=1 T 2 m(6) in terms of the transmission submatrix t of the unperturbed scattering matrix S at the Fermi energy ε F or the transmission eigenvalues T m of the M open eigenchannels (see Appendix A). The basis of the transmission eigenmodes is particularly suited to express the SGM conductance corrections. Assuming time-reversal invariance from here on, the two lowest-order corrections 10,11 are g (1) = 4π v T Tr RT Im U 21 ,(7)g (2) = − 4π 2 v 2 T Tr T 2 U 12 U 21 − R 2 U 21 U 12 + RT Re U 22 U 21 − U 21 U 11 − 4π v 2 T 2 l =1 P ∞ ε t 1 dε ε − ε F Tr RT Im U 2l (ε F ,ε)Ul 1 (ε, ε F ) .(8) R and T are real diagonal reflection and transmission submatrices appearing in the polar decomposition (A4) of S. The matrix elements Ul l m,m (ε, ε) = χ * l,ε,m (r)V T (r)χ l,ε,m (r)dr (9) are those of the tip potential between two scattering eigenfunctions, where l, m and ε label the incoming lead, the channel number, and the electron energy of the scattering eigenfunction, respectively. If the arguments of Ul l are omitted, it is understood that both are taken at ε F , and the necessary matrix element tip position dependence on r T is assumed. The limiting integration energy ε t 1 is that of the lowest transverse energy and P stands for the principal part of the integral. The traces over the N propagating modes in the leads in Eqs. (7) and (8) On a conductance plateau where the transmission is perfect, R m T m = 0 for all eigenmodes m. There, the first order contribution (7) vanishes 10,11 and the SGM response is given by v 2 T g (2) ; note that only one term above of v 2 T g (2) survives as well. The relative importance of the linear and the quadratic SGM responses when moving between conductance plateaus and conductance steps of the QPC can also be affected by temperature, which mixes the two regimes, and can lead to an increase of the SGM response with increasing temperature as it was obtained in Ref. 34. Although it is not of direct experimental relevance, the case of a local tip f (r) = δ(r) is an interesting study case. In particular, the first-order conductance correction reduces to g (1) (r T ) = 4π M m=1 R m T m Im χ * 2,εF,m (r T )χ 1,εF,m (r T ) ,(10) and the second-order correction for perfect transmission reduces to g (2) (r T ) = −4π 2 M m,m=1 \|χ 2,εF,m (r T )\| 2 \|χ 1,εF,m (r T )\| 2 ,(11) where M stands for the number of the partially open eigenchannels of the QPC in Eq. (10) and perfectly open channels in Eq. (11). These expressions can be further simplified in cases exhibiting various kinds of symmetries and/or where the geometry allows for the evaluation of the scattering wave-functions. 10,11 IV. g (1) (rT) VERSUS PLDOS IN THE CONDUCTANCE STEPS Focusing first on a QPC setup without disorder, the asymptotic form of the scattering eigenfunctions can be used everywhere in the 2DEG, except in and very close to the constriction. The form (A9) enables expressing the product of scattering eigenfunctions impinging from different leads, in the first order correction (10) due to a weak δ-potential scanned in the right of the QPC, as χ * 2,ε,m (r)χ 1,ε,m (r) = T m (+) 2 2,ε,m (r) + R m (+) 2,ε,m (r) (−) 2,ε,m (r) . g (1) (r T ) = 4π M m=1 R m Im χ 2 1,εF,m (r T ) .(14) Denoting α l,ε,m (r) as the argument of χ l,ε,m (r), Eq. (14) can be written g (1) (r T ) = 4π M m=1 R m sin[2α 1,εF,m (r T )]\|χ 1,εF,m (r T )\| 2 . (15) The sum over eigenmodes reduces to the contribution of the last one (m = M ), which is the only partially open channel having R m > 0. In the case of a single open channel (M = 1) there is a direct relation between the first-order conductance change and the PLDOS since, according to (3), g (1) (r T ) = 2R 1 sin[2α 1,εF,1 (r T )]ρ 1εF (r T ) .(16) However, in the case of M > 1, the structure of the msum in Eq. (15) does not reduce to a simple relationship with ρ 1εF (r). In a disorder-free 2DEG, the prefactor sin(2α 1,εF,1 ) of the SGM response (16) is simply sin(2k F r + α 0 ) with a constant phase α 0 , thus generating half Fermi wavelength, λ F /2, oscillations and a proportionality factor 2R 1 between the spatial oscillation amplitude of the first order conductance correction in the first step and the PLDOS. In the case of a disordered structure, Eq. (16) does not apply inside the disordered region, nevertheless if the disorder is weak and leads to small-angle forward scattering only, one can expect the structure of Eq. (16) to mostly remain. For example, the phase oscillation cannot have such a simple position-dependence strictly speaking, but a paraxial optical approximation 35 holds and a fairly regular radial phase behavior of nearly the same wavelength persists in the eigenfunctions. In these circumstances, the explicit dependence of the SGM response on the phase of the scattering eigenfunction might be helpful in characterizing properties of the fluctuating potential in the 2DEG with further analysis. In general, the first-order conductance correction in tip-strength is not proportional to the PLDOS, even for the case of a δ-tip. In fact, g (1) (r T ) is only local in the sense that Im χ 2 1,εF,m (r T ) is the local information about the eigenfunction of the unperturbed system. However, in the case of a single partial mode the PLDOS provides an upper bound for the absolute value of the former, and the sinusoid term creates a fringing effect. For one-dimensional tight-binding systems the SGM response has been expressed in terms of the real part of the local Green function 21,30 and thereby related to the LDOS. We have checked that in the case of a onedimensional chain the first-order conductance correction (10) (and therefore also the relation (16)) is consistent with the result of Refs. 21 and 30. However, (10) is more general and (16) is expected to be valid whenever there is only one single partially open mode of the QPC, without being limited to strictly one-dimensional systems. V. CORRESPONDENCE BETWEEN g (2) (rT) AND PLDOS FOR PERFECT TRANSMISSION Symmetries have been shown to play a key role in the quest of identifying SGM maps with local properties. 11 In particular, for a four-fold symmetric QPC operating in the regime of perfect transmission, the conductance change induced by a weak local tip in the absence of magnetic field has been shown to be proportional to the square of the LDOS, and also proportional to the local current density. In the same framework, it has been pointed out 36 that the correspondence with the PLDOS holds even for asymmetric QPCs, provided that the conductance is set to the first plateau, as long as the system remains time reversal invariant. An important task, undertaken in this section, is the generalization of previous results to any conductance plateau of an arbitrary QPC under the sole assumptions of time-reversal symmetry and a local tip. To describe transport within the Landauer formalism, the QPC can be treated as a scatterer centered at the origin r = 0. With the definitions of Appendix A, ϕ (−) * l,ε,m (r) = ϕ (+) l,ε,m (r), and (−) * l,ε,m (r) = (+) l,ε,m (r). Therefore, on the m-th conductance plateau, where R m = 0, χ 2,ε,m (r) = χ * 1,ε,m (r)(17) in the 2DEG on both sides of the QPC. Using this relationship in the second order correction (11) leads to g (2) (r T ) = −ρ 2 1εF (r T ) ,(18) for r T at the right of the QPC. Unlike the relation for the first step, which is linear in the PLDOS and fringed in space, perfect transmission on any plateau leads to a quadratic dependence on the PLDOS without fringing. Interestingly, no spatial symmetry is required for the correspondence (18) in the considered regime of conductance quantization. Nevertheless, a perfect conductance quantization with exact unit transmission is a regime difficult to reach in experiments with real QPCs. VI. g (2) (rT) VERSUS PLDOS NEAR PERFECT TRANSMISSION In Sec. V perfect transmission is assumed in order to establish the correspondence between the second order conductance correction and the PLDOS. Here that condition is relaxed. Beyond the unity case of perfect con-ductance quantization where all R m = 0, the first-order correction (15) is nonzero, and all terms of the secondorder correction g (2) in Eq. (8) must be considered. Begin with the situation of transmission slightly below the unity case on the M th conductance plateau, where the transmission of the highest open channel M is not perfect. The expressions of the scattering eigenstates (A9a) and (A9b) can be used to find that χ 2,ε,m (r) = 1 T m 1 + R m e 2iα1,ε,m(r) χ * 1,ε,m (r) (19) for an open mode at the right of a generic QPC. By inserting (19) into Eq. (8) (where the last term is related to a Hilbert transform of the density of states [see Appendix B]), and only keeping the lowest order terms in R m , g (2) reads g (2) (r T ) = −2πρ 1εF M m=1 \|χ 1,εF,m (r T )\| 2 (1 + 2R m {cos [2α 1,εF,m (r T )] + η εF (r T ) sin [2α 1,εF,m (r T )]}) ,(20) where η εF (r) = 1 π P ∞ ε t 1 dε ε − ε F ρε(r) 2ρ 1εF (r)(21) for positions r to the right of the QPC. Notice that the relation of the LDOS to the imaginary part of the diagonal Green function G ε (r, r) implies η εF (r) = −ReG εF (r, r)/(2πρ 1εF (r)). Taking R m = 0 for all m < M gives g (2) (r T ) = −ρ 2 1εF − 4πR M ρ 1εF \|χ 1,εF,M (r T )\| 2 {cos [2α 1,εF,M (r T )] + η εF (r T ) sin [2α 1,εF,M (r T )]} ,(22) and the small reflection amplitude is linked to the deviation from unit conductance by ∆g = R 2 M , where ∆g = M −g (0) quantifies the departure from unit transmission on the M th plateau. In the case of unit transmission one has R M = 0, and (22) reduces to (18). For completeness, in the same regime Eq. (16) can be rewritten as g (1) (r T ) = 4πR M \|χ 1,εF,M (r T )\| 2 sin[2α 1,ε,M (r T )] ,(23) which has similarities in its form with respect to the correction terms for g (2) (r T ). Recall however, the corresponding conductance correction varies linearly with the strength of the tip potential unlike for g (2) (r T ). In the case of transmission just above the unity case with low transmission T M +1 through the QPC mode M + 1, a similar procedure, assuming R m = 0 for all m ≤ M and keeping only the lowest terms in T M +1 yields g (2) (r T ) = −ρ 2 1εF + 2πT 2 M +1 (−) 2,εF,M +1 (r T ) 2 × × ρ 1εF + 4π (−) 2,εF,M +1 (r T ) 2 (1 + cos [2α 1,εF,M +1 (r T )]) 2 − 2ρ 1εF η εF (r T ) sin [2α 1,εF,M +1 (r T )] .(24) The small transmission in the QPC channel M +1 causes departures from (18) that are expected to be proportional to T 2 M +1 . However, in a real system slightly above integer dimensionless conductance, the small transmission of the M + 1 st channel can coexist with an imperfect transmission of the M th channel, ∆g = R 2 M − T 2 M +1 , and the departure from (18) has contributions from both channels, which are difficult to separate in the numerical work. To avoid this complication, we concentrate in the following on the case of positive ∆g, at positions on the conductance plateau where the opening of the next channel is exponentially suppressed and thus negligible. It is worth emphasizing a few features of the expressions contained in Eqs. (22,23). The scale of the deviations from the square of the PLDOS is greatly magnified by being proportional to the square root of ∆g as opposed to being linear. In other words the approach to the perfect transmission case is rather slow with respect to the limit ∆g → 0, and even tiny imperfections produce highly visible deviations. Nevertheless, all the deviations oscillate about zero with a wavelength on the order of λ F /2, and a spatial averaging over a region λ F /2 × λ F /2 results in a near uniform distribution of angles α over 2π, giving a means for the near elimination of the correction terms in (22). Thus, though with reduced spatial resolution, it is still possible to cleanly extract the PLDOS. The PLDOS is not proportional to the LDOS in this case, and the distinction matters. Furthermore, since the contribution of g (1) (r T ) to δg(r T ) is linearly proportional to the tip strength v T and the contribution of g (2) (r T ) quadratic, measurements with two well chosen values of v T would be sufficient to separate out the contributions from Eqs. (22) and (23); with a few more tip strength measurements per tip site, noise and other inaccuracies could be overcome in the separation as well. In the event that \|χ 1,εF,M (r T )\| 2 mostly varies slowly on the scale of λ F , then probability densities due to individual eigenstates and the spatial behavior of α could be extracted as well. Given that η(r) is related to the phase of the real part of the diagonal Green function, in an ideal situation, it could be extracted also. In order to quantify the departures of g (2) (r T ) from the perfect case, introduce the ratio between the coefficient of the second order SGM correction and the square of the PLDOS κ(r T ) = − g (2) (r T ) ρ 2 1εF (r T ) .(25) If the unperturbed conductance g (0) is just below that of M = 1, and the sum over QPC eigenmodes is restricted to m = 1, then κ(r) = 1 + 2 ∆g {cos [2α(r)] + η(r) sin [2α(r)]} . (26) The indices of α and η are omitted; it is understood that α = α 1,εF,1 and η = η εF . As mentioned above, even fairly local spatial averaging approximately yields κ = κ(r) = 1. Interest is therefore in the quantity κ − 1. Similar to the case of the firstorder SGM correction at a conductance step, discussed in Sec. IV, the above relationship provides bounds for the possible values of the ratio κ, \|κ − 1\| ≤ 2 ∆g 1 + η 2 max ,(27) where η max is the maximum value of \|η(r)\|. A priori, η max is not known, but if not extracted as described, it can be obtained by direct numerical computation of the scattering wavefunctions (see Sec. VII) or estimated from simple setups, like that of an abrupt QPC, where the analytical form of the scattering wave-functions is known. 11 The maximum value of η occurs in regions where the PL-DOS is weak, and can in general approach infinity. It's 1 2 0 0.2 0.4 g (0) ε F /t 30w 10w x y P 1 P 2 P 3 P 4 P 5 P 6 P 7 P 8 FIG. 1. The conductance of the QPC defined in a tight binding lattice with lattice parameter a and hopping t as a function of Fermi energy. The inset shows the geometry of the QPC. The width and length of the narrow channel are w = 11a and L = 19a, respectively. The points P1-P8 indicate the Fermi energies and unperturbed conductances at which the statistics of Sec. VII have been performed using tip positions inside the dashed white rectangle. actual value depends on the problem and region under consideration. In one numerical example given ahead, its maximum is of the order of 60. Another interesting quantity is the variance of κ − 1 given by σ 2 = 2∆g(1 + η 2 ) ,(28) where η 2 is the average value of η 2 in the scan region. VII. g (2) (rT) VERSUS PLDOS FOR LOCAL TIPS : SIMULATIONS In order to test our analytical approach and go beyond the above described perturbation theory we performed numerical simulations using the quantum transport package kwant 37 that is based on the recursive Green function method. 38 It can be used to calculate δg(r) as a direct subtraction, and g (1) (r T ) or g (2) (r T ) by constructing numerical derivatives with respect to v T . In our simulations the 2DEG is discretized on a tight binding network with lattice parameter a and a hopping integral t = 2 /(2m * a 2 ), m * being the electron's effective mass. We chose an abrupt constriction defined by a hardwalled square well of width w = 11a and length L = 19a attached to two semi-infinite leads, sketched in the inset of Fig. 1. In order to optimize the computational time the left lead is narrowed. Figure 1 shows the dimensionless conductance through the QPC as a function of the Fermi energy of the incoming electrons. As the latter is increased the QPC's conductance increases in steps of unit height. The structures on the plateaus are due to 20 40 the abruptness of the QPC that lead to Fabry-Perot like oscillations within the constriction. 39 60 x/a (f) (b) ρ 2 1εF t 2 a 4 ρ 2 1εF t 2 a 4 (c) (ρ 2 1εF + g (2) )t 2 a 4 (ρ 2 1εF + g (2) ) A. Local correspondence for perfect transmission In order to address this regime, consider the analytically predicted relationship (18) between the secondorder conductance correction g (2) (r T ) for a δ-tip and the PLDOS for perfect conductance. On the tight binding lattice, the δ-tip is modeled as an additional on-site energy ε T on a single site, corresponding to a tip area of a 2 and thus v T = ε T a 2 . This strength is varied so as to extract g (2) (r T ). The Fermi energies are chosen on the first and second plateaus for which the values of the unperturbed conductances g (0) are very close to perfect transmission with \|∆g\| < 10 −5 (points P 1 and P 5 in Fig. 1). The corresponding Fermi wavelengths are λ F = 16.8a and λ F = 9.4a, respectively. The resultant conductance responses are shown in Fig. 2, where g (2) (r T ) is compared to −ρ 2 1εF for the first plateau case in panels (a) and (b) and likewise for the second plateau case in panels (d) and (e). The correspondence is excellent as expected given the regime of the calculation. This is illustrated in panels (c) and (f), which show the differences, ρ 2 1εF + g (2) (r T ) , respectively, for the two plateaus. The differences are quite small as is expected and they show the λ F /2 oscillations, which are characteristic of the correction terms for imperfect transmission. B. Departures from local correspondence for imperfect transmission It is shown in Sec. VI that the precise local correspondence between the second-order SGM correction and the PLDOS squared degrades away from perfect transmission. We now present a quantitative numerical analysis Fig. 1). The corresponding departures from the quantized value are ∆g = 8 × 10 −6 , 5 × 10 −4 , 10 −3 , and 6 × 10 −3 for the black, blue, green and red points, respectively. Inset: the same data are presented after a spatial average over a disk of radius of λF/2, exhibiting a clear data collapse. of the departure from local correspondence for the example of the second conductance plateau of the QPC. Similar results can be obtained on other plateaus. Figure 3 presents the values of g (2) (r T ) and ρ 2 1εF at different points of the scanned region inside the white dashed rectangle shown in the inset of Fig. 1. The region of length 10w has been chosen so as to contain points close to the QPC and at larger distances. This region width is small as compared to the width of the 2DEG (30w), and additional lateral leads on the full length at the right of the QPC are used in order to avoid finite size effects. The data shown in Fig. 3 confirm that the exact pointby-point local correspondence is progressively broken as \|∆g\| increases. Close to the perfect transmission condition, for the case with ∆g = 8 × 10 −6 (P 5 in Fig. 1 with scans depicted in the lower panels of Fig. 2), the equivalence between −g (2) (r T ) and the square of the PLDOS is attained (black dots). For other points of the unperturbed conductance shown in Fig. 1, P 6 with ∆g = 5 × 10 −4 (blue), P 7 with 10 −3 (green) and P 8 with 6 × 10 −3 (red), the sampled points exhibit progressively wider distributions around the equivalence (18). The distributions are displayed in Fig. 4, where κ − 1 is plotted for different Fermi energies on the first plateau (P 1 , P 2 , P 3 , and P 4 in Fig. 1), labeled by the value of ∆g. In agreement with our analytical findings of the previous section, the average value of κ remains equal to one, but the width of the distribution drastically increases with ∆g within the bounds κ ± established in Eq. (27) (solid lines) using the value η max = 60 of the abrupt QPC. The probability density of (κ − 1)/σ is shown in the upper inset of Fig. 4, for the same positions on the first conductance plateau. The rescaling by the variance collapses the probability densities for all the values of ∆g to approximately a universal Gaussian form (dotted line). The analytical result of (28) for the standard deviation σ of the ratio κ from its mean value (κ = 1), is evaluated using the assumption η 2 = (η max /2) 2 , and is shown to agree with the numerical results (lower inset of Fig. 4). The possible connection of SGM response with local properties needs to be extended to the realistic situation where the QPC is surrounded by a disordered 2DEG. Though it is difficult to treat this case analytically because the asymptotic form of the scattering wavefunctions is attained only beyond the region of disorder far from the QPC, the incorporation of disorder in the numerically tackled model is straightforward. We assume the disorder to be due to randomly distributed donor atoms in a plane situated at a distance z = 10a, with a concentration of N d = 4 × 10 −4 a −2 . By taking a = 5 nm, N d is equal to 10 12 cm −2 , which is a realistic value for a high mobility 2DEG, and corresponds to elastic and transport mean free paths of 1 and 52 µm, respectively. The two vertical gray lines in Fig. 4 correspond to samplings of different disorder configurations, resulting in small departures from unit transmission, which are quantified by the values of ∆g. Thus, disordered QPCs, as well as clean ones, have departures from the local relation between −g (2) (r T ) and the PLDOS squared that are uniquely governed by the crucial parameter ∆g. C. Locally averaged correspondence for local tips Sections VI and VII B show that even small deviations from perfect conductance drastically alter the SGM-PLDOS correspondence. However, according to Eq. (26) and the calculations of the (Fig. 3) inset, the average of κ is equal to unity. The precise κ values though should fluctuate in a quasi-random way with a standard deviation scaling as the square root of ∆g. Such a behavior is the signature of the λ F /2-wavelength oscillations in the SGM response occurring in the clean case, which is modified in the presence of disorder. Nevertheless, as discussed in Sect. VI, the oscillations should self cancel once averaged over a domain of length scale as short as λ F /2 in both directions of the plane. In order to verify this interpretation, the numerically obtained values are averaged over a disk of radius of λ F /2. As illustrated in the inset of Fig. 3, the averaging results in a data collapse yielding the equivalence between −g (2) (r T ) and ρ 2 1εF (r T ) , even in the case of imperfect unit transmission. The recovery of the SGM-PLDOS correspondence upon averaging shows that there is a global structural correspondence with a characteristic length scale given by the Fermi wavelength. However, this correspondence is found for a local tip and only between the PLDOS squared and the second order correction. A finite temperature also has a tendency to reduce the fringes with period λ F /2 that are the main deviations from the SGM-PLDOS correspondence. Though the related mechanism is an energy average, very different from the spatial average proposed above, it might still be possible that a moderate temperature helps to improve the extraction of the PLDOS from SGM data. on the second conductance plateau of a QPC in a disordered 2DEG. Ordered vertically for each case, the quantities plotted are: full response δg(rT), first correction g (1) (rT), and second correction g (2) (rT). The changing nature and relative balance of the different order terms is clearly visible. The weaker tip strength is expected to be dominated by the first order term in the left column, but not so for the stronger tip strength in the right column. VIII. FULL SGM RESPONSE FOR LOCAL TIPS A priori, from an experimental point of view, the relationship between the various order terms and the full conductance change is not obvious. Even for weakly imperfect transmission somewhere on a plateau, depending on the tip strength, the full SGM response may depend not just on the leading second order term, but also crucially on the first and the other higher order terms. Thus, δg(r T ) can vary considerably as a function of the tip strength for less than perfect transmission cases, which would most often be the case in experiments. This is illustrated in Fig. 5, where δg(r T ), g (1) (r T ), g (2) (r T ) are plotted for two different tip strengths. The longer system treated here, in comparison with the simulations of Fig. 2, is numerically more demanding and thus the width of the 2DEG at the right of the QPC is limited to 20w. The specific example illustrated is on the second plateau of the quantized conductance where ∆g = 1.3 × 10 −4 using tip strengths of v T = ε F a 2 /4 and v T = 3ε F a 2 . The characteristic branching behavior of the fringes due to disorder 7 is observed. The changing nature of the full SGM response and its relationships with the linear and quadratic parts of the response are clearly seen. Continuing to restrict ourselves to the weakly invasive regime, if the goal were to extract a local quantity, in this case, the square of the PLDOS, two operations would greatly enhance the quality of the analysis. The first is to make a few measurements with different tip strengths. Depending on the accuracy of the measure- ments or ambient noise, this would allow one to separate linear, quadratic, or even higher order variations with respect to tip strength. The quadratic dependent response is the one related to the PLDOS squared; see Eq. (22). Second, one would average the data over a region of sidelength or radius λ F /2. Consider the weak tip strength case illustrated in Fig. 5. There, the first order term dominates the full SGM response δg(r T ). Nevertheless, extracting first the quadratic tip dependent part of the full response before averaging leads to a much more accurate extraction of the PLDOS squared. This is illustrated in Fig. 6. In the first row, δg(r T ) is shown with its locally averaged image to the right. In the next row, the quadratic tip dependence is deduced first, and then averaged. Finally in the bottom row, the negative of the squared PLDOS is plotted along with its average. The improvement in the correspondence of the quadratic por-tion of δg(r T ) relative to the full response to the average PLDOS is quite striking. The results shown in Fig. 6 demonstrate that the combined operations of extracting the quadratic tip dependence of δg(r T ) and λ F /2-averaging result in nearly perfect extraction of the PLDOS squared. Still, it is valuable to have a quantitative measure of the quality of this process to answer how well this works as a function of the imperfection of transmission on or near a plateau, and how well it works as a function of tip strength if one chooses just to use δg(r T ) without extracting the quadratic tip-dependence first. A good measure is given by the cross-correlation factor 29 C = \|( δg − δg)( ρ 2 − ρ 2 )\| σ δg σ ρ 2 .(29) The averages, symbolized by the overlines, are taken over Figure 8 shows an example for the case of the disordered system and tip strengths used in Fig. 5, where the saturation is reached rather quickly as v T /a 2 increases beyond the Fermi energy. Interestingly, the above dependence of δg(r T ) on the tip strength generates a criterion for the validity of perturbation theory. 10 Note that the criterion for the Born approximation in a one-dimensional scattering problem 40 v T ε F λ F is consistent with our numerical results since the linear extension of the local tip in our tight-binding model a is much smaller than λ F . In this regime, close to the perfect transmission, the second order contribution prevails, and the full SGM response to a local tip is highly correlated to the PLDOS squared even for tip strengths larger than the Fermi energy. IX. FULL SGM RESPONSE FOR NON-LOCAL TIPS The case of a local tip, discussed up to this point, is the simplest to analyze, but the existing experimental implementations of SGM setups involve extended tips. Considering the tip as a point charge at a distance d from the 2DEG, the tip profile in the plane of the 2DEG is of the form the spatial tip-extension for the smooth extended tip shape (30) (gray symbols) and for a hard-disk tip (black symbols), in a disorder-free structure. Squares and triangles represent the correlation factor between the SGM response with the unperturbed PLDOS at the tip center and at the classical tuning points, respectively. f (r) = 1 2πd 2 1 + r − r T d 2 −3/2 .(30) Numerical calculations of the electrostatic problem, treating screening within the Thomas-Fermi scheme, result in an approximately Lorentzian (Gaussian) profile when the tip-induced potential does not (does) deplete the 2DEG is 1, 16,21 . Notwithstanding, for tip strengths strong enough to produce depletion, it is observed that the main feature determining the SGM response is the diameter D of the depletion disk, and the details of the tip profile are of lesser importance. Therefore, in our numerical simulations, we adopt the tip profile (30) for all regimes, and express our results in terms of D = 2d[{v T /(2πd 2 ε F )} 2/3 − 1] 1/2 . Working in the previously established regime of strong tip strength (maximum tip potential V T (r T ) = v T /(2πd 2 ) = 2ε F ) the SGM response δg(r T ) for varying tip width d and thus different depletion diameters D is present in Fig. 9, where the unperturbed conductance and the disorder configuration is the same as in Fig. 5 (second conductance plateau with ∆g = 1.3 × 10 −4 ). For D = λ F /2 (panel (a)), the SGM scan resembles that of the δ-tip (panel (d) of Fig. 5), but with values of δg(r T ) that are one order of magnitude larger due to the tip extension. For larger tip extensions, D = λ F (panel b) and D = 2λ F (panel c), the SGM image gets more blurred and some resolution is lost. This blurring effect is more pronounced on the averaged conductance changes, as depicted in the right column panels of Fig. 9. The inset of Fig. 8 shows the cross-correlation C between the non-local SGM and the squared unperturbed PLDOS as a function of the depletion diameter D. Gray symbols correspond to the case of a tip shape of the form (30), the black ones to the case of a hard wall potential of diameter D. The squares represent cross-correlations of the SGM response with the PLDOS at the tip center, while triangles depict the results obtained when the PL-DOS is taken at the classical turning points situated at the edge of the depletion disk. Since the classical turning point is not determined uniquely in the presence of disorder, the data in this inset are for the disorder-free structure. We have checked that including disorder does not change significantly the results when the tip center is taken as the reference point for the PLDOS. For both tip shapes, (30) and hard wall, and independent of where the PLDOS is taken, the cross-correlation decreases with increasing depletion diameter D. If the PLDOS is taken at the classical turning point (triangles) instead of the tip center (squares), the SGM response becomes less correlated with the PLDOS. The classical argument of Ref. 9 that predicts that a large circular hard-wall tip does image the local properties of the unperturbed structures by reflecting back the clas-sical trajectories that hit the tip with normal incidence does not appear as a limiting case of our results. One reason could be that our numerics did not reach sufficiently large depletion disks with D λ F to observe such a behavior. 41 Another reason could be that the SGM response in the classical limit is not well correlated with the squared PLDOS as in the case of local tips, though another link to the PLDOS at the classical turning point of a large disk cannot be excluded from our study. X. CONCLUSIONS With regards to the quest of extracting information about local electronic properties in phase-coherent devices from SGM measurements, we have investigated the correspondence between the SGM response in the vicinity of a QPC and the unperturbed PLDOS. Only on the first conductance step could the PLDOS be shown to settle an upper bound for the magnitude of the first-order SGM correction. We have shown analytically that the unperturbed PLDOS squared is unambiguously related to the second-order conductance correction induced by a local tip, provided that the system is time-reversal symmetric and the QPC is tuned to perfect transmission. The second-order correction dominates the SGM response on a "perfect" conductance plateau if the tip strength is not too strong. If the QPC transmission is imperfect, the exact correspondence is broken, and the departures are quantified with a perturbation theory. It does not depend on fine details of the setup, but rather on the scale of the unperturbed conductance's deviation from perfection, ∆g. We have demonstrated that a correspondence between the locally averaged second-order SGM response and the PLDOS survives for imperfect transmission obtained when the highest propagating eigenchannel is not completely open. Numerical simulations within a recursive Green function approach have confirmed our analytical findings and shown that they also hold in the case of disordered systems. Moreover, we found that in the case of a local tip, and sufficiently small ∆g, the full SGM response is related to the PLDOS once the tip is strong enough such that the second-order conductance correction dominates. In the case of non-local tips, where the depletion disk created by the tip exceeds half the Fermi wavelength, the correspondence between the SGM response and the PLDOS established for weak local tips degrades with increasing depletion disk radius. Most SGM experiments are performed in high mobility 2DEGs in which the Fermi wavelength is smaller than the depletion disk under the tip. In that case the relationship between the SGM response and the PLDOS squared degrades and beyond a large enough radius cannot be used directly to and unambiguously extract local electronic properties. For experiments in the weakly invasive regime, the resolution of the SGM response is also limited by the width of the tip potential. 42 One way to approach the regime where the direct link is valid would be to use systems with lower Fermi energy and thus larger Fermi wavelength. In a very recent SGM experiment 43 performed using ultracold atom gases, a tightly focused laser beam played the role of the tip and could be scanned in the neighborhood of a QPC attached to two atom reservoirs. In this case a resolution better than 10 nm with a tip size well below λ F was obtained. In this regime, we expect that the relationship established between the SGM response and the LDOS is applicable. c √ k a exp [−ik − a x] φ a (y) , x > 0 , (A1b) where φ a (y) is the wave-function of the a th transverse channel along the lead and k − a the longitudinal wave number k a with an infinitesimal imaginary part necessary for incoming modes. We note c = [m * /(2π 2 )] 1/2 , with m * the effective electron mass. In writing x < 0 and x > 0 we mean the asymptotic condition in the left and right leads, respectively (see Fig. 1). In the basis of the 2N incoming modes the scattering matrix is defined by S = r t t r . (A2) The incoming modes give rise to outgoing scattering states, which in the asymptotic regions can be expressed as, Identifying (A2) and (A4), the transmission and reflexion submatrices can be expressed as t = u T 2 T u 1 , t = u T 1 T u 2 , r = −u T 1 Ru 1 , and r = u T 2 Ru 2 . Thus, t † t = u † 1 T 2 u 1 and t † t = u † 2 T 2 u 2 . Considering the vector of coefficients C 1(2)m = ([u * 1(2) ] m1 , [u * 1(2) ] m2 , ...) T of the transmission eigenmode (−) 1,ε,m (r), one can write t † tC 1m = u † 1 T 2 u 1 C 1m = T 2 m C 1m .(A6) The second equality stems from the definition of C 1m and implies that C 1m is an eigenvector of t † t with the eigenvalue T 2 m . In the same way, one finds that C 2m is an eigenvector of t † t with the same eigenvalue. The scattering eigenstates in the region x > 0 for an incoming transmission eigenmode (−) 1,ε,m (r) are obtained as tC 1m = u T 2 T u 1 C 1m . Using again the definition of C 1(2)m and the unitarity of u 1 we find tC 1m = T m C * 2m ,(A7) and similarly rC 1(2)m = ∓R m C * 1(2)m . Thus, the basis of scattering eigenfunctions is asymptotically given by The PLDOS (3), as well as the conductance corrections (7) and (8) are conveniently discussed when expressed in the basis of scattering eigenfunctions. Appendix B: Hilbert transform of LDOS In this Appendix the term of Eq. (8) containing the principal part is related with the LDOS. For a δ-tip, we can write Tr RT U 2l (ε F ,ε)Ul 1 (ε, ε F ) = ρlε(r T ) 2π Tr RT U 21 (ε F , ε F ) , and therefore −4π v 2 T 2 l=1 P ∞ ε (t) 1 dε ε − ε F Im Tr RT U 2l (ε F ,ε)Ul 1 (ε, ε F ) = g (1) (r T ) 2 1 π P ∞ ε (t) 1 dε ε F −ε ρε(r T ) .(B2) Since the LDOS vanishes forε < ε (t) 1 , the lower limit of the integral can be taken as −∞. Given that the LDOS is proportional to the imaginary part of the diagonal Green function G ε (r, r), the curly bracket at the right-hand-side represents a Hilbert transform (with respect to the energy variable) leading to the real part of G ε (r, r), as indicated in the discussion following Eq. (21). The term (B2) contributing to g (2) and fulfilling a Kramers-Kronig relation with the LDOS, is dominated by the contribution of the latter close to the Fermi energy. The emergence of the Hilbert transform of the LDOS has been signaled for the first-order SGM correction of a one-dimensional system. 21 In our case it appears in the contribution (B2) to the second-order correction (8), and it is not restricted to a one-dimensional setup. Such a contribution, also proportional to the first-order correction g (1) (r T ), is necessarily very small when the QPC operates close to the condition of conductance quantization. are dominated by the contribution from the subspace of the M open eigenmodes. ε,m (r) = χ 1,ε,m (r)/T m for x > 0 in the case of open modes (T m = 0). Thus, Eq. (10) simplifies to t 2 a 4 FIG. 2 . 42Left column: −g (2) (with the energy and length units introduced through the hopping integral t and the spatial tip extension a 2 ) vs. the tip position for the first (a) and second (d) plateaus (points P1 and P5 inFig. 1, respectively) Central column: the square of the PLDOS for the same points on the first (b) and second (e) plateau. Right column: difference between the two first columns. The QPC is situated at the upper left corner of the figures. FIG. 3 . 3Second order SGM correction vs. ρ 2 1ε F at random sampled tip positions in the scanned region for different values of the unperturbed conductance on the second plateau (points P5, P6, P7, and P8 in FIG. 4 . 4κ − 1 is plotted vs. the departure from perfect transmission ∆g, when a wide region in the right side of the QPC is sampled. The results for the clean structure of Figs. 2 and 3, ∆g = 6×10 −6 (black; P1 in Fig. 1), ∆g = 8×10 −5 (blue; P2), 7×10 −4 (green; P3) are presented, but those for ∆g = 7×10 −3 (red; P4) are out of the scale of the main figure. The data corresponding to two different disorder configurations are represented by the gray distributions. The black solid lines show the analytical bounds κ± of Eq. (27) taking ηmax = 60. Upper inset: the probability density of κ − 1. The color code is the same as in the main figure. For comparison, the dotted line shows a Gaussian probability density. Lower inset: the corresponding standard deviation vs. ∆g. The black solid line corresponds to the analytical expression (28) of σ with η 2 = (ηmax/2) 2 . FIG. 5 . 5SGM response for two tip strengths, vT = εFa 2 /4 (left column) and vT = 3εFa 2 (right column) with ∆g = 1.3 × 10 −4 FIG. 6 . 6Extracting an accurate PLDOS squared from the full SGM response in the weakly invasive regime for the disorder configuration ofFig. 5, for the weaker tip strength vT = εFa 2 /4: (a) δg(rT) on the right side of the QPC ; (b) the quadratic tip dependence portion of δg(rT); (c) the negative of the squared PLDOS, −ρ 2 1ε F . In (d), (e), and (f), respectively, the data of panels (a), (b), and (c) have been averaged over a disk of diameter λF/2. FIG. 7 . 7Cross-correlation factor(29) as a function of the strength vT of a local tip (horizontal axis) and the deviation from perfect transmission (vertical axis), on the second conductance plateau of the QPC in a disordered 2DEG of Figs. 5 and 6. the scanned area in the right of the QPC (in contradistinction to the local O(λ F /2) averages, ... , defined in Sec. VII). The standard deviations of the two quantities are the usual normalization factors of a properly normalized correlation function. Applied to δg(r T ) for a range of tip strengths and ∆g's gives the results shown inFig. 7. It shows two correlated trends. The correlation coefficient decreases with decreasing tip strength and with increasing ∆g. The value of v T for which near perfect correlation is achieved depends on the departure ∆g from perfect transmission. FIG. 8 . 8Cross-correlation factor C (29) vs. the strength of a local tip in the disordered system ofFig. 5. Inset: C vs. FIG. 9 . 9SGM response calculated using the tip shape (30) for fixed tip potential height vT/(2πd 2 ) = 2εF and varying depletion disk size D = λF/2 (a), D = λF (b), and D = 2λF (c). Panels (d), (e), and (f) show the averages of the SGM responses over a disk of radius λF/2 for the same tip sizes. of the matrix elements of the reflection r (r ) and transmission t (t ) submatrices for electrons impinging from the left (right) lead.The scattering matrix is conveniently expressed in the polar decomposition44 , which in the case of time-reversal symmetry takes the form M. A. Eriksson, R. G. Beck, M. Topinka, J. A. Katine, R. M. Westervelt, K. L. Campman, and A. C. Gossard, "Cryogenic scanning probe characterization of semiconduc-tor nanostructures," Appl. Phys. Lett. 69, 671-673 (1996). 2 M. A. Topinka, B. J. LeRoy, S. E. J. Shaw, E. J. Heller, R. M. Westervelt, K. D. Maranowski, and A. C. Gossard, ACKNOWLEDGMENTSThe authors are grateful to B. Braem, B. Brun, K. Ensslin, C. Gold, C. Gorini, T. Ihn, C. Pöltl, R. Steinacher, and G. Weick for useful discussions. Financial support from the French National Research Agency ANR through Projects No. ANR-14-CE36-0007-01 (SGM-Bal) and No. ANR-11-LABX-0058_NIE (Labex NIE within ANR-10-IDEX-0002-02) is gratefully acknowledged.Appendix A: Scattering wave-functionsIn this Appendix we recall the main concepts of scattering theory for quantum transport in view of the application to the SGM setups implemented through the text. The incoming lead modes ϕ (−) 1(2),ε,a (r) are given by2,ε,a (r) = Imaging coherent electron flow from a quantum point contact. 10.1126/science.289.5488.2323Science. 289Imaging coherent electron flow from a quantum point con- tact," Science 289, 2323-2326 (2000). Imaging coherent electron wave flow in a two-dimensional electron gas. M Topinka, B Leroy, R Westervelt, K Maranowski, A Gossard, 10.1016/S1386-9477(01)00375-7Phys. 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[ "An optimization algorithm inspired by the States of Matter that improves the balance between exploration and exploitation", "An optimization algorithm inspired by the States of Matter that improves the balance between exploration and exploitation" ]	[ "EchavarríaE Cuevas ", "A ", "Ramírez-Ortegón ", "M A ", "Erik Cuevas [email protected] \nDepartamento de Electrónica\nUniversidad de Guadalajara\nCUCEI Av. Revolución 1500\nGuadalajaraJalMéxico\n", "Alonso Echavarría \nInstitut für Nachrichtentechnik Technische\nUniversität Braunschweig\nSchleinitzstrae 2238106BraunschweigGermany\n", "Marte A Ramírez-Ortegón \nDepartamento de Electrónica\nUniversidad de Guadalajara\nCUCEI Av. Revolución 1500\nGuadalajaraJalMéxico\n" ]	[ "Departamento de Electrónica\nUniversidad de Guadalajara\nCUCEI Av. Revolución 1500\nGuadalajaraJalMéxico", "Institut für Nachrichtentechnik Technische\nUniversität Braunschweig\nSchleinitzstrae 2238106BraunschweigGermany", "Departamento de Electrónica\nUniversidad de Guadalajara\nCUCEI Av. Revolución 1500\nGuadalajaraJalMéxico" ]	[ "Applied Intelligence" ]	The ability of an Evolutionary Algorithm (EA) to find a global optimal solution depends on its capacity to find a good rate between exploitation of found-so-far elements and exploration of the search space. Inspired by natural phenomena, researchers have developed many successful evolutionary algorithms which, at original versions, define operators that mimic the way nature solves complex problems, with no actual consideration of the explorationexploitation balance. In this paper, a novel nature-inspired algorithm called the States of Matter Search (SMS) is introduced. The SMS algorithm is based on the simulation of the states of matter phenomenon. In SMS, individuals emulate molecules which interact to each other by using evolutionary operations which are based on the physical principles of the thermal-energy motion mechanism. The algorithm is devised by considering each state of matter at one different exploration-exploitation ratio. The evolutionary process is divided into three phases which emulate the three states of matter: gas, liquid and solid. In each state, molecules (individuals) exhibit different movement capacities. Beginning from the gas state (pure exploration), the algorithm modifies the intensities of exploration and exploitation until the solid state (pure exploitation) is reached. As a result, the approach can substantially improve the balance between exploration-exploitation, yet preserving the good search capabilities of an evolutionary approach. To illustrate the proficiency and robustness of the proposed algorithm, it is compared to other well-known evolutionary methods including novel variants that incorporate diversity preservation schemes. The comparison examines several standard benchmark functions which are commonly considered within the EA field. Experimental results show that the proposed method achieves a good performance in comparison to its counterparts as a consequence of its better exploration-exploitation balance.	10.1007/s10489-013-0458-0	[ "https://arxiv.org/pdf/1405.5066v1.pdf" ]	2,893,762	1405.5066	498c987e5f44b167948d4c57e3eade1683c517d8	An optimization algorithm inspired by the States of Matter that improves the balance between exploration and exploitation 2014 EchavarríaE Cuevas A Ramírez-Ortegón M A Erik Cuevas [email protected] Departamento de Electrónica Universidad de Guadalajara CUCEI Av. Revolución 1500 GuadalajaraJalMéxico Alonso Echavarría Institut für Nachrichtentechnik Technische Universität Braunschweig Schleinitzstrae 2238106BraunschweigGermany Marte A Ramírez-Ortegón Departamento de Electrónica Universidad de Guadalajara CUCEI Av. Revolución 1500 GuadalajaraJalMéxico An optimization algorithm inspired by the States of Matter that improves the balance between exploration and exploitation Applied Intelligence 4022014Please cite this article as: This is a preprint copy that has been accepted for publication in Applied Intelligence 1Evolutionary algorithmsGlobal OptimizationNature-inspired algorithms The ability of an Evolutionary Algorithm (EA) to find a global optimal solution depends on its capacity to find a good rate between exploitation of found-so-far elements and exploration of the search space. Inspired by natural phenomena, researchers have developed many successful evolutionary algorithms which, at original versions, define operators that mimic the way nature solves complex problems, with no actual consideration of the explorationexploitation balance. In this paper, a novel nature-inspired algorithm called the States of Matter Search (SMS) is introduced. The SMS algorithm is based on the simulation of the states of matter phenomenon. In SMS, individuals emulate molecules which interact to each other by using evolutionary operations which are based on the physical principles of the thermal-energy motion mechanism. The algorithm is devised by considering each state of matter at one different exploration-exploitation ratio. The evolutionary process is divided into three phases which emulate the three states of matter: gas, liquid and solid. In each state, molecules (individuals) exhibit different movement capacities. Beginning from the gas state (pure exploration), the algorithm modifies the intensities of exploration and exploitation until the solid state (pure exploitation) is reached. As a result, the approach can substantially improve the balance between exploration-exploitation, yet preserving the good search capabilities of an evolutionary approach. To illustrate the proficiency and robustness of the proposed algorithm, it is compared to other well-known evolutionary methods including novel variants that incorporate diversity preservation schemes. The comparison examines several standard benchmark functions which are commonly considered within the EA field. Experimental results show that the proposed method achieves a good performance in comparison to its counterparts as a consequence of its better exploration-exploitation balance. Introduction Global optimization [1] has delivered applications for many areas of science, engineering, economics and others, where mathematical modelling is used [2]. In general, the goal is to find a global optimum for an objective function which is defined over a given search space. Global optimization algorithms are usually broadly divided into deterministic and stochastic [3]. Since deterministic methods only provide a theoretical guarantee of locating a local minimum of the objective function, they often face great difficulties in solving global optimization problems [4]. On the other hand, evolutionary algorithms are usually faster in locating a global optimum [5]. Moreover, stochastic methods adapt easily to black-box formulations and extremely ill-behaved functions, whereas deterministic methods usually rest on at least some theoretical assumptions about the problem formulation and its analytical properties (such as Lipschitz continuity) [6]. Evolutionary algorithms, which are considered as members of the stochastic group, have been developed by a combination of rules and randomness that mimics several natural phenomena. Such phenomena include evolutionary processes such as the Evolutionary Algorithm (EA) proposed by Fogel et al. [7], De Jong [8], and Koza [9], the Genetic Algorithm (GA) proposed by Holland [10] and Goldberg [11], the Artificial Immune System proposed by De Castro et al. [12] and the Differential Evolution Algorithm Although PSO, DE and GSA are considered the most popular algorithms for many optimization applications, they fail in finding a balance between exploration and exploitation [20]; in multimodal functions, they do not explore the whole region effectively and often suffers premature convergence or loss of diversity. In order to deal with this problem, several proposals have been suggested in the literature . In most of the approaches, exploration and exploitation is modified by the proper settings of control parameters that have an influence on the algorithm´s search capabilities [47]. One common strategy is that EAs should start with exploration and then gradually change into exploitation [48]. Such a policy can be easily described with deterministic approaches where the operator that controls the individual diversity decreases along with the evolution. This is generally correct, but such a policy tends to face difficulties when solving certain problems with multimodal functions that hold many optima, since a premature takeover of exploitation over exploration occurs. Some approaches that use this strategy can be found in [21][22][23][24][25][26][27][28][29]. Other works [30][31][32][33][34] use the population size as reference to change the balance between exploration and exploitation. A larger population size implies a wider exploration while a smaller population demands a shorter search. Although this technique delivers an easier way to keep diversity, it often represents an unsatisfactory solution. An improper handling of large populations might converge to only one point, despite introducing more function evaluations. Recently, new operators have been added to several traditional evolutionary algorithms in order to improve their original explorationexploitation capability. Such operators diversify particles whenever they concentrate on a local optimum. Some methods that employ this technique are discussed in [35][36][37][38][39][40][41][42][43][44][45][46]. Either of these approaches is necessary but not sufficient to tackle the problem of the explorationexploitation balance. Modifying the control parameters during the evolution process without the incorporation of new operators to improve the population diversity, makes the algorithm defenseless against the premature convergence and may result in poor exploratory characteristics of the algorithm [48]. On the other hand, incorporating new operators without modifying the control parameters leads to increase the computational cost, weakening the exploitation process of candidate regions [39]. Therefore, it does seem reasonable to incorporate both of these approaches into a single algorithm. In this paper, a novel nature-inspired algorithm, known as the States of Matter Search (SMS) is proposed for solving global optimization problems. The SMS algorithm is based on the simulation of the states of matter phenomenon. In SMS, individuals emulate molecules which interact to each other by using evolutionary operations based on the physical principles of the thermal-energy motion mechanism. Such operations allow the increase of the population diversity and avoid the concentration of particles within a local minimum. The proposed approach combines the use of the defined operators with a control strategy that modifies the parameter setting of each operation during the evolution process. In contrast to other approaches that enhance traditional EA algorithms by incorporating some procedures for balancing the exploration-exploitation rate, the proposed algorithm naturally delivers such property as a result of mimicking the states of matter phenomenon. The algorithm is devised by considering each state of matter at one different exploration-exploitation ratio. Thus, the evolutionary process is divided into three stages which emulate the three states of matter: gas, liquid and solid. At each state, molecules (individuals) exhibit different behaviors. Beginning from the gas state (pure exploration), the algorithm modifies the intensities of exploration and exploitation until the solid state (pure exploitation) is reached. As a result, the approach can substantially improve the balance between exploration-exploitation, yet preserving the good search capabilities of an evolutionary approach. To illustrate the proficiency and robustness of the proposed algorithm, it has been compared to other well-known evolutionary methods including recent variants that incorporate diversity preservation schemes. The comparison examines several standard benchmark functions which are usually employed within the EA field. Experimental results show that the proposed method achieves good performance over its counterparts as a consequence of its better exploration-exploitation capability. This paper is organized as follows. Section 2 introduces basic characteristics of the three states of matter. In Section 3, the novel SMS algorithm and its characteristics are both described. Section 4 presents experimental results and a comparative study. Finally, in Section 5, some conclusions are discussed. States of matter The matter can take different phases which are commonly known as states. Traditionally, three states of matter are known: solid, liquid, and gas. The differences among such states are based on forces which are exerted among particles composing a material [49]. In the gas phase, molecules present enough kinetic energy so that the effect of intermolecular forces is small (or zero for an ideal gas), while the typical distance between neighboring molecules is greater than the molecular size. A gas has no definite shape or volume, but occupies the entire container in which it is confined. Fig. 1a shows the movements exerted by particles in a gas state. The movement experimented by the molecules represent the maximum permissible displacement 1 ρ among particles [50]. In a liquid state, intermolecular forces are more restrictive than those in the gas state. The molecules have enough energy to move relatively to each other still keeping a mobile structure. Therefore, the shape of a liquid is not definite but is determined by its container. Fig. 1b presents a particle movement 2 ρ within a liquid state. Such movement is smaller than those considered by the gas state but larger than the solid state [51]. In the solid state, particles (or molecules) are packed together closely with forces among particles being strong enough so that the particles cannot move freely but only vibrate. As a result, a solid has a stable, definite shape and a definite volume. Solids can only change their shape by force, as when they are broken or cut. Fig. 1c shows a molecule configuration in a solid state. Under such conditions, particles are able to vibrate (being perturbed) considering a minimal 3 ρ distance [50]. In this paper, a novel nature-inspired algorithm known as the States of Matter Search (SMS) is proposed for solving global optimization problems. The SMS algorithm is based on the simulation of the states of matter phenomenon that considers individuals as molecules which interact to each other by using evolutionary operations based on the physical principles of the thermal-energy motion mechanism. The algorithm is devised by considering each state of matter at one different exploration-exploitation ratio. Thus, the evolutionary process is divided into three stages which emulate the three states of matter: gas, liquid and solid. In each state, individuals exhibit different behaviors. 3 ρ 2 ρ 1 ρ The direction vector operator assigns a direction to each molecule in order to lead the particle movement as the evolution process takes place. On the other side, the collision operator mimics those collisions that are experimented by molecules as they interact to each other. A collision is considered when the distance between two molecules is shorter than a determined proximity distance. The collision operator is thus implemented by interchanging directions of the involved molecules. In order to simulate the random behavior of molecules, the proposed algorithm generates random positions following a probabilistic criterion that considers random locations within a feasible search space. The next section presents all operators that are used in the algorithm. Although such operators are the same for all the states of matter, they are employed over a different configuration set depending on the particular state under consideration. Direction vector The direction vector operator mimics the way in which molecules change their positions as the evolution process develops. For each n-dimensional molecule i p from the population P, it is assigned an ndimensional direction vector i d which stores the vector that controls the particle movement. Initially, all the direction vectors ( As the system evolves, molecules experiment several attraction forces. In order to simulate such forces, the proposed algorithm implements the attraction phenomenon by moving each molecule towards the best so-far particle. Therefore, the new direction vector for each molecule is iteratively computed considering the following model: 1 1 0.5 , k k i i i k gen +   = ⋅ − ⋅ +     d d a(1) where i a represents the attraction unitary vector calculated as ( ) / best best i i i = − − a p p p p , being best p the best individual seen so-far, while i p is the molecule i of population P. k represents the iteration number whereas gen involves the total iteration number that constitutes the complete evolution process. Under this operation, each particle is moved towards a new direction which combines the past direction, which was initially computed, with the attraction vector over the best individual seen so-far. It is important to point out that the relative importance of the past direction decreases as the evolving process advances. This particular type of interaction avoids the quick concentration of information among particles and encourages each particle to search around a local candidate region in its neighborhood, rather than interacting to a particle lying at distant region of the domain. The use of this scheme has two advantages: first, it prevents the particles from moving toward the global best position in early stages of algorithm and thus makes the algorithm less susceptible to premature convergence; second, it encourages particles to explore their own neighborhood thoroughly, just before they converge towards a global best position. Therefore, it provides the algorithm with local search ability enhancing the exploitative behavior. In order to calculate the new molecule position, it is necessary to compute the velocity i v of each molecule by using: (2) being init v the initial velocity magnitude which is calculated as follows: i i init v = ⋅ v d1 ( ) n high low j j j init b b v β n = − = ⋅ ∑(3) where low j b and high j b are the low j parameter bound and the upper j parameter bound respectively, whereas [0,1] β ∈ . Then, the new position for each molecule is updated by: 1 , , , rand(0,1) ( ) k k high low i j i j i j j j p p v b b + = + ⋅ ⋅ ⋅ − ρ(4) where 0.5 1 ρ ≤ ≤ . Collision The collision operator mimics the collisions experimented by molecules while they interact to each other. Collisions are calculated if the distance between two molecules is shorter than a determined proximity value. Therefore, if i q r − < p p , a collision between molecules i and q is assumed; otherwise, there is no collision, considering { } , 1, , p i q N ∈ K such that i q ≠ . If a collision occurs, the direction vector for each particle is modified by interchanging their respective direction vectors as follows: i q = d d and q i = d d(5) The collision radius is calculated by: 1 ( ) n high low j j j b b r α n = − = ⋅ ∑ (6) where [0,1] α ∈ . Under this operator, a spatial region enclosed within the radius r is assigned to each particle. In case the particle regions collide to each other, the collision operator acts upon particles by forcing them out of the region. The radio r and the collision operator provide the ability to control diversity throughout the search process. In other words, the rate of increase or decrease of diversity is predetermined for each stage. Unlike other diversity-guided algorithms, it is not necessary to inject diversity into the population when particles gather around a local optimum because the diversity will be preserved during the overall search process. The collision incorporation therefore enhances the exploratory behavior in the proposed approach. Random positions In order to simulate the random behavior of molecules, the proposed algorithm generates random positions following a probabilistic criterion within a feasible search space. For this operation, a uniform random number m r is generated within the range [0,1]. If m r is smaller than a threshold H, a random molecule´s position is generated; otherwise, the element remains with no change. Therefore such operation can be modeled as follows: 1 , 1 , rand(0,1) ( ) with probability with probability (1-) low high low j j j k i j k i j b b b H p p H + +  + ⋅ −  =    (7) where { } 1, , p i N ∈ K and { } 1, , j n ∈ K . Best Element Updating Despite this updating operator does not belong to State of Matter metaphor, it is used to simply store the best so-far solution. In order to update the best molecule best p seen so-far, the best found individual from the current k population , best k p is compared to the best individual , 1 best k − p of the last generation. If , best k p is better than , 1 best k − p according to its fitness value, best p is updated with , best k p , otherwise best p remains with no change. Therefore, best p stores the best historical individual found so-far. SMS algorithm The overall SMS algorithm is composed of three stages corresponding to the three States of Matter: the gas, the liquid and the solid state. Each stage has its own behavior. In the first stage (gas state), exploration is intensified whereas in the second one (liquid state) a mild transition between exploration and exploitation is executed. Finally, in the third phase (solid state), solutions are refined by emphasizing the exploitation process. General procedure At each stage, the same operations are implemented. However, depending on which state is referred, they are employed considering a different parameter configuration. The general procedure in each state is shown as pseudo-code in Algorithm 1. Such procedure is composed by five steps and maps the current population k P to a new population 1 k + P . The algorithm receives as input the current population k P and the configuration parameters ρ , β , α , and H, whereas it yields the new population 1 k + P . Step 1: Find the best element of the population P { } { } 1 2 ( ) max ( ), ( ), , ( ) p best best N f f f f ∈ = p P p p p p K Step 2: Calculate init v and r This is a preprint copy that has been accepted for publication in Applied Intelligence 7 Step 3: Compute the new molecules by using the Direction vector operator 3. 1.1 for (i=1; i< p N +1; i++) ( ) / best best i i i = − − a p p p p for (j=1; j<n+1; j++) 1 , , , 1 0.5 k k i j i j i j k d d a gen +   = ⋅ − ⋅ +     1 , , k i j i j init v d v + = ⋅ 1 , , , rand(0,1) ( ) k k high low i j i j i j j j p p v b b + = + ⋅ ⋅ ⋅ − ρ end for end for Step 4: Solve collisions by using the Collision operator 3. 1.2 for (i=1; i< p N +1; i++) for (j=1; j< p N +1; j++) if (( i j r − < p p ) and ( i j ≠ )) i = t d i j = d d j = d t end if end for end for Step 5: Generate new random positions by using the Random positions operator 3.1.3 for (i=1; i< p N +1; i++) if ( m r < H) then; where rand(0,1) m r ∈ for (j=1; j<n+1; j++) 1 , rand(0,1) ( ) k low high low i j j j j p b b b + = + ⋅ − end for end if end for Algorithm 1. General procedure executed by all the states of matter. The complete algorithm The complete algorithm is divided into four different parts. The first corresponds to the initialization stage, whereas the last three represent the States of Matter. All the optimization process, which consists of a gen number of iterations, is organized into three different asymmetric phases, employing 50% of all iterations for the gas state (exploration), 40% for the liquid state (exploration-exploitation) and 10% for the solid state (exploitation). The overall process is graphically described by Figure 2. At each state, the same general procedure (see Algorithm 1) is iteratively used considering the particular configuration predefined for each State of Matter. Figure 3 shows the data flow for the complete SMS algorithm. This is a preprint copy that has been accepted for publication in Applied Intelligence rand(0,1) ( ) low high low i j j j j p b b b = + ⋅ − 1, 2, , ; 1, 2, , , p j n i N = = K K(8) where j and i, are the parameter and molecule index respectively whereas zero indicates the initial population. Hence, j i p is the j-th parameter of the i-th molecule. Gas state In the gas state, molecules experiment severe displacements and collisions. Such state is characterized by random movements produced by non-modeled molecule phenomena [52]. Therefore, the ρ value from the direction vector operator is set to a value near to one so that the molecules can travel longer distances. Similarly, the H value representing the random positions operator is also configured to a value around one, in order to allow the random generation for other molecule positions. The gas state is the first phase and lasts for the 50% of all iterations which compose the complete optimization process. The computational procedure for the gas state can be summarized as follows: Step and H=0.9 being consistent with the gas state. Step 2: Apply the general procedure which is illustrated in Algorithm 1. Step 3: If the 50% of the total iteration number is completed ( This is a preprint copy that has been accepted for publication in Applied Intelligence 9 continues to the liquid state procedure; otherwise go back to step 2. Liquid state Although molecules currently at the liquid state exhibit restricted motion in comparison to the gas state, they still show a higher flexibility with respect to the solid state. Furthermore, the generation of random positions which are produced by non-modeled molecule phenomena is scarce [53]. For this reason, the ρ value from the direction vector operator is bounded to a value between 0.3 and 0.6. Similarly, the random position operator H is configured to a value near to cero in order to allow the random generation of fewer molecule positions. In the liquid state, collisions are also less common than in gas state, so the collision radius, that is controlled by α , is set to a smaller value in comparison to the gas state. The liquid state is the second phase and lasts the 40% of all iterations which compose the complete optimization process. The computational procedure for the liquid state can be summarized as follows: Step Step 5: Apply the general procedure that is defined in Algorithm 1. Step 6: If the 90% (50% from the gas state and 40% from the liquid state) of the total iteration number is completed (0.5 0.9 ) gen k gen ⋅ < ≤ ⋅ , then the process continues to the solid state procedure; otherwise go back to step 5. Solid state In the solid state, forces among particles are stronger so that particles cannot move freely but only vibrate. As a result, effects such as collision and generation of random positions are not considered [52]. Therefore, the ρ value of the direction vector operator is set to a value near to zero indicating that the molecules can only vibrate around their original positions. The solid state is the third phase and lasts for the 10% of all iterations which compose the complete optimization process. The computational procedure for the solid state can be summarized as follows: Step 7: Set the parameters Apply the general procedure that is defined in Algorithm 1. Step 9: If the 100% of the total iteration number is completed (0.9 ) gen k gen ⋅ < ≤ , the process is finished; otherwise go back to step 8. It is important to clarify that the use of this particular configuration ( 0 α = and H=0) disables the collision and generation of random positions operators which have been illustrated in the general procedure. Experimental results A comprehensive set of 24 functions, collected from Refs. [54][55][56][57][58][59][60][61], has been used to test the performance of the proposed approach. Tables A1-A4 in the Appendix A present the benchmark functions used in our experimental study. Such functions are classified into four different categories: Unimodal test functions (Table A1), multimodal test functions (Table A2), multimodal test functions with fixed dimensions (Table A3) and functions proposed for the GECCO contest ( default values which have been obtained from the Matlab © implementation for GECCO competitions, as it is provided in [59]. A detailed description of optimum locations is given in Appendix A. Performance comparison to other meta-heuristic algorithms We have applied the SMS algorithm to 24 functions whose results have been compared to those produced by the Gravitational Search Algorithm (GSA) [16], the Particle Swarm Optimization (PSO) method [17] and the Differential Evolution (DE) algorithm [13]. These are considered as the most popular algorithms in many optimization applications. In order to enhance the performance analysis, the PSO algorithm with a territorial diversity-preserving scheme (TPSO) [39] has also been added into the comparisons. TPSO is considered a recent PSO variant that incorporates a diversity preservation scheme in order to improve the balance between exploration and exploitation. In all comparisons, the population has been set to 50. The maximum iteration number for functions in Tables A1, A2 and A4 has been set to 1000 and for functions in Table A3 has been set to 500. Such stop criterion has been selected to maintain compatibility to similar works reported in the literature [4,16]. The parameter setting for each algorithm in the comparison is described as follows: 1. GSA [16]: The parameters are set to 100 o G = and 20 α = ; the total number of iterations is set to 1000 for functions 1 f to 11 f and 500 for functions 12 f to 14 f . The total number of individuals is set to 50. Such values are the best parameter set for this algorithm according to [16]. 2. PSO [17]: The parameters are set to 1 2 c = and 2 2 c = ; besides, the weight factor decreases linearly from 0.9 to 0.2. 3. DE [13]: The DE/Rand/1 scheme is employed. The crossover probability is set to 0.9 CR = and the weight factor is set to 0.8 F = . 4. TPSO [39]: The parameter α has been set to 0.5. Such value is found to be the best configuration according to [39]. The algorithm has been tuned according to the set of values which have been originally proposed by its own reference. The experimental comparison between metaheuristic algorithms with respect to SMS has been developed according to the function-type classification as follows: 1. Unimodal test functions (Table A1). (Table A2). 3. Multimodal test functions with fixed dimension (Table A3). 4. Test functions from the GECCO contest (Table A4). Multimodal test functions Unimodal test functions This experiment is performed over the functions presented in Table A1. The test compares the SMS to other algorithms such as GSA, PSO, DE and TPSO. The results for 30 runs are reported in Table 1 considering the following performance indexes: the Average Best-so-far (AB) solution, the Median Bestso-far (MB) and the Standard Deviation (SD) of best-so-far solution. The best outcome for each function is boldfaced. According to this table, SMS delivers better results than GSA, PSO, DE and TPSO for all functions. In particular, the test remarks the largest difference in performance which is directly related to a better trade-off between exploration and exploitation. Just as it is illustrated by Figure 4, SMS, DE and GSA have similar convergence rates at finding the optimal minimal, yet faster than PSO and TPSO. A non-parametric statistical significance proof known as the Wilcoxon's rank sum test for independent samples [62,63] has been conducted over the "average best-so-far" (AB) data of Table 1, with an 5% significance level. Table 2 reports the p-values produced by Wilcoxon's test for the pair-wise comparison of the "average best so-far" of four groups. Such groups are formed by SMS vs. GSA, SMS vs. PSO, SMS vs. DE and SMS vs. TPSO. As a null hypothesis, it is assumed that there is no significant difference between mean values of the two algorithms. The alternative hypothesis considers a significant difference between the "average best-so-far" values of both approaches. All p-values reported in Table 2 are less than 0.05 (5% significance level) which is a strong evidence against the null hypothesis. Therefore, such evidence indicates that SMS results are statistically significant and that it has not occurred by coincidence (i.e. due to common noise contained in the process). Table 3. SMS Multimodal test functions Multimodal functions represent a good optimization challenge as they possess many local minima (Table A2). In the case of multimodal functions, final results are very important since they reflect the algorithm's ability to escape from poor local optima and to locate a near-global optimum. Experiments using 5 f to 11 f are quite relevant as the number of local minima for such functions increases exponentially as their dimensions increase. The dimension of such functions is set to 30. The results are averaged over 30 runs, reporting the performance index for each function in Table 3 as follows: the Average Best-so-far (AB) solution, the Median Best-so-far (MB) and the Standard Deviation (SD) best-so-far (the best result for each function is highlighted). Likewise, p-values of the Wilcoxon signed-rank test of 30 independent runs are listed in Table 4. Table 3. Minimization result of benchmark functions in Table A2 with n=30. Maximum number of iterations=1000. SMS In the case of functions 8 f , 9 f , 10 f and 11 f , SMS yields much better solutions than other methods. However, for functions 5 f , 6 f and 7 f , SMS produces similar results to GSA and TPSO. The Wilcoxon rank test results, which are presented in Table 4, demonstrate that SMS performed better than GSA, PSO, DE and TPSO considering four functions 8 11 f f − , whereas, from a statistical viewpoint, there is no difference between results from SMS, GSA and TPSO for 5 f , 6 f and 7 f . The progress of the "average best-so-far" solution over 30 runs for functions 5 f and 11 f is shown by Fig. 5. TPSO over the "average best-so-far" (AB) values from Table 3 100 200 300 400 500 600 700 800 900 1000 SMS vs Multimodal test functions with fixed dimensions In the following experiments, the SMS algorithm is compared to GSA, PSO, DE and TPSO over a set of multidimensional functions with fixed dimensions which are widely used in the meta-heuristic literature. The functions used for the experiments are 12 f , 13 f and 14 f which are presented in Table A3. The results in Table 5 show that SMS, GSA, PSO, DE and TPSO have similar values in their performance. The evidence shows how meta-heuristic algorithms maintain a similar average performance when they face low-dimensional functions [54]. Fig. 6 presents the convergence rate for the GSA, PSO, DE, SMS and TPSO algorithms considering functions 12 f to 13 f . Table 5. Minimization results of benchmark functions in Table A3 with n=30. Maximum number of iterations=500. Fig. 6. Convergence rate comparison of PSO, GSA, DE, SMS and TPSO for minimization of (a) 12 f and (b) 13 f . SMS Test functions from the GECCO contest The experimental set in Table A4 includes several representative functions that are used in the GECCO contest. Using such functions, the SMS algorithm is compared to GSA, PSO, DE and TPSO. The results have been averaged over 30 runs, reporting the performance indexes for each algorithm in Table 6. Likewise, p-values of the Wilcoxon signed-rank test of 30 independent executions are listed in Table 7. According to results of Table 6, it is evident that SMS yields much better solutions than other methods. The Wilcoxon test results in Table 7 provide information to statistically demonstrate that SMS has performed better than PSO, GSA, DE and TPSO. Figure 7 Table 6. Minimization results of benchmark functions in Table A4 with n=30. Maximum number of iterations=1000. SMS vs Conclusions In this paper, a novel nature-inspired algorithm called as the States of Matter Search (SMS) has been introduced. The SMS algorithm is based on the simulation of the State of Matter phenomenon. In SMS, individuals emulate molecules which interact to each other by using evolutionary operations that are based on physical principles of the thermal-energy motion mechanism. The algorithm is devised by considering each state of matter at one different exploration-exploitation ratio. The evolutionary process is divided into three phases which emulate the three states of matter: gas, liquid and solid. At each state, molecules (individuals) exhibit different movement capacities. Beginning from the gas state (pure exploration), the algorithm modifies the intensities of exploration and exploitation until the solid state (pure exploitation) is reached. As a result, the approach can substantially improve the balance between exploration-exploitation, yet preserving the good search capabilities of an EA approach. SMS has been experimentally tested considering a suite of 24 benchmark functions. The performance of SMS has been also compared to the following evolutionary algorithms: the Particle Swarm Optimization method (PSO) [17], the Gravitational Search Algorithm (GSA) [16], the Differential Evolution (DE) algorithm [13] and the PSO algorithm with a territorial diversity-preserving scheme (TPSO) [39]. Results have confirmed a high performance of the proposed method in terms of the solution quality for solving most of benchmark functions. The SMS's remarkable performance is associated with two different reasons: (i) the defined operators allow a better particle distribution in the search space, increasing the algorithm's ability to find the global optima; and (ii) the division of the evolution process at different stages, provides different rates between exploration and exploitation during the evolution process. At the beginning, pure exploration is favored at the gas state, then a mild transition between exploration and exploitation features during liquid state. Finally, pure exploitation is performed during the solid state. Appendix A. List of benchmark functions Test function S opt f n ( )   = + − + + − +   ∑ ∑ x ( ) 1 1 4 i i x y + = + ( ) ( ) ( ) , , , 0 m i i i i m i i k x a x a u x a k m a x a x a k x a  − >   = − ≤ ≤   < − −   [ ] 50,50 n − 0 30 { } 2 2 2 2 2 9 1 1 ( ) 0.1 sin (3 ) ( 1) 1 sin (3 1) ( 1) 1 sin (2 ) n i i n n i f x x x x x π π π =     = + − + + + − +     ∑ x 1 ( ,i i i i i i i x b b x f a b b x x =   +   = − + +     ∑ x [ ]      =       A [ ]      =       P [ ] 0,1 n -3.32 6 () ( ) ( ) ( ) ( ) ( ) ( ) 2 2i i i i i F x x D D π = =     = − − − +           ∑ ∑ x ( ) 3 4 F − = x Rastringin's function   = − +     ∑ ∏ x max ( ) ( ) / i i i F F F = z z . max i F is the maximum value of the particular function i. 10 Fig. 1 . 1Different states of matter: (a) gas, (b) liquid, and (c) solid. randomly chosen within the range of [-1,1]. Cuevas, E., Echavarría, A., Ramírez-Ortegón, M.A. An optimization algorithm inspired by the States of Matter that improves the balance between exploration and exploitation, Applied Intelligence, 40(2) , (2014), 256-272. Fig. 2 : 2Evolution process in the proposed approach. Cuevas, E., Echavarría, A., Ramírez-Ortegón, M.A. An optimization algorithm inspired by the States of Matter that improves the balance between exploration and exploitation, Applied Intelligence, 40(2) , (2014), 256-272. 8 Fig. 3 : 83Data flow in the complete SMS algorithm Initialization The algorithm begins by initializing a set P of p N p is a n-dimensional vector containing the parameter values to be optimized. Such values are randomly and uniformly distributed between the pre-specified lower initial parameter bound low j b and the upper initial parameter bound high j b , just as it is described by the following expressions: 0 , Fig. 4 . 4Convergence rate comparison of GSA, PSO, DE, SMS and TPSO for minimization of (a) 1 f and (b) 3 f considering n=30. Fig. 5 . 5Convergence rate comparison of PSO, GSA, DE, SMS and TPSO for minimization of (a) 5 f and (b) 11 f considering n=30. Cuevas, E., Echavarría, A., Ramírez-Ortegón, M.A. An optimization algorithm inspired by the States of Matter that improves the balance between exploration and exploitation, Applied Intelligence, 40(2) , (2014), 256-272. Fig. 7 . 7Convergence rate comparison of PSO, GSA, DE, SMS and TPSO for minimization of (a)17 f and (b)24 f . Cuevas, E., Echavarría, A., Ramírez-Ortegón, M.A. An optimization algorithm inspired by the States of Matter that improves the balance between exploration and exploitation, Applied Intelligence, 40(2) , (2014), 256-272. the i-th row of A whereas o is a 1 n × vector whose elements are random numbers in the range [ Table A4 ) A4. In such tables, n indicates the dimension of the function, opt f the optimum value of the function and S the subset of n R . The function optimum position ( opt x ) for 1 f , 2 f , 4 f , 6 f , 7 f 10 f , 11 f and 14 f is at [ ] 0 n opt = x , for 3 f , 8 f and 9 f is at [ ] 1 n opt = x , for 5 f is at [ ] 420.96 n opt = x , for 18 f is at [ ] 0 n opt = x , for 12 f is at [ ] 0.0003075 n opt = x and for 13 f is at [ ] 3.32 n opt = − x . In case of functions contained in Table A4, the opt x and opt f values have been set to Please cite this article as: Cuevas, E., Echavarría, A., Ramírez-Ortegón, M.A. An optimization algorithm inspired by the States of Matter that improves the balance between exploration and exploitation, Applied Intelligence, 40(2) , (2014), 256-272. Table 1. Minimization result of benchmark functions of Table A1 with n=30. Maximum number of iterations=1000.GSA PSO DE TPSO AB 4.68457E-16 1.3296E-05 0.873813333 0.186584241 0,100341256 MB 4.50542E-16 7.46803E-06 4.48139E-12 0.189737658 0,101347821 1 f SD 1.23694E-16 1.45053E-05 4.705628811 0.039609704 0,002421043 AB 0.033116745 0.173618066 12.83021186 54.85755486 0.103622066 MB 1.02069E-08 0.159932758 12.48059177 54.59915941 0,122230612 2 f SD 0.089017369 0,122230612 3.633980625 4.506836836 0,006498124 AB 19.64056183 32.83253962 33399.69716 46898.34558 21.75247912 MB 26.87914282 27.65055745 565.0810149 43772.19502 28.45741892 3 f SD 11.8115879 19.11361524 43099.34439 15697.6366 14.56258711 AB 8.882513655 9.083435186 15.05362961 12.83391861 13.98432748 MB 9.016816582 9.150769929 13.91301428 12.89762202 14.01237836 4 f SD 0.442124359 0.499181789 4.790792877 0.542197802 1.023476914 Table 2 . 2p-values produced by Wilcoxon's test comparing SMS vs. PSO, SMS vs. GSA, SMS vs. DE and SMS vs. TPSO over the "average best-so-far" (AB) values from Table 4 . 4p-values produced by Wilcoxon's test comparing SMS vs. GSA, SMS vs. PSO, SMS vs. DE and SMS vs. Table A1 . A1Unimodal test functions.Test function S opt f n ( ) ( ) ( ) 5 1 418.9829 sin n i i i f n x x = = + − ∑ x [ ] 500,500 n − 0 30 ( ) ( ) ( ) 50 2 6 1 10cos 2 10 i i i f x x π = = − + ∑ x [ ] 5.12,5.12 n − 0 30 ( ) 2 7 1 1 1 cos 1 4000 n n i i i i x f x i = =   = − +     ∑ ∏ x [ ] 600,600 n − 0 30 { } 1 2 2 2 8 1 1 1 1 ( ) 10sin( ) ( 1) 1 10sin ( ) ( 1) ( ,10,100, 4) n n i i n i i i f y y y y u x n π π π − + = = Table A2 . A2Multimodal test functions.Test function S opt f n ( ) ( ) 2 2 11 2 12 2 1 3 4 Table A3. Multimodal test functions with fixed dimensions Please cite this article as: Cuevas, E., Echavarría, A., Ramírez-Ortegón, M.A. An optimization algorithm inspired by the States of Matter that improves the balance between exploration and exploitation,2 14 1 2 1 2 1 2 1.5 1 2.25 1 2.625 1 f x x x x x x = − − + − − + − − x [ ] 4.5,4.5 n − 0 2 Applied Intelligence, 40(2) , (2014), 256-272. Test function S n GECCO classification 6 2 15 1 2 ( ) 10 n i opt i f z z f = = ⋅ + + ∑ x ( ) opt osz T = − z x x : , n n osz T → for any positive integer n, it maps element-wise. ( ), osz T = a h { } 1 2 , , , n a a a = a K , { } 1 2 , , , n h h h = h K ( ) ( ) ( ) ( ) ( ) 1 2 =sign exp 0.049 sin sin , i i a h K c K c K + + where log( ) if 0 , 0 otherwise i i h h K ≠  =   1 if 0 sign( ) 0 if 0, 1 if 0 i i i i h h h h − <   = =   >  1 10 if 0 5.5 otherwise i h c >  =   and 2 7.9 if 0 3.1 otherwise i h c >  =   [ ] 5,5 , , , n h h h = h K ( ) values have been set to default values which have been obtained from the Matlab© implementation for GECCO competitions, as it is provided in[51].5 10 5 10 5 , , 2,1, , , 20,10, , 32 32 100 100 60 60   =     λ [ ] 5,5 n − 30 GECCO2005 Rotated Version of Hybrid Composition Function 16 ( ) f x The opt x and opt f Table A4 . A4Set of representative GECCO functions. 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[ "NEW EXAMPLES OF BRUNNIAN THETA GRAPHS", "NEW EXAMPLES OF BRUNNIAN THETA GRAPHS" ]	[ "Byoungwook Jang ", "Anna Kronaeur ", "Pratap Luitel ", "Daniel Medici ", "ANDScott A Taylor ", "Alexander Zupan " ]	[]	[]	The Kinoshita graph is the most famous example of a Brunnian theta graph, a nontrivial spatial theta graph with the property that removing any edge yields an unknot. We produce a new family of diagrams of spatial theta graphs with the property that removing any edge results in the unknot. The family is parameterized by a certain subgroup of the pure braid group on four strands. We prove that infinitely many of these diagrams give rise to distinct Brunnian theta graphs.	10.2140/involve.2016.9.857	[ "https://arxiv.org/pdf/1508.03014v2.pdf" ]	59,388,438	1508.03014	3ebea6fa08b7e5d6614489797457c6cacc1ff60c	NEW EXAMPLES OF BRUNNIAN THETA GRAPHS Byoungwook Jang Anna Kronaeur Pratap Luitel Daniel Medici ANDScott A Taylor Alexander Zupan NEW EXAMPLES OF BRUNNIAN THETA GRAPHS The Kinoshita graph is the most famous example of a Brunnian theta graph, a nontrivial spatial theta graph with the property that removing any edge yields an unknot. We produce a new family of diagrams of spatial theta graphs with the property that removing any edge results in the unknot. The family is parameterized by a certain subgroup of the pure braid group on four strands. We prove that infinitely many of these diagrams give rise to distinct Brunnian theta graphs. INTRODUCTION A spatial theta graph is a theta graph (two vertices and three edges, each joining the two vertices) embedded in the 3-sphere S 3 . There is a rich theory of spatial theta graphs and they show up naturally in knot theory. (For instance, the union of a tunnel number 1 knot with a tunnel having distinct endpoints is a spatial theta graph.) A trivial spatial theta graph is any spatial theta graph which is isotopic into a 2-sphere in S 3 . A spatial theta graph G ⊂ S 3 has the Brunnian property if for each edge e ⊂ G the knot K e = G \ e which is the result of removing the interior of e from G is the unknot. A spatial theta graph is Brunnian (or almost unknotted or minimally knotted) if it is non-trivial and has the Brunnian property. By far the best known Brunnian theta graph is the Kinoshita graph [8,9]. The Kinoshita graph was generalized by Wolcott [22] to a family of Brunnian theta graphs now called the Kinoshita-Wolcott graphs. They are pictured in Figure 1. Inspection shows that they have the Brunnian property. There are several approaches to showing that the Kinoshita graph (and perhaps all of the Kinoshita-Wolcott graphs) are non-trivial: Wolcott [22] uses double branched covers; Litherland [10] uses a version of the Alexander polynomial; Scharlemann [15] and Livingston [11] use representations of certain associated groups; McAtee, Silver, and Williams [12] use quandles; Thurston [21] showed that the Kinoshita graph is hyperbolic (i.e. the exterior supports a complete hyperbolic structure with totally geodesic boundary.) In this paper, we produce an infinite family of diagrams for spatial theta graphs G(A,t 1 ,t 2 ) having the Brunnian property. These graphs depend on braids A lying in a certain subgroup of the pure braid group on 4 strands and on integers t 1 ,t 2 which represent certain twisting parameters. Our main theorem shows that infinitely many braids A give rise to Brunnian theta graphs. Theorem 5.1 (rephrased). For all n ∈ Z, there exists a braid A n such that for all m ∈ Z, the graph Γ(n, m) = G(A n , −n, m) is a Brunnian theta graph. Furthermore, suppose that for a given (n, m) ∈ Z × Z, the set S(n, m) ⊂ Z × Z has the property that if (a, b) ∈ S(n, m) then Γ(a, b) is isotopic to Γ(n, m) and if (a, b), (a , b ) ∈ S(n, m) are distinct, then a + b = a + b . Then S(n, m) has at most three distinct elements. In particular, there exist infinitely many n ∈ Z such that the graphs Γ(n, 0) are pairwise non-isotopic Brunnian theta graphs. 1.1. Acknowledgements. We thank the attendees at the 2013 Spatial Graphs conference for helpful discussions, particularly Erica Flapan and Danielle O'Donnol. We are also grateful to Ryan Blair, Ilya Kofman, Jessica Purcell, and Maggy Tomova for helpful conversations. This research was partially funded by Colby College. NOTATION We work in either the PL or smooth category. For a topological space X, we let \|X\| denote the number of components of X. If Y ⊂ X then η(Y ) is a closed regular neighborhood of Y in X andη(Y ) is an open regular neighborhood. More generally,Y denotes the interior of Y . CONSTRUCTING NEW BRUNNIAN THETA GRAPHS There are two natural methods for constructing new Brunnian theta graphs: vertex sums and clasping. 3.1. Vertex sums. Suppose that G 1 ⊂ S 3 and G 2 ⊂ S 3 are spatial theta graphs. Let v 1 ∈ G 1 and v 2 ∈ G 2 be vertices. We can construct a new spatial theta graph G 1 # 3 G 2 ⊂ S 3 by taking the connected sum of S 3 with S 3 by removing regular open neighborhoods of v 1 and v 2 and gluing the resulting 3-balls B 1 and B 2 together by a homeomorphism ∂ B 1 → ∂ B 2 taking the punctures G 1 ∩ ∂ B 1 to the punctures G 2 ∩ ∂ B 2 . See Figure 2. The subscripted 3 represents the fact that we are performing the connected sum along a trivalent vertex and is used to distinguish the vertex sum from the connected sum of graphs occuring along edges of a graph (which, when both G 1 and G 2 are theta graphs, does not produce a theta graph.) An orientation on a spatial theta graph is a choice of one vertex to be the source, one vertex to be the sink, and a choice of a total order on the edges of the graph. If G 1 and G 2 are oriented theta graphs, we insist that the connected sum produce an oriented theta graph (so that the sink vertex of G 1 is glued to the source vertex of G 2 and so that the edges of G 1 # 3 G 2 can be given an ordering which restricts to the given orderings on the edges of G 1 and G 2 . Wolcott [22] showed that vertex sum of oriented theta graphs is independent (up to ambient isotopy of the graph) of the choice of homeomorphism ∂ B 1 → ∂ B 2 . FIGURE 2. A schematic depiction of the vertex sum of two spatial theta graphs If G 1 and G 2 both have the Brunnian property, then G 1 # 3 G 2 does as well since the connected sum of two knots is the unknot if and only if both of the original knots are unknots. If G 1 (say) is trivial, then G 1 # 3 G 2 is isotopic to G 2 . Similarly, if at least one of G 1 or G 2 is non-trivial then G 1 # 3 G 2 is non-trivial [22]. Consequently: Theorem 3.1 (Wolcott). If G 1 and G 2 are Brunnian theta graphs, then G 1 # 3 G 2 is a Brunnian theta graph. We say that a spatial theta graph is vertex-prime if it is not the vertex sum of two other non-trivial spatial theta graphs. The Kinoshita graph is vertex prime [1]. Using Thurston's hyperbolization theorem for Haken manifolds, it is possible to show that if G 1 and G 2 are theta graphs, then G 1 # 3 G 2 is hyperbolic if and only if G 1 and G 2 are hyperbolic. 3.2. Clasping. Clasping [18] is a second method for converting a Brunnian theta graph into another theta graph with the Brunnian property. To explain it, suppose that G is a spatial theta graph in S 3 which has been isotoped so that its intersection with a 3-ball B ⊂ S 3 consists of four unknotted arcs (as on the left of Figure 3), numbered α 1 , α 2 , α 3 , α 4 , α 5 . Assume that the first arc and the last two arcs belong to the same edge (the "red edge") of G and that the others belong to different, distinct edges of G (the "green edge" and the "blue edge"). We require that as we traverse the red edge, the arc α 1 is traversed between α 4 and α 5 . Letting e be the sub-arc of the red edge containing α 4 ∪ α 1 ∪ α 5 , we also require that there is an isotopy of e , in the complement of the rest of the graph, to an unknotted arc in B. As in Figure 3, we may then perform crossing changes to introduce a clasps between adjacent arcs. It is easily checked that this clasp move preserves the Brunnian property. Although the clasp move creates many Brunnian theta graphs, it is not clear how to keep track of fundamental properties (such as hyperbolicity) under the clasp move. Additionally, very little is known about sequences of clasp moves relating two Brunnian theta graphs. We can, however, use clasping to show that there exist Brunnian theta graphs which are not hyperbolic. Figure 4 shows an example of a Brunnian theta graph with an essential torus in its exterior. It was created by isotoping a Kinoshita-Wolcott graph to the position required to apply the clasping move via an isotopy which moved a point on one of the edges around a trefoil knot. A graph with an essential torus in its exterior is neither hyperbolic nor a trivial graph. NEW EXAMPLES OF BRUNNIAN THETA GRAPHS Besides the Kinoshita graph and vertex sums of the Kinoshita graph with itself, are there other hyperbolic Brunnian theta graphs? In this section, we give a new infinite family of examples of diagrams of spatial theta curves. In the next section we will prove that infinitely many of them are also non-trivial. These examples have the property that they are of "low bridge number". Forthcoming work [19] will show that this implies that these graphs are vertex-prime. Furthermore, since they are low bridge number it is likely that they are hyperbolic. Section 6 concludes this paper with some questions for further research. A pure n-braid representative consists of n arcs (called strands) in Q = {(x, y, z) ∈ R 3 : −1 ≤ z ≤ 1} such that the ith strand has endpoints at (i, 0, ±1) and for each arc projecting onto the z-axis is a strictly monontonic function. Two pure n-braid representatives are equivalent if there is an isotopy in Q from one to the other which fixes ∂ Q. The set of equivalence classes is PB(n). Two pure n-braid representatives can be "stacked" to create another pure n-braid representative by placing one on top of the other and then scaling in the zdirection by 1/2. Applying this operation to equivalence classes we obtain the group operation for PB(n). If σ and ρ are elements of PB(n), we let σ ρ denote the braid having a representative created by stacking a representative for σ on top of a representative for ρ and then scaling in the z-direction by 1/2. Let φ : PB(4) → PB(2) be the homomorphism which forgets the last two strands. For each A ∈ ker φ we will construct a family G(A,t 1 ,t 2 ) for t 1 ,t 2 ∈ Z of theta graphs with the Brunnian property. We will construct G(A,t 1 ,t 2 ) by placing braids into the boxes in the template shown in Figure 5. Let ρ : PB(4) → PB(6) be a monomorphism which "doubles" each of the last two strands of A ∈ PB(4) (i.e. in ρ(A) the 4th strand is parallel to the 3rd and the 6th strand is parallel to the 5th.) For a given A ∈ ker φ , we place ρ(A) into the top braid box of Figure 5. The shading indicates the doubled strands. There is more than one choice for the monomorphism ρ, as the doubled strands may be allowed to twist around each other (i.e. we may vary the framing). We will always choose the homomorphism determined by the "blackboard framing" (i.e. in our diagram the doubled strands are two edges of a rectangle embedded in the plane.) Into the second and fourth boxes from the top we place the braid A −1 . In the third box we place the element from PB(2) consisting of two strands with t 1 full twists. We use the convention that, giving the strands a downward orientation, if t 1 > 0 there are 2\|t 1 \| left-handed crossings and if t 1 < 0 there are 2\|t 1 \| right-handed twists. Into the bottom box we place t 2 full twists, using the same orientation convention as for t 1 . ρ(A) A −1 A −1 t 1 t 2 FIGURE 5. The template for the graph G A Considering the plane of projection in Figure 5 as the xy plane, the plane Π perpendicular to the plane of projection and cutting between the second and third boxes from the top functions as a "bridge plane" for G(A,t 1 ,t 2 ). Observe that if we measure the height of a point x ∈ G(A,t 1 ,t 2 ) by its projection onto the y-axis, each edge of G(A,t 1 ,t 2 ) has a single local minimum for the height function and no other critical points in its interior. This implies that Π cuts G(A,t 1 ,t 2 ) into trees with special properties. The two trees above Π have a single vertex which is not a leaf and their union is isotopic (relative to endpoints) into Π. The three trees below Π are all edges (i.e. each is a tree with two vertices and single edge) and their union can be isotoped relative to the endpoints into Π. Thus, Π is a bridge plane for G(A,t 1 ,t 2 ) and \|G(A,t 1 ,t 2 ) ∩ Π\| = 6. We might, therefore, say that G(A,t 1 ,t 2 ) has "bridge number at most 3". The precise definition of bridge number for theta graphs has been a matter of some dispute (see, for example, [14]). The forthcoming paper [19] explores bridge number for spatial graphs in detail. Theorem 4.1. For each A ∈ ker φ and t 1 ,t 2 ∈ Z, the graph G(A,t 1 ,t 2 ) has the Brunnian property. Proof. The proof is easy and diagrammatic. Color the edges coming out of the top vertex in the diagram in Figure 5 by (b)lue, (r)ed, and (v)erdant from left to right. Then the edges entering into the bottom vertex are also blue, red, and verdant from left to right. In Figure 6, we have the knots K B , K R , and K V obtained by removing the blue, red, and verdant edges respectively. Observe that the top braid box of K R and K V contains the braid A. In each of the diagrams for K B , K R , and K V we have labelled certain portions with lower case letters. We now explain those regions and why each diagram can be simplified to the standard diagram for the unknot. Consider the diagram for K B . Since the third and fourth strands of the top braid box of G(A,t 1 ,t 2 ) are parallel, we may untwist the diagram at region (a) and at region (b). At regions (c) and (d), we may also untwist at the minima. The end result is a diagram of a knot having a single crossing. The knot K B must, therefore, be the unknot. Consider the diagram for K R . At region (a) we have the trivial 2-braid since A ∈ ker φ . The braid A in the top braid box may therefore be cancelled with the braid A −1 in the third-from-the-top braid box. Finally, we may untwist the t 2 full twists in the final braid box to arrive at the standard diagram for the unknot. Consider the diagram for K V . The braids A and A −1 cancel, at which point we may untwist the t 1 full twists. We may also untwist at region (a). Thus, K V is also the unknot. K B (a) (b) (c) (d) K R A (a) A −1 t 2 K V A A −1 t 1 (a) FIGURE 6. The constituent knots K B , K R , and K V of G(A,t 1 ,t 2 ). Given a graph G(A,t 1 ,t 2 ) we can construct other theta graphs of bridge number at most 3 with the Brunnian property by using the clasping technique in such a way that we do not introduce any additional critical points in the interior of any edge, so it is highly unlikely that the template in Figure 5 encompasses all possible theta graphs of bridge number at most 3 with the Brunnian property. On the other hand, there are infinitely many braids A such that G(A, 0, 0) is a diagram of the trivial theta graph (see below), so the question as to what braids in ker φ produce non-trivial theta graphs is somewhat subtle. BRAIDS PRODUCING BRUNNIAN THETA GRAPHS In this section we produce an infinite family of braids A ∈ ker φ ⊂ PB(4) such that there exists t 1 such that for all t 2 , G(A,t 1 ,t 2 ) is Brunnian. To describe the braids A more precisely, we recall the standard generating set for PB (4). For i, j ∈ {1, 2, 3, 4} with i < j, let P i j denote the element of PB(4) obtained by "looping" the ith strand around the jth strand, as in Figure 7. Observe that P k 23 produces a twist box in the 2nd and 3rd strands with k full twists, using the sign convention from earlier. There are non-trivial braids A for which G(A, 0, 0) is trivial. For example, for every n, t 1 , and t 2 , the graphs G(P n 23 ,t 1 ,t 2 ) are all trivial. To show that there are infinitely many braids producing non-trivial graphs, let A n = P −n 23 P 13 and for m ∈ Z, let Γ(n, m) = G(A n , −n, m) (see Figure 8 for a diagram of A 2 .) Theorem 5.1. For all n, m ∈ Z, the graph Γ(n, m) is a Brunnian theta graph. Furthermore, suppose that for a given (n, m) ∈ Z × Z, the set S(n, m) ⊂ Z × Z has the properties that if (a, b) ∈ S(n, m) then Γ(a, b) is isotopic to Γ(n, m) and if (a, b), (a , b ) ∈ S(n, m) are distinct, then a + b = a + b . Then S(n, m) has at most three distinct elements. In particular, there exist infinitely many n ∈ Z such that the graphs Γ(n, 0) are pairwise distinct Brunnian theta graphs. Before proving the theorem, we establish some background. A handlebody is the regular neighborhood of a finite graph embedded in S 3 and its genus is the genus of the boundary surface. We will be considering genus 2 handlebodies. A disc D properly embedded in a handlebody H whose boundary does not bound a disc in ∂ H is called an essential disc in H. If H has genus 2 and if D ⊂ H is an essential non-separating disc, the space H \η(D) is homeomorphic to S 1 × D 2 . A knot isotopic to the core of that solid torus is called a constituent knot of H. If G ⊂ S 3 is a spatial theta graph and if H = η(G), then a disc D ⊂ H intersecting an edge e of G exactly once transversally and disjoint from the other edges of G is called a meridian disc for e. Thus, if D is a meridian disc for e, then H \η(D) is a regular neighborhood of K e . Observe that if G is a theta graph and if e ⊂ e is an edge, then any meridian disc D for e is an essential disc in the handlebody η(G), as D does not separate η(G). If G and G are spatial theta graphs such that G is isotopic to G then the isotopy can be extended to an isotopy of the handlebody η(G) to the handlebody η(G ). Furthermore, if the isotopy takes an edge e ⊂ G to an edge e ⊂ G then the isotopy takes any meridian disc for e to a meridian disc for e . On the other hand, an isotopy of η(G) to η(G ) does not necessarily correspond to an isotopy of G to G . Instead, an isotopy of η(G) to η(G ) corresponds to a sequence of isotopies and "edge slides" of G. An edge slide of an edge e ⊂ G of a graph involves sliding one end of e across edges of G (see [16].) As in Figure 9, an edge slide of a theta graph may convert a theta graph into a spatial graph that is not a theta graph. Conversely, any sequence of edge slides and isotopies of a graph G corresponds to an isotopy of η(G). Given a spatial theta graph G and an edge e, an essential non-separating disc E in η(G) is along e if it lies in a regular neighborhood of e, is not a meridian of e, if there is a meridian disc D for e such that \|D ∩ E\| (the number of components of D ∩ E) is equal to 1. Observe that if E is along e, then η(G) \η is a solid torus since E is non-separating. If E is along e, then we say that the knot which is the core of η(G) \η(E) is obtained by unzipping the edge e. Figure 10 shows two different ways of unzipping an edge. The proof of Lemma 5.5 will also be helpful in understanding the relationship between the definition of unzipping given above and the diagrams in Figure 10. The term "unzipping" is taken from Bar-Natan and D. Thurston (see, for example, [20].) It is a form of an operation also known as "attaching a band" to K e or "distance 1 rational tangle replacement" on K e . FIGURE 10. Two ways of unzipping an edge of a spatial theta graph. As is suggested by the picture, the θ -graph may be embedded in S 3 in some, potentially complicated, way. We do, however, require that the unzipping produce a knot and not a 2-component link. Since an isotopy of a handlebody in S 3 to another handlebody takes discs in the first handlebody to discs in the second and preserves the number of intersections between discs, we have: Lemma 5.2. Suppose that G and G are isotopic spatial theta graphs such that the isotopy takes an edge e of G to an edge e of G . If K ⊂ S 3 is a knot obtained by unzipping the edge e, then there is a knot K ⊂ S 3 which is obtained by unzipping the edge e such that K and K are isotopic. Rational Tangles. The key step in our proof of Theorem 5.1 is to show that unzipping Γ(n, m) does not produce any knot that can be obtained by unzipping a trivial theta graph along one of its edges. Analyzing the knots we do get will show, as a by-product, that infinitely many of the Γ(n, m) are distinct. We use rational tangles to analyze our knots. A rational tangle is a pair (B, τ) where B is a 3-ball and τ ⊂ B is a properly embedded pair of arcs which are isotopic into ∂ B relative to their endpoints. We mark the points ∂ τ ⊂ ∂ B by NW, NE, SW, and SE as in Figure 11. Two rational tangles (B, τ) and (B, τ ) are equivalent if there is a homeomorphism of pairs h : (B, τ) → (B , τ ) which fixes ∂ B pointwise. Conway [2] showed how to associate a rational number r ∈ Q ∪ {1/0} to each rational tangle in such a way that two rational tangles are equivalent if and only if they have the same associated rational number. We briefly explain the association, using the conventions of [5,Lecture 4]. Using the 3-ball with marked points as in Figure 11, we let the rational tangle R(0/1) consist of a pair of horizontal arcs having no crossings and we associate to it the rational number 0 = 0/1. The rational tangle R(1/0), consisting of a pair of vertical arcs having no crossings, is given the rational number 1/0 (thought of as a formal object.) Let h : B → B and v : B → B be the horizontal and vertical half-twists, as shown in Figure 11. Observe that the rational tangle v 2k R(0/1) is a twist box with −k full twists, using the orientation convention from earlier. Let a 1 , a 2 , . . . , a k be a finite sequence of integers such that a 2 , . . . , a k = 0. Let R(a 1 , . . . , a k ) be the rational tangle defined by R(a 1 , a 2 , . . . , a k ) = h a 1 v a 2 · · · h a k−1 v a k R(1/0) k even h a 1 v a 2 · · · v a k−1 h a k R(0/1) k odd We assign the rational number p/q = a 1 + 1 a 2 + 1 a 3 + 1 ···+ 1 a k to R(a 1 , a 2 , . . . , a k ) and we define R(p/q) = R(a 1 , . . . , a k ), with p and q relatively prime. We define the distance between two rational tangles R(p/q) and R(p /q ) to be ∆(p/q, p /q ) = \|pq − p q\|. Observe that in the 3-ball B, there is a disc D ⊂ B such that ∂ D partitions the marked points {NW, SW, NE, SE} into pairs and which separates the strands of a given rational tangle R(p/q). Indeed, given a disc D ⊂ B whose boundary partitions the marked points into pairs, there is a rational tangle R(p/q) (unique up to equivalence of rational tangles) such that D separates the strands of R(p/q). We call D a defining disc for R(p/q). If D is a defining disc for R(p/q) and D is a defining disc for R(p /q ) such that, out of all such discs, D and D have been isotoped to intersect minimally, then it is not difficult to show that ∆(p/q, p /q ) = \|D ∩ D \| (i.e. the distance between the rational tangles is equal to the minimum number of arcs of intersection between defining discs.) From a rational tangle R(p/q) we can create the unknot or a 2-bridge knot or link K (p/q) = D(R(p/q)) by taking the so-called denominator closure D of R(p/q) where we attach the point NW to the point SW and the point NE to the point SE by an unknotted pair of arcs lying in the exterior of B, as in Figure 12. Thus, the right-handed trefoil is K (1/3) and the left-handed trefoil is K (−1/3). Theorem 5.3 (Schubert [17]). Let p/q, p /q ∈ Q ∪ {1/0} with q, q > 0 and the pairs p, q and p , q relatively prime. The knot or link K (p/q) is isotopic (as an unoriented knot or link) in S 3 to the knot or link K (p /q ) if and only if q = q and either p ≡ p mod q or pp ≡ 1 mod q. On the left is the denominator closure D(R(p/q)) of the rational tangle R(p/q). On the right, we see that the right-handed trefoil is the denominator closure of the rational tangle R(1/3). from the usual convention and statement by exchanging numerators and denominators. See the discussion following Theorem 3 of [7]. 5.2. Unzipping the trivial graph. Since we want to show that each graph in a certain family of graphs is non-trivial, the following will be useful. Lemma 5.5. Suppose that G ⊂ S 3 is the trivial theta graph and that K is a knot obtained by unzipping an edge e of G. Then either K is the unknot or there exists k ∈ Z, odd such that K is a (2, k) torus knot. Proof. Let G be the trivial theta graph and let e ⊂ G be an edge. Observe that there is an isotopy of G which interchanges any two edges. Thus, we may consider G to be the union of the unit circle in R 2 with a horizontal diameter e, as in Figure 13. We may consider the neighborhood η(e) of e as a 3-ball B with a vertical disc as a meridian disc for e. The graph G intersects B in four punctures, which we label NW, NE, SW, and SE as usual. Take D to be the meridian disc for e and let E ⊂ B be a disc with boundary an essential curve in ∂ B \ G, which cannot be isotoped to be disjoint from D, and for which \|D ∩ E\| = 1. Observe that D is the defining disc for the rational tangle R(1/0). If E is the defining disc for the rational tangle R(k/ ), then 1 = ∆(1/0, k/ ) = \| \|. Consequently, the rational tangle R(k/1) consists of k horizontal half twists. Thus the knot which is the core of η(G) \η(E) is a (2, ±k) twist knot. The following corollary follows immediately from Lemmas 5.2 and 5.5. G B ∂ E τ E Corollary 5.6. Suppose that G ⊂ S 3 is a trivial spatial theta graph. Then for all edges e ⊂ G and for any knot K obtained by unzipping e there exists an odd k ∈ Z such that K is a (2, k) torus knot, i.e. K (1/k). 5.3. Proof of Theorem 5.1. The proof is similar in spirit to [22,Section 3]. We do not, however, use Wolcott's Theorem 3.11 as that theorem would require us to work with links, rather than with knots. Potentially, however, a clever use of [22,Theorem 3.11] would show that a much wider class of braids A create nontrivial graphs G(A,t 1 ,t 2 ). Our method, however, also allows us, using a result of Eudave-Muñoz concerning reducible surgeries on strongly invertible knots, to show that we have infinitely many distinct Brunnian theta graphs. Let n, m ∈ Z, and let G = Γ(n, m). To prove that G is a Brunnian theta graph, by Theorem 4.1, we need only show that G is non-trivial. Let v + be the upper vertex of G in Figure 5 and let v − be the lower vertex. Recall that we color the edges of G (from left to right at each vertex) as blue, red, and verdant. Isotope G so that the endpoint of the verdant edge V adjacent to v − is moved near v + by sliding it along the red edge, as on the left of Figure 14. This isotopy creates a diagram of G such that red edge has no crossings. Let K be the knot obtained by unzipping the red edge, as in the middle of Figure 14 (choosing the unzip so that no twists are inserted in the diagram along V ). Using the doubled strands in the top braid box, isotope K so that it has the diagram on the right of Figure 14. Inserting the braid A n into the template, as specified in Figure 5, our knot K has the diagram on the top left of Figure 15. Let 3 1 be the right-handed trefoil. Now perform the isotopies indicated in Figure 15 to see that K is the connected sum of 3 1 and the knot K (− 3 6(m + n) + 5 ) = D(R(0, −2(m + n) − 1, −1, −1, −1)). Since torus knots are prime, K is not a (2, k) torus knot for any k ∈ Z unless K (− 3 6(m+n)+5 ) is the trivial knot, that is K (− 3 6(m+n)+5 ) = K (1). By Schubert's theorem, this can only happen if 6(m + n) + 5 = 1, an impossibility. Thus, each Γ(n, m) is a Brunnian theta graph. To prove the part about distinctness, we use a theorem of Eudave-Muñoz and the Montesinos trick [13] (see also [5] for a nice explanation.) We begin by showing: Claim: If a + b = a + b , then there is no isotopy from Γ(a , b ) to Γ(a, b) which takes the red edge of Γ(a , b ) to the red edge of Γ(a, b). We prove this by contradiction. Let B ⊂ S 3 be a regular neighborhood of the red edge of Γ(a, b) and let W = S 3 \B be the complementary 3-ball. Mark the points of Γ(a, b) ∩ ∂ B by NE, SE, NW, SW so that a meridian disc for the red edge of Γ(a, b) corresponds to the rational tangle R(1/0) and the disc E along which we unzip Γ(a, b) to produce K = 3 1 #K (− 3 6(a+b)+5 ) corresponds to the rational tangle R(0/1). Let τ = K ∩W . The isotopy of Γ(a, b) to Γ(a , b ) takes B to a regular neighborhood B of the red edge of Γ(a , b ). In B there is a disc D which is along the red edge of Γ(a , b ) such that unzipping Γ(a , b ) along D produces (3 1 #K (− 3 6(a +b )+5 )). Reversing the isotopy, takes D to a disc D ⊂ B which is along e. Let R(p/q) be the rational tangle corresponding to D. The knot K = τ ∪ R(p/q)) is isotopic to the result of unzipping Γ(a , b ) along D and so K = (3 1 #K (− 3 6(a +b )+5 )). If the disc D is isotopic to the disc E, the rational tangles R(p/q) and R(0/1) are equivalent. In which case, K is isotopic to K . But this implies that a + b = a + b , a contradiction. Thus, the rational tangles R(0/1) and R(p/q) are distinct (since the discs are not isotopic). Since τ ∪ R(1/0) is the unknot in S 3 , the double branched cover of W over τ is the exterior of a strongly invertible knot L ⊂ S 3 . Since K = τ ∪ R(0/1) and K = τ ∪ R(p/q) are composite knots, the double branched covers of S 3 over K and K are reducible. In particular there are distinct Dehn surgeries on L producing reducible manifolds. The surgeries are distinct since R(0/1) is not equivalent to R(p/q). However this contradicts the fact that the Cabling Conjecture holds for strongly invertible knots [4,Theorem 4]. (Claim) For a pair (n, m) ∈ Z × Z, let S(n, m) ⊂ Z × Z be a subset with the property that for all (a, b) ∈ S(n, m), the graph Γ(a, b) is isotopic to the graph Γ(n, m) and which has the property that for all pairs (a, b), then (a, b) = (a , b ). Observe that (n, m) ∈ S(n, m). We will show that for all (n, m) ∈ Z × Z, the set S(n, m) has at most three elements. (a , b ) ∈ S(n, m) if a + b = a + b , Suppose, for a contradiction, that there exists (n, m) such that S(n, m) has at least 4 distinct elements Thus, without loss of generality, we may assume that σ 1 is a transposition. Suppose that σ 1 is the transposition (B, R,V ) → (B,V, R). Since neither σ 2 σ −1 1 nor σ 3 σ −1 1 fixes R and since σ 2 = σ 3 , the permutations σ 2 and σ 3 are the two permutations taking R to B. But then the composition σ 2 σ −1 3 takes R to R, a contradiction. The case when σ 1 is the transposition (B, R,V ) → (R, B,V ) similarly gives rise to a contradiction. Thus, for every (n, m) ∈ Z × Z, the set S(n, m) has at most three elements (including (n, m).) Define a sequence (n i ) in Z recursively. Let n 1 ∈ Z and recall that, by the above, Γ(n 1 , 0) is a Brunnian theta graph. Assume we have defined n 1 , . . . , n i so that the graphs Γ(n j , 0) for 1 ≤ j ≤ i are pairwise non-isotopic Brunnian theta graphs. Let P ⊂ Z be such that n ∈ P if and only if Γ(n, 0) is isotopic to Γ(n j , 0) for some 1 ≤ j ≤ i. Since for each j with 1 ≤ j ≤ i there are at most 3 elements n of Z such that Γ(n, 0) is isotopic to Γ(n j , 0), the set P is finite. Hence, we may choose n i+1 ∈ Z \ P. Thus, we may construct a sequence (n i ) in Z so that the graphs Γ(n i , 0) are pairwise disjoint Brunnian theta graphs. QUESTIONS AND CONJECTURES Using the software [6], and a lot of patience, it is possible to compute (approximations to) hyperbolic volumes for some of the graphs G(A,t 1 ,t 2 ). Our explorations suggest that "most" of the braids A ∈ PB(4) produce hyperbolic Brunnian theta graphs for all t 1 ,t 2 ∈ Z. Indeed, the software suggests that for a "sufficiently complicated" braid A ∈ PB(4), and for fixed t 1 ,t 2 the volume of the exterior of G(A n ,t 1 ,t 2 ) grows linearly in n. This is to be contrasted with the belief, based on the Thurston 2π-theorem, that for a fixed A and t 1 , the volumes of G(A,t 1 ,t 2 ) will converge as t 2 → ∞. Furthermore, calculations of hyperbolic volumes using [6] indicate that the graphs Γ(n, m) of Theorem 5.1 are likely not Kinoshita-Wolcott graphs. Since the calculations of hyperbolic volume are only approximate and since we can only calculate the volumes of finitely many of the graphs, we do not have a proof of that fact. These investigations raise the the following questions. On the left we have isotoped the template so that R has no crossings. The shaded boxes denote doubled strands. In the middle we have unzipped along R using a particular choice of unzipping disc. On the right, we have simplified the unzipped knot by using the parallel strands from the top braid box. (1) For what braids A ∈ ker φ is G(A, 0, 0) a Brunnian theta graph? (2) Can Litherland's Alexander polynomial (or some other algebraic invariant) prove that there are infinitely many braids A ∈ ker φ such that G(A,t 1 ,t 2 ) is a Brunnian theta graph for some t 1 ,t 2 ∈ Z? (3) Is any one of the Brunnian graphs Γ(n, m) a Kinoshita-Wolcott graph? (4) Are there infinitely many braids A such that the graph G(A, 0, 0) is a Brunnian theta graph which is not a Kinoshita-Wolcott graph? We conjecture the answer to be "yes". (5) For what A ∈ ker φ and t 1 ,t 2 ∈ Z is G(A,t 1 ,t 2 ) a hyperbolic Brunnian theta graph? We conjecture that whenever G(A,t 1 ,t 2 ) is a Brunnian theta graph, then it is hyperbolic. (6) Is it true that if G(A,t 1 ,t 2 ) is hyperbolic then G(A n ,t 1 ,t 2 ) is hyperbolic for all n ∈ N? Does the hyperbolic volume of the exterior of G(A n ,t 1 ,t 2 ) grow linearly in n? In the first step we combine the lower two twist boxes into a single twist box with m + n full twists. [6] Damien Heard, Orb, available at http://www.ms.unimelb.edu.au/~snap/orb.html. Date: October 26, 2015. FIGURE 1 . 1The Kinoshita-Wolcott graphs (figure based on [10, Figure 4]). The labels −i, − j, and −k indicate the number of full twists in each box (with the sign of −i, − j, −k indicating the direction of the twisting.) If i = j = k = 1, the graph is the Kinoshita graph. If one of i, j, k is zero, then the graph is trivial; otherwise, it is Brunnian [22, Theorem 2.1]. FIGURE 3 . 3The clasp move in the case when all three edges of the graph are involved. FIGURE 4 . 4A toroidal Brunnian theta graph. The swallow-follow torus for the doublestranded trefoil on the right is an essential torus in the exterior of the theta graph. FIGURE 7 . 7The generators for PB(4) FIGURE 8 . 8The braid A 2 . FIGURE 9 . 9An edge slide converting a theta graph into a non-theta graph. FIGURE 11 . 11The basic transformations of a rational tangle Remark 5. 4 . 4For more on Schubert's theorem, see[3, Theorem 8.7.2] or [7,Theorem 3]. Since we are using the denominator closure of rational tangles our convention and the statement of Schubert's theorem differ FIGURE 13 . 13Upper Left: The trivial graph G. Upper Right: the ball B = η(e). Lower Left: A disc E. Lower Right: The rational tangle R(−3) with defining disc E. (a 1 1, b 1 ), (a 2 , b 2 ), (a 3 , b 3 ), (n, m). Each isotopy between any two of the graphs {Γ(a 1 , b 1 ), Γ(a 2 , b 2 ), Γ(a 3 , b 3 ), Γ(n, m)} induces a permutation of the set {B, R,V } of blue, red, and verdant edges. For each i ∈ {1, 2, 3}, choose an isotopy f i from Γ(n, m) to Γ(a i , b i ) and let σ i be the induced permutation of {B, R,V }. By the claim and the definition of S(n, m), no σ i fixes R and, whenever i = j, the permutation σ i σ −1 j also does not fix R. Hence, σ i = σ j if i = j. In the permutations of the set {B, R,V }, there are exactly four that do not fix R and of those, two are transpositions. FIGURE 14 . 14FIGURE 14. On the left we have isotoped the template so that R has no crossings. The shaded boxes denote doubled strands. In the middle we have unzipped along R using a particular choice of unzipping disc. On the right, we have simplified the unzipped knot by using the parallel strands from the top braid box. FIGURE 15 . 15The isotopies showing that K is the connected sum of a right handed trefoil and K (− 3 6(m+n)+5 ). [ 7 ] 7Louis H. Kauffman and Sofia Lambropoulou, On the classification of rational knots, Enseign. Math. (2) 49 (2003), no. 3-4, 357-410. MR2028021 (2004j:57007) J Calcut, J Metcalf-Burton, arXiv:1509.07788Double branched covers of theta curves. J. Calcut and J. Metcalf-Burton, Double branched covers of theta curves, available at arXiv:1509.07788. 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[ "The Wigner rotation for photons in an arbitrary gravitational field", "The Wigner rotation for photons in an arbitrary gravitational field" ]	[ "P M Alsing \nDepartment of Physics and Astronomy\nAir Force Research Laboratory\nUniversity of New Mexico\nSpace Vehicles Directorate 3550 Aberdeen Ave87117-5776, 87131AlbuquerqueSE, Kirtland AFB, New Mexico, NM\n", "G J Stephenson Jr\nDepartment of Physics and Astronomy\nAir Force Research Laboratory\nUniversity of New Mexico\nSpace Vehicles Directorate 3550 Aberdeen Ave87117-5776, 87131AlbuquerqueSE, Kirtland AFB, New Mexico, NM\n" ]	[ "Department of Physics and Astronomy\nAir Force Research Laboratory\nUniversity of New Mexico\nSpace Vehicles Directorate 3550 Aberdeen Ave87117-5776, 87131AlbuquerqueSE, Kirtland AFB, New Mexico, NM", "Department of Physics and Astronomy\nAir Force Research Laboratory\nUniversity of New Mexico\nSpace Vehicles Directorate 3550 Aberdeen Ave87117-5776, 87131AlbuquerqueSE, Kirtland AFB, New Mexico, NM" ]	[]	We investigate the Wigner rotation for photons, which governs the change in the polarization of the photon as it propagates through an arbitrary gravitational field. We give explicit examples in Schwarzschild spacetime, and compare with the corresponding flat spacetime results, which by the equivalence principle, holds locally at each spacetime point. We discuss the implications of the Wigner rotation for entangled photon states in curved spacetime, and lastly develop a sufficient condition for special (Fermi-Walker) frames in which the observer would detect no Wigner rotation.	null	[ "https://arxiv.org/pdf/0902.1399v1.pdf" ]	118,582,507	0902.1399	c49f5fc0974f54290baab1f211fd4e5309e3ff8d	The Wigner rotation for photons in an arbitrary gravitational field 9 Feb 2009 (Dated: February 10, 2009 P M Alsing Department of Physics and Astronomy Air Force Research Laboratory University of New Mexico Space Vehicles Directorate 3550 Aberdeen Ave87117-5776, 87131AlbuquerqueSE, Kirtland AFB, New Mexico, NM G J Stephenson Jr Department of Physics and Astronomy Air Force Research Laboratory University of New Mexico Space Vehicles Directorate 3550 Aberdeen Ave87117-5776, 87131AlbuquerqueSE, Kirtland AFB, New Mexico, NM The Wigner rotation for photons in an arbitrary gravitational field 9 Feb 2009 (Dated: February 10, 2009Version: v3) We investigate the Wigner rotation for photons, which governs the change in the polarization of the photon as it propagates through an arbitrary gravitational field. We give explicit examples in Schwarzschild spacetime, and compare with the corresponding flat spacetime results, which by the equivalence principle, holds locally at each spacetime point. We discuss the implications of the Wigner rotation for entangled photon states in curved spacetime, and lastly develop a sufficient condition for special (Fermi-Walker) frames in which the observer would detect no Wigner rotation. I. INTRODUCTION Recently, there has been much interest in the study of entanglement for moving observers, both for constant velocity observers (special relativity -SR) and for arbitrarily accelerated observers (general relativity -GR). An excellent, recent review can be found in Peres and Terno [1] (and references therein). In SR and GR the important ingredient that determines the description of moving states by moving observers is how such states transform under the symmetries that govern the underlying flat or curved spacetime. The relevant concept is that of the Wigner rotation [2], which for massive particles, mixes up the spin components (along a given quantization axis) for a particle of definite spin by an O(3) rotation, and for massless particles, introduces a phase factor which is the product of a Wigner rotation angle times the helicity of the state. In this paper we investigate the transformation of photon states as they traverse trajectories in an arbitrary curved spacetime (CST), and investigate the implications for the evolution of entangled states. In a companion article [3], similar investigations were carried out on the role of the Wigner rotation on the entanglement of massive spin 1 2 particles in CST. This paper is organized as follows. In Section II we review the transformation of quantum mechanical states of massive particles under Lorentz transformations, in both flat and curved spacetime, and the consequences for entangled states. In Section III we review the current research into similar investigations for photons in flat spacetime. In Section IV we generalize the flat spacetime results of the previous section to CST and give specific examples of the Wigner rotation in the spherically symmetric Schwarzschild spacetime. In Section V we derive sufficient conditions for the existence of reference frames for observers to measure a null Wigner rotation. In Section VI we consider the consequences of the Wigner rotation on the entanglement of photon states and photon wavepackets in CST. In Section VII we present a summary and our conclusions. In the appendix we review the effect of the Wigner rotation on the rotation of the photon polarization in the plane perpendicular to its propagation direction in flat spacetime. By Einstien's equivalence principle (EP), which states that SR applies in the locally flat (Lorentz) tangent plane to a point x in CST, the flat spacetime examples presented have relevance when the Wigner rotation is generalized to photons and arbitrarily moving observers in CST. II. MASSIVE PARTICLES WITH SPIN A. Flat Spacetime In quantum field theory, the merger of quantum mechanics with SR, the particle states for massive particles are defined by their spin (as in non-relativistic mechanics) in the particle's rest frame, and additionally by their momentum. These two quantities are the Casimir invariants of the ten parameter Poincare group of SR which describes ordinary rotations, boosts and translations. The positive energy, single particle states thus form the state space for a representation of the Poincare group. For massive particles with spin j, the states are given by \| p, σ , where p is the spatial portion of the particle's 4momentum p µ , and σ ∈ {−j, −j + 1, . . . , j} are the components of the particle's spin along a quantization axis in the rest frame of the particle. For massless particles, the states are given by \| p, λ where λ indicates the helicity states of the particle (λ = ±1 for photons, λ = ±1/2 for massless fermions). Under a Lorentz transformation (LT) Λ the singleparticle state for a massive particle transforms under the unitrary transformation U (Λ) as [2] U (Λ)\| p, σ = σ ′ D (j) σ ′ σ (W (Λ, p)) \| − → Λp, σ ′ ,(1) where − → Λp are the spatial components of the Lorentz transformed 4-momentum, i.e. p ′ where p ′ µ = Λ µ ν p ν . In Eq.(1), D j σ ′ σ (W (Λ, p)) is a (2j + 1) × (2j + 1) matrix spinor representation of the rotation group O(3), and W (Λ, p) is called the Wigner rotation angle. The explicit form of the Wigner rotation in matrix form is given by W (Λ, p) = L −1 (Λp) · Λ · L(p),(2) where L(p) is a standard boost taking the standard rest frame 4-momentum k ≡ (m, 0, 0, 0) to an arbitrary 4momentum p, Λ is an arbitrary LT taking p → Λ · p ≡ Λp, and L −1 (Λp) is an inverse standard boost taking the final 4-momentum Λp back to the particle's rest frame. Because of the form of the standard rest 4-momentum k, this final rest momentum k ′ can at most be a spatial rotation of the initial standard 4-momentum k, i.e. k ′ = W (Λ, p) · k. The rotation group O(3) is then said to form (Wigner's) little group for massive particles, i.e. the invariance group of the particle's rest 4-momentum. The explicit form of the standard boost is given by [2] L 0 0 = γ = p 0 m L i 0 = p i m , L 0 i = − p i m , L i j = δ i j − (γ − 1) p i p j \| p \| 2 , i, j = (1, 2, 3),(3) where γ = p 0 /m = E/m ≡ e is the particles energy per unit rest mass. Note that for the flat spacetime metric η αβ =diag(1, −1, −1, −1), p 0 = p 0 and p i = −p i . Peres, Scudo and Terno [4] considered a free spin 1/2 particle in a normalizable state containing a distribution of momentum states. Under a LT, Eq.(1) indicates that each component of the momentum will undergo a different Wigner rotation, since the later is momentum dependent. Therefore, the reduced spin density matrix for the particle, obtained by tracing over the momentum states, will have a non-zero von Neumann entropy, which will increase as with the rapidity r of the observer with constant velocity v (with tanh r = v/c). This indicates that the spin entropy of the particle is not a relativistic scalar and thus has no invariant meaning. Alsing and Milburn [5] considered the transformation of Bell states of the form \|Ψ ± = [ \| p, ↑ \| − p, ↑ ± \| p, ↓ \| − p, ↓ ]/ √ 2, \|Φ ± = [ \| p, ↑ \| − p, ↓ ± \| p, ↓ \| − p, ↑ ]/ √ 2,(4) composed of pure momentum eigenstates. Under a Lorentz boost perpendicular to the motion of the particle, the transformed momentum p ′ is rotated by an angle θ with respect to the original momentum p, while the direction of the particle's spin is rotated slightly less by the momentum dependent Wigner angle Ω p < θ. The implication is that if the boosted observer were to orient his detectors along ± p ′ at angle ±θ there could be an apparent degradation of the violation of Bell inequalities in that inertial frame. However, if the observer were to orient his detectors along the direction of the transformed spins ±Ω p , there would again be a maximal violation of the Bell inequalities for these states in the boosted inertial frame. The entanglement of the complete bipartite state is preserved. Gingrich and Adami [6] considered a normalizable Bell state in which (analogous to Peres, Scudo and Terno) there is a distribution over momentum states. Considering a Bell state \|Φ + with the momentum in a product Gaussian distribution, a LT will transfer the spin entanglement into the momentum. If one forms the reduced 2-qubit spin density matrix, it will exhibit a Wootter's concurrence [7] which decreases with increasing rapidity. Because the quantum states contain two degrees of freedom, spin and momentum, the LT induces a spinmomentum entanglement again due to the momentum dependent Wigner rotation Eq.(1). Similar investigations were carried out for the case of photons [5,8,9], which we shall return to shortly. B. Curved Spacetime Subsequently, Terashima and Ueda [10] extended the definition of the Wigner rotation of spin 1/2 particles to an arbitrary gravitational field. The essential point in going from SR to GR is that there are in general no global inertial frames, only local frames. In fact, in GR all reference frames are allowed, locally inertial (i.e. zero acceleration trajectories -geodesics) or otherwise. Einstein's Equivalence Principle (EP) states that in an arbitrary curved spacetime, SR holds locally in the tangent plane to a given event in spacetime at the point x. The positive energy single particle states \| p(x), σ now form the state space for a local representation of the Poincare group in the locally flat Lorentz tangent plane at each spacetime point x. Of particular importance is the observer's local reference frame which can be described by set of four axes (4-vectors) called a tetrad eâ(x) withâ ∈ {0, 1, 2, 3} [11]. Three of these axesî ∈ {1, 2, 3} describe the spatial axes at the origin of the observer's local laboratory, from which he makes local measurements. The fourth axisâ = 0 describes the rate at which a clock, carried by the observer at the origin of his local laboratory, ticks (gravitational redshift effect), and is taken to be the observer's 4-velocity e0(x) ≡ u, where u is the tangent to the observer's worldline through spacetime. In curved spacetime (CST) described by coordinates x α , we can define coordinate basis vectors (CBV) e µ (x) with components e α µ (x) = δ α µ . We interpret the CBV e µ (x) as a local vector at the spacetime point x, pointing along the coordinate direction x µ . These are not unit vectors since their inner product yields the spacetime metric, e µ (x) · e µ (x) ≡ g αβ (x) e α µ (x) e β ν (x) = g µν (x). The observer's tetrad {eâ(x)} form an or-thonormal basis (ONB, denoted by carets over the indices) such that the inner product of any two basis vectors forms the flat spacetime metric of special relativity, eâ(x) · eb(x) ≡ g αβ (x) e α a (x) e β b (x) = ηâb where ηâb ≡ diagonal(1, −1, −1, −1) is the metric of SR. Note that e α a (x) are the components of the ONB vectors written in terms of the CBVs, eâ(x) = e α a (x) e α (x). That the metric g αβ (x) can be brought into the form ηâb locally at each spacetime point x (since the tetrad components e α a (x) are spacetime dependent) is an embodiment of Einstein's Equivalence Principle (EP) which states that SR holds locally at each spacetime point x in an arbitrary curved spacetime (CST). In Section IV B we will exhibit explicit examples of tetrads for various types of observers. Tensor quantities T γδ αβ (x) which "live" in the surrounding spacetime described by coordinates x α , (called world tensors, and denoted by Greek indices), transform according to general coordinate transformations (GCT) x α → x ′α (x), which simply describe the same spacetime in the new coordinates. World tensors can be projected down to the observer's local frame via the components of the tetrad and its inverse, i.e. Tĉd ab ( x) = e α a (x) e β b (x) eĉ γ (x) ed δ (x) T γδ αβ (x). Here, eâ µ (x) is the inverse transpose of the matrix of tetrad vectors e μ a (x), satisfying eâ(x) · eb(x) ≡ g αβ (x) eâ α (x) eb β (x) = ηâb, where g αβ (x) and ηâb are the inverses of the CST and flat spacetime metrics g αβ (x) and ηâb, respectively. We denote the observer's local components Tĉd ab (x) of the world tensor T γδ αβ (x) with hatted Latin indices. Objects with hatted indices transform as scalars with respect to GCTs, but as local Lorentz vectors with respect to local Loretnz transformations (LLT) Λ(x), which simply transform between different instantaneous local Lorentz frames, or instantaneous states of motion of different types of observers, at the point x (e.g. stationary, freely falling, circular orbit, etc. . . ). In particular, the local components pâ(x) of the world 4-momentum p α (x) = m u α (x) of a particle (with mass m and 4-velocity u α (x)) passing through the observer's local laboratory at the spacetime point x are given by pâ(x) = eâ α (x) p α (x) = eâ(x) · p(x). In SR, p SR = (E/c, p) is the 4-momentum of the particle, whose time component is the particle's energy and whose spatial components are its 3-momentum. In CST, p0(x) is energy of the particle with 4-momentum p as measured by the observer with tetrad {eâ(x)} as the particle passes through his local laboratory at the spacetime point x, while pî(x) are the locally measured components of the 3-momentum. Since GR allows for observer undergoing arbitrary motion (as opposed to SR which considers only zero acceleration or constant velocity observers, i.e. inertial frames), the observer's locally measured components of the 4-momentum pâ(x) depends upon the observer's state of motion, described by the motion of the axes com-prising the his local laboratory, which are given by his tetrad {eâ(x)}. As a field of vectors over the spacetime, we interpret the the tetrad {eâ(x)} as a collection of observer's located at each spacetime point x, usually of a particular type (stationary, freely falling, circular orbit, etc. . . ). We shall see explicit examples of different types of observers (tetrads) in Section IV B. As the passing (massive) particle moves in an arbitrary fashion from x α → x α ′ = x α + u α (x) dτ in infinitesimal proper time dτ , Terashima and Ueda found that the local momentum components pâ(x) would change under an infinitesimal LT, λâb(x) via pâ(x) → pâ In the instantaneous non-rotating rest frame of the particle for geodesic motion (zero local acceleration), Alsing et al [3] showed that the Wigner rotation would be zero. Further, considering the O(h) quantum correction to the non-geodesic motion of spin 1/2 particles in an arbitrary gravitational field the authors showed that in the instantaneous non-rotating rest frame of the accelerating particle (the Ferimi-Walker (FW) frame) the Wigner rotation would also be observed to be zero. ′ (x) = pâ(x)+δpâ(x) where δpâ(x) = λâb(x) pb(x) and Λâb(x) = δâb(x) + λâb(x) dτ III. PHOTONS A. Flat Spacetime In the above, we have primarily considered massive spin 1/2 particles. As briefly discussed above, analogous results have been obtained for photons in SR [5,8,9]. The fundamental difference between going from massive to massless particles is there is no rest frame for the latter. Instead, one defines a standard frame in which the photon 4-momentum takes the formk µ = (1, 0, 0, 1) where the photon travels in a predefined direction along the z-axis, and LT this to a photon of 4-momentum k of arbitrary energy k 0 and traveling in an arbitrary direction, subject to the null condition k · k ≡ k µ k µ = 0. The photon 4-momentum has the form k ↔ k µ = (k 0 , \| k\|k) where k is the spatial 3-momentum of the photon,k = k/\| k\| = n is the direction of propagation of the photon and ω ≡ k 0 = \| k\| is the frequency of the photon. Consequently, Wigner's little group for massless particles is SO(2), the group of rotations and translations in two dimensions associated with the particle's transverse plane of polarization. Under a LT Λ taking k → k ′ = Λ k the transformation of the photon helicity state \|k, λ , analogous to Eq.(1) is given by [8,9] U (Λ) \|k, λ = e i λ ψ(Λ,k) \|Λk, λ , where ψ(Λ, k) is the momentum-dependent Wigner rotation angle (phase). Due to the fact that k is a null 4vector, The Wigner angle depends only on the direction of propagation of the photonk , and not on its frequency ω, i.e. ψ(Λ, k) = ψ(Λ, n). Note that the unitary transformation in Eq.(5) does not change the helicity of the photon state, in contrast to the case for massive particles Eq.(1) in which the components of the spin are mixed up by a momentum dependent Wigner rotation. The corresponding transformation of the polarization vectors for positive and negative helicity states ǫ µ ± ǫ µ ± (k) = R(k) √ 2    0 1 ∓i 0    ,(6) is given by [8] ǫ ′ µ ± (k ′ ) ≡ D(Λ) ǫ µ ± (k) = R(Λk) R z (ψ(Λ, n)) R(k) −1 ǫ µ ± (k), = Λǫ µ ± (k) − (Λ ǫ µ ± (k)) 0 (Λ k µ ) 0 Λk µ .(7) where R(Λk) is the rotation taking the standard directionẑ tok. Here we use the (abused) shorthand notation Λk =k ′ = k ′ /\| k ′ \|. In the the appendix we illustrate several examples in flat spacetime of cases where the Wigner angle is zero, as well as cases in which it is non-zero, that will prove useful in our extension to curved spacetime below. B. Curved Spacetime In the following we extend the work of Terashima and Ueda [10] and Alsing et al [3] for massive spin 1/2 particles in curved spacetime to photons. Let k µ (x) be the photon 4-momentum, and e μ a (x) be the tetrad that defines the (timelike) observer's local laboratory. The local (covariant) components of the photon 4-momentum kâ(x), measured in the observer's laboratory, are given by projecting k µ (x) onto the observer's local axes via the tetrad, kâ(x) = e μ a (x) k µ (x). We are interested in the change δkâ(x) of the locally measured photon components as the photon moves from x µ → x ′ µ = x µ +k µ (x) δξ (where ξ is the affine parameter along the photon's trajectory). Following [3,10] we compute δkâ(x) = δe μ a (x) k µ (x) + e μ a (x) δk µ (x).(8) For the last term we have δk µ (x) = dξ ∇ k k µ (x) ≡ dξ k β (x)∇ β k µ (x) = 0,(9) where ∇ β is the Riemann covariant derivative [11] constructed from the spacetime metric. The last equality is simply the definition that the photon's trajectory is a geodesic ∇ k k = 0. Using the orthonormality of the tetrad 4-vector axes eb ν (x) e μ b (x) = δ µ ν , the first term in Eq.(8) becomes δe μ a (x) = dξ ∇ k e μ a (x) = dξ ∇ k e ν a (x) eb ν (x) e μ b (x), ≡ χb a (x) e μ b (x) dξ,(10) where we have defined the local matrix χb a (x) describing the rotation of the tetrad as χb a (x) ≡ ∇ k e ν a (x) eb ν (x).(11) Thus, the change in the local components kâ(x) of the photon's momentum as observed by the an observer with tetrad e μ a (x) is given by the LLT kâ(x) → k ′ a (x) ≡ kâ(x) + δkâ(x) = Λb a (x) kb(x) = δb a + λb a (x) dξ kb(x), (12) whose infinitesimal portion λb a (x) is given by δkâ(x) = χb a (x) kb(x) dξ ≡ λb a (x) kb(x) dξ.(13) Therefore, in CST the form of the transformation of a pure helicity photon state \|k, λ has the same form as Eq.(5) if we interpret k as the the photon wavevector as measured by the observer described by the tetrad eâ(x), i.e. with components k ↔ kâ(x) = eâ α (x) k α (x), and Λ a LLT as given in Eq.(12) for the infinitesimal motion of the photon along its geodesic from x µ → x ′ µ = x µ +k µ (x) δξ. IV. WIGNER ROTATION ANGLE FOR PHOTONS IN CURVED SPACETIME A. Derivation Since in the instantaneous non-rotating rest frame of the observer the metric is locally flat ηâb = (1, −1, −1, −1) we can follow the SR derivation of the Wigner phase ψ(Λ, k) by Caban and Rembielinski [9], with the arbitrary LT in curved spacetime given by Λâb(x) in Eq.(12) to first order in dξ. The authors' elegant derivation utilized the canonical homomorphism between SL(2, C) and the proper orthochronos homogeneous Lorentz group L ↑ + ∼ SO(1, 3). For every LLT Λ(x) ∈ L ↑ + of the photon 4-momentum kâ(x) → k ′â (x) ≡ Λâb(x) kb(x) there corresponds a transformation of the Hermetian matrix K(x) ≡ kâ(x)σâ of the form K(x) → K ′ (x) = A k ′ (x)K(x)A † k (x) where A k (x) ∈ SL(2, C) , σî are the constant Pauli matrices and σ0 is the 2×2 identity matrix. Henceforth we will drop the spacetime argument x on all local quantities for notational clarity. For the most general photon 4-momentum kâ = (k0, kî), K has the form K = k0 1 + n3 n − n + 1 − n3 ,(14) where nî = kî/k0 and n ± = n1 ± i n2. The most general element A ∈ SL(2, C) is of the form A = α β γ δ ,(15) where the entries are complex numbers subject only to the normalization condition detA = αδ − βγ = 1. We wish to relate these entries to the components of infinitesimal LLT λâb, which is antisymmetric λbâ = −λâb. Corresponding to the expression for the Wigner rota- tion W (Λ, k) = L −1 k ′ ΛL k of Eq.(1) with p → k, is the SL(2, C) transformation S(Λ, k) = A −1 k ′ AA k .(16) Here A k is the Lorentz boost that takes the standard photon 4-momentumkâ = (1, 0, 0, 1) to kâ = (k0, kî), A is an arbitrary LT Eq. (15), and A −1 k ′ is the inverse boost taking the transformed momentum k ′â back to the standard 4-momentumkâ. The most general form of S(Λ, k), the SL(2, C) element of Wigner's little group that leavesK invariant, is found by solvingK = SKS † 0 yielding S = e iψ/2 z 0 e −iψ/2 ,(17) where ψ is the Wigner angle and z is an arbitrary complex number. Following [9], we now compute S by the right hand expression in Eq.(16), using the expression for A in Eq. (15) for an arbitrary LT. The SL(2, C) element A k is obtained by solving K = A kK A † k and is given by A k = 1 2(1 + n3) 1 + n3 −n − n + 1 + n3 k0 0 0 1/ k0 ,(18) corresponding to the product of a boost in the z-direction takingk0 = 1 → k0, and a rotation takingẑ →k. Lastly, we can compute the corresponding element of A k ′ ∈ SL(2, C) which transforms K → K ′ by an arbitrary LT of the form Eq. (15). Here K ′ has the same form as Eq. (14) but with all quantities primed. The result is [9] K ′ = k ′0 1 + n ′3 n ′ − n ′ + 1 − n ′3 = A k ′ K A † k ′ ≡ k0 b c * c a − b ,(19) where we have defined a = (\|α\| 2 + \|γ\| 2 )(1 + n3) + (\|β\| 2 + \|δ\| 2 )(1 − n3) + (αβ * + γδ * )n − + (α * β + γ * δ)n + = a * b = \|α\| 2 (1 + n3) + \|β\| 2 (1 − n3) + αβ * n − + α * βn + = b * c = α * γ(1 + n3) + β * δ(1 − n3) + β * γn − + α * δn + . (20) From Eq. (19) and Eq.(20) we can deduce the transformation of the photon 4-momentum as k ′0 = a 2 k0, n ′3 = 2b a − 1, n ′ + = 2c a(21) where the quantities a, b, c in Eq.(20) depend only on the LT parameters α, β, γ, δ of A k and the direction of the photon n ± , n3, but not on the photon frequency k0. Forming the inverse of Eq.(19) and substituting it and Eq.(18) into the right hand side of Eq.(16), and subsequently equating the result to Eq.(17), yields e iψ(Λ,k)/2 = 1 a b(1 + n3) [α(1 + n3) + βn + ]b + [γ(1 + n3) + δn + ]c * .(22) Note that the expression for the arbitrary complex number z is not needed since it does not enter into the expression for the transformation of the photon helicity state Eq.(5). Finally, to relate the entries of Eq.(15) to the infinitesimal LLT λâb, we expand K ′ = AKA † in terms of kâ and k ′â ≡ Λâb kb and expand Λâb as in Eq.(12). Multiplying by a general Pauli matrix and using the relation where tr is the matrix trace. Since we are interested in an infinitesimal LLT we further expand A as A = α β γ δ ≡ I +Ã dξ,Ã = αβ γ −α ,(24) which satisfies det(A) = 1 to first order in dξ. A straightforward calculation leads tõ α = 1 2 λ03 − iλ12 , β = 1 2 λ01 + λ31 − i λ02 + λ23 γ = 1 2 λ01 − λ31 + i λ02 − λ23 ,(25) with λâb(x) = χâb(x) given in terms of the observer's tetrad by Eq.(11). Using the above expressions forÃ we can form the entries of A, and using the expression for a, b, c from Eq.(20) determine the infinitesimal Wigner rotation angleψ(Λ, n) when we expand Eq.(22) to O(dξ) as e iψ(Λ,k)/2 ∼ 1 + iψ(Λ, k) dξ/2.(26) Finite Wigner rotations can be built up as a time ordered integration of infinitesimal Wigner rotations over the geodesic trajectory x(ξ) of the photon (obtained by solving Eq. (9)) via exp iψ(Λ, n)/2 = T exp i ψ Λ, n(ξ) dξ/2 (27) where n(ξ) = n(x(ξ)), Λ µ ν (ξ) = Λ µ ν (x(ξ) ) and T is the time order operator. B. Examples of the Wigner rotation angle in the Schwarzschild metric We now consider some specific examples of the Wigner rotation angle ψ(Λ, k) as computed from Eq.(20) and Eq.(22) in the static, spherically symmetric Schwarzschild spacetime, and compare and contrast the results with SR, which holds locally at each spacetime point. The Schwarzschild metric is given by ds 2 = (1−r s /r) c 2 dt 2 − dr 2 (1 − r s /r) +r 2 (dθ 2 +r 2 sin 2 θ dφ 2 ) (28) where r s = 2GM/c 2 is the Schwarzschild radius of the central gravitating object of mass M (e.g. for the Earth r s⊕ = 0.89 cm, and for the Sun r s⊙ = 2.96 km). Henceforth, we use units where G = c = 1. Since the metric is independent of φ, orbital angular momentum is conserved [12], so without loss of generality we consider photon orbits in the equatorial plane θ = π/2. Radially infalling photons The 4-momentum for a radially in-falling photon (satisfying ∇ k k = 0) with zero orbital angular momentum is given by k µ (x) ≡ k t (x), k r (x), k θ (x), k φ (x) = ω 1/(1 − r s /r), −1, 0, 0 ,(29) satisfying k · k ≡ g µν k µ k ν = 0. The constant (of the motion) ω is the frequency of the photon as measured by a stationary observer (discussed below) at spatial infinity (r → ∞). We now consider several different types of (massive) observers defined by stating their associated tetrads. An obvious first choice is to consider a stationary observer who sits at fixed spatial coordinates (r, θ, φ ) whose tetrad is given by Since we are considering motion in the equatorial plane, we have oriented our coordinate system so that the local 3 axis points along the (increasing) radial direction, the2 axis points along the (increasing) φ direction, then the1 axis direction (normal to the equatorial plane) points in the (increasing) θ direction. Note that the world components of the tetrad vectors e α a (x) are given in the order α = (0, 1, 2, 3) ↔ (t, r, θ, φ), while the ordering of our observer's local axes are given in the orderâ = (0,1,2,3) ↔ (t,θ,φ,r) corresponding to the observer's local time and x,ŷ,ẑ axes. A stationary observer must exert an acceleration a defined by a = ∇ u u in order to oppose the gravitational attraction of the central mass M and remain at a fixed spatial location. (Note that in SR a stationary observer undergoes zero acceleration since there is no gravitational field, i.e. M = 0). In general, the magnitude of the local acceleration experienced by the observer in his local frame is a ≡ \|\|a\|\| ≡ (−a · a) 1/2 (where the minus sign results from the fact that a is a spacelike vector). For the stationary observer, the local acceleration is given by a stat = (M/r 2 ) (1 − r s /r) −1/2 , which approaches the usual Newtonian form of M/r 2 as r → ∞. The fact that lim r→rs a stat = ∞ indicates that for r < r s , i.e. inside the event horizon of the black hole, the observer can no longer remain stationary and is inexorably drawn into the singularity of the black hole. From the above tetrad we can compute the local components kâ(x) of the photon's 4-momentum k µ (x) as measured in the observer's local frame with kâ(x) = eâ µ (x) k µ (x). Here, eâ µ (x) is the inverse transpose of the matrix of tetrad vectors e μ a (x). The unit vector nî, used in Eq.(20) and Eq.(22), which describes the direction of the photon as measured in the observer's local frame is given by nî = kî/(kî kî) 1/2 where kî kî = 3 i=1 (kî) 2 is the ordinary Euclidean flat spacetime dot product of the spatial 3-vector portion kî of the local 4-vector kâ. For the radially infalling photon that we consider, we have nî = (0, 0, −1 ). The general formula for the Wigner rotation angle ψ(Λ, n) is given in Eq.(22). Since all relevant quantities in this formula are a function of the spacetime point x, we are interested in computing the infinitesimal Wigner rotation angleψ(Λ, k) as defined in Eq.(26). This requires the infinitesimal version of the entries {α, β, γ, δ} of the SL(2, C) matrix A in Eq.(24) which describes the LLT. We defined these infinitesimal entries as {α,β,γ,δ} through the SL(2, C) matrixÃ in the same equation, which described the infinitesimal LLT. From Eq.(25), {α,β,γ,δ} are related to the 4 × 4 infinitesimal LLT λâb, which from Eq.(13) is equal to the matrix χâb describing the rotation of the tetrads. Finally, χâb can be computed from Eq.(11) given the observer's tetrad and the photon 4-momentum. Thus, combining all these infinitesimal contributions into the right hand side of Eq.(22) and equating this to Eq.(26) allows us to extract the Wigner rotation angle ψ(Λ, n) to first order in dξ. Applying the above prescription to the radially infalling photon and the stationary observer leads to the result ψ(Λ, n) = 0. This is an expected result since as the photon moves along its radial geodesic it is continually boosted in the same direction. As in the flat spacetime case, an stationary observer in the path of the photon detects no Wigner rotation of the photon's polarization [9]. As discussed above, the stationary observer in a static metric undergoes a non-zero acceleration to remain at a fixed spatial location. The general relativistic observer that is locally analogous to the inertial (constant velocity) observer of SR is the freely falling frame (FFF). The FFF is defined by the tetrad eâ(x) satisfying the condition ∇ u eâ = 0.(31) Since e0(x) = u(x) is the 4-velocity of the observer's geodesic, theâ =0 equation in Eq.(31) is just the geodesic equation, stating that the FFF observer experience zero local acceleration \|\|a F F F \|\| = 0. The remaininĝ a =î equations state that the spatial tetrad axes eî(x) are parallel transported along the observer's geodesic. In the Schwarzschild metric, the tetrad for a radially FFF observer is given by (e F F F 0 ) µ (x) = (1 − r s /r) −1 , −(r s /r) 1/2 , 0, 0 = e F F F t , (e F F F 3 ) µ (x) = −(r s /r) 1/2 (1 − r s /r) −1 , 1, 0, 0 = e F F F r , (e F F F 1 ) µ (x) = (0, 0, 1/r, 0) = e F F F θ , (e F F F 2 ) µ (x) = (0, 0, 0, 1/r) = e F F F φ ,(32) Following the above prescription with the FFF tetrad, we again find that ψ(Λ, n) = 0. Since SR holds at each spacetime point, we could have invoked Einstien's equivalence principle to deduce this last result. Photon with nonzero angular momentum A general photon orbit in the Shcwarzschild metric obeys the radial "energy" equation [13] 1 b 2 ph = dr dξ 2 + W ef f (r), W ef f (r) = 1 r 2 1 − r s r ,(33) where ξ is the affine parameter along the photon geodesic, and k µ = dx µ /dξ. In Eq.(33), the quantity 1/b 2 ph acts as an effective energy. Here b ph = \|l ph /e ph \| is the ratio of the orbital angular momentum l ph and the energy e ph of the photon, both of which are constant since the Schwarzschild metric is independent of t and φ, respectively. For r ≫ r s the quantity b has the interpretation of the impact parameter of the photon with respect to M situated at r = 0. The most general infalling photon geodesic (starting at spatial infinity) in the equatorial plane (k θ = dθ/dξ = 0 ) is given by k µ (x) =   1 (1 − r s /r) , − 1 − b 2 ph (1 − r s /r) r 2 1/2 , 0, b ph r 2   ,(34) which reduces to the radial infalling photon in the equatorial plane Eq.(29), for b ph → 0. Unlike the flat spacetime case of SR, the orbit of photon in Eq.(33) is curved, and there even exits an unstable circular orbit for b 2 ph = 27M 2 for which W ef f has a maximum value. For stationary metrics, one can interpret the the bending of light in an optical-mechanical analogy in which the metric can be viewed as a spatially varying index of refraction, that takes on a value of unity at spatial infinity and is infinite at the event horizon [14]. For 1/b 2 ph < 1/(27M 2 ) there is a turning point in the photon orbit, and the photon will again escape to infinity. For 1/b 2 ph > 1/(27M 2 ) the photon spirals into the event horizon and is captured by the black hole. An observer (massive) traveling on a geodesic in the equatorial plane satisfies the corresponding radial "energy" equation [13] E ≡ e 2 obs − 1 2 = 1 2 dr dτ 2 + V ef f (r), V ef f (r) = − M r + − l obs 2r 2 − M l 2 obs r 3 ,(35) where the constants of the motion e obs = (1 − r s /r) dt/dτ and l obs = r 2 sin 2 θ dφ/dτ (here θ = π/2) can be interpreted, at large r, as the observer's energy per unit mass and orbital angular momentum per unit mass. Here τ is the observer's proper time defined from the metric Eq.(28) as c dτ = ds. In general, the FFF satisfying ∇ u eâ = 0 depends separately on both e obs and l obs . To keep the algebra manageable, in the following we will consider the observer's geodesic to lie in the equatorial plane (u θ = dθ/dτ = 0 with e obs = 1 but arbitrary orbital angular momentum l obs . This FFF tetrad, which we denote by e F F F (l) a (x) is given by (e F F F (l) 0 ) µ (x) = 1 (1 − r s /r) , u r , 0, l obs r 2 = e F F F (l) t , (e F F F (l) 3 ) µ (x) = − r s r cos Φ(r) (1 − r s /r) , − r s r u r cos Φ(r) − l obs (1 − r s /r) sin Φ(r) √ r s r , 0, l obs cos Φ(r) √ r s r 3 − u r sin Φ(r) √ r s r = e F F F (l) r , (e F F F (l) 1 ) µ (x) = (0, 0, 1/r, 0) = e F F F (l) θ , (e F F F (l) 2 ) µ (x) = − r s r sin Φ(r) (1 − r s /r) , l obs (1 − r s /r) cos Φ(r) √ r s r + r r s u r sin Φ(r), 0, − u r cos Φ(r) √ r s r + l obs sin Φ(r) √ r s r 3 = e F F F (l) φ ,(36) where the radial component of the observer's 4-velocity u r = dr/dτ and the angle of rotation Φ(r) of the spatial tetrads in the equatorial plane are given by u r (r) = − r s r − l 2 obs r 2 (1 − r s /r) 1/2 , dΦ(r) dr = − l obs 2 r 2 u r (r) . The photon and observer geodesics given by Eq.(34) and Eq.(36), respectively, both lie in the equatorial plane (θ = π/2). From the discussion in the appendix of the Wigner rotation in flat spacetime, we can invoke the EP to associate the observer's local (spatial tetrad) axes (1,2,3) with the inertial axesx,ŷ,ẑ used in the appendix to discuss the Wigner rotation in the flat spacetime of SR. For both the photon and observer geodesics in the equatorial plane2-3 (eφer) corresponding to the SR y-ẑ plane used in the appendix, the Wigner rotation is identically zero, ψ(Λ, n) = 0 (see the discussion in the appendix following Fig.(2)). This is borne out by a GR calculation utilizing Eq.(34) and Eq.(36). Following the appendix, to obtain a non-zero Wigner angle in flat spacetime, we considered the situation with the photon moving in thex-ŷ with the observer moving in theŷ-ẑ plane (see the discussion in the appendix of Fig.(4)). Invoking the EP this corresponds in the GR case to the photon's geodesic being in the1-2 or eθeφ plane, while the observer's geodesic remains in the equatorial plane,2-3, or eφer plane. We can modify the photon geodesic lying in the equatorial plane θ = π/2 Eq.(34), to a geodesic lying in the plane φ = π/2 with the expression k µ (x) =   1 (1 − r s /r) , − 1 − b 2 ph (1 − r s /r) r 2 1/2 , b ph r 2 , 0   , ≈ 1 (1 − r s /r) , −1, b ph r 2 , 0 + O(b 2 ph ).(38) In the last line of Eq.(38) we have expanded the photon 4-vector to first order in b ph , to keep the algebra manageable in the following calculation. Similarly, expanding Eq.(36) to first order in l obs we obtain (e F F F (l) 0 ) µ (x) = 1 (1 − r s /r) , − r s r , 0, l obs r 2 = e F F F t , (e F F F (l) 3 ) µ (x) = r s r 1 (1 − r s /r) , 1, 0, − 2l obs √ r s r 3 = e F F F r , (e F F F (l) 1 ) µ (x) = (0, 0, 1/r, 0) = e F F F θ , (e F F F (l) 2 ) µ (x) = −l obs r 1 (1 − r s /r) , l obs (2 − r s /r) √ r s r , 0, 1 r = e F F F φ ,(39) which satisfies the FFF tetrad conditions Eq.(31) and e · g · e T = η to correction terms of O(l 2 obs ). Computingψ(Λ, k) in Eq.(26) we find a non-zero (infinitesimal) Wigner rotation angle. ψ(Λ, k) = b ph l obs r 3 1 + 3 2 r r s − 5 4 r s r .(40) Note that the above result is proportional to both b ph and l obs . If b ph = 0, the photon would be radial in the plane φ = π/2 and would therefore only intersect the observer's curved geodesic in the equatorial plane θ = π/2 at r = 0. Similarly, if l obs = 0, the observer's geodesic would be radial in the equatorial plane, and intersect the photons curved geodesic again at r = 0. For both b ph and l obs non-zero we have the trajectories as illustrated in Fig.(1). Note that in general the photon and and observer's geodesic intersect at one spacetime point (the two-toned circle in the figure). The observer at this instant in time and this location, observing the photon traversing his local laboratory, will measure the nonzero infinitesimal Wigner angle in Eq.(40). V. GENERAL FRAMES IN WHICH THE WIGNER ROTATION ANGLE IS ZERO A. Conditions We now explore the condition under which the observer will observe zero Wigner rotation of the photon's In Schwarzschild coordinates x α = (t, r, θ, φ), the photon geodesic lies in the plane φ = π/2, while the observer's geodesic lies in the θ = π/2 equatorial plane. The photon geodesic has a nonzero impact parameter b ph , while the observer's geodesic has nonzero angular momentum l obs . The two-toned colored circle shows the intersection of photon and observer geodesics. Compare with the SR flat spacetime case illustrated in Fig.(4), which by the EP, holds at the intersection point. In the case of massive particles considered by Alsing et al [3], the photon momentum k would be replaced by the massive particle's 4-velocity u = e0. The corresponding condition ∇ u e ν a (x) = 0 defines the instantaneous nonrotating rest frame of the particle traveling on a geodesic (zero acceleration -if one ignores the particle's spin), i.e. the observer's local laboratory rides along with the passing particle. Since there is no rest frame for a photon, Eq.(41) describes something different, and it is not immediately obviously that a solution for the observer's local laboratory, described by the tetrad e ν a (x), exists. We have found solutions to Eq.(41), which describe a class of observers situated at each spacetime point x, which are in fact Fermi-Walker frames (FWF). The FWF is the instantaneous non-rotating rest frame of a particle experiencing arbitrary non-zero acceleration a = ∇ u u, and is defined by the following equation ∇ u s = (u · s) a − (a · s) u.(42) A vector s satisfying Eq.(42) is said to be Fermi Walker transported. If s = u, Eq.(42) reproduces the definition of the acceleration a. In general, s is any one of the three orthonormal spatial axes eî a (x) of the observer's tetrad that is orthogonal to u, and Eq.(42) is a generalization of parallel transport when a = 0. Equation (42) defines a locally non-rotating frame in the sense that if s is orthogonal to the instantaneous osculating plane defined by u and a then ∇ u s = 0, i.e. s is parallel transported along the world line with tangent u. B. Sufficient condition for existence of zero Wigner rotation angle frames That solutions of Eq.(41) can be found that also satisfy Eq.(42) is not readily obvious since the later FW transport equation is a statement solely about the observer and therefore is independent of the photon 4-momentum k. To show that solutions of Eq.(41) are compatible with Eq.(42) let s be any of the three orthonormal spatial vectors of the tetrad, and u = et be the observer's 4-velocity. Hence, u · u = 1, s · s = −1, and u · s = 0. Equation (27) is then restated as ∇ k e ν a (x) = 0, ⇒ ∇ k u = 0, ∇ k s = 0.(43) As a integrability condition, we take ∇ k of the FW transport equation Eq.(42) and use Eq.(43) repeatedly along with the orthogonality of u and s, noting that a = ∇ u u. This yields the equation ∇ k ∇ u s + [(∇ k ∇ u u) · s] u = 0.(44) Projecting Eq.(44) onto the the particle's 4-velocity by taking its dot product with u yields (∇ k ∇ u s) · u + (∇ k ∇ u u) · s = 0.(45) Equation (45) can be obtained independently by differentiating the orthogonality condition s · u = 0, first with respect to ∇ u and then with respect to ∇ k and making repeated use of Eq.(43). Similarly, by projecting Eq.(44) onto an arbitrary spatial vector s ′ of the tetrad we obtain ∇ k ∇ u s · s ′ = 0.(46) Since we could have equivalently written down Eq.(44) using s ′ and then subsequently projected onto s, Eq.(46) must also hold with these two vectors reversed, i.e. ∇ k ∇ u s ′ · s = 0.(47) Since s ′ · s = −1 if s ′ = s, and zero otherwise, applying first ∇ u and then ∇ k to the relation s ′ · s = constant, and again repeatedly using Eq.(43), yields ∇ k ∇ u s · s ′ + ∇ k ∇ u s ′ · s = 0.(48) One possible solution of Eq.(48) is that each separate term is identically zero as in Eq.(46) and Eq.(47). This shows that a solution of Eq.(41) has the FW transport equation Eq.(42) as a sufficient condition, though not a necessary condition. For example, in Schwarzschild spacetime we can explicitly solve for the tetrad for a radially accelerating FWF observer for case of a radially infalling photon, both in the equatorial plane such that ψ = 0 (though, as discussed in the appendix, the Wigner angle is identically zero in this case when both the photon and observer geodesics are in the equatorial plane). Finally, note that nowhere in the above argument have we made use of the fact that k is a null vector. VI. ENTANGLEMENT CONSIDERATIONS A. Photon Helicity States As discussed in Section III A, in CST the transformation of a photon state \|k, λ of pure helicity λ has the same form as the SR flat spacetime form U (Λ) \|k, λ = e i λ ψ(Λ, n) \|k ′ , λ ,(49) if we interpret k = (k0, \| k\| n) , ( n · n = 1) as the the photon wavevector as measured by the observer described by the tetrad eâ(x), i.e. with components k ↔ kâ(x) = eâ α (x) k α (x), and Λ a LLT as given in Eq.(12) kâ(x) → k ′â (x) ≡ kâ(x) + δkâ(x) = Λâb(x) kb(x) = δâb + λâb(x) dξ kb(x), for the infinitesimal motion of the photon along its geodesic from x µ → x ′ µ = x µ + k µ (x) δξ. As discussed earlier, the Wigner rotation angle is a function of the LLT λâb(x), the direction of propagation n of the photon, but not its frequency ω = \| k\|, i.e. ψ = ψ(Λ, n). Eq.(49) holds for an infinitesimal LLT with ψ(Λ, n) computed from Eq.(22) and Eq.(26). For finite motion of the photon along its geodesic from x µ → x ′ µ the time ordered expression for the the Wigner angle in Eq.(27) must be employed, and Eq.(49) generalized to U (Λ) \|kâ(x), λ = T [e iλ Rψ (Λ(ξ), n(ξ))dξ ] × \| T [e Rλâb (Λ(x(ξ)), k(ξ))dξ ] kb(x), λ .(50) The term outside the ket is the time ordered product of the infinitesimal Winger rotationψ(Λ(ξ), n(ξ)) along the photon's geodesic parameterized by ξ, while the term inside the ket is the time ordered product of the LLT λâb Λ(x(ξ)), k(ξ) along the observer's geodesic through which the photon passes. The time ordered integration is over the extent of ξ for which the photon passes through the observer's local laboratory. A more relevant way to interpret Eq.(50) is as follows. Rather than considering a single observer with tetrad eâ(x) and following his motion through spacetime , we consider eâ(x) as describing an infinite set of observers distributed throughout spacetime at points x (e.g. the class of stationary observers with tetrad Eq.(30), each at a fixed spatial location). Then, as the photon passes through the local laboratories of this set of observers with tetrad eâ(x),λâb Λ(x(ξ)), k(ξ) describes the LLT applied to each local laboratory at x, as the observer measures the local 4-momentum kâ(x) of the photon at x. Thus, the evolution of the photon helicity state in Eq.(50) is that measured by this class of observers (vs a single observer). Suppose we have a bipartite photon helicity Bell state at the spacetime point x of the form \|Φ(x) = \|kâ 1 (x), λ 1 \|kb 2 (x), λ 2 ± \|kâ 1 (x), λ 2 \|kb 2 (x), λ 1 . (51) As a specific example, one might consider two photons of helicity λ 1 and λ 2 , one traveling inward and the other traveling outward along a common geodesic (e.g. a ± k r (x) in Eq.(38)). The infinitesimal evolution of this state along the photon's trajectories is then U (Λ) \|Φ(x) = e iλ1 ψ(Λ, n1) \|k ′â 1 (x), λ 1 e iλ2 ψ(Λ, n2) \|k ′b 2 (x), λ 2 ± e iλ2 ψ(Λ, n1) \|k ′â 1 (x), λ 2 e iλ1 ψ(Λ, n2) \|k ′b 2 (x), λ 1 , = \|k ′â 1 (x), λ 1 \|k ′b 2 (x), λ 2 ± e −i(λ1−λ2) (ψ(Λ, n1)−ψ(Λ, n2)) \|k ′â 1 (x), λ 2 \|k ′b 2 (x), λ 1 ,(52) where in the last line we have dropped an overall phase. As we let these photons separate macroscopically we have to apply the time ordering operations as in Eq.(50) U (Λ) \|Ψ(x) = T [e iλ1 Rψ (Λ(ξ), n1(ξ))dξ ] \| T [e Rλâĉ (Λ(ξ), k(ξ))dξ ] kĉ 1 (x), λ 1 T [e iλ2 Rψ (Λ(ξ), n2(ξ))dξ ] \| T [e Rλbd (Λ(ξ), k(ξ))dξ ] kd 2 (x), λ 2 ± T [e iλ2 Rψ (Λ(ξ), n1(ξ))dξ ] \| T [e Rλâĉ (Λ(ξ), k(ξ))dξ ] kĉ 1 (x), λ 2 T [e iλ1 Rψ (Λ(ξ), n2(ξ))dξ ] \| T [e Rλbd (Λ(ξ), k(ξ))dξ ] kd 2 (x), λ 1(53) The lesson of the above expression is that over some macroscopic (non-infinitesimal) evolution of the photon trajectories, the relative phase between the product photon states (as in Eq.(52)) depends on the class of local laboratories (tetrads) of the observers that the photons pass through. That is, we need to know the set of observers along the trajectory of the photons in order to determine the relative phase, since the state of motion of these observers (tetrads) determines the locally measured components of the direction of the photon n 1 (x(ξ)) and n 2 (x(ξ)), and hence the observer measured infinitesimal Wigner rotation anglesψ(Λ(ξ), n j (ξ)) at each spacetime point x(ξ). For example, as discussed in Section IV B, for any motion of both the photon and observer in the equatorial plane, the phase factor in Eq.(52) is unity and the Bell state in Eq.(52) and Eq.(53) retains its original form of Eq.(51). B. More General Photon States In the above we have considered pure photon helicity states with a definite 4-momentum k ↔ kâ. In general, we can form wave packets states over a distribution of 4momenta, and a linear combination of the helicities. The most general photon state has the form [15] \|ψ = λ=±1 d k ψ λ (k) \|k, λ ,(54)dk = 1 (2π)3 θ(k0)δ (3) ( k − k ′ ) δ λ ′ λ = d 3 k (2π) 3 2k0 , λ=±1 d k \|ψ λ (k)\| 2 = 1. k ′ , λ ′ \|k, λ = (2π) 3 2k0 δ (3) ( k − k ′ ) δ λ ′ λ ≡δ (3) ( k − k ′ ) δ λ ′ λ , wheredk is the invariant Lorentz integration measure, and we have used the covariant normalization convention for the inner product of helicity states. A linear polarization state (LPS) of definite 4-momentum k = (k0, \| k\| n) (where ω = \| k\|) and polarization angle φ is given by a linear combination of the two photon helicity states of the form \|k, φ ≡ \|(k0, \| k\| n), φ = 1 √ 2 λ=±1 e iλφ \|k, λ .(55) In Eq.(55) the polarization angle φ is defined (as described in detail in the appendix) in the standard photon frame in which the photon propagates along theẑaxis and the transverse polarization vectors lie in thex-ŷ plane, one with angle φ with respect to thex-axis, and the other with angle φ + π/2. The polarization angle φ is independent of the photon 4-momentum. The most general LPS with definite direction n is given by the photon wave packet \|g, φ, n = 1 √ 2 λ=±1 e iλφ d\| k\| g(\| k\|) \|(k0, \| k\| n), λ . We can form a reduced helicity density matrix of the state \|g, φ, n by projecting out the momenta ρ red (g, φ, n) = d k k\|g, φ, n g, φ, n\|k = λλ ′ 1 2 e i(λ−λ ′ )φ \|λ λ ′ \| = 1 2 1 e i2φ e −i2φ 1 ,(57) where the rows and columns of the matrix are labeled by the helicity index in the order λ = {1, −1}. In flat spacetime, Caban and Rembielinski [9] showed that under a SR LT this LPS remains a LPS, with a new polarization angle φ ′ = φ + ψ(Λ, n) via U (Λ)\|g, φ, n = \|g ′ , φ, n ′ , g ′ (\| k\|) = 2 a g 2\| k\| a , = 1 √ 2 λ=±1 e iλ(φ+ψ(Λ, n)) d\| k\| g ′ (\| k\|) \|(k0, \| k\| n ′ ), λ ,(58) where n and g(\| k\|) are the original photon propagation direction and wave packet momentum distribution, and n ′ and g ′ (\| k\|) are the corresponding transformed quantities. The authors showed that the reduced helicity density matrix for transformed state U (Λ)\|g, φ, n transforms properly under LTs, i.e. ρ ′ red = U (Λ)ρ red (g, φ, n)U † (Λ) = 1 2 1 e i2(φ+ψ(Λ, n)) e −i2(φ+ψ(Λ, n)) 1 = ρ red (g ′ , φ + ψ(Λ, n)), n ′ ), U (Λ) λλ ′ = e iλψ(Λ, n) δ λλ ′(59) The fact that the LPS admits a covariant description of the reduced density matrix in terms of helicity degrees of freedom is related to the fact that LTs do not create entanglement between the helicity and momentum directions [9], as indicated by the matrix elements of U (Λ) in the last line of Eq.(59) arising from fundamental transformation law Eq.(49). This is very different from the situation for massive particles in which the action of U (Λ) Eq.(1) does entangle momentum and spin [6]. These considerations remain true in CST, where by the EP, SR holds locally at each spacetime point if we interpret kâ(x) = eâ α (x) k α (x) as the components of the photon 4-momentum as measured by an observer at x with tetrad eâ(x). The transformation of g → g ′ in Eq.(58) arose from applying the unitary transformation Eq.(49) to the helicity states, making a change of integration variable from \| k\| to \| k ′ \| and using the frequency transformation law k ′0 = ak0/2 from Eq.(21) (with a final relabeling of \| k ′ \| to \| k\|). Note that the transformed polarization angle φ ′ = φ + ψ(Λ, n) involves the Wigner angle ψ(Λ, n) evaluated at the original propagation direction of the photon n, while the kets on the right hand side are evaluated at the transformed photon direction n ′ . Since ψ(Λ, n) depends only on the direction of the photon and not on its frequency ω = \| k\|, the phase factor e iλ ψ , resulting from the unitary transformation of the photon helicity states, can be pulled outside the integral. A general two particle photon state takes the form \|Ψ = λ1,λ2 d k 1d k 2 g λ1λ2 (k 1 , k 2 )\|k 1 , λ 1 \|k 2 , λ 2 , λ1,λ2 d k 1d k 2 \|g λ1λ2 (k 1 , k 2 )\| 2 = 1.(60) For the choice of the distribution function g λ1λ2 (k 1 , k 2 ) g λ1λ2 (k 1 , k 2 ) = 1 √ 2 e iλ1φ1 e iλ2φ2 δ λ1λ2 f (k 1 , k 2 ),(61) where φ 1 and φ 2 constant, momentum independent polarization angles, the state \|Ψ is fully entangled in helicity and entangled in momentum if f is non-factorizable, i.e. f (k 1 , k 2 ) = f 1 (k 1 ) f 2 (k 2 ). In analogy with the single particle state Eq.(57), the helicity reduced density matrix obtained by tracing the pair of momenta is ρ red (φ 1 , φ 2 ) = 1 2     1 0 0 e i2(φ1+φ2) 0 0 0 0 0 0 0 0 e −i2(φ1+φ2) 0 0 1     ,(62)ρ ′ red (φ 1 , φ 2 ) = 1 2    1 0 0 (ρ ′ red ) 11,−1−1 0 0 0 0 0 0 0 0 (ρ ′ red ) * 11,−1−1 0 0 1    , (ρ ′ red ) 11,−1−1 = d k 1d k 2 \|f ′ (k 1 , k 2 )\| 2 e i2Φ( n1, n2) Φ( n 1 , n 2 ) = φ 1 + ψ(Λ, n 1 ) + φ 2 + ψ(Λ, n 2 ).(63) In Eq.(63) f ′ is the transformed distribution function (analogous to g ′ in Eq.(58)) that depends on the untransformed frequencies \| k j \|, but the transformed directions n ′ j . Sincedk j ∝ d 3 k j = δΩ nj d\| k j \| \| k j \| 2 involves an integration over the untransformed photon directions n j , (where n ′ j and n j are related by Eq.(21)), the integral over δΩ nj is in general very complicated, mixing up the photon directions, but again without entangling with the helicity. Without the factors of ψ nj in the argument of the phase of (ρ ′ red ) * 11,−1−1 Eq.(63) reduces to Eq.(62), showing that analogous to Eq.(59), the reduced helicity density matrix ρ ′ red (φ 1 , φ 2 ) transforms covariantly under LLTs. In CST, it is possible that the observer (tetrad) changes for each LLT along the trajectory of the photons, from which the local Wigner angles ψ ni (x(ξ)) are measured by (massive) observers. VII. SUMMARY AND CONCLUSIONS The Wigner rotation for a photon can be envisioned as the rotation of the transverse linear polarization vectors, in the plane perpendicular to the direction of propagation of the photon, resulting from a Lorentz transformation Λ between observers. The natural quantum state description of the photon is in terms of helicity states \|k, λ , λ = ±1, k = (k0, \| k\| n), in which the corresponding induced unitary transformation U (Λ) introduces a phase factor, dependent upon Λ and the propagation direction of the photon n, without changing the helicity, i.e. U (Λ)\|k, λ = e iλψ(Λ, n) \|k ′ , λ . In the flat spacetime of special relativity Λ transforms between a special class of observers, namely inertial observers for which the acceleration of the observer is zero (constant velocity observers). Such observers are global in the sense that they are position independent and exist over the whole of the flat spacetime. In going to curved spacetime (CST) where general relativity applies, all types of observers, in arbitrary states of motion, are allowed. The motion of these observers is now reduced to a local description, encapsulated in an orthonormal tetrad eâ(x) that describes the four axes (three spatial and one temporal) that defines the observer's local laboratory at the spacetime point x from which he makes measurements. For example, the photon 4-momentum k α (x) existing in a CST described by coordinates x α , has components kâ(x) in the observer's local laboratory given by kâ(x) = eâ α (x) k α (x). By the equivalence principle, the laws of special relativity apply in this local laboratory (local tangent plane to the curved spacetime), at the spacetime point x. Therefore, we can compute the Wigner rotation angle in CST by applying the calculational procedure appropriate for flat spacetime to the observer's instantaneous local laboratory. The quantum state of the photon is described by the local helicity state \|kâ(x), λ and the observer by the tetrad eâ(x). The instantaneous Wigner rotation angle ψ(Λ, n), arising from a local Lorentz transformation as the photon traverses infinitesimally along its geodesic, now depends on the propagation direction of the photon ( n)î = kî/\| k\| as measured locally by the observer at x. In this work we have developed the local Wigner rotation for photons in an arbitrary CST. We have given specific examples in the case of Schwarzschild spacetime and compared these with the results from flat spacetime. The difference in the CST case is that an explicit description of the observer, via his tetrad, is needed to compute the local Wigner angle. That is, the locally measured Wigner rotation angle is observer dependent, which we develop explicitly. In terms of a local helicity state de-scription of the quantum photon states, the induced local Lorentz transformation that gives rise to the local Wigner angle as the photon traverses its geodesic, does not entangle the photon direction with the helicity, in contrast to the case for spin-momentum entanglement that occurs for massive particles. We have also developed a sufficient condition for observers who would measure zero Wigner rotation and have shown that such observers can be in Fermi-Walker frames, i.e. the instantaneous non-rotating rest frame of the accelerating observer. velocity of the frame we are transforming to and ξ defined by tanh ξ = −\| v\|/c is the rapidity parameter of the boost. We take the two transverse polarization vectors to lie along thex andŷ axes, so that thex polarization vector has a polarization angle φ = 0. The boost rotates the propagation vectork and thex polarization vector in thex-ẑ plane counterclockwise about theŷ-axis by some boost dependent angle ϑ [5,8], leaving theŷ polarization vector unchanged. To determine φ ′ we rotatek ′ back to theẑ-axis clockwise about theŷ-axis, undoing the original rotation, and thus returning all the vectors to their original orientations. Therefore, φ ′ = φ and thus the Wigner angle is zero. It is straightforward to show that for the photon traveling along theẑ-axis, a boost along any direction yields a zero Wigner angle. In general fork not alongẑ, we can determine the polarization angles φ and φ ′ as follows. Given a photon propagating along the directionk with polarization vectors in a plane perpendicular to this vector, we find the polarizations vectors in the standard frame by applying the rotation Rk ×ẑ (θ) to the triad, where cos θ =k 3 which takesk →ẑ. A pure boost along theê rotates the photon propagation direction by a boost dependent angle ϑ counterclockwise about the axisê ×k,k ′ = Rê ×k (ϑ)k ′ . To determine the transformed polarization angle φ ′ we rotatek ′ counter clockwise along the directionk ′ ×ẑ by the angle θ ′ where cos θ ′ =k ′ 3 , which takesk ′ →ẑ, and the transformed polarization vectors to thex-ŷ plane. From the above discussion we can also infer that the Wigner angle is zero ifê andk both lie in thex-ẑ plane, or both lie in theŷ-ẑ plane. In the latter case, if the first polarization vector also lies in theŷ-ẑ plane orthogonal tok, and the second polarization vector lies along thê x-axis, the a boost in theŷ-ẑ plane will rotate the triad (k,ǫ 1 ,ǫ 2 =x) about thex-axis. Therefore, when we undo this rotation about thex-axis in order to calculate φ ′ , the triad is returned to its original orientation, so that φ ′ = φ, implying a zero Wigner angle, analogous to Fig.(2). However, ifê andk both lie in thex-ŷ plane, the Wigner angle is non-zero, as illustrated in Fig.(3) for the case of (k=x,ǫ 1 =ŷ,ǫ 2 =ẑ). Here the initial polarization angle forǫ 1 is φ = π/2. For a boost in thex-ŷ plane, as indicated in the figure,k is rotated counter clockwise about theẑ-axis by the boost dependent angle ϑ, yielding a transformed polarization vectorǫ ′ 1 oriented at angle φ ′ = π/2−ϑ with respect to thex-axis. Upon rotatingk ′ (aboutǫ ′ 1 ) to theẑ-axis,ǫ ′ 1 is left invariant, maintaining the relation φ ′ = π/2 − ϑ, or equivalent, a Wigner angle of ψ(Λ, n) = −ϑ. In Fig.(4) we illustrate one last case that is relevant to the discussion in the main body of the text when we consider an example of a non-zero Wigner angle in curved Schwarzschild spacetime. Here in flat spacetime, we consider the case when the photon directionk is in thex-ŷ plane at some angle ϕ to thex-axis, and the boost directionê is in theŷ-ẑ plane at some polar angle θ with respect to theẑ-axis. We choose the first polarization vector to lie along theẑ-axis, and the second polarization vector to lie in thex-ŷ orthogonal tok with polarization angle φ = ϕ − π/2, A boost alongê induces a rotation of the triad (k,ǫ 1 =ẑ,ǫ 2 ) about the axis (ê ×k orthogonal to thek-ê plane by some boost dependent angle ϑ. This pushesk ′ below thex-ŷ plane and pullsǫ ′ 2 above thex-ŷ plane. Upon rotatingk ′ back to theẑ-axis by a rotation about the directionk ′ ×ẑ, ǫ ′ 2 is returned to thex-ŷ plane asǫ ′ 2 . For a boost of infinitesimal angle δϕ as illustrated in Fig.(4), the polarization angle of the transformed polarization vector is φ ′ = φ + ψ(Λ, n) with the non-zero Wigner angle given to O(δϑ) as ψ(Λ, n) = δϑ sin θ cos ϕ = δϑk 3 . tr σbσĉA + σĉσbA † , online) Example of a nonzero Wigner angle measured by an observer in Schwarzschild spacetime. polarization. From Eq.(13) the change δkâ(x) in the local components of the photon wavevector is zero if λb a (x) = χb a (x) = 0. From Eq.(11) a sufficient condition for this to occur is ∇ k e ν a (x) = 0. (41) where the normalization condition for g λ1λ2 in Eq.(60) has been used, and the rows and columns of the matrix are labeled by the double helicity indices λ 1 λ 2 = {11, 1− 1, −11, −1−1}. Under the action of an infinitesimal LLT U(Λ) = U (Λ) ⊗ U (Λ) the reduced helicity density matrix for the state U(Λ) \|Ψ is online) Example of a zero Wigner angle ψ: photon propagation directionk along the z-axis, boostê along the z-axis. In fact, the Wigner angle ψ is identically zero for any boost directionê if the photonk travels along the z-axis. online) Example of a non-zero Wigner angle ψ: photon propagation directionk along the x-axis, boostê in thex-ŷ plane. AcknowledgmentsPMA wishes to acknowledge the support of the Air Force Office of Scientific Research (AFOSR) for this work.APPENDIX A: WIGNER ROTATION IN FLAT SPACETIME: EXAMPLESIn this appendix we give explicit examples illustrating the operational meaning of the Wigner rotation in flat spacetime in terms of its effect on the polarization vectors for photons.As given in Eq.(6), the polarization vectors for positive and negative helicity states ǫ µ ± (k) (right and left circular polarization) with propagation 4-vector k = (k0, \| k\| n) are given bywith the components of the column vector labeled by the Cartesian coordinates x µ = (t, x, y, z). Here R(k) is the rotation that takes the standard directionẑ-axis to the photon propagation directionk = k/\| k\| (= n). Under a LT Λ, the polarization vector transforms as ǫ µHere we use the typical "abuse of notation" denoting Λk for the transformed photon directionk ′ = k ′ /\| k ′ \|, where k ′ is the 3-vector portion of the transformed photon 4momentum k ′ µ = Λ µ ν k ν . Thus R(Λk) is the rotation taking the standard directionẑ tok ′ .From Eq.(A1) we can construct a linear polarizationwhere the polarization angle φ is defined by the angle the LPVǫ µ φ (ẑ) makes with thex-axis when the photon propagates along theẑ-axis in this standard frame, i.e. k =ẑ (denoted by tildes over vectors). The LPV in the standard frameǫ µ φ (ẑ) is obtained by rotating the polarization vector ǫ µ φ (k) propagating in the directionk, by the transformation that takesk back toẑ, i.e. by the rotation R −1 (k).After a LT Λ, the new LPV is given byAgain, the transformed LPVǫ ′µ φ ′ (ẑ) in the standard frame (k ′ =ẑ) is obtained by rotating the polarization vector ǫ ′µ φ ′ (Λk) propagating in the directionk ′ , by the rotation that takesk ′ back toẑ, i.e. R −1 (Λk). The angle that ǫ ′µ φ ′ (ẑ) makes with thex-axis in the standard frame defines the transformed polarization angle φ ′ .By multiplying Eq.(A2) by R −1 (Λk) and inserting Eq.(A4) and Eq.(A5) we obtaiñwhich upon comparing the arguments of the trigonometric functions yieldsThis states that the effect of a LT Λ is a rotation of the polarization angle φ → φ ′ in the standard frame by the Wigner angle ψ(Λ, n), i.e. a rotation of the standard polarization vectors once we bring them back to the standard frame in which the photon propagates along thê z-axis, (k =ẑ), and the standard polarization vectors lie in thex-ŷ plane. It is in this standard frame that we most easily measure the polarization angles φ and φ ′ and therefore determine the Wigner angle ψ(Λ, n). In the following we illustrate a few sample cases in which the Wigner angle is zero, and non-zero. InFig.(2)we consider the photon to be traveling along theẑ-axis, and consider a boost along thex-axis. We denote the boost direction byê ≡ v/\| v\| where v is the : angle on polarizati the Christoffel connection Γ λ αβ (x) is obtained from the metric as Γ λ αβ = 1 2 g λσ (∂α g βλ + ∂ β g αλ − ∂ λ g αβ ).[12] The force free (zero acceleration) equations of motion of a (massive or massless) particle can be obtained from the Euler-Lagrange equations d/dσ(∂L/∂ẋ α ) − ∂L/∂x α = 0, where L = ds = (g αβ (x)ẋ αẋβ ) 1/2 is the effective Lagrangian, andẋ a = dx α /dσ with σ the affine paramter along the particle's geodesic trajectory. If the metric is independent of the coordinate x α 0 ≡ xo, the quantity ∂L/∂ẋo is a constant of the motion along the geodesic. The latter expression can be written as ζ · u = constant where u α =ẋ α is the 4-velocity of the particle, i.e. the tangent to its geodesicc, and ζ α = δ α xo is called a Killing vector, and reprsents an isometry of the metric. The Schwarzschild metric Eq.(28) is independent of the coordinates t and φ. The quantity e = ζ · u = (1 − rs/r)ṫ for ζ α = δ α t represents the particle's energy (per unit mass for m = 0), and l = ζ · u = r 2 sin 2 θφ for ζ α = δ α . A Peres, D R Terno, Rev. Mod. Phys. 7693A. Peres and D.R. Terno, Rev. Mod. Phys. 76, 93 (2004). S Weinberg, Quantum Theory of Fields. CambridgeCambridge Univ. PressIS. Weinberg, Quantum Theory of Fields: Vol-I, Cam- bridge Univ. Press, Cambridge, 62-74 (1995). Spininduced non-geodesic motion, gyroscopic precession, Wigner rotation and EPR correlations of massive spin-1 2 particles in a gravitational field, submitted to. P M Alsing, G J StephensonJr, P Kilian, arXiv:0902.1396Phys. Rev. A. quant-phP.M. Alsing, G.J. Stephenson Jr. and P. Kilian Spin- induced non-geodesic motion, gyroscopic precession, Wigner rotation and EPR correlations of massive spin-1 2 particles in a gravitational field, submitted to Phys. Rev. A, February (2009); arXiv:0902.1396 [quant-ph]. . A Peres, P F Scudo, D R Terno, Phys. Rev. Lett. 88230402A. 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Freeman and Co., San Francisco, CA, 207-216, 327-332 (1973); ∇ β kµ = ∂ β kµ−Γ λ φ represents the particle's orbital angular momentum (per unit mass for m = 0). S M Carroll, Spacetime, Addison Geometry, San Wesley, Francisco, Ca, S.M. Car- roll, Spacetime and Geometry, Addison Wesley, San Fran- cisco, CA, 483-494 (2004). ∇ β kµ = ∂ β kµ−Γ λ φ represents the particle's orbital angular momentum (per unit mass for m = 0). . James B Hartle, Gravity, Addison-Wesley, San Francisco, CAJames B. Hartle, Gravity, Addison-Wesley, San Fran- cisco, CA, 193-204, 298-302, 440-441 (2003). The Classical Theory of Fields, 4th. L D Landau, E M Lifshitz, Pergamon PressN.Y.L.D. Landau and E.M. Lifshitz, The Classical Theory of Fields, 4th. Ed., Pergamon Press, N.Y., 291-294 (1975); . J Evans, K K Nandi, A M Islam ; P, ; J Alsing, P M Evans, S Alsing, K K Giorgetti, Nandi, Gen. Rel. and Grav. P.M. Alsing, J.C. Evans and K.K. Nandi64Am. J. Phys.J. Evans, K.K. Nandi and A. Islam, Am. J. 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[ "QUANTITATIVE COMPARISONS OF MULTISCALE GEOMETRIC PROPERTIES", "QUANTITATIVE COMPARISONS OF MULTISCALE GEOMETRIC PROPERTIES" ]	[ "Jonas Azzam ", "Michele Villa " ]	[]	[]	We generalize some characterizations of uniformly rectifiable (UR) sets to sets whose Hausdorff content is lower regular (and in particular, do not need to be Ahlfors regular). For example, David and Semmes showed that, given an Ahlfors d-regular set E, if we consider the set B of surface cubes (in the sense of Christ and David) near which E does not look approximately like a union of planes, then E is UR if and only if B satisfies a Carleson packing condition, that is, for any surface cube R,We show that, for lower content regular sets that aren't necessarily Ahlfors regular, if β E (R) denotes the square sum of β-numbers over subcubes of R as in the Traveling Salesman Theorem for higher dimensional sets [AS18], thenWe prove similar results for other uniform rectifiability critera, such as the Local Symmetry, Local Convexity, and Generalized Weak Exterior Convexity conditions. En route, we show how to construct a corona decomposition of any lower content regular set by Ahlfors regular sets, similar to the classical corona decomposition of UR sets by Lipschitz graphs developed by David and Semmes.CONTENTS2010 Mathematics Subject Classification. 28A75, 28A78, 28A12.	10.2140/apde.2021.14.1873	[ "https://arxiv.org/pdf/1905.00101v2.pdf" ]	141,481,801	1905.00101	a76c77a2df3c8f429361bf50f4833615f827be90	QUANTITATIVE COMPARISONS OF MULTISCALE GEOMETRIC PROPERTIES 18 Jun 2020 Jonas Azzam Michele Villa QUANTITATIVE COMPARISONS OF MULTISCALE GEOMETRIC PROPERTIES 18 Jun 2020 We generalize some characterizations of uniformly rectifiable (UR) sets to sets whose Hausdorff content is lower regular (and in particular, do not need to be Ahlfors regular). For example, David and Semmes showed that, given an Ahlfors d-regular set E, if we consider the set B of surface cubes (in the sense of Christ and David) near which E does not look approximately like a union of planes, then E is UR if and only if B satisfies a Carleson packing condition, that is, for any surface cube R,We show that, for lower content regular sets that aren't necessarily Ahlfors regular, if β E (R) denotes the square sum of β-numbers over subcubes of R as in the Traveling Salesman Theorem for higher dimensional sets [AS18], thenWe prove similar results for other uniform rectifiability critera, such as the Local Symmetry, Local Convexity, and Generalized Weak Exterior Convexity conditions. En route, we show how to construct a corona decomposition of any lower content regular set by Ahlfors regular sets, similar to the classical corona decomposition of UR sets by Lipschitz graphs developed by David and Semmes.CONTENTS2010 Mathematics Subject Classification. 28A75, 28A78, 28A12. A set E ⊆ R n is said to be d-rectifiable if it can be covered up to Hausdorff d-measure zero by Lipschitz images of R d . While classifying rectifiable sets is a classical problem dating back to Besicovitch, starting in the late 80's, geometric measure theorists and harmonic analysts began to study rectifiability in a quantitative manner with an eye on applications to harmonic analysis, particularly singular integrals, analytic capacity, and harmonic measure. Much of this work has focused on classifying when Ahlfors regular sets are uniformly rectifiable, which was initiated by David and Semmes in their seminal texts [DS91,DS93]. Recall that a set E ⊆ R n is Ahlfors d-regular if there is A > 0 so that (1.1) r d /A ≤ H d (B(ξ, r) ∩ E) ≤ Ar d for ξ ∈ E, r ∈ (0, diam E) and is uniformly rectifiable (UR) if it has E has big pieces of Lipschitz images (BPLI), i.e. there are constants L, c > 0 so for all ξ ∈ E and r ∈ (0, diam E), there is an L-Lipschitz map f : R d → R n such that H d (f (R d ) ∩ B(ξ, r)) ≥ cr d . Characterizations of uniformly rectifiable sets laid out in the aforementioned texts and in later papers have been indispensable for several important problems in harmonic analysis [NTV14] and harmonic measure [HM14,HLMN17,AHM+17]. On one hand, being uniformly rectifiable may imply some nice estimates on a set's multiscale geometry that can be useful for a particular problem (such as [AHM+17]); conversely, if one is trying to establish that a set has some (uniformly) rectifiable structure in a given problem, the settings of that problem may more easily imply the criteria for one characterization of uniform rectifiability than another (as in [HM14,NTV14,HLMN17]). We will define some of these criteria here. Let D denote the Christ-David cubes for E (see Theorem 2.2 below for their definition and the relevant notation we will use below). We say a family of cubes C satisfies a Carleson packing condition if there is a constant C so that for all R ∈ D, Q∈C Q⊆R ℓ(Q) d ≤ Cℓ(R) d . By Theorem 2.2, for each cube Q ∈ D, there is a ball B Q centered on and containing Q of comparable size. Given two closed sets E and F , and B a set we denote d B (E, F ) = 2 diam B max sup y∈E∩B dist(y, F ), sup y∈F ∩B dist(y, E) For C 0 > 0, and ǫ > 0, let BWGL(C 0 , ǫ) = {Q ∈ D\| d C 0 B Q (E, P ) ≥ ǫ for all d-planesP }. BWGL stands for the bilateral weak geometric lemma. David and Semmes showed in [DS93] that E is UR if and only if for every C 0 ≥ 1 there is ǫ > 0 sufficiently small so that BLWG(C 0 , ǫ) satisfies a Carleson packing condition (with constant depending on ǫ). Another important classification from [DS93] is bilateral approximation uniformly by planes (BAUP): for R ∈ D and ǫ > 0, let BAUP(C 0 , ǫ) = {Q ∈ D\| d C 0 B Q (E, U) ≥ ǫ, U is a union of d-planes}. (1.2) David and Semmes showed that E is UR if and only if BAUP(C 0 , ǫ) satisfies a Carleson packing condition each C 0 > 1 and ǫ > 0 small enough (depending on C 0 ). This was a key tool in [HLMN17] in showing that the weak-A ∞ condition for harmonic measure implies UR, and also was key in Nazarov, Tolsa, and Volberg's solution to David and Semmes' conjecture in codimension 1 [NTV14]. The focus on Ahlfors regular sets is due to the fact that Hausdorff measure on the set is rather well behaved, and so techniques like stopping-time arguments on dyadic cubes in the Euclidean setting often translate over to this non-smooth setting. The motivation of the current paper, however, is to try and obtain similar estimates on multiscale geometry that exist for uniformly rectifiable sets, but instead for sets that are not Ahlfors regular. Not all quantitative results on rectifiability are in the Ahlfors regular setting. The classical example is the Analyst's Traveling Salesman Theorem stated below, which will serve as a model for the kind of results we are after. [Sch07]) Let n ≥ 2. There is a C = C(n) such that the following holds. Let E ⊂ R n . Then there is a connected set Γ ⊇ E such that (1.4) H 1 (Γ) n diam E + Q∈∆ Q∩E =∅ β 1 E,∞ (3Q) 2 diam(Q). Conversely, if Γ is connected and H 1 (Γ) < ∞, then (1.5) diam Γ + Q∈∆ Q∩Γ =∅ β 1 Γ,∞ (3Q) 2 diam(Q) n H 1 (Γ). This was first shown by Jones in [Jon90] in the plane, then Okikiolu in R n [Oki92], and finally in Hilbert space by Schul [Sch07], though the statement is different than above. There are also some partial and complete generalizations that hold for curves in other metric spaces [DS17, DS19, FFP07, LS16a, LS16b, Li19]. An analogue for d-dimensional of the second half of Theorem 1.1 is false due to Fang (see [AS18] for a proof). David and Semmes, however, coined a different definition of a β-number in terms of which they gave a classification of uniformly rectifiable sets. In [AS18], the first author and Schul altered their definition to get a version of Theorem 1.1 for higher dimensional sets, which we describe now. For a set E, a ball B, and a d-dimensional plane L, define β d,p E (B, L) = 1 r d B 1 0 H d ∞ ({x ∈ B ∩ E\| dist(x, L) > tr B })t p−1 dt 1 p where r B is the radius of B, and set β d,p E (B) = inf{β d,p E (B, L)\|L is a d-dimensional plane in R n }. If E is Ahlfors d-regular and we replace H d ∞ with H d , this is the βnumber David and Semmes used. However, the d-dimensional traveling salesman will be stated for lower content regular sets. Definition 1.2. A set E ⊆ R n is said to be (c, d)-lower content regular in a ball B if H d ∞ (E ∩ B(x, r)) ≥ cr d for all x ∈ E ∩ B and r ∈ (0, r B ). We can now state the result from [AS18]. It is phrased slightly differently from there, but we justify the reformulation in the appendix. Theorem 1.3. Let 1 ≤ d < n and E ⊆ R n be a closed set. Suppose that E is (c, d)-lower content regular and let D denote the Christ-David cubes for E. Let C 0 > 1. Then there is ǫ > 0 small enough so that the following holds. Let 1 ≤ p < p(d) where (1.6) p(d) := 2d d−2 if d > 2 ∞ if d ≤ 2 . For R ∈ D, let BWGL(R) = BWGL(R, ǫ, C 0 ) = Q∈BWGL(ǫ,C 0 ) Q⊆R ℓ(Q) d . and β E,A,p (R) := ℓ(R) d + Q⊆R β d,p E (AB Q ) 2 ℓ(Q) d . Then for R ∈ D, (1.7) H d (R) + BWGL(R, ǫ, C 0 ) ∼ A,n,c,p,C 0 ǫ β E,A,p (R). Since all these values are comparable for all admissible values of A and p, below we will simply let β E (R) := β E,3,2 . The presence of BWGL(R, ǫ, C 0 ) may seem odd, but it disappears in some natural situations. It is just zero if E is ǫ-Reifenberg flat, for example (c.f. [DT12] for this definition). When d = n − 1 and E is satisfies Condition B, we have that for any cube R ⊆ E, BWGL(R, ǫ, C 0 ) H d (R). In an upcoming paper, the second author will show that this same estimate occurs for the higher codimensional generalized Semmes surfaces introduced by David in [Dav88] (check there for these definitions). In these scenarios, we then have the more natural looking estimate (more closely resembling (1.4)) β E (R) ∼ A,n,c,ǫ H d (R). Even in the general case, the higher dimensional Traveling Salesman Theorem above says that BWGL(R, ǫ, C 0 ) has some meaning if we compute the sum for a non-Ahlfors regular set: even though it does not necessarily satisfy a Carleson packing condition, it is comparable to the square sum of β's for any lower regular set. This opens the question of whether the same holds for sums over other cube families for which a Carleson packing condition would characterize UR sets in the Ahlfors regular setting. In this paper, we show this is indeed the case for a large class of the original UR characterizations developed by David and Semmes. A consequence of our results is the following (see Section 6 for its proof): Theorem 1.4. Let E ⊆ R n be a (c, d)-lower content regular set with Christ- David cubes D. For R ∈ D, define BAUP(R, C 0 , ǫ) = Q⊆R Q∈BAUP(C 0 ,ǫ) ℓ(Q) d . For all R ∈ D, C 0 > 1, and ǫ > 0 small enough depending on C 0 and c, (1.8) H d (R) + BAUP(R, C 0 , ǫ) ∼ C 0 ,ǫ,c β E (R). We mention one other geometric criteria studied by David and Semmes which we consider: The Local Symmetry' (LS) property is defined as follows. Given ǫ > 0, let LS(R, ǫ, α) be the sum of ℓ(Q) d over those cubes in R for which there are y, z ∈ B Q ∩ E so that dist(2y − z, E) ≥ ǫr. Theorem 1.5. Let E ⊆ R n be a (c, d)-lower content regular set and D its Christ-David cubes. Then for ǫ > 0 small enough (depending on c), and R ∈ D, (1.9) β E (R) ∼ c,ǫ H d (R) + LS(R, ǫ). This may be surprising, since the Local Symmetry condition is dimensionless, that is, the integer d does not appear in the definition at all, and in fact it could be that, in the "good" cubes not featured in the sum, E could be very not flat and quite close in the Hausdorff distance to a (d + 1)dimensional cube, say, whereas the β-numbers measure the distance to a d-dimensional plane and would be large for these cubes. However, with the assumption that H d (R) is finite, this prevents there being too many cubes where E is symmetric but looks like a (d + 1)-dimensional surface (and this is natural considering that the proof in [DS91] connecting LS to flatness of the set uses the Ahlfors regularity of the sets they consider). Our method for extending these results is quite flexible: the other characterizations of UR for which we prove analogous statements like those are the Local Convexity (LCV) and Generalized Weak Exterior Convexity (GWEC) conditions, although one could also consider other suitable characterizations in [DS93] as well. In fact, our main result is a general test for when a geometric criteria that guarantees uniform rectifiability (like BAUP or BWGL) also implies a result of the form Theorem 1.4. Its statement is a bit lengthy to give here, so we postpone it to Section 4. Loosely speaking, by a geometric criteria P, we mean a way of splitting up the surface cubes of a set E into "good" and "bad" cubes, the good cubes being those cubes near which E satisfies some condition that is trivially satisfied for a d-dimensional plane, like being close in the Hausdorff distance to a plane or union of planes. We say it guarantees UR if, whenever we have an Ahlfors regular set, a Carleson packing condition on the bad cubes implies UR. Our result, Lemma 4.5 below, states that if we have a geometric criterion that guarantees UR and it is, in some sense, continuous in the Hausdorff metric, then a result like Theorem 1.4 hold with BAUP replaced by P. The main lemma that we use may be of independent interest, and has a few forthcoming applications to other problems (see [Azz, Vil]) . For the reader familiar with uniform rectifiability, this result says that we can perform a Coronization of lower regular sets by Ahlfors regular sets in a way similar to how David and Semmes construct Coronizations of uniformly rectifiable sets by Lipschitz graphs (see [ (1) We have (1.10) R∈T op(k 0 ) ℓ(R) d c,d H d (Q 0 ). (2) Given R ∈ Top(k 0 ) and a stopping-time region T ⊆ Tree(R) with maximal cube T , let F denote the minimal cubes of T and d F (x) = inf Q∈F (ℓ(Q) + dist(x, Q)) (1.11) For C 0 > 4 and τ > 0, there is a collection C of disjoint dyadic cubes covering C 0 B T ∩ E so that if E(T ) = I∈C ∂ d I, where ∂ d I denotes the d-dimensional skeleton of I, then the following hold: (1.12) C 0 B T ∩ E ⊆ I∈C I ⊆ 2C 0 B T . (c) E is close to E(T ) in C 0 B T in the sense that (1.13) dist(x, E(T )) τ d F (x) for all x ∈ E ∩ C 0 B T . (d) The cubes in C satisfy (1.14) ℓ(I) ∼ τ inf x∈I d F (x) for all I ∈ C . The last inequality says that the cubes in C are distributed in a sort of Whitney fashion. In particular, if two cubes in C are adjacent, then they have comparable sizes. Observe that the constants don't depend on k 0 . The presence of k 0 is an artifact of the proof, but in applications we will take k 0 → ∞. 1.2. Outline. In Section 3, we prove the Main Lemma and show that a general lower regular set can be approximated by Alhfors regular sets. In Section 4, we show how, if the sum of cubes where a geometric criteria like the BAUP is finite, then we can actually make these Ahlfors regular sets uniformly rectifiable. Using a result of David and Semmes, we know that the sum of β's will be finite for these sets, and then that will imply the β's for the original set are summable by approximation. After that, we apply our works to get results similar to the Traveling Salesman, but with BWGL replaced by other geometric criteria. In Section 5, we show the same result holds with BWGL replaced by the Local Symmetry and Local Convexity conditions. In Section 6, we consider the BAUP condition and prove Theorem 1.4, and in Section 7, we study the GWEC. 1.3. Acknowledgements. We'd like to thank Raanan Schul for his useful conversations and encouragement and Matthew Hyde for carefully proofreading the manuscript. 2. PRELIMINARIES 2.1. Notation. We will write a b if there is C > 0 such that a ≤ Cb and a t b if the constant C depends on the parameter t. We also write a ∼ b to mean a b a and define a ∼ t b similarly. For sets A, B ⊂ R n , let dist(A, B) = inf{\|x − y\| \| x ∈ A, y ∈ B}, dist(x, A) = dist({x}, A), and diam A = sup{\|x − y\| \| x, y ∈ A}. 2.2. Dyadic cubes. Let I denote the dyadic cubes in R n , and for k ∈ Z, let I k be those dyadic cubes of sidelength 2 −k . Give a dyadic cube I 0 , we will write I (I 0 ) to denote the subfamily of dyadic cubes which are contained in I 0 . Given some m ∈ Z, we set I m := ∞ k=m I k , that is, I is the family of dyadic cubes with side length at least 2 m . We will also write I m (I 0 ) := I m ∩ I (I 0 ). Finally, given a dyadic cube I, we denote by n(I) the integer number so that ℓ(I) = 2 n(I) . For a cube I ∈ I , we write ∂ d I to denote the d-dimensional skeleton of I. For a dyadic cube I, its d-dimensional skeleton is just the union of its d-dimensional faces. Remark 2.1. We may also use the notation I m to mean the family of cubes with side length ℓ(I) = 2 −m . 2.3. Christ-David Cubes. We recall the following version of "dyadic cubes" for metric spaces, first introduced by David [Dav88] but generalized in [Chr90] and [HM12]. Theorem 2.2. Let X be a doubling metric space. Let X k be a nested sequence of maximal ρ k -nets for X where ρ < 1/1000 and let c 0 = 1/500. For each n ∈ Z there is a collection D k of "cubes," which are Borel subsets of X such that the following hold. (1) For every integer k, X = Q∈D k Q. (2) If Q, Q ′ ∈ D = D k and Q ∩ Q ′ = ∅, then Q ⊆ Q ′ or Q ′ ⊆ Q. (3) For Q ∈ D, let k(Q) be the unique integer so that Q ∈ D k and set ℓ(Q) = 5ρ k(Q) . Then there is ζ Q ∈ X k so that (2.1) B X (ζ Q , c 0 ℓ(Q)) ⊆ Q ⊆ B X (ζ Q , ℓ(Q)) and X k = {ζ Q \| Q ∈ D k }. For a cube Q ∈ D k , we put Child(Q) := {Q ′ ∈ D k+1 \| Q ′ ⊂ Q} . (2.2) Definition 2.3. A collection T ⊆ D is a stopping-time region or tree if the following hold: (1) There is a cube Q(T ) ∈ T that contains every cube in T . ( 2) If Q ∈ T , R ∈ D, and Q ⊆ R ⊆ Q(T ), then R ∈ T . (3) Q ∈ T and there is Q ′ ∈ Child(Q)\T , then Child(Q) ⊂ T c . PROOF OF THE MAIN LEMMA Let E and Q 0 be as in the Main Lemma. Notice that Q 0 is also a lower regular set, although it may not be closed, but we will not need that. We split the proof into a few subsections. 3.1. Frostmann's Lemma. The first step of the proof follows the proof Frostmann's lemma, but with some extra care. Let I 0 = [0, 1] n . Without loss of generality, we assume that Q 0 ⊂ I 0 and that diam(Q 0 ) ≥ ℓ(I 0 ). For k ∈ Z, let I k (Q 0 ) = {I ∈ I k \| I∩Q 0 = ∅}, I k (Q 0 ) = k j=0 I j , I (Q 0 ) = I ∞ (Q 0 ) and E k = I∈I k (Q 0 ) I. Let m ∈ N (we will choose it later). First let µ m m = H n \| Em 2 (n−d)m . In this way, µ m m (I) = ℓ(I) d for all I ∈ I m (Q 0 ). We define a set of cubes Bad (which depends on m) as follows. First, we immediately add I m (Q 0 ) to Bad. Next, for each I ∈ I m−1 (Q 0 ), if µ m m (I) > 2ℓ(I) d , then we add I to Bad and define µ m−1 m \| I = ℓ(I) d µ m m \| I µ m m (I) < 1 2 µ m m \| I Otherwise, we set µ m−1 m \| I = µ m m \| I . Inductively, suppose we have defined µ k+1 m for some integer k < m. For I ∈ I k (E), if µ k+1 m (I) > 2ℓ(I) d , then place I ∈ Bad and set (3.1) µ k m \| I = ℓ(I) d µ k+1 m \| I µ k+1 m (I) < 1 2 µ k+1 m \| I . Otherwise, we set (3.2) µ k m \| I = µ k+1 m \| I . Finally, we put I 0 ∈ Bad. Given a cube I ∈ I , recall that n(I) is the integer such that ℓ(I) = 2 −n(I) . Moreover, for J ∈ I m (Q 0 ) (so that ℓ(J) ≥ 2 −m ), we let b(J) be the numbe of cubes from Bad properly containing J. Now, again with J ∈ I m (Q 0 ), let I 0 , ..., I b(J) ∈ Bad be all the bad cubes containing J, sot hat I j ⊃ I j+1 (note that this is consistent with how we defined I 0 before). With this notation, we see that b(I j ) = j for all j and if a dyadic cube I ∈ Bad, then I = I b(I) . Let now I ∈ Bad and J ∈ I m (Q 0 ) so that J ⊂ I. We write µ n(I) m (J) = µ n(I b(I) ) m (J) (3.1) < 1 2 µ n(I b(I) )+1 m (J) (3.2) = 1 2 µ n(I b(I)+1 ) m (J) < · · · (3.3) · · · < 1 2 b(J)−b(I) µ n(I b(J ) ) m (J) = 1 2 b(J)−b(I) µ m m (J) = ℓ(J) d 2 b(J)−b(I) . Finally, observe that since Q 0 is (c, d)-lower content regular, if J ∩ Q 0 = ∅ and J ∈ I m (Q 0 ), then (3.4) ℓ(J) d c H d ∞ (3J ∩ Q 0 ) ≤ H d (3J ∩ Q). and the cubes {3J \| J ∈ I m (Q 0 )} have bounded overlap. Thus, I∈Bad ℓ(I) d = I∈Bad µ n(I) m (I) = I∈Bad J ∈Im(Q 0 ) J ⊆I µ n(I) m (J) (3.5) (3.3) < I∈Bad J ∈Im(Q 0 ) J ⊆I 2 −b(J)+b(I) ℓ(J) d = J∈Im(Q 0 ) ℓ(J) d I∈Bad I⊇J 2 −b(J)+b(I) J∈Im(Q 0 ) ℓ(J) d (3.4) c J∈Im(Q 0 ) H d (3J ∩ Q 0 ) d H d (Q 0 ). For I ∈ Bad, let µ I := µ n(I) m \| I . Note that by construction, for each J ⊆ I, we have that µ I (J) ≤ 2ℓ(J) d , and thus this also holds for all dyadic cubes J, even when J ⊇ I or J ∩ I = ∅. In particular, since any ball B(x, r) can be covered by boundedly many dyadic cubes of size comparable to r, we obtain that (3.6) µ I (B(x, r)) r d for all x ∈ R n , r > 0. Moreover, µ I (I) = ℓ(I) d . Remark 3.1. Ideally what we'd like to do at this stage is, for each I ∈ Bad, find the maximal bad cubes I j ∈ Bad properly contained in I and define a set like E I = j ∂ d I j where ∂ d J is the d-dimensional skeleton of a cube J. Then one can use µ I to show that E I is an Ahlfors regular set. However, the collection E I will not be suitable for the applications we have in mind, since we need that the sizes of the cubes whose skeletons form E I don't vary too wildly (that is, adjacent cubes should have comparable sizes). This is why more work is needed. 3.2. Trees. For I ∈ Bad, we will let Tree(I) be those cubes in I contained in I for which the smallest cube from Bad that they are properly contained in is I, and we will let Stop(I) be those cubes from Bad in Tree(I) properly contained in I. (3.7) 2 d−n−1 ℓ(J) d ≤ µ I (J) ≤ 2ℓ(J) d Proof. Note that by construction, for I ∈ Bad, and because there are 2 ndyadic cubes J ⊆ I with ℓ(J) = ℓ(I)/2, µ n(I)+1 m (I) = J ∈I n(I)+1 J ⊆I µ n(I)+1 m (J) ≤ J ∈I n(I)+1 J ⊆I 2ℓ(J) d = J ∈I n(I)+1 J ⊆I 2 1−d ℓ(I) d ≤ 2 n−d+1 ℓ(I) d and for J ∈ Stop(I), µ n(I)+1 m (J) = µ n(J) m (J) = ℓ(J) d . Thus, 2ℓ(J) d ≥ µ I (J) = µ n(I) m (J) = µ n(I)+1 m (J) ℓ(I) d µ n(I)+1 m (I) ≥ 2 d−n−1 ℓ(J) d . Let M > 1, we will choose it later. For Q ∈ D(k 0 ) and I ∈ I (Q 0 ), we write Q ∼ I if (3.8) MB Q ∩ I = ∅ and ρℓ(I) ≤ ℓ(Q) < ℓ(I) where ρ is as in Theorem 2.2. Observe that for m large enough, (3.9) {I ∈ I (Q 0 ) : I ∼ Q for some Q ∈ D(k 0 )} ⊆ I m (Q 0 ). Indeed, If Q ∈ D(k 0 ), this means ℓ(I) ≥ ρℓ(Q) ≥ 5ρ k 0 +1 > 2 −m for m large enough, and now we just recall Remark 3.2. We now perform the following stopping-time algorithm on the cubes D(k 0 ). Note that for each R ∈ Top, if R 1 is its parent, then R 1 ∈ Stop(R ′ ) for some cube R ′ , and so there is I R ∈ Bad with I R ∼ R ′′ for some sibling R ′′ ∈ Child(R 1 ). In particular, the map R → I R maps boundedly many cubes to one cube, and so For R ∈ D(k 0 ) contained in Q 0 , we let Stop(R) denote the set of maximal cubes in R from D(k 0 ) that are either in D k 0 or have a child Q for which there is I ∈ Bad such that Q ∼ I. Observe that if R ∈ D k 0 , then Stop(R) = {R}.(3.10) R∈Top ℓ(R) d M I∈Bad ℓ(I) d (3.5) N H d (Q 0 ). The collection Top is our desired collection and {Tree(R) \| R ∈ Top} are the desired stopping-time regions for the Main Lemma and (1.10) now follows from (3.10). It remains to verify items (2) of the Main Lemma, which will be the focus of the next two subsections. We will first need a lemma about our trees: Lemma 3.4. Let R ∈ Top and S(R) := {I ∈ I (Q 0 )\| Q ∼ I for some Q ∈ Tree(R)}. Then there is N 0 n,M 1 and J 1 (R), ..., J N 0 (R) ∈ Bad so that S(R) ⊆ Tree(J 1 (R)) ∪ · · · ∪ Tree(J N 0 (R)). Proof. Consider the cubes I 1 , ..., I N 0 in I (Q 0 ) of maximal size so that I j ∼ R) (note that N 0 here depends only on n and M). Then for m large enough, each I j is contained in Tree(J j ) for some J j ∈ Bad by (3.9). Now let I ∈ S(R), so by definition there is Q ∈ Tree(R) satisfying (3.8), and for a dyadic cube I ∈ I , then I ⊆ I j ⊆ J j for some j. If I ∈ Tree(J j ), then there is J ∈ Stop(J i ) so that I J ⊆ I j ⊆ J j . Since I j ∼ R, ℓ(I j ) < ℓ(R), so we must have ℓ(J) < ℓ(R). Thus, if Q ′ is the maximal ancestor of Q with ℓ(Q ′ ) < ℓ(J), then ℓ(Q ′ ) < ℓ(R), and so Q ′ R. Since Q ∈ Tree(R), this implies Q ′ ∈ Tree(R). Since Q ∼ I and ρℓ(J) ≤ ℓ(Q ′ ) < ℓ(J) by the maximality of Q ′ , we also have Q ′ ∼ J. So the parent Q ′′ ⊆ R of Q ′ must be in Stop(R),d F (I) := inf x∈I d F (x) = inf S∈F (ℓ(S) + dist(I, S)) . (3.12) We define C F to be the set of maximal cubes I ∈ I (Q 0 ) for which (3.13) ℓ(I) < τ d F (I). The following lemmas are quite standard and appear in different forms depending on the scenario in which they are being applied (depending on between which kinds of cubes, dyadic or not, that d F is computing), see for example [DS91,Lemma 8.7]. We include their proofs below for completeness. Lemma 3.5. Let I, I ′ ∈ I . Then, d F (I) ≤ 2ℓ(I) + dist(I, I ′ ) + 2ℓ(I ′ ) + d F (I ′ ). (3.14) Proof. Let x, y ∈ I and x ′ , y ′ ∈ I ′ . Let also Q ∈ F ; we have d F (x) ≤ \|x − y\| + \|y − y ′ \| + \|y ′ − x ′ \| + dist(x ′ , Q) + ℓ(Q), (3.15) simply by triangle inequality and the definition of d F . Clearly, \|y − y ′ \| ≤ dist(I, I ′ ); moreover, infimising first over all Q ∈ F and then over all x ′ ∈ I, we obtain (3.14). Lemma 3.6. Let I ∈ C F ; then τ 2 d F (I) ≤ ℓ(I) < τ d F (I). (3.16) Proof. By (3.13), ℓ(I) < τ d F (I), and by definition it is a maximal cube satisfying this inequality. Hence ifÎ is the parent of I, there is a point z ∈Î with τ d F (z) ≤ 2ℓ(I). The fact that d F is 1-Lipschitz gives the remaining inequality. The following lemma says that if two cubes in C F are close to each other, then they have comparable size. Lemma 3.7. Let I, J ∈ C F and recall that C F depends on a parameter τ . Let 0 < η < 1 be another small parameter. If η −1 J ∩ η −1 I = ∅, (3.17) for τ −1 > 2 √ n/η, ℓ(I) ∼ ℓ(J). (3.18) Proof. It suffices to show that for all y ∈ η −1 J, τ −1 ℓ(J) ∼ d F (y) (3.19) Since d F is 1-Lipschitz, we see that \|d F (J) − d F (y)\| ≤ η −1 diam(J) = √ n η ℓ(J) . Hence if τ −1 > 2 √ n/η, d F (y) ≥ d F (J) − √ n η ℓ(J) ≥ τ −1 − √ n η ℓ(J) ≥ 1 2τ ℓ(J) (3.20) On the other hand, again using the fact that d F is 1-Lipschitz, we see that d F (y) (η −1 + τ −1 )ℓ(J) τ −1 ℓ(J). (3.21) 3.4. Constructing an Ahlfors regular set with respect to a tree. Let R ∈ Top and T ⊆ Tree(R) be a stopping-time region, let T denote the maximal cube in T , F be the set of minimal cubes of T (that is, those cubes in T that don't properly contain another cube in T ). Observe that since all the cubes we are working with come from D(k 0 ) and the number of these cubes in Q 0 is finite, the infimum d F is attained, and so for each I ∈ I there is Q I ∈ F so that (3.22) d F (I) = ℓ(Q I ) + dist(Q I , I). Let C 0 > 4 and set T = {Q ∈ D \| ℓ(Q) = ℓ(T ), Q ∩ C 0 B T = ∅}, (3.23) C = {I ∈ C F \| I ∩T = ∅}, (3.24) andÊ := I∈C ∂ d I. (3.25) This setÊ will be our desired E(T ) as in the statement of the Main Lemma (we just writeÊ for short). Proof. Note that by (3.16), and because Q I ∈ D(k 0 ), for I ∈ C , ℓ(I) ≥ τ 2 d F (I) ≥ τ 2 ℓ(Q I ) ≥ 5τ 2 ρ k 0 , and for τ small enough, ℓ(I) < τ d F (I) ≤ τ (ℓ(T ) + dist(I, T )) ≤ τ (C 0 + 1)ℓ(T ) < 1 5 ℓ(Q 0 ) = 1. Thus, (3.26) follows for m large enough from these two inequalities. Remark 3.9. Note that we definitely don't have that C ⊆ I m (Q 0 ), since some cubes in C are actually disjoint from Q 0 . This will cause some difficulties later. Proof. Firstly, as C 0 B T ∩ E ⊆T , we immediately have the first containment, so we just need to show the second containment. Note that if I ∈ C , then I ∩ Q = ∅ for some Q ∈ D with ℓ(Q) = ℓ(T ) and Q ∩ C 0 B T = ∅. Thus, dist(I, T ) ≤ dist(Q, T ) + diam Q ≤ C 0 ℓ(T ) + 2ℓ(T ) < (C 0 + 2)ℓ(T ). Thus, diam I = √ nℓ(I) < √ nτ d F (I) ≤ √ nτ (dist(I, T )+ℓ(T )) < √ nτ (C 0 +3)ℓ(T ) so for τ > 0 small, diam I ≤ C 0 2 ℓ(T ). Thus, I ⊆ (3C 0 /2+2)B T ⊆ 2C 0 B T , which proves the lemma. Lemma 3.11. Part (c) of the Main Lemma holds. Proof. Let x ∈ E ∩ C 0 B T ⊆T . By part (b), there is I so that x ∈ I ∈ C ⊆ C F . By definition, ∂ d I ⊆Ê, and so dist(x,Ê) ≤ diam I ≤ √ nℓ(I) < √ nτ d F (I) ≤ √ nτ d F (x). Moreover, (1.14) follows from (3.16). Thus, to prove the Main Lemma, all that remains to be shown is the following lemma. Proof. Let x ∈Ê and 0 < r < diamÊ ≤ 2C 0 ℓ(T ). We define C (x, r) = {I ∈ C \| I ∩ B(x, r) = ∅}. We split into three cases, and in each case we prove first the upper estimate for being Ahlfors regular and then the lower estimate. Case 1: 2r ≤ d F (x). Since d F is Lipschitz, this means d F (y) ≥ d F (x) − \|x − y\|, and so if I ∈ C (x, r), y ∈ I is so that d F (I) = d F (y), and z ∈ I ∩ B(x, r), then \|z − y\| ≤ diam I = √ nℓ(I), and so ℓ(I) (3.16) ∼ τ d F (I) = τ d F (y) ≥ τ (d F (x) − \|x − y\|) ≥ τ (2r − \|x − z\| − \|z − y\|) τ (r − √ nℓ(I)) and so for τ ≪ √ n we have ℓ(I) τ r. This implies #C (x, r) n,τ 1, and so it is not hard to show that H d (Ê ∩ B(x, r)) ∼ n,τ 1. Case 2: 8ℓ(T ) > 2r > d F (x). Before we proceed, we record a few estimates. First, for I ∈ C (x, r), if 2r > d F (x), (3.27) τ −1 ℓ(I) (3.16) < d F (I) ≤ d F (y) ≤ d F (x) + \|x − y\| < 2r + r = 3r Next, note that for all I ∈ C , ℓ(Q I ) ≤ d F (I). Let Q ′ I be the largest cube in T containing Q I so that ℓ(Q ′ I ) ≤ d F (I). Lemma 3.13. If x ∈Ê and d F (x) < 2r < 24ℓ(T ), then (3.28) ℓ(Q ′ I ) ∼ τ ℓ(I). Proof. If Q ′ I = T , then Q I = T , so (3.29) ℓ(T ) ≤ d F (I) (3.27) < 3r ℓ(T ) and so ℓ( Q ′ I ) = ℓ(T ) ∼ τ ℓ(I). Otherwise, if ℓ(Q ′ I ) < ℓ(T ), then ℓ(Q ′ I ) ∼ d F (I) (3.16) ∼ τ ℓ(I) by maximality of Q ′ I (indeed, if ℓ(Q ′ I ) < ρd F (I), then its parent Q ′′ I satisfies ℓ(Q ′′ I ) < d F (I) and Q ′′ I ∈ T since Q ′ I T , but this contradicts the maximality of Q ′ I ). This proves the lemma. Recall (3.26) and let C 1 (x, r) = {I ∈ C (x, r) : I ∩ Q 0 = ∅} = C (x, r) ∩ I m (Q 0 ), C 2 (x, r) = C (x, r)\C 1 (x, r). Lemma 3.14. If x ∈Ê and d F (x) < 2r < 24ℓ(T ), then (3.30) I∈C 1 (x,r) ℓ(I) d r d . Proof. We need an estimate like ℓ(I) d H d ∞ (I ∩ Q 0 ), but this may not necessarily be true: of course I ∩ Q 0 = ∅ since I ∈ C 1 (x, r), but it could be that I only intersects Q 0 at a corner of I so H d ∞ (I ∩ Q 0 ) could be very small compared to ℓ(I) d . To overcome this, we associate to I a neighboring dyadic cube that does intersect E in a large set. Let Nei(I) be the set of dyadic cubes J ⊆ 3I with ℓ(J) = ℓ(I). Then ℓ(I) d H d ∞ (3I ∩ Q 0 ) ≤ J∈Nei(I) H d ∞ (J ∩ Q 0 ). Hence there is I ′ ∈ Nei(I) so that by (3.26)), this implies I ′ ∼ Q ′ I , and so I ′ ∈ S(R) (where S(R) is as in Lemma 3.4). In particular, there is J i = J i (R) so that I ′ ∈ Tree(J i ) by Lemma 3.4. We will use this fact shortly, but we need one more estimate: We now claim that (3.31) H d ∞ (I ′ ∩ Q 0 ) ℓ(I) d . Since I ′ ⊆ 3I, we know that dist(I ′ , Q ′ I ) ≤ diam I+dist(I, Q I ) ≤ √ nℓ(I)+d F (I) (3.16) τ −1 ℓ(I) ∼ ℓ(Q ′ I ). As ℓ(I) ∼ τ ℓ(Q ′ I ), for M ≫ τ −1 large enough, MB Q ′ I ∩ I ′ = ∅, and since Q ′ I ∈ T ⊆ Tree(R) and I ′ ∈ I m (Q 0 ) (because I ′ ∩ Q 0 = ∅ and I ∈ I mI∈C 1 (x,r) 1 I ′ 1 B(x,2r) . Indeed, if y ∈ I ′ 1 ∩ · · · ∩ I ′ ℓ for some distinct I 1 , ..., I ℓ ∈ C 1 (x, r), then the I j are disjoint and y ∈ 3I 1 ∩ · · · ∩ 3I ℓ , so Lemma 3.7 implies they have sizes all comparable to I 1 and are also contained in 9I 1 (assuming I 1 is the largest). Thus if \|A\| denotes the Lebesgue measure of a set A, ℓ\|I 1 \| ∼ ℓ i=1 \|I i \| = ℓ i=1 I i ≤ \|9I 1 \| which implies ℓ 1, thus, I∈C 1 (x,r) 1 I ′ 1. Finally, note that diam I = √ nℓ(I) (3.16) < τ √ nd F (I) (3.27) < 3 √ nτ r and since I and I ′ touch, dist(x, I ′ ) ≤ diam I + r < (3 √ nτ + 1)r, so for τ > 0 small enough, I ′ ⊆ B(x, 2r). Thus, (3.31) follows. Thus, H d (Ê ∩ B(x, r)) I∈C 1 (x,r) ℓ(I) d I∈C 1 (x,r) H d ∞ (I ′ ∩ Q 0 ) ≤ I∈C 1 (x,r) N 0 i=1 J ∈Stop(J i ) J ⊆I ′ (diam J) d (3.7) I∈C 1 (x,r) N 0 i=1 J ∈Stop(J i ) J ⊆I ′ µ J i (J) ≤ I∈C 1 (x,r) N 0 i=1 µ J i (I ′ ) (3.31) N 0 i=1 µ J i (B(x, 2r)) r d . This proves (3.30). Lemma 3.15. If x ∈Ê and d F (x) < 2r < 8ℓ(T ), then (3.32) I∈C 2 (x,r) ℓ(I) d r d . Proof. For I ∈ C 2 (x, r), letQ I denote the child of Q ′ I containing the center of Q ′ I . We claim that the cubes {Q I : I ∈ C 2 (x, r)} have bounded overlap. Indeed, suppose there were I 1 , ...., I ℓ ∈ C 2 (x, r) distinct and a point y ∈ ℓ j=1Q I j . We can assume thatQ I 1 is the largest, and since they are all cubes, this impliesQ I 1 ⊇Q j for all j. Since (3.33) dist(I j ,Q I 1 ) ≤ dist(I j ,Q I j ) ≤ dist(I j , Q I j ) ≤ d F (I j ) (3.16) τ −1 ℓ(I j ) and the I j are disjoint, and because ℓ(I j ) (3.28) ∼ τ ℓ(Q ′ I j ) ∼ ℓ(Q I j ) , for given ǫ > 0, there can be at most boundedly many I j (depending on ǫ and τ ) for which diam I j ≥ ǫℓ(Q I 1 ). For the rest of the j, we have that dist(I j ,Q I 1 ) (3.33) τ −1 ℓ(I j ) < ǫ τ ℓ(Q I 1 ), so for ǫ > 0 small enough, and recalling that ρ < c 0 /2 in Theorem 2.2, this implies I j ⊆ c 0 B Q ′ I j . Since I j ∩T = ∅ and the balls {c 0 B Q : Q ∈ D k } are disjoint for each k by Theorem 2.2, this means ∅ = I j ∩ Q ′ I j ⊆ I j ∩ Q 0 , and so I j ∈ C 1 (x, r), which is a contradiction since we assumed I j ∈ C 2 (x, r). Thus, there are no other j, and so ℓ ǫ 1. This finishes the proof that the sets {Q I : I ∈ C 2 (x, r)} have bounded overlap. Fix I ∈ C 2 (x, r) and let J ∈ C so that J ∩ c 0 2 BQ I = ∅. Then ℓ(J) < τ d F (J) ≤ τ ℓ(Q I ), so for τ small enough, J ⊆ c 0 BQ I . Thus, if {J i I } L I i=1 = {J ∈ C : J ∩dist(I, J i I ) ≤ dist(I,Q I ) ≤ dist(I, Q I ) ≤ d F (I) < 2r, hence J i I ∈ C 1 (x, 3r ). Now we have by our assumptions that d F (x) < 2r < 2 · (3r) = 3 · (2r) < 3 · 8ℓ(T ) = 24ℓ(T ). Thus, (3.30) holds for 3r in place of r, and so I∈C 2 (x,r) ℓ(I) d ∼ I∈C 2 (x,r) ℓ(Q I ) d ∼ c I∈C 2 (x,r) H d ∞ c 0 2 BQ I I∈C 2 (x,r) J ∈C J ∩ c 0 2 BQ I =∅ ℓ(J) d = I∈C 2 (x,r) L I i=1 ℓ(J i I ) d J∈C 1 (x,3r) d ℓ(J) d r d where we used the bounded overlap property in the penultimate inequality. Thus, combining the two previous lemmas, we have that H d (Ê ∩ B(x, r)) 2 i=1 I∈C i (x,r) ℓ(I) d r d . Now to complete the proof in this case, we need to show the reverse estimate. Let I ∈ C (x, r/2). Then (3.27) implies that for τ small enough, I ⊆ B(x, r). Moreover, since I ∈ C , I ∩ Q = ∅ for some Q ⊆T with ℓ(Q) = ℓ(T ). If I ∈ C (x, r/2) is the cube so that x ∈ ∂ d I, then for τ small, dist(x, Q) ≤ diam I (3.27) ≤ 3 √ nτ r < r 4 Thus, there is y ∈ Q ∩ B(x, r/4), and so we can find a subcube Q ′ ⊆ B(x, r/2) ∩ Q containing y so that ℓ(Q ′ ) ∼ r and the cubes from C (x, r/2) cover Q ′ . Thus, H d (B(x, r) ∩Ê) ≥ I∈C (x,r/2) H d (∂ d I) ∼ I∈C (x,r/2) ℓ(I) d H d ∞ (Q ′ ) ℓ(Q ′ ) d ∼ r d . Case 3: 2C 0 ℓ(T ) > r > 4ℓ (T ). Note that by the previous case, H d (B(x, r) ∩ 2B T ∩Ê) ≤ H d (2B T ∩Ê) ℓ(T ) d r d . So to prove upper regularity, we just need to verify H d (B(x, r) ∩Ê\2B T ) r d . If I ∩ B(x, r)\2B T = ∅, and if y ∈ I\2B T , ℓ(I) ∼ τ d F (I) ≥ τ (d F (y) − diam I) ≥ τ dist(y, T ) − τ √ nℓ(I) ≥ τ ℓ(T ) − τ √ nℓ(I) and so for τ small enough, τ 2 ℓ(T ) ≤ ℓ(I) Moreover, since I ∩ B(x, r) = ∅, x ∈Ê, and T ⊆ J∈C F J, ℓ(I) < τ d F (I) ≤ τ (ℓ(T ) + dist(I, T )) < τ (ℓ(T ) + diam I + 2r + dist(x, T )) < τ (ℓ(T ) + √ nℓ(I) + 4C 0 ℓ(T ) + diamÊ) τ (ℓ(T ) + ℓ(I)) So for τ > 0 small enough, we also have ℓ(I) τ ℓ(T ), hence ℓ(I) ∼ τ ℓ (T ). There can only be at most boundedly many disjoint cubes I ∈ C with ℓ(I) ∼ τ ℓ(T ), and so H d (Ê ∩ B(x, r)\2B T ) ℓ(T ) d ∼ r d . For the lower bound, if x ∈Ê ∩ 2B T , then r > 4ℓ(T ) implies by the previous case that H d (Ê ∩ B(x, r)) ≥ H d (Ê ∩ 2B T ) ℓ(T ) d ∼ r d . Alternatively, if x ∈Ê\2B T , then by the arguments above, if I ∈ C contains x, then ℓ(I) ∼ τ ℓ(T ) ∼ τ r, so for τ small enough, I ⊆ B(x, r). Thus, H d (Ê ∩ B(x, r)) ≥ H d (∂ d I) ∼ ℓ(I) d ∼ r d . This completes the proof. This finishes the proof of the Main Lemma. A GENERAL LEMMA ON QUANTITATIVE PROPERTIES We now want to apply the approximation by Ahlfors regular sets obtained in the previous section to derive quantitative bounds on the sum of the β coefficients. The method we present is quite easy and general. The idea is the following: let us pick one of the quantitative properties described by David and Semmes. For example, the BAUP (which stands for bilateral approximation by union of planes) (see [DS93], II, Chapter 3), the GWEC (generalised weak exterior convexity) (see [DS93], II, Chapter 3), or the LS (local symmetry), see [DS91], Definition 4.2. On each cube R ∈ Top, we run a stopping time on Tree(R) where we stop whenever we meet a cube which does not satisfy the chosen property. By doing so, we obtain a new tree and consequently a new approximating Ahlfors regular set. This time, however, this set will turn out to be uniformly rectifiable exactly because it approximates E at those scales where E is very well behaved. Let us try to make all this precise. Definition 4.1 (Quantitative property). By a quantitative property (QP) P of E we mean a finite set of real numbers {p 1 , ..., p N } with 0 < p 1 ≤ 1 together with two subsets of E × R + = E × (0, ∞) G P = G P (p 1 , ..., p N ) and B P = B P (p 1 , ..., p N ), which depend on {p 1 , ..., p N }, such that G P ∪ B P = E × R + and G P ∩ B P = ∅. (4.1) We will call {p 1 , ..., p N } the parameters of P. If we want to specify the subset E upon which we are applying a quantitative property P, we may write, for example, G P E , or B P E . Let us give a few examples of quantitative properties described in the book [DS93]: BWGL: The so-called 'Bilateral Weak Geometric Lemma' (BWGL) is a quantitative property. Given a real number ǫ > 0, for each pair (x, r) ∈ E × R + , BWGL asks whether there exists a plane P so that d B(x,r) (E, P ) < ǫ. If one such a plane exists, then we put (x, r) ∈ G BWGL ; if not, then (x, r) ∈ B BWGL . This is clearly a partition of E × R + . Hence BWGL is a QP with parameter ǫ. LS: The 'Local Symmetry' (LS) property is defined as follows. Given ǫ > 0, for each pair (x, r) ∈ E × R + , we say (x, r) ∈ B LS (ǫ, α) if there are y, z ∈ B(x, r) ∩ E so that dist(2y − z, E) ≥ ǫr. LCV For the quantitative property 'Local Convexity' (LCV), we define B LCV to be those (x, r) ∈ E × R + for which there are y, z ∈ B(x, r) ∩ E such that dist((y + z)/2, E) ≥ ǫr. WCD: Let two positive numbers C 0 and ǫ be given. The 'Weak Constant Density' (WCD) condition asks the following: for (x, r) ∈ E × R + , does a measure µ x,r exists, such that spt(µ x,r ) = E; µ x,r is Ahlfors d − regular with constant C 0 ≥ 1; \|µ x,r (y, s) − s d \| ≤ ǫt d for all y ∈ E ∩ B(x, r) and 0 < s ≤ r. If one such a measure µ x,r exists, then we put (x, r) ∈ G WCD (C −1 0 , ǫ). If not, then (x, r) ∈ B WCD (C −1 0 , ǫ). This is clearly a partition of E × R + and so WCD is a QP with parameters (C −1 0 , ǫ). BP: Let us give one more example. Let 1 ≥ θ > 0 be a positive real number. The 'Big Projection' (BP) condition asks if for a pair (x, r), there exists a d-dimensional plane P such that \|Π P (B(x, r) ∩ E)\| ≥ θr d , where Π P is the standard orthogonal projection onto P and \| · \| is the d-dimensional Lebesgue measure on P . We put (x, r) ∈ G BP (θ) if this is the case; otherwise (x, r) ∈ B BP (θ). Thus BP is a QP with parameter θ > 0. Definition 4.2. Fix a (small) parameter ǫ 1 > 0 and two (large) constants C 1 , C 2 ≥ 1 and let P be a quantitative property with parameters {p 1 , ..., p N }. We say that P is (ǫ 1 , C 1 , C 2 )-continuous, if there exist positive constants 0 < c 1 , ..., c N < ∞ depending on ǫ 1 and C 1 such that the following holds. Let E 1 and E 2 be two subsets of R n and let B = B(x B , r B ) be a ball so that B is centered on E 1 ; (x B , r B ) ∈ G P E 1 (p 1 , ..., p N ); d C 2 B (E 1 , E 2 ) < ǫ. If B ′ = B(x B ′ , r B ′ ) is a ball so that B ′ is centered on E 2 ; C 2 B ′ ⊂ B; r B ′ ≥ r B C 1 , then (x B ′ , r B ′ ) ∈ G P E 2 (c 1 p 1 , ..., c N p N ). (4.2) Remark 4.3. In particular a continuous quantitative property is monotonic (or stable) in the following sense; take a set E and a ball B centered on E with (x B , r B ) ∈ G P E (p 1 , ..., p N ). If we assume that P is continuous and we take E 1 = E 2 = E in Definition 4.2, then we see that (x B ′ , r B ′ ) ∈ G P E (c 1 p 1 , ..., c N p N ) whenever C 2 B ′ ⊂ B and r B ′ ≥ r B C 1 . Let us look at our concrete examples of QP, and see whether they are continuous, and thus stable. • One can quite easily check that BWGL, LS, LCV, and BP are stable quantitative properties. • On the other hand, the WCD is not. Definition 4.4 (QP guaranteeing uniform rectifiability). We say a QP (with parameters p 1 , ..., p N ) guarantees uniform rectifiability for Ahlfors d-regular sets with constant C 1 if, whenever A is Ahlfors d-regular with constant C 1 and 1 B P (p 1 ,...,p N ) dr r dH d \| A is a Carleson measure on A × R + , (4.3) then A is a uniformly rectifiable set. Conversely, if A is uniformly rectifiable, then we say a QP (with parameters p 1 , ..., p N ) is guaranteed by uniform rectifiability if the measure in (4.3) is a Carleson measure for the parameters (p 1 , ..., p N ). Let us go back to our examples. • In the two monographs [DS91] and [DS93], David B∩E R 0 1 B BWGL (ǫ) (x, r) dr r dH d (x) ≤ C(ǫ)r d B , (4.4) for all balls B centered on E with r B ≤ diam(E). In general, one may have that C(ǫ) → ∞ as ǫ → 0. On the other hand, it suffices to find a sufficiently small ǫ > 0 for which (4.4) holds to prove that A is uniformly rectifiable. • The property BP, on the other hand, does not guarantee uniform rectifiability. The standard 4-corner Cantor set is purely unrectifiable but still satisfy the Carleson measure condition above since it has large projections in some directions (although of course not many directions), see [Dav91, Part III Chapter 5]. Let now P be a continuous quantitative property with parameters {p 1 , ..., p N }. For a cube Q 0 ∈ D, we let B P (Q 0 ) = B P (Q 0 , p 1 , ..., p N ) (4.5) := Q ∈ D \|Q ⊂ Q 0 ; (ζ Q , ℓ(Q)) ∈ B P ; G P (Q 0 ) = G P (Q 0 , p 1 , ..., p N ) := D(Q 0 ) \ B P . Thus we put P(Q 0 , p 1 , ..., p N ) := P(Q 0 ) := Q∈B P Q 0 ℓ(Q) d . The following is the main result of this section. In later sections, we will show how the comparability results (Theorems 1.4 and 1.5) follow as corollaries. Lemma 4.5. Let E ⊂ R n be a (c, d)-lower content regular set, and let 0 < ǫ < 1, C 2 ≥ 1, and C 1 > 4C 2 /ρ. There is C ′ 0 depending on c so that the following holds. Let P be a QP of E with parameters {p 1 , ..., p N } such that P is (ǫ, C 1 , C 2 )-continuous. with constants c 1 , ..., c N (4.6) P guarantees (and is guaranteed by) UR for C ′ 0 -Ahlfors d-regular sets (4.7) for parameters c 1 p 1 , ..., c N p N ; (4.8) Then for any Q 0 ∈ D β E (Q 0 ) c,C 1 ,ǫ H d (Q 0 ) + P(Q 0 , c 1 p 1 , ..., c N p N ). (4.9) The proof of Lemma 4.5 will take up the rest of this section. Let us get started by first modifying the tree structure of Top(k 0 ), as in the statement of the Main Lemma by introducing a further stopping condition which is related to the QP P. Let R ∈ Top(k 0 ) and R ′ ∈ Tree(R). Let Stop(R ′ ) be the maximal cubes in Tree(R) that are either in Stop(R) or contain a child in B P (Q 0 ), and let Tree(R ′ ) be the subfamily of cubes Q ∈ Tree(R) contained in R that are not properly contained in a cube from Stop(R ′ ). In other words, Tree(R ′ ) is a pruned version of Tree(R), where we cut whenever we found a cube Q ∈ B P D . Let Next 0 (R) = {R} and for j ≥ 0, if we have defined Next k (R), let Next j+1 (R) = R ′ ∈Next j Q∈ Stop(R ′ ) Child(Q). This process terminates at some integer K R since Tree(R) is finite. Enumerate Next j = {Q j i } i j i=1 . Lemma 4.6. Let R ∈ Top(k 0 ) and let 0 ≤ j ≤ K R and 1 ≤ i ≤ i j . Then there exists a constant c 1 < 1 so that Q∈ Tree(Q j i ) β d,2 E (3B Q )ℓ(Q) d ≤ C(c 1 , τ, n, C 0 )ℓ(Q j i ) d . (4.10) To prove Lemma 4.6, we will need the following Lemma from [AS18]. Lemma 4.7 ([AS18], Lemma). Let 1 ≤ p < ∞ and E 1 , E 2 lower content d-regular subsets of R n ; let moreover x ∈ E 1 and choose a radius r > 0. Then if y ∈ E 2 is so that B(x, r) ⊂ B(y, 2r), we have β p,d E 1 (x, r) β p,d E 2 (y, 2r) + 1 r d E 1 ∩B(x,2r) dist(y, E 2 ) r p dH d ∞ (y) 1 p . (4.11) QUANTITATIVE COMPARISONS OF MULTISCALE GEOMETRIC PROPERTIES 27 Let R, j, i as above. Let E Q j i = E i,j be the Ahlfors regular set obtained from the Main Lemma for Tree(Q j i ) and d Q j i be the function defined in (1.11), where, in this instance, F = Stop(Q j i ) and T = Tree(Q j i ). Specifically, for C 0 > 4, as in (3.23), we set T i,j := Q ∈ D \| ℓ(Q) = ℓ(Q j i ), Q ∩ C 0 B Q j i = ∅ ; following (3.24), we then put C i,j := I ∈ I \| I ∩T = ∅ and I is maximal with ℓ(I) < τ d Q j i (I) . Then E i,j := I∈C i,j ∂ d I. (4.12) It follows from the Main Lemma that E i,j is Ahlfors d-regular. Lemma 4.8. Let k 0 , τ > 0, R, j and i as above. Then E i,j is uniformly rectifiable. We want to use the fact that P guarantees uniform rectifiability and that it is continuous. We will show that there exist constants c 1 , ..., c N such that the measure 1 B P (c 1 p 1 ,...,c N p N ) (x, r) dt t dH d (x) (4.13) is Carleson on E i,j × R + . We test this measure on a ball B centered on E i,j and with radius r B . Note that B∩E i,j ητ d Q j i (x) 0 1 B P (c 1 p 1 ,...,c N p N ) (x, r) dr r dH d (x) n,d r d B . (4.14) holds automatically: indeed, for any x ∈ E i,j and whenever 0 < r ≤ ητ d Q j i (x), B(x, r) ∩ E i,j is just a finite union of d-dimensional planes, and the number of planes in this union is bounded above by a universal constant only depending on n and d. Therefore B(x, r) ∩ E i,j is uniformly rectifiable and thus (4.14) holds. Also, using the Ahlfors regularity of E i,j , it is immediate to see that B∩E i,j η −1 d Q j i (x) ητ d Q j i (x) 1 B P (c 1 p 1 ,...,c N p N ) (x, r) dr r dH d (x) τ,η r d B . (4.15) Let us check that B∩E i,j τ 10 ℓ(Q j i ) η −1 d Q j i (x) 1 B P (c 1 p 1 ,...,c N p N ) (x, r) dr r dH d (x) τ,η r d B . (4.16) Lemma 4.9. Let (x, r) ∈ E i,j × R + be such that η −1 d Q j i (x) ≤ r ≤ τ ℓ(Q j i ). (4.17) Then, for η > 0 sufficiently small (depending only on n), there exists a cube P in Tree(Q j i ) so that B P ⊂ B(x, r). Proof. For this proof, we put Q = Q j i . Let I x be the cube in C Q containing x, so ℓ(I x ) ∼ τ d Q (x). Let P * be the minimiser of d Q (x). Note that (4.18) dist(x, P * ) ≤ d Q (x) ≤ ηr. Let us look at two distinct cases. Case 1. Suppose first that d Q (x) = ℓ(P * ) + dist(x, P * ) ≤ 2ℓ(P * ). (4.19) Then we immediately obtain that ℓ(P * ) ≤ d Q (x) ≤ 2ℓ(P * ) and therefore that ℓ(P * ) ∼ τ −1 ℓ(I x ). (4.20) But (4.19) also implies that dist(x, P * ) ≤ ℓ(P * ) (4.21) Now, because of the assumption (4.17), we see that (using also (4.20)) r ≥ η −1 d Q (x) ≥ η −1 d Q (I x ) ∼ η −1 τ −1 ℓ(I x ) ∼ η −1 ℓ(P * ), and so, because (4.21) and (4.18), we have for η small B P * ⊂ B(x, r). Case 2. Suppose now that d Q (x) = ℓ(P * ) + dist(x, P * ) ≤ 2 dist(x, P * ). Then we have dist(x, P * ) ∼ d Q (x) ≤ Cηr. Also, by (4.17), it holds that ℓ(P * ) ≤ d Q (x) ≤ ηr. This implies, for η > 0 sufficiently small, that also in this case we have B P * ⊂ B(x, r). Lemma 4.10. There exist constants (c 1 , ..., c N ) such that the following holds. Let (x, r) ∈ E i,j × R + be such that c 1 p 1 , ..., c N p N ). Proof. We know from Lemma 4.9 that if (x, r) satisfies (4.22), then there exists a cube P * ∈ Stop(Q j i ) such that B P * ⊂ B(x, r). Thus, there must exist an ancestor P * ∈ Tree(Q j i ) of P * so that ρℓ( P * ) ≤ 4C 2 r < ℓ( P * ), (4.23) and thus so that B(x, C 2 r) ⊂ B P * , and since C 1 > 4C 2 /ρ, we also have r ≥ ℓ( P * )/C 1 . But recall that if P * ∈ Tree(Q j i ), then we must have, by definition, that (ζ P * , ℓ( P * )) ∈ G P (p 1 , ..., p N ). η −1 d Q j i (x) ≤ r ≤ τ ℓ(Q j i ). (4.22) Then (x, r) ∈ G P E i,j ( Let us check that d C 2 B P * (E i,j , E) < τ. By (1.13), if y ∈ E ∩ C 2 B P * dist(y, E i,j ) τ d Q i,j (y) ≤ τ (dist(y, P * ) + ℓ( P * )) C 2 τ ℓ( P * ). That for any x ∈ E i,j ∩ C 2 B P * we have dist(x, E) τ ℓ( P * ) follows in the same way, since any such x is contained in a dyadic cube I touching E so that ℓ(I) < τ d Q j i (I) ≤ 8ητ ℓ( P * ). Choosing τ in the construction of C i,j appropriately (depending on ǫ and C 2 ), the lemma follows from the (ǫ, C 1 , C 2 )-continuity of P. Proof of Lemma 4.8. We have shown that there exist constants c 1 , ..., c N such that, for any pair (x, r) ∈ E i,j × R + with η −1 d Q j i (x) ≤ r ≤ τ ℓ(Q j i ) we have (x, r) ∈ G P E i,j (c 1 p 1 , ..., c N p N ). (4.24) Thus the integral in (4.16) equals to zero. Now, we also see that, trivially B∩E i,j diam(E i,j ) τ 10 ℓ(Q j i ) 1 B P (c 1 p 1 ,...,c N p N ) (x, r) dr r dH d (x) τ r d B . (4.25) This together with the previous estimates (4.14), (4.15) and (4.16) proves that the measure 1 B P (c 1 p 1 ,...,c N p N ) (x, r) dr r dH d \| E i,j (x) is a Carleson measure on E i,j × R + ; then, because P guarantees uniform rectifiability with the appropriate parameters and it is (ǫ, C 1 , C 2 )-continuous, E i,j is uniformly rectifiable. Note that all the constants involved depend only on n, d, τ, η (and c 0 ); in particular, they are all independent of Q j i , R and k 0 . Proof of Lemma 4.6. We want to apply Lemma 4.7 with E 1 = E, E 2 = E i,j and p = 2. For Q ∈ Tree(Q j i ), recall that ζ Q denotes the center of Q. By (1.13), we know that dist(z Q , E i,j ) τ d Q j i (z Q ) ≤ τ ℓ(Q), and in particular, if we denote by x ′ Q the point in E i,j which is closest to x Q , we see that B Q := B(ζ Q , ℓ(Q)) ⊂ B(x ′ Q , 2ℓ(Q)) =: B ′ Q for τ small enough. Hence for each cube Q ∈ Tree(Q j i ) the hypotheses of Lemma 4.7 are satisfied and we may write Q∈ Tree(Q j i ) β 2,d E (3B Q ) 2 ℓ(Q) d Q∈ Tree(Q j i ) β 2,d E i,j (6B ′ Q ) 2 ℓ(Q) d + Q∈ Tree(Q j i ) 1 ℓ(Q) d 6B Q ∩E dist(x, E i,j ) ℓ(Q) 2 dH d ∞ (x) := I 1 + I 2 . We first look at I 1 . We apply Theorem 2.2 to E i,j ; let us denote the cubes so obtained by D E i,j . Note that for each P ∈ Tree(Q j i ) with P ∈ D(k 0 ), x ′ P belongs to some cube P ′ ∈ D E i,j so that ℓ(P ′ ) ∼ ℓ(P ); hence there exists a constant C 1 ≥ 1 so that 6B ′ P ⊂ C 1 B P ′ . (4.26) This in turn implies that β p,d E i,j (6B ′ P ) p,n,d,C 1 β p,d E i,j (C 1 B P ′ ). Hence, P ∈Tree(Q j i ) P ∈D(k 0 ) β 2,d E i,j (6B ′ P ) 2 ℓ(P ) d p,n,d,C 1 P ′ ∈D E i,j ℓ(P ′ ) ℓ(Q j i ) β 2,d E i,j (C 1 B P ′ ) 2 ℓ(P ′ ) d . (4.27) Since E i,j is uniformly rectifiable, we immediately have that I 1 ℓ(Q j i ) d by the main results of [DS91] (in particular, see (C3) and (C6) in [DS91, Chapter 1]). Let us now worry about I 2 . We put It is clear that Approx(Q j i ) := maximal S ∈ D(k 0 ) \| there is I ∈ C i,Q j i ⊂ S∈Approx(Q j i ) S, (4.29) since C i,j covers Q j i . Now let x ∈ Q j i . We claim that there exists a cube S ∈ Approx(Q j i ) so that dist(x, E i,j ) ≤ Cℓ(S). (4.30) By (4.29), we see that if x ∈ Q j i , then there exists an S ∈ Approx(Q j i ) so that x ∈ S. But, then, by definition, there exists an I ∈ C i,j such that ℓ(I) ≤ ℓ(S) and I ∩ S = ∅. Thus dist(x, E i,j ) ≤ diam I + dist(x, I) ℓ(S). We now estimate I 2 as follows: first, 1 ℓ(Q) d 6B Q ∩E dist(x, E i,j ) ℓ(Q) 2 dH d ∞ (x) (4.30) S∈Approx(Q j i ) S∩6B Q =∅ S ℓ(S) 2 ℓ(Q) d+2 dH d ∞ S∈Approx(Q j i ) S∩6B Q =∅ ℓ(S) 2+d ℓ(Q) 2+d . (4.31) Hence we obtain that I 2 (4.31) Q∈ Tree(Q j i ) S∈Approx(Q j i ) S∩6B Q =∅ ℓ(S) d+2 ℓ(Q) 2 S∈ApproxQ j i S∩6B Q j i ℓ(S) d+2 Q∈ Tree(Q j i ) S∩6B Q =∅ 1 ℓ(Q) 2 . (4.32) Note that the number of cubes Q ∈ Tree(Q j i ) which belong to a given generation and such that S ∩ 6B Q = ∅ is bounded above by a constant C which depends on n. Indeed, if S ∩ 6B Q = ∅, then we must have that dist(Q, S) ≤ 6ℓ(Q). Moreover, because S ∈ Approx(Q j i ) and using Lemma 3.6, we see that, if I ∈ C i,j is so that I ∩ S = ∅ and ℓ(S) ∼ ℓ(I) (as in (4.28)), ℓ(S) ∼ ℓ(I) ∼ τ d Q j i (I) τ (ℓ(Q)+dist(I, Q)) ≤ τ ℓ(Q)+τ (6ℓ(Q)+2ℓ(S)) so for τ small enough, ℓ(S) ℓ(Q). Thus we can sum the interior sum in (4.32): Q∈Tree(Q j i ) S∩6B Q =∅ 1 ℓ(Q) 2 τ 1 ℓ(S) 2 . Finally, we see that I 2 τ,n S∈Approx(Q j i ) S∩6B Q j i ℓ(S) d+2 ℓ(S) 2 = S∈Approx(Q j i ) S∩6B Q j i ℓ(S) d . (4.33) Now, by definition of Approx(Q j i ), the last sum in (4.33) is bounded above by a constant times I∈C i,j ℓ(I d ) H d   I∈C i,j ∂ d I   = H d (E i,j ) ℓ(Q j i ) d , where we also used the Ahlfors regularity of E i,j . This proves (4.10). Proof of Lemma 4.5. Let Q 0 ∈ D as in the statement of the Lemma. Then we see that R∈Top(k 0 ) K R j=0 Q∈Next j (R) P ∈ Tree(Q) β 2,d E (3B P ) 2 ℓ(P ) d (4.10) τ R∈Top(k 0 ) K R j=0 Q∈Next j (R) ℓ(Q) d . (4.34) Note that for 1 ≤ j ≤ K R , if Q ∈ Next j (R), then there is a sibling Q ′ of Q so that (ζ Q ′ , ℓ(Q ′ )) ∈ B P . Also recall that we put Next 0 (R) = {R}. Then any cube appearing in the sum (4.34), either belongs to Top(k 0 ) (whenever it belongs to Next 0 (R)), or is adjacent to a cube in B P (Q 0 , p 1 , ..., p N ), as defined in (4.5). Thus we see that (4.34) τ   R∈Top(k 0 ) ℓ(R) d + P(Q 0 , p 1 , ..., p N )   (1.10) τ H d (Q 0 ) + P(Q 0 ). Note that all these estimates were independent of k 0 . Sending k 0 to infinity and recalling (A.3) (and recalling that ℓ(Q 0 ) d c H d (Q 0 )) gives the estimate (4.9). APPLICATIONS: THE DIMENSIONLESS QUANTITIES LS AND LCV Here we give a proof of Theorem 1.5 Proof. First, it is not hard to show that there is c > 0 so that if Q ∈ G BWGL E (Q 0 , cǫ), then for any children Q ′ of Q, since ℓ(Q ′ ) = ρℓ(Q) < 1 4 ℓ(Q), we have Q ′ ∈ G LS E (ǫ). Using this fact, we get H d (R) + LS(R, ǫ) ≤ H d (R) + BWGL(R, cǫ) β E (R) and so we just need to prove the reverse inequality. First we show that for all C > 1 and ǫ > 0 is small depending on C and B ∈ G LS E (ǫ) and E ′ is another lower d-regular set so that d 4B (E, E ′ ) < ǫ, then any ball B ′ with 4B ′ ⊆ B centered on E ′ with r B ′ ≥ r B /C, we have that B ′ ∈ G LS E ′ (cǫ) for some c > 0, and so LS is (ǫ, C, 4)-continuous for all C > 1 and ǫ > 0 sufficiently small depending on C. Let x ′ , y ′ ∈ E ′ ∩B ′ , then there are x, y ∈ E with \|x−x ′ \|, \|y −y ′ \| < 4ǫr B . For ǫ > 0 small depending on C, since r B ′ ≥ r B /C, x, y ∈ 3 2 B ′ , and so 2x − y ∈ 3B ′ ⊆ B. Since B ∈ G LS (ǫ), there is ξ ∈ E so that \|2x − y − ξ\| < ǫr B . For ǫ > 0 small enough, since 2x − y ∈ 3 2 B ′ , ξ ∈ 4B ′ ⊆ B, thus there is ξ ′ ∈ E ′ with \|ξ − ξ ′ \| < 4ǫr B . Thus, dist(2x ′ −y ′ , E ′ ) ≤ \|2x ′ −y ′ −ξ ′ \| ≤ \|2x−y−ξ\|+\|x−x ′ \|+\|y−y ′ \|+\|ξ−ξ ′ \| < 16ǫr B . Hence, B ′ ∈ G LS E ′ (16ǫ). Thus, for ǫ > 0 small enough, Lemma 4.5 implies the second half of (5.1). This completes the proof. Another dimensionless quantity is the LCV. This can be proven in much the same way, so we omit the proof. Theorem 5.1. Let E ⊆ R n be a lower d-regular set and D its Christ-David cubes. Then for ǫ > 0 small enough, and R ∈ D, (5.1) β E (R) ∼ H d (R) + LCV(R, ǫ). APPLICATION: THE BAUP In this section, we show that we can apply Lemma 4.5 to the quantitative property BAUP (recall the definition (1.2)). Namely, we will show that BAUP is (ǫ, C 1 , C 2 )-continuous. That BAUP guarantees rectifiability is due to David and Semmes, see [DS93], Proposition 3.18. Let ǫ 0 > 0 and C 0 ≥ 1 be given. Let us first define the actual partition that BAUP determines. We put G BAUP (ǫ 0 , C 0 ) = G BAUP := (x, r) ∈ E × R + \| there is a family F of d-planes s.t. d B(x,C 0 r) (E, ∪ P ∈F P ) < ǫ 0 B BAUP (ǫ 0 , C 0 ) = B BAUP := E × R + \ G BAUP . Lemma 6.1. Let ǫ 0 > 0, C 0 ≥ 1, and consider the quantitative property BAUP with parameters (ǫ 0 , C 0 ). If C 1 ≥ 1, C 2 > 2C 0 , ǫ 0 is small enough (depending on C 2 and C 1 ), and 0 < ǫ 1 ≤ ǫ 0 then BAUP is (ǫ 1 , C 1 , C 2 )continuous. Proof. Let us consider two subsets E 1 , E 2 or R n . From Definition 4.2, we take a ball B = B(x B , r B ) centered on E 2 and so that, first, (x B , r B ) ∈ G BAUP E 1 (ǫ 0 , C 0 ), and second, d C 2 B (E 1 , E 2 ) < ǫ 1 , (6.1) where C 2 and ǫ 1 ≤ ǫ 0 will be determined later with respect to C 0 and ǫ 0 . Thus, there is a union of d-dimensional planes F so that d C 0 B (E 1 , F ) < ǫ 0 . Next, we consider a ball B ′ = B(x ′ B , r ′ B ) centered this time on E 2 with C 2 B ′ ⊆ B and so that r ′ B ≥ r B C 1 . We want to show that for any such a ball B ′ , d C 0 B ′ (E 2 , F ) < c 1 ǫ 0 r ′ B . (6.2) for some constant c 1 to be determined. Let y ∈ E 2 ∩ C 0 B ′ . Since 2B ′ ⊆ C 2 B ′ ⊂ B, we have 2C 0 B ′ ⊆ C 0 B ⊆ C 2 B, so we can use (6.1) to find an x ∈ E 1 so that \|x − y\| < ǫ 1 C 2 r B . Since ǫ 0 ≤ 1, x ∈ E 1 ∩ 2C 0 B ′ ⊆ C 0 B, and because (x B , r B ) ∈ G BAUP E 1 (ǫ 0 , C 0 ), it holds that dist(x, F ) < ǫ 0 C 0 r B . Now, because ǫ 1 ≤ ǫ 0 , we have that sup y∈E 2 ∩C 0 B ′ dist(y, F ) ≤ ǫ 1 C 2 r B + ǫ 0 C 0 r B ≤ (2C 2 C 1 ) ǫ 0 r ′ B Next, for q ∈ F ∩ C 0 B ′ , we look at dist(q, E 2 ); note in particular that q ∈ F ∩ C 0 B and thus, because d C 0 B (E 1 , F ) < ǫ 0 , there is an x ∈ E 1 with \|x − q\| ≤ ǫ 0 C 0 r B . Moreover, choosing C 2 > 2C 0 , since ǫ 0 ≤ 1, we also have that x ∈ 2C 0 B ⊆ C 2 B, and thus dist(x, E 2 ) < C 2 ǫ 1 r B . All in all, we obtain that sup q∈E 2 ∩C 0 B ′ dist(q, E 2 ) ≤ \|x − q\| + dist(x, E 2 ) ≤ C 0 ǫ 0 r B + C 2 ǫ 1 r B ≤ (2C 2 C 1 ) ǫ 0 r ′ B . This implies (6.2) with c 1 = 2C 1 C 2 ; thus BAUP is (ǫ 1 , C 1 , C 2 )-continuous, whenever ǫ 1 ≤ ǫ 0 , and C 2 is sufficiently large, with respect to the parameter C 0 . We can now prove Theorem 1.4. Firstly, note that we immediately have BAUP(Q 0 , C 0 , ǫ) ≤ BWGL(Q 0 , C 0 , ǫ) β E (Q 0 ). Furthermore, since BAUP(C 0 , ǫ) guarantees and is guaranteed by UR for all ǫ > 0 sufficiently small depending on C 0 by [DS93, Theorem III.3.18]. Since it is also (ǫ, C 1 , C 2 )-continuous for C 2 > 2C 0 and all C 1 ≥ 1 and ǫ > 0 sufficiently small, we have, for all C 0 ≥ 1 and ǫ small enough (depending on C 0 ) β E (Q 0 ) H d (Q 0 ) + BAUP(Q 0 , C 0 , ǫ). APPLICATION: THE GWEC Let us give one last example of quantitative property which can be handled within the framework of Lemma 4.5. For a parameter ǫ 0 > 0, we put in B GWEC all the pairs (x, r) ∈ E × R + for which there exists an (n − d − 1)dimensional sphere S satisfying the following three conditions. S ⊂ B(x, r) and dist(S, E) > ǫ 0 r; (7.1) S can be contracted to a point inside {y ∈ B(x, r) \| dist(y, E) > ǫ 0 r} ; (7.2) ch(S) ∩ E = ∅, (7.3) where ch(S) is the convex hull of S. We then put G GWEC (ǫ 0 ) := E × R + \ B GWEC (ǫ 0 ). We want to check that we can apply Lemma 4.5 with this quantitative property. That the GWEC guarantees uniform rectifiability is Theorem 3.28 in [DS93]. All that's left to do is to prove that GWEC is continuous. Lemma 7.1. The quantitative property GWEC with parameter ǫ 0 > 0 is (ǫ 1 , C 1 , C 2 )-continuous, for all C 1 ≥ 3, for all C 2 ≥ 1 and whenever ǫ 1 is sufficiently small with respect to ǫ 0 , C 1 , and C 2 . Proof. Let E 1 and E 2 be two subsets of R n . Let B = B(x B , r B ) be a ball centered on E 1 so that (x B , r B ) ∈ G GWEC E 1 (ǫ 0 ) and d C 2 B (E 1 , E 2 ) < ǫ 1 C 2 r B . (7.4) We want to find a constant c 1 so that, for any ball B ′ = B(x ′ B , r ′ B ) centered on E 2 and with 2B ′ ⊂ B and r ′ B ≥ r B /C 1 , we have that (x ′ B , r ′ B ) ∈ G GWEC E 2 (c 1 ǫ 0 ). We argue by contradiction. Suppose that for some c 1 (to be determined), we can find a sphere S ′ as in (7.1), (7.2) and (7.3) for the ball B ′ . We will construct a sphere S for B satisfying the same three conditions: this will contradict the hypothesis that B is a good ball. Letŷ ∈ E 2 ∩ ch(S ′ ); note that in particularŷ ∈ B(x ′ B , r ′ B ) ⊂ B, and thus we can find a pointx ∈ E 1 with \|ŷ −x\| < ǫ 1 C 2 r B (using (7.4)). If W ′ is the (n − d)-dimensional plane which contains S ′ , we put W = W ′ + (x −ŷ). Hence we let S denote the sphere in W with center center(S ′ ) + (x −ŷ) and radius equal to that of S ′ . We claim that S satisfies (7.1), (7.2) and (7.3) relative to the pair (x B , r B ). Note first that S ⊂ N 2C 2 ǫ 1 r B (S ′ ). (7.5) We show that dist(E 1 , N 2C 2 ǫ 1 r B (S ′ )) > ǫ 0 r B . Let s ′ ∈ S ′ and y ∈ E 1 be closest to each other. Since s ′ ∈ B, we must have y ∈ 2B. Let s ∈ S be closest to s ′ , so \|s − s ′ \| < ǫ 1 C 2 r B . Let y ′ ∈ E 2 be closest to y; then as y ∈ 2B, \|y − y ′ \| < ǫ 1 C 2 r B ; then we have that dist(E 1 , N 2C 2 ǫ 1 r B (S ′ )) = \|y − s ′ \| ≥ \|y ′ − s\| − \|s ′ − s\| − \|y − y ′ \| ≥ dist(E 2 , S) − 2ǫ 1 C 2 r B ≥ c 1 ǫ 0 r ′ B − 2ǫ 1 C 2 r B ≥ c 1 C 1 ǫ 0 r B − 2ǫ 1 C 2 r B . Now, choosing ǫ 1 small enough (depending on ǫ 0 ) and c 1 sufficiently large (depending on C 1 ), it follows that dist(E 1 , S) ≥ dist(E 1 , N 2C 2 ǫ 1 r B (S ′ )) > ǫ 0 r B . This proves (7.1) for (x B , r B ). We now need to show that we can contract S to a point inside the set {y ∈ B(x B , r B ) \| dist(y, E 1 ) > ǫ 0 r B }. To see this, we use (7.5): if we denote by Q t the contraction of S ′ to a point, then dist(Q t , E 2 ) > c 1 ǫ 0 r B . Denote by {T t } 0≤t≤1 the homotopy T t (x) = x + t(ŷ −x), so that T 0 (S) = S, T 1 (S) = S ′ and T t (S ′ ) is a (n − d − 1)-dimensional sphere lying in the ch(S ∪ S ′ ). Then we see that T t (S) ⊂ N 2C 2 ǫ 1 r B (S ′ ), so dist(T t (S), E 1 ) ≥ ǫ 0 r B . Thus, putting T t (x) := T 2t (x) for 0 ≤ t ≤ 1 2 Q 2t−1 for 1 2 ≤ t ≤ 1, we see that T t is the desired contraction; this settles (7.2). Moreover, (7.3) holds from the definition of S. But this implies that (x B , r B ) belongs to B GWEC E 1 (ǫ 0 ). This is impossible, and so no sphere S ′ satisfying (7.1) to (7.3) can exists, and therefore (x ′ B , r ′ B ) ∈ G GWEC E 2 (c 1 ǫ 0 ) for c 1 appropriately chosen (depending on C 1 ), and ǫ 1 sufficiently small. We can now apply Lemma 4.5 (and use the fact that GWEC(Q 0 , ǫ) BWGL(Q 0 , cǫ) β E (Q 0 ) for some c > 0), to obtain the following corollary. Corollary 7.2. Let E be lower content regular, let Q 0 ∈ D. Then β(Q 0 ) ∼ H d (Q 0 ) + GWEC(Q 0 , ǫ). APPENDIX A. THE TRAVELING SALESMAN THEOREM In this section we prove Theorem 1.1. We begin by recalling the original Traveling Salesman Theorem for higher dimensional sets from [AS18, Theorem 3.2 and 3.3]. Theorem A.1. Let 1 ≤ d < n and E ⊆ R n be a closed. Suppose that E is (c, d)-lower content regular and let D denote the Christ-David cubes for E. Let (1) Let C 0 > 1 and A > max{C 0 , 10 5 } > 1, p ≥ 1, and ǫ > 0 be given. For R ∈ D, let Then for R ∈ D, (A.1) H d (R) + BWGL(R, ǫ, C 0 ) A,n,c,C 0 ǫ ℓ(R) d + Q⊆R β d,p E (AB Q ) 2 ℓ(Q) d . Furthermore, if the right hand side of (A.1) is finite, then E is drectifiable. (2) For any A > 1 and 1 ≤ p < p(d), there is C 0 ≫ A and ǫ 0 = ǫ 0 (n, A, p, c) > 0 such that the following holds. Let 0 < ǫ < ǫ 0 . Then β E,3,p (R) BWGL(R, ǫ, C 0 ) β E,A,1 (R) β E,3,1 (R) β E,3,2 (R). This completes the proof of Theorem 1.1. 1 .. 1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2. Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1. Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Background. y, L)\|y ∈ E ∩ B} where L ranges over d-planes in R n . Thus, β d E,∞ (B) diam(B) is the width of the smallest tube containing E ∩ B. Theorem 1.1. (Jones: R 2 [Jon90]; Okikiolu: R n [Oki92],Schul DS91, Chapter 2]). For the definitions of Christ-David cubes and stopping-time regions, see Section 2.3. Main Lemma. Let k 0 > 0, τ > 0, d > 0 and E be a set that is (c, d)-lower content regular. Let D k denote the Christ-David cubes on E of scale k and D = k∈Z D k . Let Q 0 ∈ D 0 and D(k 0 ) = k 0 k=0 {Q ∈ D k \|Q ⊆ Q 0 }. Then we may partition D(k 0 ) into stopping-time regions T ree(R) for R from some collection T op(k 0 ) ⊆ D(k 0 ) with the following properties: (a) E(T ) is Ahlfors regular with constants depending on C 0 , τ, d, and c. (b) We have the containment Remark 3. 2 . 2Observe that Stop(I) ⊆ Tree(I), and while the collections {Tree(I) : I ∈ Bad} do not form a disjoint partition of I m , they do cover I m , and they only intersect at the top cubes and stopped cubes. Lemma 3. 3 . 3For I ∈ Bad and J ∈ Stop(I), We then let Tree(R) be those cubes contained in R that are not properly contained in any cube from Stop(R), so in particular, Stop(R) ⊆ Tree(R). Let Next(R) be the children of cubes in Stop(R) that are also in D(k 0 ) (so this could be empty). Now let Top 0 = {Q 0 }, and for R ∈ Top k , we let Top k+1 = R∈Top k Next(R), that is, Top k+1 are the children of the cubes in Stop(R) for each R ∈ Top k . Let Top = k≥0 Top k . but this contradicts Q being in Tree(R). We let J i (R) = J i and this proves the lemma.3.3. Smoothing. We follow the "smoothing" process of David and Semmes (c.f. [DS91, Chapter 8]). Fix 0 < τ < 1. For a finite family of cubes F ⊂ D, define the following smoothing function: for a point x ∈ R n , set d F (x) := inf S∈F (ℓ(S) + dist(x, S)) , (3.11) Lemma 3 . 8 . 38For m large enough, (3.26) C ⊆ I m . Lemma 3. 10 . 10Part (b) of the Main Lemma holds. Lemma 3 . 12 . 312Part (a) of the Main Lemma holds, that is, the setÊ is Ahlforsd-regular. I = ∅}, since theQ I have bounded overlap, so do the cubes {J i I : i = 1, ..., L I , I ∈ C 2 (x, r)}. For I ∈ C (x, r) and i = 1, ..., L I , j s.t. I ∩ S = ∅ and ρℓ(S) ≤ ℓ(I) ≤ ℓ(S) . 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School Of Mathematics, Jcmb Edinburgh, Kings Build-Ings, Mayfield Road, Edinburgh, Scotland 3jz, UNIVERSITY OF EDINBURGH, JCMB, KINGS BUILD-INGS, MAYFIELD ROAD, EDINBURGH, EH9 3JZ, SCOTLANDE-mail address: m.villa-2 ''at" sms.ed.ac.ukSCHOOL OF MATHEMATICS, UNIVERSITY OF EDINBURGH, JCMB, KINGS BUILD- INGS, MAYFIELD ROAD, EDINBURGH, EH9 3JZ, SCOTLAND. E-mail address: j.azzam ''at" ed.ac.uk SCHOOL OF MATHEMATICS, UNIVERSITY OF EDINBURGH, JCMB, KINGS BUILD- INGS, MAYFIELD ROAD, EDINBURGH, EH9 3JZ, SCOTLAND. E-mail address: m.villa-2 ''at" sms.ed.ac.uk	[]
[ "TWO-COVER DESCENT ON HYPERELLIPTIC CURVES", "TWO-COVER DESCENT ON HYPERELLIPTIC CURVES" ]	[ "Nils Bruin ", "Michael Stoll " ]	[]	[]	We describe an algorithm that determines a set of unramified covers of a given hyperelliptic curve, with the property that any rational point will lift to one of the covers. In particular, if the algorithm returns an empty set, then the hyperelliptic curve has no rational points. This provides a relatively efficiently tested criterion for solvability of hyperelliptic curves. We also discuss applications of this algorithm to curves of genus 1 and to curves with rational points.	10.1090/s0025-5718-09-02255-8	[ "https://arxiv.org/pdf/0803.2052v2.pdf" ]	937,422	0803.2052	a2e5c3da916451ef162fa491ad6512ee0e6edcde	TWO-COVER DESCENT ON HYPERELLIPTIC CURVES 21 Oct 2008 Nils Bruin Michael Stoll TWO-COVER DESCENT ON HYPERELLIPTIC CURVES 21 Oct 2008 We describe an algorithm that determines a set of unramified covers of a given hyperelliptic curve, with the property that any rational point will lift to one of the covers. In particular, if the algorithm returns an empty set, then the hyperelliptic curve has no rational points. This provides a relatively efficiently tested criterion for solvability of hyperelliptic curves. We also discuss applications of this algorithm to curves of genus 1 and to curves with rational points. Introduction In this paper we consider the problem of deciding whether an algebraic curve C over a number field k has any k-rational points. We assume that C is complete and non-singular. A necessary condition for C(k) to be non-empty is that C has a rational point over every extension of k. In particular, for any place v of k, the curve C should have a rational point over the completion k v of k at v. For curves of genus 0, this is sufficient as well: if a genus 0 curve C has a rational point over every completion k v of k, then C(k) is non-empty. A k v -point of C is referred to as a local point of C at v and a k-point of C is called a global point. For a genus 0 curve C, having a local point everywhere (at all places of k) implies having a global point. Genus 0 curves are said to obey the local-to-global principle for points. The local-to-global principle is important from a computational point of view. One can decide in finite time whether a curve has points everywhere locally: for any curve C over a number field k, the set C(k v ) is nonempty for all places v outside an explicitly determinable finite set S. For the remaining places v ∈ S, one can decide in finite time if C(k v ) is non-empty as well. See [4] for some algorithms, in particular a quite efficient algorithm for hyperelliptic curves. Thus, whether a curve has points everywhere locally can actually be decided in finite time. It is well known that curves of genus greater than 0 do not always obey the local-to-global principle. Most proofs of this phenomenon are based on the fact that curves of positive genus can have unramified Galois-covers. In fact, if a curve C of positive genus has a rational point P , then the Abel-Jacobi map allows us to consider C as a non-singular subvariety of its Jacobian variety Jac(C). Since the map π : Jac(C) → Jac(C), Q → nQ + P is unramified, the pull-back π * (C) yields an unramified cover of C that has a rational point mapping to P . More generally, an n-cover of C is a cover that is isomorphic to one of this form over an algebraic closure k of k. Thus, if one can show that an algebraic curve C does not have n-covers that have a rational point, then it follows that C has no rational points. Even though C may have points everywhere locally, it is possible that on every cover, rational points can be ruled out by local conditions. The major content of this paper is inspired by the following well-known observation. Let φ : D → C be an unramified cover of a curve C over a number field k that is Galois over an algebraic closure k of k. It is a standard fact, going back to Chevalley and Weil [12] that there is a finite collection of twists φ δ : D δ → C of the given cover φ such that any rational point on C has a rational pre-image on one of the covers D δ . We call such a set of covers a covering collection. Furthermore, at least in principle, such a finite collection of covers is explicitly computable given a cover φ : D → C. Thus, one approach for testing solvability of a curve C that has points everywhere locally is: (1) Fix n ≥ 2. (2) Construct an n-cover D of C. If no such covers exist, then C(k) is empty. (3) Determine a covering collection associated to D → C. (4) Test each member of the covering collection for local solvability. If none of the members has points everywhere locally then no curve in the covering collection has any rational points, and C has no rational points either. Each of the covers might have a local obstruction to having rational points, while the underlying curve C has none. Thus the procedure sketched above can actually prove that a curve does not obey the local-to-global principle for points. See [25] for a detailed discussion, including the theoretical background and some links to the Brauer-Manin obstruction against rational points. In the present article we discuss a relatively efficient way of carrying out the procedure sketched above for hyperelliptic curves. We consider unramified covers D/C over k such that for an algebraic closure k of k we have that Aut k (D/C) ≃ Jac(C) [2](k) as Gal(k/k)-modules. These are exactly the two-covers of C over k. We write Cov (2) (C/k) for the set of isomorphism classes of two-covers of C over k. Our claim to efficiency stems from the fact that we avoid explicitly constructing a covering collection. In fact, the method can be described without any reference to unramified covers, see Section 2. Instead, we determine an abstract object that (almost) classifies the isomorphism classes of two-covers. See Section 3 for how to construct the covers explicitly from the information provided by the algorithm. We observe that for any field extension L/k there is a well-defined map C(L) → Cov (2) (C/L) by sending a point to the two-cover that has an L-rational point above it. For completions k v /k, we show that this map is locally constant: If two points P 1 , P 2 ∈ C(k v ) lie sufficiently close, then they lift to the same two-cover. This allows us to explicitly compute the k v -isomorphism classes of two-covers that have points locally at v. We can then piece together this information to obtain the global isomorphism classes of two-covers that have points everywhere locally. We define Sel (2) (C/k) ⊂ Cov (2) (C/k) to be the set of everywhere locally solvable two-covers of C. The algorithm computes a related object, that we denote by Sel (2) fake (C/k), which is a quotient of Sel (2) (C/k). It classifies everywhere locally solvable two-covers, up to an easily understood equivalence. Since any curve C with a rational point admits a globally and hence everywhere locally solvable two-cover, Sel (2) fake (C/k) = ∅ implies that C(k) is empty. A priori, the elements of Sel (2) fake (C/k) represent two-covers that have a model of a certain form (described in Section 3), but we will prove that every two-cover that has points everywhere locally does have such a model, see Theorem 3.4 below. In Section 7 we illustrate how the algorithm presented in Sections 4, 5, and 6 can be used to show that a hyperelliptic curve has no rational points. This method was also used in a large scale project [8] to determine the solvability of all genus 2 curves with a model of the form y 2 = f 6 x 6 + · · · + f 0 , where f i ∈ {−3, −2, −1, 0, 1, 2, 3} for i = 0, . . . , 6. In Section 10 we describe some statistics that illustrate how frequently one would expect that Sel (2) fake (C/k) = ∅ for curves of genus 2. Section 8 shows how information obtained on Sel (2) fake (C/k) can be used for determining the rational points on curves if C(k) is non-empty. This complements Chabauty-methods as described in [2,3,15]. In earlier articles, the selection of the covers requiring further attention was done by ad-hoc methods. Here we describe a systematic and relatively efficient approach. In Section 9 we describe the results of applying the method to curves of genus 1. We find (unsurprisingly) that we recover well-known algorithms for performing 2-descents and second 2descents on elliptic curves [10,16]. A practical benefit of this observation is that to our knowledge, nobody has bothered to implement second two-descent over arbitrary number fields, whereas our implementation in MAGMA [1] (which is available for download at [9]) can immediately be used. We give an example by exhibiting an elliptic curve over Q( √ 2) with a non-trivial Tate-Shafarevich group. 2. Definition of the fake 2-Selmer group Let k be a field of characteristic 0 and let C be a non-singular projective hyperelliptic curve of genus g over k, given by the affine model C : y 2 = f n x n + · · · + f 0 = f (x), where f is square-free. If n can be chosen to be odd, then C has a rational Weierstrass point. This is a special situation and, by not placing any Weierstrass point over x = ∞, we see that we lose no generality by assuming that n is even, in which case n = 2g + 2. In practice, however, computations become considerably easier by taking n odd if possible. The construction below can be adapted to accommodate for odd n. See Remark 2.1 below. From here on we assume that n is even unless explicitly stated otherwise. We consider the algebra A = k[x]/(f (x)) and we write θ for the image of x in A, so that f (θ) = 0. We consider the subset of A * modulo squares and scalars (elements of k * ) with representatives in A * that have a norm in k * that is equal to f n modulo squares: H k = {δ ∈ A * /A * 2 k * : N A/k (δ) ∈ f n k * 2 }. The set H k might be empty (but see Question 7.2). As we will see, this implies that C has no rational points. This follows from the fact that we can define a map C(k) → H k . First, we define the partial map: µ k : C(k) → H k (x, y) → x − θ . Here and in the following, we write x − θ instead of the correct, but pedantic, (x − θ)A * 2 k * ; we hope that no confusion will result. This definition of µ does not provide a valid image for any point (x 1 , 0) ∈ C(k). For any such point, we can write f (x) = (x − x 1 )f (x) and we define: µ k ((x 1 , 0)) = (x 1 − θ) +f (θ). Furthermore, if f n ∈ k * 2 then the desingularisation of C has two points, say ∞ + , ∞ − above x = ∞. We define µ(∞ + ) = µ(∞ − ) = f n , where f n ≡ 1 modulo k * 2 . Remark 2.1. While we lose no generality by assuming that f (x) is of even degree, for computational purposes it is often preferable to work with odd degree f (x) as well. We can use the definitions above if we replace the definition of H k by H k = {δ ∈ A * /A * 2 : N A/k (δ) ∈ f n k * 2 }. In this case, there is a unique point ∞ above x = ∞. We define µ(∞) = f n If K is a field containing k (we will consider a number field k with a completion K), the natural map A → A ⊗ k K induces the commutative diagram C(k) µ k G G H k ρK C(K) µK G G H K If k is a number field and v is a place of k, we write µ kv = µ v , µ k = µ and ρ kv = ρ v . For a number field k, we define Sel (2) fake (C/k) = {δ ∈ H k : ρ v (δ) ∈ µ v (C(k v )) for all places v of k} ⊂ H k . It is then clear that µ k (C(k)) ⊂ Sel (2) fake (C/k). In particular, Sel (2) fake (C/k) = ∅ implies that C does not have k-rational points. Geometric interpretation of H k In this section we give a geometric and Galois-cohomological interpretation of the set H k and the map µ k we defined in Section 2. The material in this section is not essential for the other sections in this text. Definition 3.1. Let C be a non-singular curve of genus g over a field k. A non-singular absolutely irreducible cover D of C is called a two-cover if D/C is unramified and Galois over a separable closure k of k and Aut k (D/C) ≃ (Z/2Z) 2g . We denote the set of isomorphism classes of 2-covers of C over k by Cov (2) (C/k). This definition is motivated by the fact that if C can be embedded in Jac(C), then such a cover can be constructed by taking D to be the pull-back of C along the multiplication-by-two morphism [2] : Jac(C) → Jac(C). Furthermore, over k, all two-covers of C are isomorphic, and Aut k (D/C) ≃ Jac(C) [2](k) as Gal(k/k)-modules in a canonical way. Let C be a curve and suppose that D 1 , D 2 are two-covers of C over k. Let φ : D 1 → D 2 be an isomorphism of C-covers over k. We can associate a Gal(k/k)-cocycle to this via ξ : Gal(k/k) → Aut k (D 2 /C) = Jac(C)[2](k) σ → φ σ • φ −1 The cocycle ξ is trivial in H 1 (k, Jac(C) [2]) precisely if D 1 and D 2 are isomorphic as C-covers over k. Furthermore, given a cocycle ξ, one can produce a twist of D ξ of a given cover D: (2) (C/k), then the above construction gives a bijection Cov (2) (C/k) → H 1 (k, Jac(C) [2]). Theorem 3.2 ([22, Chapter X, Theorem 2.2]). If (D → C) ∈ Cov We now interpret the set H k defined in Section 2 in terms of two-covers. Using the notation from the previous section, consider δ ∈ A * . There are unique quadratic forms Q δ,i (u) ∈ k[u 0 , . . . , u n−1 ] such that the identity below holds in A[u 0 , . . . , u n−1 ]: δ(u 0 + u 1 θ + · · · + u n−1 θ n−1 ) 2 = n−1 i=0 Q δ,i (u)θ i . We consider the projective variety over k in P n−1 described by D δ : Q δ,2 (u) = · · · = Q δ,n−1 (u) = 0 This curve D δ is a degree 2 n−1 cover of P 1 via the function x(u) = − Q δ,0 (u) Q δ,1 (u) . Furthermore, if f n N A/k (δ) = v 2 for some v ∈ k, we can define a function y(u) = v N A[u]/k[u] n−1 i=0 u i θ i (Q δ,1 (u)) n/2 which gives us morphisms φ δ,± , depending on the choice of v, represented by φ δ,± : D δ → C (u 0 : · · · : u n−1 ) → (x(u), ±y(u)) It is proved in [2,6] that the cover D δ /C is a two-cover. Furthermore, if δ 1 , δ 2 represent distinct classes in H k , the covers D δ1 and D δ2 are not isomorphic over k. For a fixed δ, the two morphisms φ δ,+ and φ δ,− show that D δ is a two-cover of C in two ways. These are related via the hyperelliptic involution on C, denoted by ι : C → C (x, y) → (x, −y) . With this notation, we have φ δ,− = ι • φ δ,+ . The hyperelliptic involution induces a map ι * : Cov (2) (C/k) → Cov (2) (C/k) via ι * (φ) = ι • φ. Since elements of H k define two-covers up to ι * we have: H k ⊂ Cov (2) (C/k)/ ι * . Whether ι * is the identity depends on whether there exists an isomorphism D → D over k such that the diagram below commutes. D φ 2 2 @ @ @ @ @ @ @ ? A A j f c _ [ X T D ι•φ~~~~~~~C We do not need it here, but we quote a result from [18] (Thm. 11.3) that characterizes whether ι * acts trivially on H 1 (k, Jac(C) [2]). Proposition 3. 3. Let C be as defined above. Then ι * acts trivially on H 1 (k, Jac(C) [2]) if and only if C has an odd degree Galois invariant and ι-symmetric divisor class. A curve C : y 2 = f (x), where f is square-free of degree n, has such a divisor class precisely • when n is odd, or • when n ≡ 0 (mod 4) and f has an odd degree factor, or • when n ≡ 2 (mod 4) and f has an odd degree factor or is the product of two quadratically conjugate factors. If k is a number field, we define the 2-Selmer set of C/k to be the set of everywhere locally solvable 2-covers: Sel (2) (C/k) = (φ : D → C) ∈ Cov (2) (C/k) : D(k v ) = ∅ for all places v of k. We define the fake 2-Selmer set to be the everywhere locally solvable 2-covers of the form D δ : Sel (2) fake (C/k) := H k ∩ Sel (2) (C/k)/ ι * . This is consistent with the definition given in Section 2: if P ∈ C(k) and δ = µ(P ), then D δ has a rational point that maps to P . Therefore, if δ ∈ H k restricts to an element in µ kv (C(k v )) for all places v of k, then D δ has a k v -rational point for each v. This argument also shows that if H k is empty then C(k) is empty. More precisely, the set H k classifies two-covers of the form D δ , so if it is empty, this represents an obstruction against the existence of a two-cover of this specific form. We will now show that every two-cover D → C such that D has points everywhere locally can be realized as a cover D δ . Theorem 3.4. Let φ : D → C be a two-cover such that D has points everywhere locally. Then there is δ ∈ Sel (2) fake (C/k) such that φ is isomorphic to φ δ,+ or φ δ,− . In particular, Sel fake (C/k) = Sel (2) (C/k)/ ι * . Proof. We first note that all divisors φ * (P ) on D, where P is a Weierstrass point on C, are linearly equivalent. This is a geometric statement, so we can assume k to be algebraically closed; then φ ≃ φ 1,+ , and on D 1 , it is easy to see that the divisors in question all are hyperplane sections (for example, if P = (θ, 0), then φ * (P ) is given by the vanishing of u 0 + θu 1 + · · · + θ n−1 u n−1 ). Let us denote the class of all these divisors by W . Since the Galois action maps Weierstrass points to Weierstrass points, W is defined over k. The assumption that D has points everywhere locally implies that every k-rational divisor class contains a k-rational divisor. So W induces a projective embedding D → P N such that the pullbacks of Weierstrass points on C are hyperplane sections. Now consider the function x − θ ∈ A(C). Its pull-back to D has divisor 2φ * ((θ, 0)) − V , where V is a k-rational effective divisor whose class is twice that of a hyperplane section. In terms of the coordinates on P N (projective N -space over theétale algebra A), this means that we can write (x − θ) • φ = δ ℓ 2 q with a constant δ ∈ A * , a linear form ℓ with coefficients in A, and a quadratic form q with coefficients in k. Taking norms (recall that f n N A/k (x − θ) = y 2 ), we find that (y • φ) 2 = f n N A/k (δ) N A/k (ℓ) q n/2 2 , so that δ represents an element of H k . We can write the linear form ℓ as ℓ = ℓ 0 + ℓ 1 θ + · · · + ℓ n−1 θ n−1 where the ℓ i are linear forms with coefficients in k. We obtain a map D → D δ ⊂ P n−1 , given by (ℓ 0 : · · · : ℓ n−1 ), which is the desired isomorphism. Computing the local image of µ at non-archimedean places In this section, we assume that k is a non-archimedian complete local field of characteristic 0. We write O for the ring of integers inside k, ord for the discrete valuation and \|.\| for the absolute value on k. Furthermore π will be a uniformizer. We will assume that O/(π) is finite of characteristic p and that R ⊂ O is a complete set of representatives for O/(π). Let f (x) ∈ O[x] be a square-free polynomial and suppose that f (x) = g 1 (x) · · · g m (x) is a fac- torisation into irreducible polynomials with g i ∈ O[x]. If we write L i = k(θ i ) = k[x]/(g i (x)) then A ≃ L 1 ⊕ · · · ⊕ L m and A * /A * 2 ≃ (L * 1 /L * 2 1 ) × · · · × (L * m /L * 2 m ). The following definitions and lemma allow us to find µ(C(k)) without computation in many cases. Definition 4.1. Let L be a local field and let δ ∈ L * . We say the class of δ in L * /L * 2 is unramified if the extension L( √ δ)/L is unramified. If the residue characteristic p is odd, this just means that ord L (δ) is even. Definition 4.2. Let A be anétale algebra over a local field k and suppose that A ≃ L 1 ⊕ · · · ⊕ L m is a decomposition of L into irreducible algebras. Then we say that δ ∈ A * /A * 2 is unramified if the image of δ in each of L * i /L * 2 i is unramified. We say that an element of A * /A * 2 k * is unramified if it can be represented by an unramified δ ∈ A * /A * 2 . Lemma 4.3. Suppose that f ∈ O[x] , that the residue characteristic p is odd, that ord k (disc(f )) ≤ 1 and that the leading coefficient of f is a unit in O. Then µ(C(k)) consists of unramified elements. Furthermore, if in addition q = #O/(π) satisfies √ q + 1 √ q > 2(2 2g (g − 1) + 1) then µ(C(k)) consists exactly of the unramified elements of H k ⊂ A * /A * 2 k * . (Compare Prop. 5.10 in [24] for a similar statement in the context of 2-descent on the Jacobian of C.) Proof. Let L/k be a splitting field of f and let θ 1 , . . . , θ n be the roots of f . Since the leading coefficient of f is a unit, we have θ i ∈ O L . We extend ord = ord k to L by writing ord(y) = 1 e(L/k) ord L (y), where e(L/k) is the ramification index of L/k. First we prove that e(L/k) ≤ 2. Note that, since ord(disc(f )) ≤ 1, we have e(k[θ i ]/k) ≤ 2. If e(k[θ i ]/k) = 1 for all i, then e(L/k) = 1. Therefore, suppose that e(k[θ 1 ]/k) = 2. Write f (x) = (x − θ 1 )f (x). Then disc(f ) = disc(f )(f (θ 1 )) 2 If e(k[θ 1 ]/k) = 2 then ord(f (θ 1 )) > 0, so in fact ord(f (θ 1 )) ≥ 1 2 . But then, from 1 = ord(disc(f )) = ord(disc(f )) + 2 ord(f (θ)) it follows that ord(disc(f )) = 0 and hence that L/k[θ i ] is unramified for i ≥ 2, so e(L/k) ≤ 2 and ord takes values in 1 2 Z on L * . This allows us to conclude that the roots of f (x) are p-adically widely spaced. From 1 ≥ ord(disc(f )) = ord i<j (θ i − θ j ) 2 = 2 i<j ord(θ i − θ j ) it follows that ord(θ i − θ j ) = 0 for all but at most one pair {i, j}, say {1, 2}. If L/k is ramified then we have ord(θ 1 − θ 2 ) = 1 2 and (x − θ 1 )(x − θ 2 ) is an irreducible quadratic factor of f (x) over k. We now consider a point (x, y) ∈ C(k) with x ∈ O. Since f (x) is a square in k and the leading coefficient is a unit, we have that 2 \| n i=1 ord(x − θ i ) Note, however, that ord(θ i − θ j ) ≥ min(ord(x − θ i ), ord(x − θ j )). A priori, we could still have ord(x − θ 1 ) = ord(x − θ 2 ) = 1 2 (note that these orders must be equal because x − θ 1 and x − θ 2 are Galois-conjugate) and ord(x − θ i ) = 0 for i = 3, . . . , n, but then f (x) has odd valuation and hence is not a square in k. It follows that ord(x − θ i ) = 0 holds for all but at most one i and therefore that all of ord(x − θ i ) are even. This proves that µ(x, y) is unramified for any point ( x, y) ∈ C(k) with x ∈ O. If x / ∈ O, then ord(x) < 0 ≤ ord(θ i ) for all i, so (since p is odd) (x − θ i )/x is a square in k(θ i ). So µ(x, y) is in A * 2 k * , i.e., trivial. It follows that the image µ(C(k)) ⊂ H k is unramified. Conversely, if ord(disc(f )) = 0 and δ ∈ H k is unramified, then D δ as defined in Section 3 can be presented by a model with good reduction (by taking δ to be a unit). Since D δ is an unramified cover of degree 2 2g over a genus g curve C, we can compute using the Riemann-Hurwitz formula that genus(D δ ) = 2 n−3 (n − 4) + 1 = 2 2g (g − 1) + 1. The Weil bounds for the number of points on a nonsingular curve over a finite field of cardinality q imply that if q satisfies the inequality stated in the lemma, then the reduction of D δ has a nonsingular point. Hensel's lifting theorem tells us that D δ (k) is non-empty and therefore that δ ∈ µ(C(k)). If ord(disc(f )) = 1 and δ ∈ H k is unramified, we claim that the reduction D δ of D δ is a singular curve of genus 2 2g−2 (2g − 3) + 1, with a unique singularity at which 2 2g−1 branches meet. If the desingularization of D δ has more than 2 2g−1 points, then D δ must have a non-singular point, so via Hensel's lemma, D δ (k) is non-empty and δ ∈ µ(C(k)). Again, from the Weil bounds it follows that this is the case if √ q + 1 √ q > 2 2 2g−2 (2g − 3) + 1 + 2 2g−1 √ q . It is straightforward to check that for g > 0, this is a weaker condition than the one stated in the lemma. We now prove the claim. By taking δ to be a unit, we see that we can construct D δ by applying the construction of D δ over the residue class field F = O/(π). The reduction of f has a unique double root in F and otherwise simple roots in an algebraic closure of F. We can assume the double root to be at x = 0. Since the statement is geometric, we assume that F is algebraically closed. Let θ 2 , . . . , θ n−1 be the simple roots. We obtain equations defining D δ by eliminating X and Z from the following system X = δ 0 z 2 0 , −Z = z 0 (δ 1 z 0 + 2δ 0 z 1 ) X − θ j Z = δ j z 2 j , j ∈ {2, . . . , n − 1}. Here the first pair of equations is obtained from the component F[x]/(x 2 ) of the algebra F[x]/(f (x)) in the following way: if t is the image of x in F[x]/(x 2 ) , we write elements of this algebra in the form a 0 + a 1 t. We get the first two equations by setting X − Zt = (δ 0 + δ 1 t)(z 0 + z 1 t) 2 and comparing coefficients. Substituting the expressions for X and Z into the second set of equations, we obtain z 0 δ 0 z 0 + θ j (δ 1 z 0 + 2δ 0 z 1 ) = δ j z 2 j , j ∈ {2, . . . , n − 1} . It can be easily checked that the only singular point of this curve is where all variables but z 1 vanish. Projecting away from this point, we obtain a smooth curve in P n−2 that is the complete intersection of n − 3 = 2g − 1 quadrics and therefore has genus 2 2g−2 (2g − 3) + 1. Since (away from z 0 = 0) we can reconstruct z 1 from the remaining coordinates, this projection is a birational map, hence the (geometric) genus of D δ is as given. The points on the smooth model that map to the singularity on D δ have z 0 = 0 (this is where the function z 1 /z 0 is not defined on the smooth model), and it can be checked that there are exactly 2 2g−1 = 2 n−3 such poins (z 0 = 0, and the ratios of the squares of the other n − 2 coordinates are fixed and nonzero). Hence the smooth points of D δ are in bijection with the remaining points of the smooth model. See also [2, Section 3.1] for a more in-depth discussion of this model of D δ and [6] for a characterisation of D δ as a maximal elementary 2-cover of P 1 , unramified outside {θ 1 , . . . , θ n }. In other cases, when a prime divides the discriminant of f more than once, the residue field is too small or the leading coefficient of f is not a unit or k has even residue characteristic, we have to do some computations to find the image of µ. In principle, one could construct H k as a finite set, enumerate all D δ and test each of these for k-rational points. We present a more efficient algorithm that instead enumerates points from C(k) up to some sufficient precision. Computational models for complete local fields usually consist of computing in the finite ring O/π e for some sufficiently large e, which is usually referred to as the precision. The following definition allows us to elegantly state precision bounds. The variable ǫ is an indeterminate. ord : O [ǫ] → Z a i ǫ i → min i ord(a i ) It follows that ord a i x i ≥ ord a i ǫ i for all x ∈ O, and hence that if f (x) ∈ O[x] and v = ord(f (x 1 +π e ǫ)−f (x 1 )), then the value of f (x 1 ) is determined in O/π v by the value of x 1 in O/π e . Lemma 4.4. Suppose g(x) ∈ k[x] is an irreducible polynomial and that for some e ∈ Z ≥0 and x 0 ∈ O, we have ord(g(x 0 + ǫπ e ) − g(x 0 )) > ord(g(x 0 )) Let L = k[θ] = k[x]/(g(x) ). Then for any x 1 ∈ x 0 + π e+ord(4) O we have (x 0 − θ)(x 1 − θ) ∈ L * 2 Proof. (Compare [24], Lemma 6.3.) Using that g(x) = g 0 N L/k (x − θ), where g 0 is the leading coefficient of g(x), we have ord N L/k x 0 + ǫπ e − θ x 0 − θ − 1 > 0. Writing ord L for the valuation on L, we have that ord L (π) is the ramification index of L/k and that ord L x 0 + ǫπ e − θ x 0 − θ − 1 > 0. With some elementary algebra we see that if x 1 ∈ x 0 + π e+ord(4) O then x 1 − θ x 0 − θ ∈ 1 + π ord k (4)+1 O L ⊂ L * 2 and hence that (x 0 − θ)(x 1 − θ) is a square in L. This lemma forms the basis for a recursive algorithm that determines the image of µ for points (x 1 , y 1 ) ∈ C(k), with x 1 ∈ x 0 + π e O. A similar procedure is described in [24, page 270]. There are a few differences: • we fully describe the algorithm for places with even residue characteristic as well, • we do not place extra assumptions on the Newton polygon of f (x), • the polynomial f (x) does not change upon recursion. The algorithm in [24] applies variable substitutions to f (x). This will usually involve a lot of arithmetic with the polynomial coefficients of f to a relatively high π-adic precision. We therefore expect that our algorithm will run slightly faster than [24], especially for small residue fields. We first give an informal outline of the algorithm. We build up the possible x 1 ∈ x 0 + π e O, one π-adic digit at the time. At each stage, we make sure that f (x 1 ) is indistinguishable from a square (step 4 below). After finitely many steps Lemma 4.4 guarantees that the digits we have fixed for x 1 , determine the image of x 1 − θ ∈ L * /L * 2 . We then add that value to the set W , unless x 1 lies close to the x-coordinate of a Weierstrass point. Hence, the purpose of the routine below is not to return a useful value, but to modify a global list W such that all values of µ(C(k) ∩ x −1 (x 0 + π e O)) outside those corresponding to Weierstrass points are appended to W . When first called, G 0 contains the irreducible factors of f . This set gets adjusted upon recursion. The parameter c 0 is an auxiliary parameter that plays a role in keeping track of whether the conditions of Lemma 4.4 are met when ord(4) = 0. Its value is irrelevant if G 0 contains at least two polynomials or at least one polynomial of degree larger than 1. Recall that R is a complete set of representatives of O/(π) in O. define SquareClasses(x 0 , e, G 0 , c 0 ): 1. for r ∈ R: 2. x 1 := x 0 + π e r 3. v 1 = ord(f (x 1 )); E 1 := ord(f (x 1 + ǫπ e+1 ) − f (x 1 )) 4. if E 1 ≤ v 1 or (2 \| v 1 and f (x 1 )/π v1 ∈ (O/π E1−v1 ) * 2 ): 5. G 1 := {g ∈ G 0 : ord(g(x 1 + ǫπ e+1 ) − g(x 1 )) ≤ ord(g(x 1 ))} 6. if G 1 = ∅ or (G 1 = {g} and deg(g) = 1): 7. if G 0 = G 1 : c 1 := ord(4) else c 1 := c 0 − 1 8. if c 1 = 0: 9. if G 1 = ∅: Add the class of µ(x 1 ) to W . 10. return 11. call SquareClasses(x 1 , e + 1, G 1 , c 1 ) Explanation: ad 1. We split up x 0 + π e O into smaller neighbourhoods x 1 + π e+1 O ad 3. Here E 1 is the precision to which f (x 1 ) is determined: E 1 is the largest integer such that f (x 1 + π e+1 O) ⊂ f (x 1 ) + π E1 O. ad 4. We only need to consider neighbourhoods that may contain a point (x 1 , y 1 ) ∈ C(k). This is only the case if f (x 1 ) is a square up to the precision to which it is determined. The sets (O/π E1−v1 ) * 2 are only needed for 1 ≤ E 1 − v 1 ≤ ord(4) + 1 and can be precomputed. ad 5. The correctness of this algorithm hinges on Lemma 4.4. We let G 1 be the subset of G 0 for which the lemma does not apply yet. ad 6. Note that for any path in the recursion, in finitely many steps, the value of any g ∈ G 0 on x 1 + π e+1 O is determined up to a sufficiently high precision to be distinguished from 0, or x 1 is a good approximation to the root of exactly one degree 1 element of G 0 . ad 7. Informally, Lemma 4.4 states that the value of g(x 1 ) in L * /L * 2 is determined if at least ord(4) π-adic digits of x 1 beyond the ones needed to distinguish g(x 1 ) from 0 are known. Since every recursion in 11 has the effect of fixing another digit, we need a device to count ord(4) more iterations. If G 1 is different from G 0 , then we have just gained a digit that helps to establish that g(x 1 ) = 0 for some G ∈ G 0 , and hence we should initialize c 1 = ord(4), to count the full ord(4) digits that still need to be added to x 1 . Otherwise, we have just determined one more step, so we should set c 1 = c 0 − 1. ad 8. If c 1 = 0 then the conditions of Lemma 4.4 are satisfied for all g / ∈ G 1 : If c 1 = 0 and g i / ∈ G 1 then ( x 1 − θ i )(x 2 − θ i ) is a square in L * i for all x 2 ∈ x 1 + π e+1 O. Therefore, if G 1 = ∅, then all P ∈ C(k) with x(P ) ∈ x 1 + π e+1 O have the same image for x(P ) − θ i in L * i /L * 2 i and therefore, µ(P ) = x 1 − θ in A * /A * 2 . In addition, we know that f (x 1 ) = f 0 i N Li/k (x 1 − θ i ) is a square due to the test in step 4. This verifies that such points P do exist and thus that x 1 − θ represents an element of µ(C(k)). Alternatively, suppose that G 1 contains one polynomial, of degree 1. We write G 1 = {g j (x)} with g j (x) = a(x − θ j ). For any point P ∈ C(k) with x(P ) ∈ x 1 + π e+1 O, we have that f (x(P )) is a square. However, since f (x) = f 0 (x − θ j ) i =j N Li[x]/k[x] (x − θ i ) and (x(P ) − θ i )(x 1 − θ i ) is a square in L i for all i = j, we see that the square class of x(P )−θ j , if non-zero, must be constant too and that all such points P have µ(P ) = µ ((θ j , 0)). Therefore, if we take care to record the images of all degree 1 Weierstrass points of C beforehand, these points are taken care of. ad 11. If the test in step 6 does not hold true, or if c 1 = 0 then we cannot guarantee that µ is constant for all P ∈ C(k) with x(P ) ∈ x 1 + π e+1 O. In this case, we call the same routine again, to refine our search. As remarked for step 6, the condition there will be satisfied after finitely many recursion steps and then after at most ord(4) steps, we will have c 1 = 0 as well. define LocalImage(f ): 1. Let g 1 · · · · · g m = f be a factorization into irreducible polynomials. 2. A := k[θ] = k[x]/f (x); H := A * /A * 2 3. W := (x 1 − θ) + f (x) x−x1 x=x1 in H : x 1 a root of f (x) in k 4. µ : x → x − θ in H 5. G := {g 1 , . . . , g m } 6. call SquareClasses(0, 0, G, −1) 7. if n is even:f := f 0 x n + · · · + f n elsef := f 0 x n+1 + · · · + f n x 8. LetG consist of a factorisation off into irreducibles. 9.μ : x → x(1 − xθ) 10. if n is odd or f n is a square: Add 1 to W 11. if n is even and f n is a square: Add x toG 12. call SquareClasses(0, 1,G, −1) while usingf andμ instead of f and µ. 13. return W Explanation: ad 2. The algorithm we describe determines µ(C(k)) for µ : C(K) → A * /A * 2 at no extra cost. If n is even, we need to take the image under the map A * /A * 2 → A * /A * 2 k * in order to find a proper interpretation of the computed set. ad 3. We initialize W with the images of the degree 1 Weierstrass points under µ. Thus, when we call SquareClasses and find ourselves with c 1 = 0 and G 1 non-empty, then the possible image under µ has already been accounted for. ad 4. We initialize µ with the definition that works for most points, for use in SquareClasses. (See step 10 for why we are explicit about this here.) ad 6. We now call SquareClasses to add to W the images µ(P ) of points P ∈ C(k) with x(P ) ∈ O. Given that G consists of the full factorization of f , the value of c 0 passed to SquareClasses is irrelevant. We pass the dummy value of −1. ad 7. Note that for the remaining points, we have 1/x(P ) ∈ πO. Therefore, by making a change of variables z = 1/x and w = y/x ⌈n/2⌉ , we are left with finding the images of the points on w 2 = f 0 z n + · · · + f n if n is even or w 2 = f 0 z n+1 + · · · + f n z if n is odd, with z ∈ πO under µ : z → (1/z − θ) = z(1 − zθ) modulo squares, for non-Weierstrass points, except for ∞ + and ∞ − . ad 10. If n is odd or f n is a square, then there are points P ∈ C(k) with x(P ) = ∞ and hence z(P ) = 0. We know that for such points, µ(P ) = 1, so we add that value to W . ad 11. If n is even and f n is a square, then ∞ + and ∞ − are rational points. However, the definition ofμ does not yield the correct value for these points, since z(∞ ± ) = 0. As a workaround, add x toG, so that the recursive search does not try to evaluate step 9 of SquareClasses for these points. The correct value has already been added to W in step 10. ad 12. We now call SquareClasses to add to W the images µ(P ) of points P ∈ C(k) with 1/x(P ) ∈ πO. The nature ofG ensures that the value passed to c 0 is irrelevant, so we pass a dummy value of −1. Together with steps 3, 6, and 10, this guarantees that after this, W equals µ(C(k)). Computing the local image of µ at real places If k is a completion of a number field at a complex place, then A * = A * 2 for all A = k[x]/f (x) with f (x) a square-free polynomial. Furthermore C(k) is non-empty for all curves C. In this case, µ(C(k)) = H k = {1}, so there is nothing to do. Now suppose that k = R and that f (x) = (x − θ 1 ) · · · (x − θ r )g(x) where θ 1 > θ 2 > · · · > θ r are the real roots of f (x) and g(x) is a polynomial with no roots in R. Then A * /A * 2 = (R * /R * 2 ) r ≃ (Z/2Z) 2 and µ : C(R) → R * /R * 2 × · · · × R * /R * 2 (x, y) → (x − θ 1 , · · · , x − θ r ) Due to the ordering on the Note that, if n is even, we have to quotient out by the subgroup generated by (−1, . . . , −1), consisting of the image of R * in A * /A * 2 . θ i , we have that if x − θ i < 0 then x − θ j < 0 for j ≤ i. For a point P ∈ C(R) we have f (x(P )) ≥ 0, so if f n > 0 While the computation of µ(C(R)) is quite straightforward, the use of this information in computing Sel (2) fake (C/k) for some number field k is one of the most error-prone parts due to precision issues. See Remark 6.1 for more details. Computing the fake Selmer set In this section, let k be a number field. We consider the algebra A = k[x]/(f (x)). Let S be the finite set of places p satisfying one of: • p is infinite • p has even residue characteristic • f has coefficients that are not integral at p • the leading coefficient of f is not a unit at p • ord p (disc(f )) > 1 We write H k (S) ⊂ H k for the elements δ ∈ H k such that ρ p (δ) is unramified according to Definition 4.2 for all places p / ∈ S. The first part of Lemma 4.3 asserts that Sel (2) fake (C/k) ⊂ H k (S). It is a standard fact from algebraic number theory that the subgroup A(2, S) ⊂ A * /A * 2 of elements that are unramified outside S is finite, so H k (S) is a finite set. This set can be computed, see the explanation of the FakeSelmerSet algorithm below. Let T be the union of S with the set of primes p for which √ q + 1 √ q ≤ 2(2 2g (g − 1) + 1), where q := #O k /pO k . The second part of Lemma 4.3 guarantees that for any prime p / ∈ T we will have ρ p (H k (S)) ⊂ µ p (C(k p )). Hence Let S ′ ⊃ S contain generators for the 2-parts of the class groups of k and the simple factors of A. We abuse notation slightly by writing A * S ′ for the S ′ -unit subgroup of A * . It is easy to verify that A(2, S ′ ) = A * S ′ /A * 2 S ′ and that A(2, S) ⊂ A(2, S ′ ). Determining A(2, S) ⊂ A * S ′ /A * 2 S ′ is a matter of F 2 -linear algebra. In practice, determining A * S ′ is the bottle-neck in these computations, because it requires finding the class groups and unit groups of the number fields constituting A. We write G for the classes representable by A(2, S). It is clear that Sel (2) fake (C/k) ⊂ G and that G is an explicitly computable finite group. ad 7. In all cases, the norm map induces a well-defined homomorphism G → k * /k * 2 , because N A/k (k * ) ⊂ k * 2 if 2 \| deg(f ). Furthermore, it may happen that W is empty in this step. Since µ(C(k)) ⊂ Sel (2) fake (C/k) ⊂ W , this implies that C(k) is empty. ad 8. As remarked in Lemma 4.3, we may obtain information at primes of good reduction, if the size of the residue field is small. The probability that these larger primes make a difference is rather small, and in practice they often don't. The theoretical size of T grows extremely quickly: If C is of genus 2 then T should include all primes of norm up to 1153 and for genus 3 all primes up to norm 66553. If it is infeasible to work with the full set T , one can work with a smaller set of primes. The set we compute can then be strictly larger than Sel (2) fake (C/k). ad 11. In practice, W will be a rather small set and the only reason we want to compute W p is to reduce the size of W in step 17. Especially for large residue fields, LocalImage can be extremely expensive. By integrating steps 11 through 17, one can detect early if W p is big enough to cover all of ρ(W ). In that case, one does not have to compute the rest of W p and can continue with the next p. This makes an immense difference in running time in practice. ad 12. Note that the implementation of LocalImage only produces a set of representatives in A * p /A * 2 p for µ p (C(k p )). We still have to quotient out by k * p if 2 \| deg(f ). ad 16. Since G is a finite group, ρ p can simply be computed by computing the images of the generators. However, one should take care that the generators of G in A are represented by S-units. Algorithms naturally find these with respect to a factor basis and writing the generators in another form may be prohibitively expensive. One should instead compute the images of the factor basis and take the appropriate linear combinations in an abstract representation of the multiplicative groups. Remark 6.1. As is noted in Section 5, the computation of µ v (C(k v )) is quite straightforward for real places v. The difficulty is in computing the map ρ v : H k (S) → H kv . Any first approach would probably involve representing H k (S) using generators of the ring of S-units in A k . Their images in real completions can lie very close to 0, making it necessary to compute very high precision approximations to their real embeddings. As is remarked ad 16 above, a better approach is to determine the signs of a factor basis and use the fact that H kv lies in a group to compute the images of H k (S). Proving non-existence of rational points One of the most important applications of computing Sel (2) fake (C/k) is that, if it is empty, we can conclude that C(k) is empty. This may even be the case if C does have points everywhere locally, and so it allows us to detect failures of the local-to-global Principle. Example 7.1. Consider the hyperelliptic curve C : y 2 = 2x 6 + x + 2. Then C(Q) is empty, but C has points everywhere locally. Proof. It is straightforward to check that C does have points everywhere locally. In this case, A = Q[x]/(2x 6 + x + 2), which is a number field. Write O for the ring of integers in A. As it turns out, we have the prime ideal factorisation 2O = pq 5 . We have disc(2x 6 + x + 2) = 2 4 · 11 · 271169, so we know that Sel (2) fake (C/Q) ⊂ H k (S) with S = {2, ∞}. The ideal class group of O is Z/2 and p and q are not principal ideals. Hence, there is no S-unit u ∈ A such that N A/Q (u) ∈ 2Q * 2 . Thus, in step 7 of FakeSelmerSet, we will find that W is empty and thus that Sel (2) fake (C/Q) = ∅ and therefore C(Q) is empty. In this example, H Q (S) is empty, so there are no everywhere locally solvable two-covers of C. However, H Q is non-empty (the element 2(θ 5 − θ 4 + θ 2 − θ + 1) ∈ A has norm 18, hence represents an element of H Q , for example), so C does have two-covers of the form D δ . This raises the following question, to which we do not yet have an answer. Question 7.2. Can a hyperelliptic curve over a number field k be everywhere locally solvable and yet have H k empty? Another example that is worthwhile to illustrate, is that small primes of good reduction can still yield information in the fake Selmer group calculation. C : y 2 = −x 6 + 2x 5 + 3x 4 − x 3 + x 2 + x − 3 This curve has points everywhere locally over Q, has good reduction outside 2 and 35783887, and has no rational points. One can show this by proving that Sel (2) fake (C/Q) is empty, but one needs to consider this curve locally at 73. In particular, this shows that C has an unramified degree 16 cover over Q 73 , with good reduction and no F 73 -points in the special fiber. Proof. In this case the algebra A is a number field with trivial class group. We write O for its ring of integers. We have disc(−x 6 + 2x 5 + 3x 4 − x 3 + x 2 + x − 3) = 2 2 · 35783887 so in step 2 of FakeSelmerSet, we find that S = {2, ∞}. Since the class group of A is represented by prime ideals above S, we have A(2, S) = O * S /O * 2 S , so generators are represented by a system of fundamental units together with generators of the prime ideals above 2: α N A/Q (α) u 0 −1 1 u 1 θ 5 − 2θ 4 − 4θ 3 + 2θ 2 + 3θ − 1 1 u 2 θ 5 − 3θ 4 + θ 2 − 2θ + 2 −1 u 3 θ 4 − θ 3 − 1 −1 p 2 θ − 1 −2 q 2 2/(θ − 1) 2 16 In Step 7 we find that W is represented by {u 2 , u 3 , u 1 u 2 , u 1 u 3 }. For p = ∞, the set W does not get reduced. Note that A has only two real embeddings, corresponding to θ → 0.85 and θ → 2.94. Since all representatives in W have norm −1, we see that the real embeddings of these elements will be of opposite sign, and since we are working modulo Q * , we can choose which is positive. On the other hand, since the leading coefficient of f is negative, the algorithm in Section 5 predicts that µ(C(R)) = {(−1, 1)}, which corresponds to the description above. For p = 2 the set W does get reduced. We find that if (x, y) ∈ C(Q 2 ) then x ∈ 2 2 + O(2 3 ). It is only for δ = u 1 u 3 = −θ 5 − θ 4 + 1 that we have that δ ∈ (4 − θ + O(2 3 ))Q * 2 modulo squares in (A ⊗ Q 2 ) * , so if there is a point P 0 ∈ C(Q) then µ(P 0 ) = u 1 u 3 . For p = 73, we find in step 17 of FakeSelmerSet that u 1 u 3 does not map into the image of C(Q 73 ) and hence that C(Q) is empty. Applications to curves with points Let k be a number field and let C : y 2 = f (x) be a curve of genus at least 2 over k. Even if C(k) is non-empty, the set Sel (2) fake (C/k) still contains useful information. If rank(Jac C (k)) < genus(C) and Jac C (k) is actually known, then one can use explicit versions of Chabauty's method [11,13,14] to compute a bound on #C(k). In fact, if one combines this with Mordell-Weil sieving [5,7,19], then one would expect that one should be able to arrive at a sharp bound [17]. If rank(Jac C (k)) ≥ genus(C) then one can try to pass to covers. One chooses an unramified Galois cover D/C. By the Chevalley-Weil theorem [12], the rational points of C are covered by the rational points of finitely many twists D δ /C of D/C. For hyperelliptic curves C, a popular choice is the 2-cover D δ described in Section 3 [2,3,6]. The genus of D δ is much larger than the genus of C. This means that it is possible that rank(Jac D δ (k)) < genus(D δ ) for all relevant δ and thus that Chabauty's method can be applied to each D δ . The curve D δ is usually of too high genus to do computations with directly. However, over k, the curve D δ covers many hyperelliptic curves besides C. These arise from factorisations f (x) = g(x)h(x), where at least one of g, h has even degree. Suppose that g(x) is monic and that its field of definition is L/k, i.e., g(x), h(x) ∈ L[x]. We write E γ : γy 2 1 = g(x) E ′ γ : (1/γ)y 2 2 = h(x) It is straightforward to see that, for every δ, there is a value of γ = γ(δ) ∈ L * /L * 2 such that, over L, we have the diagram D δ Õ Õ % % 3 3 3 3 3 3 3 3 3 3 3 3 3 E γ x 2 2 A A A A A A A A C x E ′ γ x~} } } } } } } } P 1 Note that D δ (k) maps to C(k) and from there to P 1 (k). Therefore, D δ (k) maps to {P ∈ E γ (L) : x(P ) ∈ P 1 (k)} and in order to find which of those points correspond to points in C(k) we only have to find which points in P 1 (k) ∩ x(E γ (L)) lift to C(k). There is ample literature on how to perform this last step [2,3,15]. However, the problem of finding the relevant values for γ has largely been glanced over. This is mainly because in any particular situation, it is quite easy to write down a finite collection of candidates for γ and then test, for every place p of k and any extension q of p to L, for each value if x(E γ (L q )) ∩ P 1 (k p ) is non-empty. In fact, this is quite doable for the fibre product E γ × P 1 E ′ γ too (see [2, Appendix A]). However, the smallest set of values for γ we can hope to arrive at through local means, is {γ(δ) : δ ∈ Sel(2) fake (D/k)}. Also, note that the degree of L over k will usually be larger than the degree of A. For instance, if C : y 2 = f (x) where f (x) is a sextic with Galois group S 6 over k, the the field L over which f factors as a quadratic times a quartic is of degree 15, while A is of degree 6. Hence, from a computational point of view it is interesting to avoid as much computation as possible in L. The map δ → γ(δ) is in fact straightforward to compute, given representatives δ ∈ A = k[x]/(f (x)). Let L[Θ] = L[x]/(g(x)). Then there is a natural k-algebra homomorphism j : A → L[Θ] given by θ → Θ. Using these definitions we have γ(δ) = N L[Θ]/L (j(δ)). While the degree of L[Θ] is probably quite high, we only need to compute a norm with respect to it. This is not such an expensive operation. The following example illustrates how the computation of the fake 2-Selmer set fits in with the standard methods for determining the rational points on hyperelliptic curves. Example 8.1. Let C : y 2 = 2x 6 + x 4 + 3x 2 − 2. then C(Q) = {(±1, ±2)}. Proof. First we observe that C covers two elliptic curves, v 2 1 = 2u 3 1 + u 2 1 + 3u 1 − 2 and v 2 2 = −2u 3 2 + 3u 2 2 + u 2 + 2, but each of these curves has infinitely many rational points. When we apply FakeSelmerSet to this curve, we find that Sel (2) fake (C/Q) is represented by {−1 − θ, 1 − θ}, so it is equal to µ(C(Q)). Putting L = Q(α), where α 2 + α + 2 = 0, we obtain the factorization: C : y 2 = f (x) = (2x 2 − 1)(x 2 − α)(x 2 + α + 1) We choose g(x) = (x 2 − 1 2 )(x 2 − α) and h(x) = 2(x 2 + α + 1). We find that γ(1 − θ) = γ(−1 − θ) = 1 2 (1 − α) and write E : y 2 1 = (1 − α)(2x 2 − 1)(x 2 − α) . Any point (x, y) ∈ C(Q) must correspond to a point (x, y 1 ) ∈ E(L) with x ∈ Q. The curve E is isomorphic over L to the elliptic curvẽ E : v 2 3 = u 3 + (1 − α)u 2 + (2 − 9α)u + (16 − 2α), which hasẼ(L) ≃ (Z/2) × Z. This makes methods as described in [2,15] applicable and a p-adic argument at p = 5 proves that x(E(L)) ∩ P 1 (Q) = {±1}. Applications to genus 1 curves In this section we illustrate how the computation of Sel (2) fake (C/k) yields interesting results even when C is a genus 1 double cover of P 1 . We recover well-known algorithms for doing 2-descents and second 2-descents on elliptic curves. To our knowledge, nobody took the effort yet of implementing second descent on elliptic curves over number fields, whereas our implementation [9] in MAGMA for computing Sel (2) fake (C/k) does work if k is a general number field. We illustrate the use by giving an example of an elliptic curve over Q( √ 2) with non-trivial Tate-Shafarevich group. If we have a genus 1 curve of the form E : y 2 = f (x) = x 3 + a 2 x 2 + a 4 x + a 6 , then Sel (2) fake (E/k) = Sel (2) (E/k) is equal to the usual 2-Selmer group of E. In this case, the algorithm presented in Section 6 could be improved by using the fact that the sets computed in LocalImage are groups of known size. One recovers an algorithm to compute the 2-Selmer group of an elliptic curve very similar to [10]. Following Section 3, for every δ ∈ Sel (2) (E/k), we can write down an everywhere locally solvable cover D δ . The model we obtain is the intersection of two quadrics in P 3 . The pencil of quadrics cutting out D δ contains a singular quadric Q 0 over k, however. Since D δ (k p ) is non-empty for every p, the Hasse principle for conics mandates that Q 0 contains a line over k and thus that the lines on Q 0 form a P 1 over k. By sending a point on D δ to the line on Q 0 that goes through that point, we realise D δ as a double cover of a P 1 and we obtain a model of the form C δ : Y 2 = F 4 X 4 + F 3 X 3 + · · · + F 0 where we know that C δ is everywhere locally solvable. Note that C δ itself is again a curve of genus 1. We can compute Sel (2) fake (C δ /k) in this case as well. The data obtained is the same as that obtained by doing a second 2-descent, along the lines of [16]. In this case too, the algorithm could be optimised a bit by observing that Sel (2) (C δ /k) maps surjectively to the fiber over δ with respect to Sel (4) (E/k) → Sel (2) (E/k); the fibers of the map Sel (2) (C δ /k) → Sel (4) (E/k) are isomorphic to E(k) [2]/2E(k) [4] (see section 6.1.3 in [23]). Thus Sel (2) (C δ /k) is either empty or has a known cardinality. Similarly, in LocalImage, the fact that the set W carries a µ kp (E(k p ))-action and is of known cardinality can speed up the computation immensely. Example 9.1. Let α = √ 2 and consider E : y 2 = x 3 + (2 − 2α)x + (2 − 9α) Then X(E/Q(α)) is non-trivial. Proof. We find that δ = θ 2 + (8 − 4α)θ + 13 − 6α is in Sel (2) fake (E/Q(α)). The corresponding cover of E can be given by the model C : Y 2 = −(2α + 3)X 4 + (4α + 6)X 3 − (18α + 24)X 2 + (16α + 24)X + 2α + 4 . It turns out that Sel (2) fake (C/k) is empty. Hence, it follows that δ ∈ Sel (2) (E/k) is not in the image of µ(E(k)) and therefore represents a non-trivial member of X(E/Q(α)) [2]. In fact, we have shown that δ represents an element of X(E/Q(α)) that is not divisible by 2. Efficiency of two-cover descent It is natural to ask how often we should expect that Sel (2) fake (C/k) is empty if C(k) is. This is equivalent to determining what proportion of curves C over k have an everywhere locally solvable two-cover (see Theorem 3.4). To quantify this question, let us limit ourselves to curves C of genus 2 and k = Q. We define a naïve concept of height on the set of genus 2 curves over Q so that we can define the proportion we are interested in as a limit. y 2 = f (x) where f (x) = f 6 x 6 + · · · + f 0 ∈ Z[x] and \|f i \| ≤ D for i = 0, . . . , 6 Note that (especially when D is large) an isomorphism class of a curve may be represented by many different models in M (D). It would be perhaps more natural to order the curves by \|disc(f )\| rather than maximal absolute value of the coefficients. Our motivation for partitioning the set of genus 2 curves by M (D) is that one can easily sample uniformly from M (D), whereas this is much more complicated otherwise. A related question has a definite answer. In [20,21] it is proved that there is a well-defined proportion of genus 2 curves over Q that have points everywhere locally. In fact, if one actually computes the local densities involved, one arrives at . To this end, we tested if Sel (2) fake (C/Q) = ∅ for a fairly large number of curves, sampled from M (D) for various D. For D = 1, 2, 3 we considered all of M (D) (see [8]) and our results are unconditional for all but 42 of the roughly 200 000 isomorphism classes of curves involved. In the table below, these curves are counted according to isomorphism class. For D = 4, . . . , 60 and for D = 100 we have sampled curves C from M (D) uniformly randomly and computed the following: • Whether the curve has a small rational point (whose x-coordinate is a rational number with numerator and denominator bounded by 10 000 in absolute value), • Whether the curve is everywhere locally solvable, • Whether Sel (2) fake (C/Q) = ∅, where for reasons of efficiency, the class group information needed to compute A(2, S) was only verified subject to the generalized Riemann hypothesis. This places each curve in one of the four categories • C(Q v ) = ∅ for some place v • Sel (2) fake (C/Q) = ∅ (but C does have points everywhere locally) • C(Q) contains a small point • It is unknown whether C has a rational point or not. See Table 1 for the statistics and Figure 1 for a graph of the data. It should be noted that the samples for the various D are not completely independent: Any curve sampled from M (D) that happened to have all its coefficients bounded by D ′ < D was also included as sample from M (D ′ ). We make a number of observations. (1) The proportion of curves with a local obstruction against rational points tends to a value near 15% remarkably quickly. (2) As D increases, the proportion of curves in M (D) with a small rational point decreases in the expected way. The jumps that can be observed at D = 4, 9, 16, . . . come from additional possibilities for points at infinity or with x = 0 that occur when the leading or trailing coefficient is a square. fake (C/Q) non-empty decreases more slowly. Figure 1 clearly shows that, at least for C ∈ M (100), testing if Sel (2) fake (C/Q) = ∅ is a very useful criterion to decide if C(Q) is empty, with less than 15% of undecided curves. (4) The data is inconclusive on a possible limit value for the proportion of curves C ∈ M (D) with Sel (2) fake (C/Q) = ∅ as D → ∞, but it suggests that it might be somewhere between 65% and 85%. It would be very interesting to find out if this limit exists and what its approximate value might be. What makes this likely to be hard is the subtle interplay between local and global information that determines the size of Sel (2) fake (C/Q). then µ(C(R)) = {(1, . . . , 1), (−1, −1, 1, . . . , 1), . . .}, which is to say, all vectors consisting of an even number of −1 entries followed by 1 entries. Conversely, if f n < 0 then µ(C(R)) = {(−1, 1, . . . , 1), (−1, −1, −1, 1, . . . , 1), . . .}, vectors consisting of an odd number of −1 entries followed by 1 entries. Example 7 . 3 . 73Consider the hyperelliptic curve lim D→∞ #{C ∈ M (D) : C(Q p ) = ∅ for all p} #M (D) ≈ 85% On the other hand, one would expect that the proportion of curves that actually have a rational point vanishes as D grows: lim D→∞ #{C ∈ M (D) : C(Q) = ∅} #M (D) ? = 0 . Heuristic considerations suggest that the quantity under the limit should be of order D −1/2 . We are interested in the question whether the following limit exists and what might be its value: lim D→∞ #{C ∈ M (D) : Sel Figure 1 . 1Proportion chart of obstructions (3) The proportion of curves with Sel Table 1 . 1Statistics on genus 2 curvesrational points not ELS 2-cover obstr. remaining Sel(2)fake (C/k) = {δ ∈ H k (S) : ρ p (δ) ∈ µ p (C(k p )) for all p ∈ T }. This gives us a way to compute the fake Selmer-set explicitly.define FakeSelmerSet(f ):total 1 45 3 0 401 449 2 2823 1096 29 14116 18064 3 29403 27786 1492 137490 196171 10 5903 9915 1546 20242 37606 20 2020 4748 1012 5393 13173 30 2717 6675 1579 5959 16930 40 4025 10648 2682 7963 25318 50 18727 51831 13538 34269 118365 60 1589 4571 1278 2547 9985 100 8106 24063 7045 10786 50000 The Magma algebra system. I. The user language. The MAGMA computer algebra system is described in Wieb Bosma, John Cannon, and Catherine Playoust. 24The MAGMA computer algebra system is described in Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235-265. Dissertation. N R Bruin, Chabauty methods and covering techniques applied to generalized Fermat equations. Amsterdam; Leiden13311042University of LeidenStichting Mathematisch Centrum Centrum voor Wiskunde en InformaticaN. R. Bruin, Chabauty methods and covering techniques applied to generalized Fermat equations, CWI Tract, vol. 133, Stichting Mathematisch Centrum Centrum voor Wiskunde en Informatica, Amsterdam, 2002. Disserta- tion, University of Leiden, Leiden, 1999. MR 1916903 (2003i:11042) Chabauty methods using elliptic curves. Nils Bruin, 27-49. MR 2011330J. Reine Angew. Math. 56211051Nils Bruin, Chabauty methods using elliptic curves, J. Reine Angew. Math. 562 (2003), 27-49. 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[ "THEÉTALE FUNDAMENTAL GROUPOID AS A TERMINAL COSTACK ILIA PIRASHVILI", "THEÉTALE FUNDAMENTAL GROUPOID AS A TERMINAL COSTACK ILIA PIRASHVILI" ]	[]	[]	[]	Let X be a noetherian scheme. We denote by Π 1 (X) the fundamental groupoid. In this paper we prove that the assignments U ↦ Π 1 (U ) is the 2-terminal costack over the site ofétale coverings of X.	10.1215/21562261-2017-0041	[ "https://arxiv.org/pdf/1412.5473v2.pdf" ]	117,649,451	1412.5473	6dbcb40863bcafda6327a905fe9f8ef9eb2c0cd9	THEÉTALE FUNDAMENTAL GROUPOID AS A TERMINAL COSTACK ILIA PIRASHVILI 17 Dec 2014 THEÉTALE FUNDAMENTAL GROUPOID AS A TERMINAL COSTACK ILIA PIRASHVILI 17 Dec 2014 Let X be a noetherian scheme. We denote by Π 1 (X) the fundamental groupoid. In this paper we prove that the assignments U ↦ Π 1 (U ) is the 2-terminal costack over the site ofétale coverings of X. Introduction In a recent paper [6], we showed that for a good topological space X, the assignment U ↦ Π 1 (U ), U ⊂ X, being the topological fundamental groupoid, was the 2-terminal costack over X. In this paper, we show that the analogue of this theorem holds for theétale fundamental groupoid over a noetherian scheme as well. This paper is part of my PhD thesis at the University of Leicester under the supervision of Dr. Frank Neumann. He introduced me to the fundamental groupoid and insipred much of this paper, for which I would like to thank him. Basic definitions and general results in 2-mathematics 2.1. Limits, 2-limits, colimits, 2-colimtis. s First let us fix some notations. For functors and natural transformations A F / G B G1 / G G2 / G C T / G D , α ∶ G 1 ⇒ G 2 . one denotes by α⋆F and T ⋆α the induced natural transformations G 1 F ⇒ G 2 F , T G 1 ⇒ T G 2 . Let I be a category and let F ∶ I → Cat be a covariant 2-functor from the category I to the 2-category of categories. For an element i ∈ I we let F i be the value of F at i. For a morphism ψ ∶ i → j we let ψ * ∶ F i → F j be the induced functor. For any i ψ → j ν → k, one has the natural transformation µ ψ,ν ∶ ν * ψ * → (νψ) * satisfying the coherent relations. Recall the construction of the category 2-lim i F i called the 2-limit of F. 2-Limits of categories. Objects of the category 2-lim i F i are collections (x i , ξ ψ ), where x i is an object of F i , while ξ ψ ∶ ψ * (x i ) → x j for ψ ∶ i → j is a morphism of the category F j satisfying the the following condition: For any i ψ → j ν → k, the following diagram ν * (ψ * (x i )) ν * (ξ ψ ) / G µ ψ,ν ν * (x j ) ξν (νψ) * (x i ) ξ ψν / G x k commutes. A morphism from (x i , ξ ψ ) to (y i , η ψ ) is a collection (f i ), where f i ∶ x i → y i is a morphism of F i such that for any ψ ∶ i → j, the following ψ * (x i ) ξ ψ / G ψ * (fi) x j fj ψ * (y i ) η ψ / G y j is a commutative diagram. 2.1.2. 2-Colimits of categories. Dually, for a (covariant) 2-functor G ∶ I → Cat one can define the 2-colimit of G as follows: Let ψ ∶ i → j a morphism in I and ψ * ∶ G i → G j be its induced functor. For a composition i ψ → j ν → k, one has the natural transformation µ ψ,ν ∶ ν * ψ * → (νψ) * . Then 2-colim i G i is a category together with a family of functors α i ∶ G i → 2 − colim i G i and natural transformations λ ψ ∶ α j ψ * ⇒ α i , satisfying the following condition: For any i ψ → j ν → k, the following diagram α k (νψ) * λ νψ / G α i (α k ν * )ψ * λν * * ψ * / G α k * µ ψ,ν O y α j ψ * λ ψ O y commutes. Furthermore, one requires that for any category G, the canonical functor κ ∶ Hom Cat (2 − colim i G i , G) → lim i Hom Cat (G i , G) is an equivalence of categories. Here the functor κ is given by κ(χ) = (χ ○ α i , χ i ⋆ λ ψ ). It is wellknown that 2-colim exists and is unique up to an equivalence of categories, see, [1, pp. 192-193]. Proposition 2.1. Let I be a category, A ∶ I → Cat a 2-functor and denote L = 2-lim i∈I A i . • Assume that finite limits exist in every category A i and that the maps A i → A j preserve finite limits. Then finite limits exist in L and the canonical maps L → A i respect finite limits as well. • Assume that finite colimits exist in every category A i and that the maps A i → A j preserve finite colimits. Then finite colimits exist in L and the canonical maps L → A i respect finite colimits as well. Proof. Let C be a finite category and A ∶ C → L a functor. To show that it has a limit, recall first the construction of the 2-limit. We have that any object A in L is a collection {(A i , α ψ ∶ ψ * (A i ) → A j )} such that the α ij are compatible. So for any object c ∈ C, the object A c , being A(c) can be seen as a compatible collection {(A ci , α ψ ∶ ψ * c (A ci ) → A cj )}. Hence for every i, we can take the limit of the {A ci } with respect to c, which we denote by P i . The fact that these are compatible and form an element in L, which we denote by P , comes from the universality of the P i 's. To show the universality of P , one takes an other element Q in L and since Q is again a collection of elements {(Q ci , α ψ ∶ ψ * c (Q ci ) → Q cj )}, we get an map Q → P . Universality follows again from universality of the P i 's. For colimits, the proof is analogous to the above one. Proposition 2.2. Let I be a filtered category, A ∶ I → Cat a 2-functor and denote L = 2-colim i∈I A i . • Assume that finite limits exist in every category A i and that the maps A i → A j preserve finite limits. Then finite limits exist in L and the canonical maps L → A i respect finite limits as well. • Assume that finite colimits exist in every category A i and that the maps A i → A j preserve finite colimits. Then finite colimits exist in L and the cannonical maps L → A i respect finite colimits as well. Proof. Let C be a category and A ∶ C → L a functor. By the definition of the 2-colimit for filtered system, every A c can be thought to be in one of the A i 's. Hence by the finiteness of C and the fact that I is filtered, we can find a single A j such that the whole diagram A ∶ C → L can be represented in it. We then take the colimit of A in any such A j , which gives us an element in L and we denote it by P . Note that it does not depend on our choice of A j . To show that it is indeed the colimit in L, consider an other object Q ∈ L such that we have compatible maps Q → A c for all c ∈ C. Again by the definition of the 2-colimit, Q is in one of the A i 's and we can again find a category A k such that the diagram A ∶ C → L, P and Q, as well as all the morphisms are inside it and hence we will have a map Q → P . Uniqueness follows trivially as well. The proof for the colimit is analogous to the above one. Stacks and Costacks. 2.2.1. Stacks. Let X be a site and F ∶ X op → Cat a 2-functor where Cat is the 2-category of categories. This is called a fibered category over X. If we have two fibred categories over X, then a morphism between them is called a fibered functor. The following important fact holds. Lemma 2.3. Let F be a fibered category over X and let A, B be objects in F(U ). Then the assignment V ↦ Hom F(V ) (i * (A), i * (B)) for any morphism i ∶ V → U defines a presheaf on X U . It is denoted by Hom F (A, B). Definition 2.4 (Prestack). If the presheaf Hom F (A, B) is in addition a sheaf, then we say F is a prestack. Let X be a site and F ∶ X op → Cat a 2-functor. Let U be an object in X and U = {U i → U } a covering of U . Then we can consider the following diagram: ⊓ i∈I F(U i ) / G / G ⊓ i,j∈I F(U ij ) / G / G / G ⊓ i,j,k∈I F(U ijk ) . We denote its limit by lim(U, F) and its 2-limit by 2-lim(U, F). Note that the last part ⊓ i,j,k∈I F(U ijk ) does not factor in the limit, only the 2-limit. Also note that the 2-limit is usually called the descent data. But since we will use 2-limits and 2-colimits throughout this paper, it is preferable to call it the 2-limit. Costacks. Let X be a site and F ∶ X → Cat a 2-functor where Cat is the 2-category of categories. We call this a cofibred category over X. Let X be a site and F ∶ X → Cat a 2-functor. Let U be an object in X and U = {U i → U } a covering of U . Then we can consider the following cosimplicial sequence: ⊓ i∈I F(U i ) ⊓ i,j∈I F(U ij ) o o o o ⊓ i,j,k∈I F(U ijk ) o o o o o o where U ij = U i × U U j . We denote its colimit by colim(U, F) and its 2-colimit by 2-colim(U, F). Note that the last part ⊓ i,j,k∈I F(U ijk ) does not factor in the colimit, only the 2-limit. Definition 2.7 (Costack). A cofibered category F over X is called a costack if for all objects U of X and for all coverings U of U , the functor F(U ) ← 2-colim(U, F) is an equivalence of categories. Alternatively, we can define a costack using stacks. Definition 2.8. Let F be a cofibered category over X. We say that F is a costack, if for every category C, the assignment U ↦ Hom Cat (F(U ), C) is a stack. Note that if our category took values in groupoids, than it would be enough to check it for every groupoid. The 2-category of Galois Categories It should be noted that the following definition of a Galois category differs from the standard definition of a Galois category. Definition 3.1. A (finite-connected) Galois category is a category C together with a set of covariant functors {F i ∶ C → FSets} i∈I , where I is a finite set, satisfying the following axioms: (1) Finite projective limits exist in C. (2) Finite inductive limits exist in C. (3) Any morphism u ∶ Y → X in C factors as Y u ′ → X ′ u ′′ → X ′′ , where u ′ is a strict epimorphism and u ′′ is a monomorphism which is an isomorphism onto a direct summand of X. If I can be chosen to be a one element set, then C is connected. This is equivalent to the standard definition of a Galois category. However, from now on, Galois category will refer to Def. 3.1. For more on connected Galois categories, see [2]. Definition 3.2. Let {F i ∶ C → FSets} i∈I and {G j ∶ D → FSets} j∈J be two Galois categories. A morphism of galois categories consists of a map f ∶ J → I, a collection of functors ϕ j ∶ C → D, j ∈ J, preserving finite limits and finite colimits and isomorphisms λ j , j ∈ J, as given in the following diagram D Gj " 4 ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❴ ❴ ❴ ❴ k s λj C ϕj o o F f (j) \| \| ② ② ② ② ② ② ② ② FSets . For simplicity, we will sometimes just write {ϕ j ∶ F f (j) → G j }. To define composition, we need to define the composition of the λ j 's. So say we now have E H k ' 9 ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ⑤ ⑤ ⑤ ⑤ z Ò λ k,φ k D G f (k) φ k o o ❇ ❇ ❇ ❇ ] e λ k,ϕ k C ϕ k o o Fg (i) w w ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ FSets . Define λ k,φ k ○ λ k,ϕ k (x) = λ k,φ k (ϕ(x)) ○ λ k,ϕ k (x) . In more detail we have λ k,φ k ○ λ k,ϕ k (x) ∶ F (x) λ k,ϕ k (x) → G(ϕ(x)) λ k,φ k (ϕ(x)) → E(φ ○ ϕ(x)). It is easily verified that the above construction is strictly associative. Definition 3.3. A 2-morphism between {ϕ j ∶ F f (j) → G j } and {φ j ∶ F f (j) → G j } is a collection of natural transformations C φj / G ϕj / G ζj ✤✤ ✤✤ D such that additionally the following diagram F x λ k,ϕ k (x) / G λ k,φ k (x) # 5 ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ G(ϕ(x)) ζ(x) G(φ(x)) commutes. This shows that we can talk about the (strict) 2-category of Galois categories. We will denote it by GCat. Hence we can now talk about 2-functors with values in GCat, as well as stacks, prestacks etc. A stack with values in the 2-category of Galois categories will be referred to as a Galois stack. If we have two 2-functors with values in Galois categories, then we will call a morphism between them that respects the structures a Galois functor. In the case of fibered categories, we will keep the notation and refer to a morphism between two fibered categories that preserves the Galois structures a Galois transformation, even though it would technically be a Galois functor. Note that throughout the paper, whenever we are dealing with 2-functors in Galois categories, we will ignore the functors in Sets. This is to keep the notations simple, but in actuality the functors are of course part of the structure. Let G be a groupoid and assume that π 0 (G) is trivial and equipped with a discrete topology. Then we say that G is a profinite groupoid if for every object X of G, the group Aut(X) is a profinite group and for every morphism x → y, the associated group homomorphism Aut(y) → Aut(x) respects the profinite structures. Proof. This equivalence is given by associating to a profinite groupoid G, the Galois category Hom Cat (G, Sets). On functors and natural transformations, the 2-functor is defined in the obvious way by composition. The fact that Hom Cat (G, Sets) is a Galois category and that it defines an equivalence is easy to check. Let X be a site and F ∶ X → Groupoids a covariant 2-functor. Then we denote by F S the contravariant 2-functor given by U ↦ Hom Cat (F(U ), Sets). Now let E ∶ X → Groupoids and F ∶ X → Groupoids be two covariant 2-functors and F ∶ E → F a natural transformation. Then it is clear that F S ∶ F S → E S is a Galois transformation. But indeed the above proposition shows that the reverse is also true. Hence we have the following as well. Corollary 3.5. Let X be a site. Then the 2-category of fibred functors over X with values in Galois categories, and morphisms and 2-morphism preserving the Galois structure, is anti-equivalent to the 2-category of cofibered functors over X with values in Groupoids. Lemma 3.6. Let X be a site and F ∶ X → GCat a contravariant 2-functor. ThenF again takes values in GCat, whereF is the stackification of F. Proposition 3.7. Let X be a site, F a contravariant 2-functor from X to GCat and S be a Galois stack. Let F ∶ F → S be a Galois transformation. ThenF ∶F → S is a Galois transformation, whereF is the associated stack of F andF the associated 2-functor of F . Proof. To prove this, we will use definition 2.6. First observe that Des(U, F) is a 2-limit, hence using Prop.2.1, we know that it respects finite limits and finite colimits, hence respects Galois transformations. Since the colimit in 2-colim U Des(U, F) is taken with respects to coverings, and hence a filtered system, it too respects Galois 2-transformations by 2.2. Since stackification is obtained purely through 2-limits and filtered 2-colimits, the proposition is proven. The Van-Kampen theorem for the etalé fundamental groupoid Let X be a noetherian scheme. We denote by FEC(X) the site of finite etalé coverings of X and for an etalé covering Y → X, we denote by FEC(Y X) coverings of Y such that the diagram Z / G Ỹ~⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ X commutes. It is a well known result that the category of finite etalé coverings is equivalent to a Galois category. This result is well known. For example [4] discuses this to some extent. Equivalently, this can be stated as the following. For the proof see the proposition on page 140 in [3]. Proof. The proof of this statement follows from Lemma 4.4. For any covering Z ∈ Cov(Y X) we have 2-colim(Z, Π 1 ) → Π 1 (Z), where 2-colim(Z, Π 1 ) denotes the 2-colimit of Π 1 (Z ⊗ X Y ) Π 1 (Z ⊗ X Y ⊗ X Y ) o o o o Π 1 (Z ⊗ X Y ⊗ X ⊗ X Y ) o o o o o o Hence get the associated functor Hom Cat (Π 1 (Z), Sets) → Hom Cat (2-colim(Z, Π 1 ), Sets). Since Hom is left exact, we have Hom Cat (2-colim(Z, Π 1 ), Sets) ≅ 2-lim(Z, Hom Cat (Π 1 , Sets), where 2lim(Z, Hom Cat (Π 1 , Sets) denotes the 2-limit of Hom Cat (Π 1 (Z ⊗ X Y ), Sets) / G / G Hom Cat (Π 1 (Z ⊗ X Y ⊗ X Y ), Sets) / G / G / G Hom Cat (Π 1 (Z ⊗ X Y ⊗ X ⊗ X Y ), Sets). Since by Lemma 4.2 the functor Hom Cat (Z, Sets) → 2-lim(Z, Hom Cat (Π 1 , Sets) is an equivalence of categories, by lemma 4.4 2-colim(Z, Π 1 ) → Π 1 (Z), where 2-colim(Z, Π 1 ) is an equivalence of categories as well, proving the assertion. The 2-Terminal Costack Definition 5.1. Let C be a 2-category. We say that P is the 2-terminal object of C, if for any other obect C ∈ C, Hom Cat (C, P) is equivalent to the 1-point category. Theorem 5.2. Let X be a noetherian scheme. Then the assignment U ↦ Π 1 (U ), U ∈ X is the 2-terminal costack over the site ofétale coverings of X. To prove this theorem, we first need a few other results. Proof. First of all, observe that we can replace the category of sets, with the category of constant presheaves with values in sets. Next we consider the associated prestack of this 2-functor. To do so, we keep the objects the same and replace the presheaves Hom U (a, b) by its sheafification. We claim that this is equivalent to the category of constant sheaves on U . The fact that the objects of these two categories are equivalent is clear. To see that the hom sets are isomorphic, first observe that the sheafification of Hom U (a, b) is Hom(u , Hom(a, b)), which is isomorphic to (b a ) π0(U) = b aπ0(U) . Denote by a the constant sheaf with value a. We have Hom Sheaf (a, b) = Hom P resheaf (a, b) = Hom P resheaf (a, b). Since b is given by U ↦ Hom(U, B), Hom P resheaf (a, b) = Hom(a, Hom(U, b)). This now is isomorphic to (b π0(U) ) a = b π0(U)a , proving the assertion. Let A be a covariant 2-functor. Recall that we denoted by A S the contravariant 2-functor given by U ↦ Hom(A(U ), FSets). Proof of Thm. 5.2. We have already shown that the assignment U → Π 1 (U ) forms a costack. Hence to prove this theorem, we essentially have to show that for every costack C we have an essentially unique map C → Π 1 . Denote by P the covariant assignment U ↦ pt. It is clear that we have a map C → P and hence a map P S → C S which is a Galois transformation by definition. Since C was a costack, C S is a stack and hence the map P S → C S factors through the stackification of P S , which is U ↦ LCS(U ), where LCS(U ) denotes the category of localy constant sheafs on U . In the case of noetherian schemes, LCS(U ) is equivalent to Hom(Π 1 (U ), FSets). Hence, using Prop. 3.6 and 3.7, we know that there exists an essentially unique map C → Π 1 , proving the result. Definition 2. 5 ( 5Stack). A fibered category F over X is called a stack if for all objects U of X and for all coverings U of U , the functor F(U ) → 2-lim(U, F) is an equivalence of categories.Definition 2.6 (Direct Stackification). Let F ∶ X op → Cat be a 2-functor. Define F ′ (U ) ∶= 2colim U Des(U, F) and then we iterate it 3 times. I.e. define F ′′ (U ) ∶= 2-colim U Des(U, F ′ ) and finallyF = F ′′′ (U ) ∶= 2-colim U Des(U, F ′′ ).It is a general result thatF is the associated stack of F. ( 4 ) 4Every F i is right exact. (5) Every F i is left exact. (6) Let {u ∶ Y → X} be a morphism in C. Then u is an isomorphism if and only if F i (u) isan isomorphism for all i ∈ I. Proposition 3 . 4 . 34The 2-category of Galois categories is equivalent to the 2-category of profinite groupoids. Proposition 4 . 1 . 41The 2-functor FEC ∶ FEC(X) → GCat, where FEC denotes the category of finite etalé coverings, given by Y ↦ FEC(Y X), forms a stack. Lemma 4. 2 . 2The 2-functor FEC ∶ FEC(X) → GCat given by Y ↦ Hom Cat (Π 1 (Y ), Sets), where Π 1 (Y ) denotes the etalé fundamental groupoid of Y , forms a stack. Theorem 4. 3 ( 3Van-Kampen Theorem). Let X be a noetherian scheme. Then the assignment Y ↦ Π 1 (Y ) defines a costack on the site of finite etalé coverings of X. Lemma 4 . 4 . 44Let G → H be a functor between groupoids. Assume that the functor Hom Cat (H, Sets) → Hom Cat (G, Sets) is an equivalence of categories. Then G → H is an equivalence of categories as well. Lemma 5 . 3 . 53Consider the constant 2-functor s ∶ U ↦ FSets, with morphisms chosen in the contravariant way. Then the associated prestack of it is given by s ∶ U ↦ CS(U ), where CS(U ) denotes the category of constant sheafs on U . Corollary 5 . 4 . 54Consider the constant 2-functor s ∶ U ↦ FSets, with morphisms chosen in the contravariant way. Then the associated stackŝ is given byŝ(U ) = LCS(U ), where LCS(U ) denotes the category of locally constant sheaves on U . Loop spaces, characteristic classes and geometric quantization. J.-L Brylinski, Progress in Mathematics. v. 107. Birkhaüser. J.-L. Brylinski. Loop spaces, characteristic classes and geometric quantization. Progress in Mathematics. v. 107. Birkhaüser. Boston, Basel, Berlin. 1993. Arithmetic and Geometry around Galois Theory. A Cadoret, Progress in Mathematics. v. 304. Birkhaüser. A. Cadoret. Arithmetic and Geometry around Galois Theory. Progress in Mathematics. v. 304. Birkhaüser. 2013. Ergebnisse der Mathematik und ihrer Grenzgebiete. Peter Gabriel, & Michel Zisman, Springer-Verlag35Calculus of fractions and homotopy theoryPeter Gabriel & Michel Zisman. Calculus of fractions and homotopy theory. Ergebnisse der Mathematik und ihrer Grenzgebiete. Band 35. Springer-Verlag. 1967. Quadratic and Hermitian Forms over Rings. Grundlehren der mathematischen Wissenschaften. M-A Knus, Springer-Verlag294M-A Knus. Quadratic and Hermitian Forms over Rings. Grundlehren der mathematischen Wissenschaften. v.294. Springer-Verlag. 2013. Journal of pure and applied algebra. R Street, v. 23Two dimensional sheaf theoryR. Street. Two dimensional sheaf theory. Journal of pure and applied algebra. v.23 (251-270). 1982. I Pirashvili, arXiv:[email protected] Fundamental groupoid as a terminal costack. Leicester, LE1 7RH, UK E-mail addressDepartment of Mathematics, University of Leicester, University RoadI. Pirashvili. The Fundamental groupoid as a terminal costack. arXiv:1406.4419. 2014. Department of Mathematics, University of Leicester, University Road, Leicester, LE1 7RH, UK E-mail address: [email protected]	[]
[ "Pruned Graph Neural Network for Short Story Ordering", "Pruned Graph Neural Network for Short Story Ordering" ]	[ "Melika Golestani [email protected] ", "Zeinab Borhanifard [email protected] ", "Farnaz Tahmasebian [email protected] ", "Heshaam Faili [email protected] ", "\nSchool of Electrical and Computer Engineering\nCollege of Engineering\nUniversity of Tehran\nTehranIran\n", "\nEmory University\nAtlantaGAUSA\n" ]	[ "School of Electrical and Computer Engineering\nCollege of Engineering\nUniversity of Tehran\nTehranIran", "Emory University\nAtlantaGAUSA" ]	[]	Text coherence is a fundamental problem in natural language generation and understanding. Organizing sentences into an order that maximizes coherence is known as sentence ordering. This paper is proposing a new approach based on the graph neural network approach to encode a set of sentences and learn orderings of short stories. We propose a new method for constructing sentence-entity graphs of short stories to create the edges between sentences and reduce noise in our graph by replacing the pronouns with their referring entities. We improve the sentence ordering by introducing an aggregation method based on majority voting of state-of-the-art methods and our proposed one. Our approach employs a BERT-based model to learn semantic representations of the sentences. The results demonstrate that the proposed method significantly outperforms existing baselines on a corpus of short stories with a new state-of-the-art performance in terms of Perfect Match Ratio (PMR) and Kendall's Tau (τ ) metrics. More precisely, our method increases PMR and τ criteria by more than 5% and 4.3%, respectively. These outcomes highlight the benefit of forming the edges between sentences based on their cosine similarity. We also observe that replacing pronouns with their referring entities effectively encodes sentences in sentence-entity graphs.	10.48550/arxiv.2203.06778	[ "https://arxiv.org/pdf/2203.06778v1.pdf" ]	247,446,769	2203.06778	0ce5828ab07669377a2e056b4310ae99c2afa614	Pruned Graph Neural Network for Short Story Ordering Melika Golestani [email protected] Zeinab Borhanifard [email protected] Farnaz Tahmasebian [email protected] Heshaam Faili [email protected] School of Electrical and Computer Engineering College of Engineering University of Tehran TehranIran Emory University AtlantaGAUSA Pruned Graph Neural Network for Short Story Ordering Text coherence is a fundamental problem in natural language generation and understanding. Organizing sentences into an order that maximizes coherence is known as sentence ordering. This paper is proposing a new approach based on the graph neural network approach to encode a set of sentences and learn orderings of short stories. We propose a new method for constructing sentence-entity graphs of short stories to create the edges between sentences and reduce noise in our graph by replacing the pronouns with their referring entities. We improve the sentence ordering by introducing an aggregation method based on majority voting of state-of-the-art methods and our proposed one. Our approach employs a BERT-based model to learn semantic representations of the sentences. The results demonstrate that the proposed method significantly outperforms existing baselines on a corpus of short stories with a new state-of-the-art performance in terms of Perfect Match Ratio (PMR) and Kendall's Tau (τ ) metrics. More precisely, our method increases PMR and τ criteria by more than 5% and 4.3%, respectively. These outcomes highlight the benefit of forming the edges between sentences based on their cosine similarity. We also observe that replacing pronouns with their referring entities effectively encodes sentences in sentence-entity graphs. Introduction Text coherence is a fundamental problem in natural language generation and understanding. A coherent text adheres to a logical order of events which facilitates better understanding. One of the subtasks in coherence modeling, called sentence ordering, refers to organizing shuffled sentences into an order that maximizes coherence (Barzilay and Lapata, 2008). Several downstream applications benefit from this task to assemble sound and easy-to-understand texts, such as extraction-based multi-document summarization (Barzilay and El-hadad, 2002;Galanis et al., 2012;Nallapati et al., 2017;Logeswaran et al., 2018), natural language generation (Reiter and Dale, 1997), retrieval-based question answering , concept-to-text generation (Konstas and Lapata, 2012), storytelling (Fan et al., 2019;Hu et al., 2020;Zhu et al., 2020), opinion generation (Yanase et al., 2015), conversational analysis (Zeng et al., 2018), image captioning (Anderson et al., 2018), recipe generation (Chandu et al., 2019), and discourse coherence (Elsner et al., 2007;Barzilay and Lapata, 2008;Guinaudeau and Strube, 2013). In early studies, researchers modeled sentence structure using hand-crafted linguistic features (Lapata, 2003; Barzilay and Lee, 2004;Elsner et al., 2007;Barzilay and Lapata, 2008), nonetheless, these features are domain-specific. Therefore recent studies employed deep learning techniques to solve sentence ordering tasks (Golestani et al., 2021;Logeswaran et al., 2018;Gong et al., 2016;Li and Jurafsky, 2017;. (Cui et al., 2018) used graph neural networks, called ATTOrderNet, to accomplish this task. They used a self-attention mechanism combined with LSTMs to encode input sentences. Their method could get a reliable representation of the set of sentences regardless of their input order. In this representation, an ordered sequence is generated using a pointer network. Since ATTOrderNet is based on fully connected graph representations, it causes to build an association among some irrelevant sentences, which introduces massive noise into the network. Furthermore, since a self-attention mechanism only uses the information at the sentence level, other potentially helpful information such as entity information is missed. To overcome these drawbacks, Yin et al. (2019) developed the Sentence-Entity Graph (SE-Graph), which adds entities to the graph. While in the AT-TOrderNet, every node is a sentence representation, SE-Graph consists of two types of nodes: sen-arXiv:2203.06778v1 [cs.CL] 13 Mar 2022 tence and entity 1 . Moreover, the edges come in two forms called SS and SE: • SS: this edge is between sentence nodes that share a common entity, • SE: this edge connects a sentence and an entity within the sentence labeled with the entity's role. However, the introduced methods perform poorly for short story reordering tasks. The SE-graph solution seems effective for long texts but not for short stories. In this paper, we suggest modifications to the introduced graph methods to improve the performance for short story reordering tasks. Some issues arise in short stories: First, entities are often not repeated in multiple sentences in a short story or text; instead, pronouns refer to an entity. To address this problem, we improve the semantic graph by replacing the pronouns with their corresponding entities. Another issue is a high correlation between the sentences in a short story along with a high commonality of entities across sentences, which leads us to end up with a complete graph in most cases. Our solution is moving towards a Pruned Graph (PG). As with the ATTOrderNet and SE-Graph networks, the PG architecture consists of three components: 1. Sentence encoder based on SBERT-WK model (Wang and Kuo, 2020), 2. Graph-based neural story encoder, and 3. Pointer network based decoder. With PG, the network nodes and SE edges are created based on SE-Graph, in a way that the first and third components are the same. However, in the story encoding phase, after generating the nodes and SE edges based on SE-Graph method, the pruning phase is started on SS edges. This pruning process is defined as follow: each sentence edged out its neighbors with the first and second most cosine similarities (Rahutomo et al., 2012). This method alleviates some problems of the previous two methods in the case of organizing short story sentences. It is noteworthy that pronouns are replaced with entities during pre-processing. Finally, we present a method based on majority voting to combine our proposed graph networkbased method with the state-of-art methods to benefit from each. Contributions of this study are as follows: 1 the entity should be common to at least two sentences 1. Proposing a new method based on graph networks to order sentences of a short stories corpus by: (a) suggesting a new method for creating the edges between sentences, (b) creating a better sentence-entity graph for short stories by replacing pronouns in sentences with entities, (c) Moreover, taking advantage of BERTbased sentence encoder. 2. Using majority voting to combine sentence ordering methods. 2 Related Work Sentence Ordering In early studies on sentence ordering, the structure of the document is modeled using hand-crafted linguistic features (Lapata, 2003;Barzilay and Lee, 2004;Elsner et al., 2007;Barzilay and Lapata, 2008). Lapata (2003) encoded sentences as vectors of linguistic features and used data to train a probabilistic transition model. Barzilay and Lee (2004) developed the content model in which topics in a specific domain are represented as states in an HMM. Some other like Barzilay and Lapata (2008) utilize the entity-based approach, which captures local coherence by modeling patterns of entity distributions. Other approaches used a combination of the entity grid and the content model Elsner et al. (2007) or employed syntactic features (Louis and Nenkova, 2012) in order to improve the model. However, linguistic features are incredibly domain-specific, so applying these methods across different domains can decrease the performance. To overcome this limitation, recent works have used deep learning-based approaches. Li and Hovy (2014) proposes a neural model of distribution of sentence representations based on recurrent neural networks. In (Li and Jurafsky, 2017), graphbased neural models are used to generate a domainindependent neural model. Agrawal et al. (2016) introduced a method that involves combining two points elicited from the unary and pairwise model of sentences. used an LSTM encoder and beam search to construct a pairwise model. Based on a pointer network that provides advantages in capturing global coherence, Gong et al. (2016) developed an end-to-end approach that predicts order of sentences. In another work, by applying an encoder-decoder architecture based on LSTMs and attention mechanisms, Logeswaran et al. (2018) suggested a pairwise model and established the gold order by beam search. In (Pour et al., 2020), we presented a method that does not require any training corpus due to not having a training phase. We also developed a framework based on a sentence-level language model to solve the sentence ordering problem in (Golestani et al., 2021). Moreover, in several other studies, including (Cui et al., 2018) and (Yin et al., 2019), graph neural networks are used to accomplish this task, as explained in the following. Graph Neural Networks in NLP Graph neural networks (GNN) have shown to be effective in NLP applications, including syntactic dependency trees (Marcheggiani and Titov, 2017), neural machine translation (Beck et al., 2018), knowledge graphs , semantic graphs , sequence-to-graph learning (Johnson, 2017), graph-to-sequence learning (Beck et al., 2018), sentence ordering (Yin et al., 2019), and multi-document summarization (Christensen et al., 2013;Yasunaga et al., 2017). In particular, text classification is a common application of GNNs in natural language processing. A GNN infers document labels based on the relationships among documents or words (Hamilton et al., 2017). Christensen et al. (2013) used a GNN in multi-document summarization. They create multi-document graphs which determine pairwise ordering constraints of sentences based on the discourse relationship between them. Kipf and Welling (2017) proposed Graph Convolutional Networks (GCN), which is used in Yasunaga et al. (2017) to generate sentence relation graphs. The final sentence embeddings indicate the graph representation and are utilized as inputs to achieve satisfactory results on multi-document summarization. Another method is presented in Marcheggiani and Titov (2017) where a syntactic GCN is developed with a CNN/RNN as sentence encoder. The GCN indicates syntactic relations between words in a sentence. In a more recent work, Yin et al. (2019) proposed a graph-based neural network for sentence ordering, in which paragraphs are modeled as graphs where sentences and entities are the nodes. The method showed improvement in evaluation metrics for sentence ordering task. In this work, we explore the use of GRN for NLP tasks, es-pecially to perform sentence-ordering on a corpus of short stories. Baselines This section introduces ATTOrderNet (Cui et al., 2018) and SE-Graph (Yin et al., 2019), which achieve state-of-the-art performances and serve as baseline for our work. ATTOrderNet ATTOrderNet introduced in (Cui et al., 2018) is a model using graph neural networks for sentence ordering. The model includes three components as follows: a sentence encoder based on Bi-LSTM, a paragraph encoder based on self-attention, and a pointer network-based decoder. In the sentence encoder, sentences are translated into distributional representations with a word embedding matrix. Then a sentence-level representation using the Bi-LSTM is learned. An average pooling layer follows multiple self-attention layers in the paragraph encoder. The paragraph encoder computes the attention scores for all pairs of sentences at different positions in the paragraph. Therefore, each sentence node is connected to all others where the encoder exploits latent dependency relations among sentences independent of their input order. Having an input set of sentences, the decoder aims to predict a coherent order, identical to the original order. In this method, LSTM-based pointer networks are used to predict the correct sentence ordering from the final paragraph representation. Based on the sequence-to-sequence model, the pointer network-based decoders predict the correct sentence sequence (Sutskever et al., 2014). Specifically, input tokens are encoded using the pointer network as summary vectors 2 , and the next token vector is decoded repeatedly. Finally, the output token sequence is derived from the output token vector. SE-Graph SE-Graph, similarly to ATTOrderNet, consists of three components: 1. a sentence encoder based on Bi-LSTM, 2. a paragraph encoder, 3. a pointer network based decoder Nevertheless, the difference between SE-Graph and ATTOrderNet is only in the encoder paragraph component, described in the following. In contrast to the fully connected graph representations explored by ATTOrderNet, Yin et al. (2019) represented input paragraphs as sentence-entity graphs. The SE-Graph includes two types of nodes: sentence and entity. The entity should be common to at least two sentences to be considered as a node of the graph. There are also two types of edges: SS edges that connect sentence nodes with at least a common entity, and SE edges that link a sentence with an entity within that and with a label of the entity's role. SE edges are labeled based on the syntactic role of the entity in the sentence, such as a subject, an object, or other. When an entity appears multiple times in a sentence with different roles, the role that has the highest rank is considered. The highest rank of roles is the subject role; after that are the object roles. SE-Graph framework utilizes a GRN-based paragraph encoder that integrates the paragraphlevel state along with the sentence-level state. Methodology In this section, first, the problem is formulated, second the dataset is introduced and explains why this dataset is suitable for the sentence ordering task. Then two methodologies are proposed. The first proposed method, called Pruned Graph, is based on graph networks, and the second is based on the majority voting to combine the outputs of three different models. Problem Formulation Consider S(O) is a set of n unordered sentences taken from a coherent text: O = s 1 , s 2 , . . . , s n , s(o 1 ) > s(o 2 ) > · · · > s(o n )(1) The goal of sentence ordering is to find a permutation of sentences of O like S(o ), s(o 1 ) > s(o 2 ) > . . . > s(o n )(2) that corresponds to the gold data arrangement. In other words, sentence ordering aims to restore the original orders: s(o 1 * ) > s(o 2 * ) > . . . > s(o n * )(3) Where S(o * ) represents the original or gold order. As a result a correct output leads to S(o ) = S(o * ). Based on the above definition and notions we propose our sentence ordering method. Dataset In this paper, we used a corpus of short stories, called ROCStories (Mostafazadeh et al., 2016). It contains 98,162 commonsense stories, each with exactly five sentences and an average word count of 50. Mostafazadeh et al. (2017) created ROCStories corpus for a shared task called LSDSem, in which models are supposed to predict the correct ending to short stories. 3,742 of the stories have two options for the final sentence. It is worth noting that humans generated all of the stories and options. We can learn sequences of daily events from this dataset because it contains some essential characteristics: The stories are rich with causal and temporal relations among events, which makes this dataset a highly useful resource for learning narrative structure across a wide range of events. The dataset consists of a comprehensive collection of daily and non-fictional short stories useful for modeling the coherence of a text (Mostafazadeh et al., 2016). Due to these features, ROCStories can be used to learn sequences of sentences. Thus, the corpus is useful for organizing sentences in a text. Pruned Graph Sentence Ordering (PG) We propose a neural network based on the pruned graph for arranging the sentences of short stories, a modified version of the ATTOrderNet (Cui et al., 2018) and Sentence-Entity Graph (Yin et al., 2019). The PG method consists of three components: sentence encoder, story encoder, and decoder. In order to be a fair comparison, we used the same decoder as ATTOrderNet and SE-Graph. Due to space limitations, here we explain our sentence encoder and our story encoder. The Sentence encoder uses BERT encoding to encode sentences, while Story encoder uses a graph neural network for encoding stories. Sentence Encoder: SBERT-WK We use fine-tuned pre-trained SBERT-WK model to encode sentences. BERT contains several layers, each of which captures a different linguistic characteristic. SBERT-WK found better sentence representations by fusing information from different layers, Wang and Kuo (2020). The system geometrically analyzes space using a deep contextual model that is trained on both word-level and sentence-level, without further training. For each word in a sentence, it determines a unified word representation then computes the final sentence embedding vector based on the weighted average based on the word importance of the word representations. Even with a small embedding size of 768, SBERT-WK outperforms other methods by a significant margin on textual similarity tasks (Wang and Kuo, 2020). Story Encoder To use graph neural networks for encoding stories, input stories should be represented as graphs. We propose a pruned graph (PG) representation instead of SE-Graph (Yin et al., 2019) for encoding short stories. Nodes in PG are composed of sentences and entities. We replace pronouns with the entities they refer to since entities are not often repeated from one sentence to another during a short story 3 , we will go into more detail in the experiments. We consider all nouns of an input story as entities at first. After that, we eliminate entities that do not occur more than once in the story. We can formalize our undirected pruned graphs as G = (V s , V e , E), where V s indicates the sentencelevel nodes, V e denotes the entity-level nodes, and E represents edges. Edges in PG graphs are divided into two types: SS and SE. The SS type links two sentences in a story that have the highest or secondhighest value of cosine similarity with each other; and the SE type links a sentence with an entity within that with a label of the entity's role. Equation 4 shows the formula for calculating the cosine similarity, where CosSim is cosine similarity and Emb s i represents vector of sentence i. CosSim(Emb s i , Emb s j ) = Emb s i * Emb s j \|\|Emb s i \|\|\|\|Emb s j \|\| (4) SE edges are labeled according to the syntactic role of the entity in the sentence, such as a subject, an object, or other. The role that has the highest rank in an instance of an entity appearing multiple times is considered. The ranking is as follows: subject role, object roles, and other. The use of referring entities rather than pronouns is crucial. Thus, sentence nodes are linked to both sentence and entity nodes, whereas an entity node is not connected to any other entity nodes. For graph encoding, we use GRN , which has been found effective for various kinds of graph encoding tasks. GRN used in our PG is the same 3 we use the Stanford's tool (Lee et al., 2011) as GRN in (Yin et al., 2019), so we do not explain it. Majority Voting We combine the output of three methods to achieve better results in majority voting. Since the stories in Rocstories all have five sentences, there are 20 possible pair sentence orderings as follow: 1. s 1 s 2 or 2. s 2 s 1 , 3. s 1 s 3 or 4. s 3 s 1 , 5. s 1 s 4 or 6. s 4 s 1 , 7. s 1 s 5 or 8. s 5 s 1 , 9. s 2 s 3 or 10. s 3 s 2 , 11. s 2 s 4 or 12. s 4 s 2 , 13. s 2 s 5 or 14. s 5 s 2 , 15. s 3 s 4 or 16. s 4 s 3 , 17. s 3 s 5 or 18. s 5 s 3 , 19. s 4 s 5 or 20. s 5 s 4 Each suggested order for a story includes 10 of the above pair orderings, either of the two pair orderings that have an "or" between them. Through majority voting 4 , we can combine the outputs of three separate methods to generate a final order. According to the number of occurrences in each of the three output arrangements, we assign scores to each of the 20 possible pairings. As a result, each of these possible pairings is scored between 0 and 3. 0 indicates that this pairing does not appear in any of the three methods' outputs, while 3 indicates that it appears in all of them. In the end, all pairs with a greater score of 1 occur in the final orderings 5 . Indeed, these are ten pairs 6 , and with the chosen pairs, the sentences of the story are arranged uniquely. In the following subsection, we are proving that majority voting is a valid way to combine the outputs generated from three different methods for arranging sentences. By using contradiction, we demonstrate the validity of the majority voting method for combining three distinct methods of sentence ordering to arrange two sentences. Assuming the majority voting of three methods fails to create an unique order, then two orders are possible, s 1 s 2 , and s 2 s 1 . In the first case, s 1 appears before s 2 in two or more outputs of the methods, and in the second case, s 2 appears before s 1 in two or more outputs. Due to the three methods, this assumption causes a contradiction. To ordering more than two sentences, it can be proved by induction. Experiment Evaluation Metrics We use two standard metrics to evaluate the proposed model outputs that are commonly used in previous work: Kendall's tau and perfect match ratio, as described below. • Kendall's Tau (τ ) Kendal's Tau (Lapata, 2006) measures the quality of the output's ordering, computed as follows: τ = 1 − (2 * # of Inversions) N * (N − 1)/2(5) Where N represents the sequence length (i.e. the number of sentences of a story, which is always equal to 5 for ROCStories), and the inversions return the number of exchanges of the predicted order with the gold order for reconstructing the correct order. τ is always between -1 and 1, where the upper bound indicates that the predicted order is exactly the same as the gold order. This metric correlates reliably with human judgments, according to Lapata (2006). • Perfect Match Ratio (PMR) According to this ratio, each story is considered as a single unit, and a ratio of the number of correct orders is calculated. Therefore no penalties are given for incorrect permutations (Gong et al., 2016). PMR is formulated mathematically as follows: P M R = 1/N N 1 o i = o * i(6) where o i represents the output order and o * i indicates the gold order. N specifies the sequence length. Since the length of all the stories of ROCStories is equal to 5, N in this study is always equal to 5. PMR values range from 0 to 1, with a higher value indicating better performance. Contrast Models We compare our PG to the state of the arts, namely the following: 1. LSTM + PtrNet (Gong et al., 2016), 2. LSTM + Set2Seq (Logeswaran et al., 2018), 3. ATTOrderNet (Cui et al., 2018), 4. SE-Graph (Yin et al., 2019), 5. HAN (Wang and Wan, 2019), 6. SLM (Golestani et al., 2021), 7. Rank-TxNet ListMLE (Kumar et al., 2020), 8. Enhancing PtrNet + Pairwise (Yin et al., 2020), 9. B-TSort (Prabhumoye et al., 2020. We teach LSTM + Ptr-Net, ATTOrderNet, SE-Graph, and B-TSort on the ROCStories. The following is a brief description of the mentioned methods, but We refer to Section 3 to explain ATTOrderNet and SE-Graph. Gong et al. (2016) proposes LSTM + PtrNet as a method for ordering sentences. In this end-toend method, pointer networks sort encrypted sentences after decoding them by LSTM. Logeswaran et al. (2018) recommended LSTM + Set2Seq. Their method encodes sentences, learns context representation by LSTM and attention mechanisms, and utilizes a pointer network-based decoder to predict sentences' order. A transformer followed by an LSTM was added to the sentence encoder in (Wang and Wan, 2019) to capture word clues and dependencies between sentences; and so on, HAN is developed. In (Golestani et al., 2021), we developed the Sentence-level Language Model (SLM) for Sentence Ordering, consisting of a Sentence Encoder, a Story Encoder, and a Sentence Organizer. The sentence encoder encodes sentences into a vector using a fine-tuned pre-trained BERT. Hence, the embedding pays more attention to the sentence's crucial parts. Afterward, the story encoder uses a decoder-encoder architecture to learn the sentencelevel language model. The learned vector from the hidden state is decoded, and this decoded vector is utilized to indicate the following sentence's candidate. Finally, the sentence organizer uses the cosine similarity as the scoring function in order to sort the sentences. An attention-based ranking framework is presented in (Kumar et al., 2020) to address the task. The model uses a bidirectional sentence encoder and a self-attention-based transformer network to endcode paragraphs. In (Yin et al., 2020), an enhancing pointer network based on two pairwise ordering prediction modules, The FUTURE and HISTORY module, is employed to decode paragraphs. Based on the candidate sentence, the FU-TURE module predicts the relative positions of other unordered sentences. Although, the HIS-TORY module determines the coherency between the candidate and previously ordered sentences. And lastly, Prabhumoye et al. (2020) designed B-TSort, a pairwise ordering method, which is the current state-of-the-art method for sentence ordering. This method benefits from BERT and graph-based networks. Based on the relative pairwise ordering, graphs are constructed. Finally, the global order is derived by a topological sort algorithm on the graph. Setting For a fair comparison, we follow Yin et al. (2019)'s settings. Nevertheless, we use SBERT-WK's 768dimension vectors for sentence embedding. Furthermore, the state sizes for sentence nodes are set to 768 in the GRN; The Batch size is 32. In preprocessing, we use Stanford's tool (Lee et al., 2011) to replace pronouns with the referring entities. Results In this paper, we propose a new method based on graph networks for sentences ordering short stories called Pruned Graph (PG). In order to achieve this, we propose a new method for creating edges between sentences (by calculating the cosine similarity between sentences), and we create a better sentence-entity graph for short stories by replacing pronouns with the relevant entities. Besides, to make a better comparison, we also teach the following cases: 1. All nodes in the graph are of the sentence type, and the graph is fully connected. In other words, we train ATTOrderNet on ROC-Stories 7 . 2. The nodes include sentence and entity nodes, and each sentence's node has the edge over all other sentences' nodes (semi fully connected SE-Graph 8 ). 3. The network comprises sentence and entity nodes, and every two sentences with at least one entity in common are connected (SE-Graph 9 ). 7 (Cui et al., 2018) did not train ATTOrderNet on the ROC-Stories dataset. 8 Entity nodes are not connected to all nodes. 9 We train SE-Graph on ROCStories since (Yin et al., 2019) did not. Replacing pronouns with the relevant entities in SE-Graph (SE-Graph + Co-referencing). 5. Similar to PG, but each sentence is connected to a sentence with the highest cosine similarity (semi P G 1 ). 6. Similar to PG; however, each sentence is connected to three other sentences based on their cosine similarity (semi P G 3 ). 7. Pruned Graph with a Bi-LSTM based sentence encoder 10 (P G ). Note that in the above methods, where the graph also contains the nodes of the entity, there is an edge between a sentence and an entity within it 11 . Table 1 reports the results of Pruned Graph (PG) and the above seven methods. To get the training, validation, and testing datasets, we randomly split ROCStories into 8:1:1. Therefore, the training set includes 78,529 stories, the validation set contains 9,816 stories, and the testing set consists of 9,817 stories. As shown in table 1, our PG beats all seven other methods. The results show that all three of our innovations to the graph-based method have improved the performance. Based on our analysis, the SBERT-WK sentence encoder is more beneficial than the Bi-LSTM. Our experiences also find that using referring entities instead of pronouns is helpful to create a more effective sentence-entity graph. Additionally, it indicates connecting each sentence to two others using cosine similarity is efficient to encode a story. Table 2 reports the results of the proposed method of this paper in comparison with competitors. When compared with ATTOrdeNet, PG improved the Tau by over 8.5% as well as PMR by 13.5%. Furthermore, the Tau is increased by 10.8% and the PMR by more than 16.8% compared to SE-Graph. PG outperforms the state-of-the-art on ROCStories with a more tthan 1.8% increase in pmr and a more than 3.9% improvement in τ . Finally, we merged the outputs of the three methods using the majority voting method, including Enhancing PtrNet, B-TSort, and Our Pruned Graph. Table 3 shows the results of the combination, which improves the PMR and τ criteria by more than 5% and 4.3% on ROCStories, respectively. conclusion This paper introduced a graph-based neural framework to solve the sentence ordering task. This framework takes a set of randomly ordered sentences and outputs a coherent order of the sentences. The results demonstrate that SBERT-WK is a reliable model to encode sentences. Our analysis examined how the method is affected by using a Bi-LSTM model in the sentence encoder component. In addition, we found that replacing pronouns with their referring entities supplies a more informative sentence-entity graph to encode a story. The experimental results indicate that our proposed graph-based neural model significantly outperforms on ROCStories dataset. Furthermore, we recommend a method for combining different Method τ PMR combination 0.8470 0.5488 Table 3: Results of combining of Enhancing PtrNet, B-TSort, and Pruned Graph using majority voting methods of sentence ordering based on majority voting that achieves state-of-the-art performance in PMR and τ scores. In future, we plan to apply the trained model on sentence ordering task to tackle other tasks including text generation, dialogue generation, text completion, retrieval-based QA, and extractive text summarization. Table 1: Reporting the results of the proposed network called Pruned Graph in comparison with the seven methods mentioned aboveMethod τ PMR ATTOrderNet 0.7364 0.4030 Fully connected SE-Graph 0.7300 0.3927 SE-Graph 0.7133 0.3687 SE-Graph + Co-ref 0.7301 0.3981 PG+ 1 SS 0.7534 0.4349 PG + 3 SS 0.7379 0.4100 PG + Bi-LSTM-based sentence encoder 0.7852 0.4769 Pruned Graph 0.8220 0.5373 Model τ PMR LSTM+PtrNet 0.7230 0.3647 LSTM+Set2Seq 0.7112 0.3581 ATTOrderNet 0.7364 0.4030 SE-Graph 0.7133 0.3687 HAN 0.7322 0.3962 SLM 0.7547 0.4064 RankTxNet ListMLE 0.7602 0.3602 Enhancing PtrNet + Pairwise 0.7681 0.4600 B-TSort 0.8039 0.4980 Our Pruned Graph 0.8220 0.5373 Table 2 : 2Results of our PG compared to baselines and competitors The paragraph vector is nonetheless influenced by the permutations of input sentences. For example, either s1s2 or s2s1 occurs, and without a doubt, the co-occurrence of these is a vast and impossible contradiction.5 Suppose the outputs of the three methods for arranging sentence1 (s1) and sentence2 (s2) are: Method 1: s1s2, Method 2: s1s2, and Method 3: s2s1. Therefore, the order s1s2 gets two points and the order s2s1 gets one, so s1s2 applies to the final output. 6 either of the two pair orderings that have an "or" between them. To demonstrate the advantages of the PG's BERT-based sentence encoder, this component is considered exactly like the sentence encoder of SE-Graph and ATTOrderNet. 11 Entity nodes can only have a link to sentence nodes. Sort story: Sorting jumbled images and captions into stories. 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[ "Dynamic generation of entangling wave packets in XY spin system with decaying long range couplings", "Dynamic generation of entangling wave packets in XY spin system with decaying long range couplings" ]	[ "S Yang \nDepartment of Physics\nNankai University\n300071TianjinChina\n", "Z Song \nDepartment of Physics\nNankai University\n300071TianjinChina\n", "C P Sun \nDepartment of Physics\nNankai University\n300071TianjinChina\n\nInstitute of Theoretical Physics\nChinese Academy of Sciences\n100080BeijingChina\n" ]	[ "Department of Physics\nNankai University\n300071TianjinChina", "Department of Physics\nNankai University\n300071TianjinChina", "Department of Physics\nNankai University\n300071TianjinChina", "Institute of Theoretical Physics\nChinese Academy of Sciences\n100080BeijingChina" ]	[]	We study the dynamic generation of spin entanglement between two distant sites in an XY model with 1/r 2 -decay long range couplings. Due to the linear dispersion relation ε(k) ∼ \|k\| of magnons in such model, we show that a well-located spin state can be dynamically split into two moving entangled local wave packets without changing their shapes. Interestingly, when such two wave packets meet at the diametrically opposite site after the fast period τ = π, the initial well-located state can be recurrent completely. Numerical calculation is performed to confirm the analytical result even the ring system of sizes N up to several thousands are considered. The truncation approximation for the coupling strengths is also studied. Numerical simulation shows that the above conclusions still hold even the range of the coupling strength is truncated at a relative shorter scale comparing to the size of the spin system.	10.1007/s11433-008-0002-0	[ "https://arxiv.org/pdf/quant-ph/0606029v1.pdf" ]	46,450,919	quant-ph/0606029	e7df923169b0d94f33f2cb1dde41ce6dcfb2ae37	Dynamic generation of entangling wave packets in XY spin system with decaying long range couplings 4 Jun 2006 S Yang Department of Physics Nankai University 300071TianjinChina Z Song Department of Physics Nankai University 300071TianjinChina C P Sun Department of Physics Nankai University 300071TianjinChina Institute of Theoretical Physics Chinese Academy of Sciences 100080BeijingChina Dynamic generation of entangling wave packets in XY spin system with decaying long range couplings 4 Jun 2006arXiv:quant-ph/0606029v1numbers: 0367-a0367Lx0365Ud7510Jm We study the dynamic generation of spin entanglement between two distant sites in an XY model with 1/r 2 -decay long range couplings. Due to the linear dispersion relation ε(k) ∼ \|k\| of magnons in such model, we show that a well-located spin state can be dynamically split into two moving entangled local wave packets without changing their shapes. Interestingly, when such two wave packets meet at the diametrically opposite site after the fast period τ = π, the initial well-located state can be recurrent completely. Numerical calculation is performed to confirm the analytical result even the ring system of sizes N up to several thousands are considered. The truncation approximation for the coupling strengths is also studied. Numerical simulation shows that the above conclusions still hold even the range of the coupling strength is truncated at a relative shorter scale comparing to the size of the spin system. INTRODUCTION In quantum information processing, it is also crucial to generate entangled qubits, which can be used to perfectly transfer a quantum state over long distance. For optical system this task has been completed long time ago, but for a solid state systems it remains a great challenge both in experiment and theoretical setup to create quantum entanglement by a solid state device. Recently many proposals to entangle distant spins have been proposed based on various physical mechanisms [1, 2,3,4]. Most protocols for accomplishing quantum state transfer in a spin array base on the fixed inter-qubit couplings [5,6,7,8,9,10,11,12,13,14,15,16,17,18]. The simplest coupled spin system with uniform nearest neighbor (NN) coupling has been studied by the pioneer work [6]. The paradigm to generate maximal entanglement and to perform perfect qubit-state transmission over arbitrary distance is the protocols using the pre-engineered inhomogeneous NN couplings [7,14]. Among them, the practical scheme realizes the transmission of Gaussian wave packet as flying qubit via the spin system with uniform NN couplings [8,16]. The advantages of such scheme rest with its fast transfer, which means that the period of transfer time is proportional to the distance, and the higher fidelity for longer distance. We also notice that there is a current interest in studying systems with longrange inter-qubit interactions [20,21,22]. In this paper, we revisit the issue of entanglement generation and quantum state transfer between two distant qubits in a qubit array with long-range inter-qubit interactions. An alternative way to construct a perfect medium for quantum state transfer may require the long-range interactions beyond NN interactions, but they should be required to decay rapidly in order to avoid the direct connecting coupling between two distant qubits that trivially causes quantum entanglement. Now we ( ) k ε π − π k FIG. 1: (color online) Schematic illustration for the ideal dispersion relation ε(k) ∼ \|k\| of a system which is shown to be a perfect entangler. Such kind of system can be realized by the XY model with pre-engineered long-range couplings. propose a novel protocol based on a pre-engineered XYmodel with long-range 1/r 2 -decay interactions. It is found that such model has the same function as that of the modulated NN coupling spin model [7], offering significant advantages over other protocols in the tasks perfectly transferring quantum state and generating entanglement between two sites over longer distance. This paper is organized as follows, in Sec. II, we consider a model with linear dispersion relation and show that it ensures the perfect state transfer and creation of entanglement between spatial separated qubits. In Sec. III we propose a spin model with 1/r 2 -decay interactions that process the desired dispersion relation. In Sec. IV we perform numerical calculation to confirm some of the obtained analytical results and the validity of the truncation approximation. 2: (color online) Schematic illustration of the generation process of two distant entangled qubits. At t = 0 the initial state is a single-spin flip on the saturated ferromagnet. At t > 0 the initial δ-pulse magnon is separated into two components which have maximal entanglement. At t = τ , the recurrence time, the two sub-wave packets meet together and the initial state recurrent. i (a) 0 t = i (b) i-r i+r 0 t > FIG. FORMALISM Pre-engineered model with linear dispersion relation Usually, we regard a 1/2 spin as a qubit, and a coupled spin system as a qubit array respectively. The simple coupled qubit array is usually described by the spin-1/2 XY model. Our proposal makes the quantum spin array behave as a perfect spin-networks, in which a spin flip at any site (or the superposition of local single-spin-flip states) can evolve into two entangled, local wave packets. Consider the prescribed spin-1/2 XY -model on ring system with N sites. The Hamiltonian reads H = 2 i,r J r (S x i S x i+r + S y i S y i+r ) (1) or H = i,r J r (S + i S − i+r + S − i S + i+r ),(2) where S x i , S y i and S z i are Pauli matrices for the site ith, and S ± i = S x i ± iS y i ; J r is the coupling strength for the two spins separated by the distance r. Since the Hamiltonian conserves spin, i.e., [ i S z i , H] = 0, the dynamics can be reduced to that in some invariant subspaces. Thus we can only concentrate on the single excitation subspace hereafter. If a single-site flipped state can be correctly transferred, a qubit state should also be transferred correspondingly because the saturate ferromagnetic state with all spins up state \|0 ≡ N i=1 \|↑ i is an eigenstate of the Hamiltonian. Actually, in the single-spin-flip subspace or the single excitation (magnon) subspace, the basis states are denoted by the single-site flipped state (or δ-pulse) \|i = S − i \|0 , i ∈ [1, N ] . Therefore, if state \|i can evolve to state \|j after a period of time τ , \|j = exp(−iHτ ) \|i , we have (α \|↑ j + β \|↓ j ) N l =j \|↑ l = e −iHτ (α \|↑ i + β \|↓ i ) N l =i \|↑ l ,(3) i.e., a qubit state α \|↑ + β \|↓ can be transferred from i to j. Furthermore, any single-magnon state can be expressed as \|ψ = i A i S − i \|0 ≡ i A i \|i(4) and \|ψ = k D k S − k \|0 ≡ k D k \|k(5) respectively in spatial {\|i } and momentum {\|k } spaces. Here, we have used the spin-wave operator S − k = 1 √ N j e −ikj S − j ,(6)with discrete momentum k = 2πn/N , n ∈ [−N/2, N/2 − 1]. We begin with assumption that there exists an optimal distribution of J r , which ensures the single-magnon spectrum possessing a linear dispersion relation, i.e. H s = k ε k \|k k\| (7) ε k = N 2π \|k\| as illustrated in Fig. 1. In the following, we first show that such kind of systems can perform perfect state transfer and long range entanglement generation, and then we can provide a practical example that satisfies this linear dispersion relation. Time evolution of wave packets in the linear dispersion regime In order to investigate the dynamics of generating entanglement in the spin system with linear dispersion relation mentioned above, we concentrate on the case where the initial state is single-site flipped state S − i \|0 , i ∈ [1, N ]. Intuitively, a well localized state has equal probability amplitudes with respect to momentum eigenstates \| ± \|k\| , due to the inverse Fourier transformation S − j = 1 √ N k e ikj S − k N −→ ∞ − −−−−− → 1 √ N N 2π π −π e ikj S − k dk = √ N 2π π 0 (e ikj S − k + e −ikj S − −k )dk.(8) Thus it should be split into two classes of waves with ± \|k\| driven by the Hamiltonian with linear singlemagnon dispersion relation (7). If the superposition of the two classes of waves are still well-localized (or form two wave packets) in real space, it will lead to the entanglement of the two wave packets. Generally we first consider a wave packet dynamics with an initial state \|ψ(N A , 0) = S − NA \|0 = 1 √ N k e ikNA S − k \|0 ,(9) which is a single flip at site N A . The time evolution starting with this state can be calculated as \|ψ(N A , t) = e −iHt \|ψ(N A , 0) = \|φ − (N A , t) + \|φ + (N A , t)(10) where \|φ ± (N A , t) = ∓ √ N 2π ∓π 0 e ik(NA± N 2π t) dk \|k(11) denote the two wave packets with explicit forms in the real space \|φ ± (N A , t) = ∓ √ N 2π ∓π 0 e ik(NA± N 2π t) dk 1 √ N j e −ikj \|j = 1 2π j ∓e ∓iπ(NA−j± N 2π t) ± 1 i N A − j ± N 2π t \|j(12) We will show that such two wave packets are localized and nonspreading, with velocity v ± = ±N/2π, respectively. Actually, from Eq. (12) one have \|φ ± (N A , t + t 0 ) = φ ± (N A ± N 2π t, t 0 ) (13) or e −iHt \|φ ± (N A , t 0 ) = T N 2π t \|φ ± (N A , t 0 )(14) where T a acts as translational operator for the states \|φ ± (l, t) : T a \|φ ± (l, t) ≡ \|φ ± (l ± a, t) . In general, the spatial coordinate is treated as discrete variable while the temporal coordinate is continuous. In order to make T a to be operable, the translational spacing a should be integer. In the following, we only consider the states at discrete instant t, which ensures a to be integer, i.e., N t/2π = [N t/2π] (the integer part of N t/2π). In terms of l ± (t) = N A − j ± N 2π t ,(15) the wave packets (12) can be re-written as \|φ ± (N A , t) = 1 2π j { 1 il ± (t) [∓e ∓iπl±(t) ± 1]} \|j . (16) Obviously, Eq. (14) shows that state \|φ ± (N A , t) is shape-invariant with velocity v ± = ±N/2π. Now we show the locality of these states. The projections j \|φ ± (N A , t) =    1 2 , l ± (t) = 0 ∓ i πl±(t) , odd l ± (j, t) 0, even l ± (t) = 0(17) of \|φ ± (N A , t) onto the localized state \|j shows that the shape-invariant states are well-localized. And each wave packet has the probability amplitude \|φ ± (N A , t)\| 2 ≃ 1 4 + 2 π 2 ∞ n=0 1 (2n + 1) 2 = 1 2 ,(18) which indicates that the initial state (9) is splited into two wave packets completely. On the other hand, the dispersion (7) requires the translational invariance of the Hamiltonian since the momentum is the conserved quantity, which leads to the periodicity of the states \|φ ± φ ± (N A ± N 2π t, t) = φ ± (N A ± N 2π t ∓ m ± N, t) ,(19) where m ± = 1, 2, · · · , are integers. Obviously, when τ = (m + + m − )π, the initial state \|ψ(N A , 0) = S − NA \|0 evolves to \|ψ(N A , τ ) = S − NA− N 2 (m+−m−) \|0 ,(20) which is just the recurrence of \|ψ(N A , 0) on the positions N A − N (m + − m − )/2 at instants (m + + m − )π. The physics of this phenomenon can be understood as the interference of two wave packets (12). It also accords with the prediction for the system with spectrum-symmetry matching condition (SSMC) [14,18]. In the single magnon invariant subspace, the single flip states {\|j } at site j constitute the complete basis. Then the above conclusion can be applied to any states in this subspace. Consider an arbitrary initial state \|Φ(0) = j A j \|j = j A j \|ψ(j, 0) , which is a coherent superposition of single flip states. At time t, it evolves to \|Φ(t) = e −iHt \|Φ(0) = \|Φ + (t) + \|Φ − (t) ,(21) where \|Φ ± (t) = j A j \|φ ± (j, t)(22) represent two invariant-shape states. Accordingly, at instants τ = (m + + m − )π, the final state is a translation of \|Φ(0) \|Φ(τ ) = T N 2 (m+−m−) \|Φ(0) .(23) Furthermore, during the period of time t = τ , wave packets \|Φ ± (t) are still well localized in space if the initial state \|Φ(0) is local. Then the revival of a wave packet can be used to implement perfect quantum state transfer. Entanglement of two separated spins Now we turn our attention on the entanglement of the two separated spins induced by the wave packets (12). The reduced density matrix of a state \|Φ(t) for two spins located at sites i and j [23,24] has the form ρ ij =     v + ij 0 0 0 0 w ij z ij 0 0 z ij w ij 0 0 0 0 v − ij    (24) with respect to the standard basis vectors \|↑↑ , \|↑↓ , \| ↓↑ , and \|↓↓ . Here, the matrix elements are v ± ij = 1 4 + 1 2 Φ(t)\| [2S z i S z j ± (S z i + S z j )] \|Φ(t) , w ij = 1 4 − Φ(t)\| S z i S z j \|Φ(t) , z ij = 1 2 Φ(t)\| (S + i S − j + S − i S + j ) \|Φ(t) .(25) Correspondingly, the concurrence of two spins located at sites i and j for a state \|Φ(t) can be calculated by C ij = max 0, 2 \|z ij \| − v + ij v − ij ,(26) Since the state we concern is in the invariant subspace with S z = N/2 − 1, we have v + ij = 0 and then the concurrence reduces to C ij = Φ(t)\| (S + i S − j + S − i S + j ) \|Φ(t) .(27) Consider an initial state having reflectional symmetry with respect to a point N A , which has the form \|Φ(0) = f NA (0) \|N A + N/2 j=1 f j (0)[\|N A + j +(−1) R \|N A − j ],(28) where R=even (odd) represents the parity of the state under the reflection. The concurrence between two spins at N A + j and N A − j for the state \|Φ(t) is C(j, t) = C NA+j,NA−j = Φ(t)\| (S + NA+j S − NA−j + S − NA+j S + NA−j ) \|Φ(t) = 2 (−1) R f * j (t)f j (t) = 2 \|f j (t)\| 2 .(29) Obviously, the total concurrence j>0 C(j, t) = 1 due to the normality of the wave function \|Φ(t) . Usually, the total concurrence is regarded as the concurrence of two wave packets. Taking the single flip at N A in the form (9) to be the initial state as an example, straightforward calculation shows that non-zero concurrences are C(j, t 0 ) = 1 2 C(j, t n ) = 2 (2n − 1) 2 π 2 . (30) For this to occur, one requires that t 0 = 2πj/N , t n = 2π[j + (2n − 1)]/N , n = 1, 2, .... It indicates that the concurrence with magnitudes (30) for two spins at N A +j and N A − j can be generated at the moment t 0 , t 1 , t 2 , ..., t n . MICROSCOPIC LONG-RANGE INTERACTION MODEL In this section, we consider the possibility to realize the above scheme in a system which possesses the linear dispersion relation based on an one-dimensional arrangement of spins (qubits) coupled by long-range interactions. Actually, using the identity \|2ξ\| = N 2 − ∞ r=odd 4N r 2 π 2 cos rπ(2ξ) N ,(31) we have N 2π \|k\| ≈ J 0 + N/2−1 r=odd 2J r cos(kr),(32) where J 0 = N 4 , J r = N r 2 π 2 .(33) Then the Hamiltonian matches the case (7) in single-magnon invariant subspace can be rewritten as H = h 0 + N/2−1 r=odd J r h r , h 0 = N J 0 , h r = i (S + i S − i+r + h.c.).(34) Notice that, although such model involves the long range interaction, J r decays rapidly as r increases. So we call this as "microscopic long-range interactions". The above analytical analysis shows that such microscopic long-range interaction can lead to nontrivial long-range entanglement and QST. NUMERICAL RESULTS AND TRUNCATION APPROXIMATION In the previous sections, it is found that the microscopic long-range interaction can lead to nontrivial longrange entanglement and QST for large N limit system. In this section, numerical simulations are performed for finite systems to illustrate the result we obtained above and investigate the application of it. The numerical exact diagonalization method is employed to calculate the time evolution of the single flip state (9) and the Gaussian wavepacket in the following form \|ψ G (N A , α) = 1 √ Ω i e − α 2 2 (NA−i) 2 S − i \|0(35) in the finite N system. Here, Ω is the normalized factor and N A , α determine the center and the shape of wavepacket. In Fig. 3 and 4, the time evolution of a single flip state (9) and a Gaussian wave packet (35) of α = 0.1 in a N = 100 ring system are plotted. It shows that the local initial states split into two wave packets in both cases and the wave packets keep their shapes without spreading. Similarly, numerical simulations are also employed for the open chain systems. The similar conclusions are also obtained for open chain system which will be discussed in the following. A natural question is that what role the long-range coupling plays for the peculiar behavior of the propagation of the local wave in such a system. As shown above, the long-range hopping coupling decays rapidly. Then if the size of system is large enough, it is believed that the too long range coupling can be neglected. Thus, the "long range coupling" in the form (33) can be regarded as the relative local, or "microscopic long-range" coupling. In order to investigate this problem, or the boundary between the so called "long range" and "microscopic long- range" couplings, we consider the truncated Hamiltonian H = h 0 + r=odd, <r0 J r h r ,(36) where r 0 is the truncation distance. From the above analysis, the initial state \|ψ(N A , 0) should recurrence at the positions N A at instant 2π, if the system is perfect for the quantum state transfer. The autocorrelation \|A(t)\| = \| ψ(N A , 0) \|ψ(N A , t) \| is appropriate quantity to investigate the role that r 0 plays. During the period of time ∼ [0, 3π], the maxima of autocorrelations \|A max \| = max{\|A(t)\|} of the state (9) in the systems with N = 500, 1000, 1500, 2000, and r 0 = 10, 20, ..., 100 are calculated numerically by the exact diagonalization method. In Fig. 5, the dependence of the truncation distance r 0 on the quantity \|A max \| is plotted. It shows that \|A max \| approaches to 1 when r 0 is around 90, which is called the critical truncation distance or the boundary between the "long range" and "microscopic long-range" couplings, for the systems with different N . It indicates that in the case of N ≫ r 0 , the wave packets still travel without spreading. By making use of this observation, we find that although the interactions between spins are "long range", the 1/r 2 -distribution of coupling strength allow us to limit the maximal interaction range while minimizing the degradation of the quantum coherence obtained for ideal model. Then we have the conclusion that the 1/r 2 -decay coupling can be regarded as local coupling, or microscopic long-range coupling. In other word, the robust long-range entanglement between two distant qubits is not due to the direct long range coupling interaction between them. From the above observation, we find that, for large size system the long-range coupling (r 0 90) can be neglected. Then in thermodynamic limit, a ring system is equivalent to a more practical system, open chain system. To demonstrate this, the numerical simulation is employed to investigate the concurrence C(l, r 0 , t) between two far separated sites N/2 ± l (l ∼ N/2) for the N -site system with different truncation r 0 . In Fig. 6, plots of C(l, r 0 , t) for the systems with N = 1000, l = 400 and different r 0 are presented. It shows that for an open chain system, long range entanglement between two distant qubits can be achieved via the time evolution of a single flip state. The maximal entanglement created by such system is 0.5, as measured by the two-point concurrence. The results for different truncation approximations show that range of LR coupling can be taken in a small scale due to the 1/r 2 -decay of coupling constants. SUMMARY In summary, the system with long-range coupling is investigated analytically and numerically. It is found that the 1/r 2 -decay long-range coupling model can exhibit approximately linear dispersion ε ∼ \|k\|. The dynamics of such model possesses a novel feather that an initial local wave packet can be separated into two entangled local wave packets. Furthermore, during the traveling period each wave packets can keep their shapes without spreading. Numerical simulation indicates that there exists a critical truncation distance r 0 , which limits the range of the interaction but not affects the generation of entanglement between two distant qubits in the distance l ≫ r 0 . This model open up the possibility to realize the solidstate based entangler for creating two entangled but spatially separated qubits. FIG. 3 :FIG. 4 : 34(color online) Time evolution of a δ-pulse in a N = 100 ring system obtained by numerical simulation. It shows that the two sub-waves are local and keep the shape. (color online)Time evolution of a Gaussian wave packet of α = 0.1 in a N = 100 ring system obtained by numerical simulation. It shows that the two sub-waves are local and keep the shape. FIG. 5 : 5(color online) The maximal autocorrelation functions of the initial δ-pulse state in the N -site systems with the truncation distance r0. It demonstrates that critical r0, at which \|Amax\| start to approach to 1, do not depend on N strongly. FIG. 6 : 6(color online) The time-dependent concurrences of two spins separated by the distance 2l = 800 in the chain system with N = 1000, r0 = 10, 15, 20, 30 and 500. The results for r0 = 500 is in agreement with the analytical analysis. This work is supported by the NSFC with grant Nos. 90203018, 10474104 and 60433050. It is also funded by the National Fundamental Research Program of China with Nos. 2001CB309310 and 2005CB724508. [a] emails: [email protected] and [email protected] [b] Internet www site: http://www.itp.ac.cn/˜suncp [1] A. T. Costa Jr., S. Bose, Y. Omar, e-print quant-ph/0503183; T. Boness, S. Bose, and T. S. Monteiro, Phys. Rev. Lett. 96, 187201 (2006). . M Paternostro, M S Kim, E Park, J Lee, Phys. Rev. A. 7252307M. Paternostro, M. S. Kim, E. Park and J. Lee, Phys. Rev. A 72, 052307 (2005). . S Yang, Z Song, C P Sun, quant-ph/0602209S. Yang, Z. Song and C.P. Sun, e-print quant-ph/0602209. . Bing Chen, Z Song, C P Sun, quant-ph/0603033Bing Chen, Z. Song, and C.P. 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[ "CAN WE PROVE STABILITY BY USING A POSITIVE DEFINITE FUNCTION WITH NON SIGN-DEFINITE DERIVATIVE?", "CAN WE PROVE STABILITY BY USING A POSITIVE DEFINITE FUNCTION WITH NON SIGN-DEFINITE DERIVATIVE?" ]	[ "Iasson Karafyllis \nDepartment of Environmental Engineering\nTechnical University of Crete\n73100ChaniaGreece\n" ]	[ "Department of Environmental Engineering\nTechnical University of Crete\n73100ChaniaGreece" ]	[]	Novel criteria for global asymptotic stability are presented. The results are obtained by a combination of the "discretization approach" and the ideas contained in the proof of the original Matrosov's result. The results can be used for the proof of global asymptotic stability by using continuously differentiable, positive definite functions which do not have a negative semi-definite derivative. Illustrating examples are provided.	10.1093/imamci/dnr035	[ "https://arxiv.org/pdf/0907.4131v1.pdf" ]	46,657,956	0907.4131	2ac4b4bdf03ca55bf1d848b72cd9279ef9e5c86c	CAN WE PROVE STABILITY BY USING A POSITIVE DEFINITE FUNCTION WITH NON SIGN-DEFINITE DERIVATIVE? Iasson Karafyllis Department of Environmental Engineering Technical University of Crete 73100ChaniaGreece CAN WE PROVE STABILITY BY USING A POSITIVE DEFINITE FUNCTION WITH NON SIGN-DEFINITE DERIVATIVE? 1uniform robust global asymptotic stabilityuniform robust global exponential stabilityLyapunov methods Novel criteria for global asymptotic stability are presented. The results are obtained by a combination of the "discretization approach" and the ideas contained in the proof of the original Matrosov's result. The results can be used for the proof of global asymptotic stability by using continuously differentiable, positive definite functions which do not have a negative semi-definite derivative. Illustrating examples are provided. Introduction Lyapunov's direct method has been proved to be irreplaceable for the stability analysis of nonlinear systems. However, the main difficulty in the application of Lyapunov's direct method is to find a Lyapunov function for a given dynamical system. Most positive definite functions will not have a negative definite derivative for a given dynamical system and therefore cannot be used for stability analysis by using Lyapunov's direct method. There are two ways to relax the requirement of a negative definite derivative: 1) By using the Krasovskii-La Salle principle (see [9,12,13,16]) or by using Matrosov's theorem (see [22]). The original result by Matrosov has been generalized recently in various directions (see [11,12,14,15,16,24]). However, in order to able to apply all available results it is necessary to have a positive definite (Lyapunov) function with negative semi-definite derivative or to assume uniform Lyapunov and Lagrange stability (which can be shown by a positive definite function with negative semi-definite derivative). It should be noted that the main idea in the proof of the original Matrosov's result is the division of the state space into two regions: in the first region (the "bad region") the non-positive derivative of the Lyapunov function can be arbitrarily small in absolute value while in the second region (the "good region") the derivative of the Lyapunov function has a negative upper bound. The proof is accomplished by showing that the solution cannot stay in the "bad region" forever and by estimating the time that the solution spends in the "good region". Recently, in [8] a different approach was proposed for a Lyapunov function which can have positive derivative in certain regions of the state space: by using the derivative of auxiliary functions the methodology guarantees that the solution enters the "good region" after a finite time and remains bounded. The idea of switching between system modes with negative and positive derivative of a Lyapunov function has been also used recently in the stability analysis of hybrid systems in [18,19]. However, in hybrid systems the time period for which the derivative of the Lyapunov function is positive is determined by the switching signal and it is not necessary to estimate it. 2) By using the "discretization approach" (see [1], the Appendix in [2] and recent generalizations in [17,20,21] as well as the proof of the main result in [6]), which does not require a negative definite derivative. Instead, the discretization approach requires that the difference of the Lyapunov function )) 0 ( ( )) ( ( x V T x V − is negative definite, where ) (t x denotes the solution of the dynamical system and 0 > T is a fixed time. Therefore, in this approach the Lyapunov function can even have a positive derivative in certain regions of the state space. However, the main difficulty in the application of this approach is the estimation of the difference of the Lyapunov function )) 0 ( ( )) ( ( x V T x V − . The application of this approach to feedback stabilization problems gave very important results in [2] (see also recent extensions in [7]). The purpose of the present work is to combine the above approaches and to provide useful global stability criteria that use a positive definite function with a non sign-definite derivative. The results are developed for the autonomous uncertain case D d x x d f x n ∈ ℜ ∈ = , ) , ( & (1.1) where ) (t x is the state and l D t d ℜ ⊂ ∈ ) ( is a time-varying disturbance. However, the obtained results can be extended to the local case or the time-varying case. The key idea is the idea used in the proof of the original Matrosov's result described above concerning the division of the state space into two regions: the "good region", where the derivative of the Lyapunov function has a negative upper bound and the "bad region" where the derivative of the Lyapunov function can be positive. The first step is to show that the solution of (1.1) cannot stay in the "bad region" forever. Additional technical difficulties arise since we have to guarantee that the solution remains bounded while it stays in the "bad region". The second step is to estimate the difference of the Lyapunov function )) 0 ( ( )) ( ( x V T x V − , where 0 > T is chosen appropriately so that the solution is in the "good region". Finally, by extending the discetization approach, we can guarantee robust global asymptotic stability or robust global exponential stability (Theorem 3.1, Corollary 3.5, Theorem 3.7 and Corollaries 3.8, 3.9, 3.10). The structure of the paper is as follows. Section 2 provides the definitions of the notions used in the paper and some preliminary results that generalize the discretization approach. The results of Section 2 are interesting, since are necessary and sufficient conditions for robust global asymptotic stability. In Section Notations Throughout this paper we adopt the following notations: * For a vector n x ℜ ∈ we denote by x its usual Euclidean norm and by x′ its transpose. * We say that an increasing continuous function . , * By ) ( A C j ( ) ; ( Ω A C j ), where 0 ≥ j is a non-negative integer, n A ℜ ⊆ ,) (x V ∇ denotes the gradient of V at n x ℜ ∈ , i.e., ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ ∂ ∂ = ∇ ) ( ),..., ( ) ( 1 x x V x x V x V n . We say that a function + ℜ → ℜ n V : is positive definite if 0 ) ( > x V for all 0 ≠ x and 0 ) 0 ( = V . We say that a continuous function + ℜ → ℜ n V : is radially unbounded if the following property holds: "for every 0 > M the set } ) ( : { M x V x n ≤ ℜ ∈ is compact". Preliminary Results Throughout this paper we assume that system (1.1) satisfies the following hypotheses: (H1) l D ℜ ⊂  is compact. (H2) The mapping n n x d f x d D ℜ ∈ → ∋ ℜ × ) , ( ) , ( is continuous with 0 ) 0 , ( = d f for all D d ∈ . (H3) There exists a symmetric positive definite matrix n n P × ℜ ∈ such that for every compact set n S ℜ ⊂ it holds that +∞ < ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ≠ ∈ ∈ − − ′ − y x S y x D d y x y d f x d f P y x , , , : )) , ( ) , ( ( ) ( sup 2 . Hypothesis (H2) is a standard continuity hypothesis and hypothesis (H3) is often used in the literature instead of the usual local Lipschitz hypothesis for various purposes and is a generalization of the so-called "one-sided Lipschitz condition" (see, for example [23], page 416 and [3], page 106). Notice that the "one-sided Lipschitz condition" is weaker than the hypothesis of local Lipschitz continuity of the vector field ) , ( x d f with respect to n x ℜ ∈ . It is clear that hypothesis (H3) guarantees that for every D n M d x × ℜ ∈ ) , ( 0 , there exists a unique solution ) (t x of (1.1) with initial condition 0 ) 0 ( x x = corresponding to input D M d ∈ . We denote by ) ; , ( 0 d x t x the unique solution of (1.1) with initial condition n x x ℜ ∈ = 0 ) 0 ( corresponding to input D M d ∈ . Occasionally, we will use the following hypothesis for system (1.1): (H4) For every compact set n S ℜ ⊂ it holds that +∞ < ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ≠ ∈ ∈ − − y x S y x D d y x y d f x d f , , , : ) , ( ) , ( sup . instead of hypothesis (H3). Hypothesis (H4) is more demanding than hypothesis (H3) in the sense that the implication (H4) ⇒ (H3) holds. We next continue by recalling the notion of Uniform (Robust) Global Asymptotic Stability. { } ε δ ≤ ∈ ≤ ≥ D M d x t d x t x , ,0 ; ){ } ε τ ≤ ∈ ≤ ≥ D M d s x t d x t x , , ; ) ; , ( sup 0 0 (Uniform Attractivity for bounded sets of initial states) For disturbance-free systems we say that n ℜ ∈ 0 is uniformly globally asymptotically stable (UGAS) for system (1.1). It should be noted that the notion of uniform robust global asymptotic stability coincides with the notion of uniform robust global asymptotic stability presented in [10]. We next provide the notion of global exponential stability (see also [9]). 0 ≥ t , D n M d x × ℜ ∈ ) , ( 0 : 0 0 ) exp( ) ; , ( x t M d x t x σ − ≤ The following result is a generalization of the discretization approach for the autonomous case (1.1). ( ) ) ( )) ; , ( ( 0 0 x V a d x t x V ≤ , )] ( , 0 [ 0 x T t ∈ ∀ (2.1) ( ) ) ( ) ( )) ; , ( ( min 0 0 0 )] ( , 0 [ 0 x V q x V d x t x V x T t − ≤ ∈ (2.2) Then n ℜ ∈ 0 is URGAS for (1.1). Moreover, if 0 ) ( > ≡ r x T , Ms s a = : ) ( , qs s q = : ) ( , where 0 , > r M , ) 1 , 0 ( ∈ q and there exist constants 2 1 0 K K < < with 2 2 2 1 ) ( x K x V x K ≤ ≤ for all n x ℜ ∈ then n ℜ ∈ 0 is Robustly Globally Exponentially Stable for (1.1). Proof: Let KL ∈ σ be the function with the following property. Property (P): if ∞ = ≥ 0 } 0 { i i V is a sequence with ( ) i i i V q V V − ≤ +1 then ) , ( 0 i V V i σ ≤ for all 0 ≥ i . The existence of KL ∈ σ which satisfies property (P) is guaranteed by Lemma 4.3 in [4]. Define 1 ) 0 ( = T . Let n x ℜ ∈ 0 , D M d ∈ arbitrary and define the following sequences: and consequently inequality (2.4) holds as well in this case. Therefore, property (P) guarantees that ) , , ( 1 d P x t x x i i i i τ = + , ) ( i i x T T = , i i i t + = + τ τ 1 , ) ( i i x V V = , 0 ≥ i (2.3a) with 0 0 = τ , where D M d P ∈ τ is defined by ( ) ) ( ) ( τ τ + = t d t d P for 0 ≥ t and ] , 0 [ i i T t ∈ satisfies )) ; , ( ( min )) ; , ( ( ] , 0 [ d P x t x V d P x t x V i i i i T t i i τ τ ∈ = (2.) , ( 0 i V V i σ ≤ for all 0 ≥ i (2.5) where KL ∈ σ is the function involved in property (P). Inequality (2.1), definitions (2.3a) and the semigroup property guarantee that ( ) i i i V a d P x t x V d x t x V i ≤ − = )) ; , (( ))( ) i i i V a d P x t x V d x t x V i ≤ − = )) ; , ( ( )) ; , ( ( 0 τ τ for all ] , [ 1 + ∈ i i t τ τ for the case 0 = i x as well. Since ∞ = ≥ 0 } 0 { i i V is non-increasing (a consequence of (2.4)), we obtain ( ) ) ( )) ; , ( ( 0 0 x V a d x t x V ≤ for all ) sup , 0 [ i t τ ∈ (2.6) Next we show that ( ) ) ( )) ; , ( ( 0 0 x V a d x t x V ≤ for all 0 ≥ t (2.7) It should be noticed that Robust Lyapunov and Lagrange stability follows directly from inequality (2.7). For the proof of inequality (2.7) we distinguish two cases: Case 1: +∞ < i τ sup . By virtue of inequality (2.5) we obtain that 0 lim = i V and consequently ( ) 0 )) ; , ( ( lim 0 sup = − → d x t x V i t τ . This implies that ( ) 0 ) ; , ( lim 0 sup = − → d x t x i t τ , which implies 0 ) ; , ( 0 = d x t x for all i t τ sup ≥ . Therefore inequality (2.7) is a consequence of (2.6) and the fact that 0 )) ; , ( ( 0 = d x t x V for all i t τ sup ≥ . Case 2: +∞ = i τ sup . In this case inequality (2.7) is a direct consequence of inequality (2.6). We next show Robust Attractivity. Exploiting the fact that ) ; ( 0 + ℜ ℜ ∈ n C V is a continuous, positive definite and radially unbounded function, it suffices to show that for every 0 > ε , 0 ≥ R there exists 0 ) , ( ≥ R T ε such that ( ) ε ≤ ) ; , ( 0 d x t x V for all ) , ( R T t ε ≥ , n x ℜ ∈ 0 with R x ≤ 0 and D M d ∈ . Let 0 > ε , 0 ≥ R , n x ℜ ∈ 0 with R x ≤ 0 and D M d ∈ be arbitrary. By virtue of (2.7) and the semigroup property follows that if ) ( 1 ε − ≤ a V i for some 0 ≥ i then we have ε ≤ )) ; , ( ( 0 d x t x V for all i t τ ≥ . Define { } ) ( : 0 min : 1 ε − ≤ ≥ = a V i J i and { } R x x V R B ≤ = : ) ( max : ) ( . Let + ∈ Z R N ) ( ε such that ) ( )) ( ), ( ( 1 ε σ ε − ≤ a R N R B and notice that inequality (2.5) and the fact that ) ( ) ( 0 R B x V ≤ implies ) (R N J ε ≤ . Next suppose that 1 ≥ J . Since ) ( 1 ε − ≥ a V i for all 1 − ≤ J i (a consequence of definition { } ) ( : 0 min : 1 ε − ≤ ≥ = a V i J i ) , we get from (2.3a) and the facts that ∞ = ≥ 0 } 0 { i i V is non-increasing (a consequence of (2.4)) and ) ( ) ( 0 R B x V ≤ : ) ( 1 R T T t i i i i i i + ≤ + ≤ + = + τ τ τ τ for all 1 − ≤ J i where { } ) ( ) ( ) ( ) ( : ) ( sup : ) ( 1 1 R B a x V a x T R T + ≤ ≤ = − − ε ε ε . Therefore ) ( 1 R T i i ε τ ≤ + , for all 1 − ≤ J i and therefore inequality ) (R N J ε ≤ implies ) ( ) ( R T R N J ε ε τ ≤ . It follows that ε ≤ )) ; , ( ( 0 d x t x V for all ) ( ) ( R T R N t ε ε ≥ . The above conclusion holds as well in the case 0 = J , namely we have ε ≤ )) ; , ( ( 0 d x t x V for all ) ( ) ( R T R N t ε ε ≥ . Thus for every 0 > ε , 0 ≥ R there exists 0 ) ( ) ( ) , ( ≥ = R T R N R T ε ε ε such that ( ) ε ≤ ) ; , ( 0 d x t x V for all ) , ( R T t ε ≥ , n x ℜ ∈ 0 with R x ≤ 0 and D M d ∈ . Finally, for the case 0 ) ( > ≡ r x T , Ms s a = : ) ( , qs s q = : ) ( , 2 2 2 1 ) ( x K x V x K ≤ ≤ for all n x ℜ ∈ , where 0 , > r M , ) 1 , 0 ( ∈ q and 2 1 0 K K < < , we notice that inequality (2.4) implies 0 ) 1 ( V q V i i − ≤ for all 0 ≥ i . Therefore, using the inequality ( ) i V a d x t x V ≤ )) ; , ( ( 0 for all ] , [ 1 + ∈ i i t τ τ , in conjunction with definition Ms s a = : ) ( gives 0 0 ) 1 ( )) ; , ( ( MV q d x t x V i − ≤ for all ] , [ 1 + ∈ i i t τ τ . The previous inequality combined with the inequalities 2 2 2 1 ) ( x K x V x K ≤ ≤ for all n x ℜ ∈ gives 0 1 2 0 ) exp( ) ; , ( x K M K i d x t x σ − ≤ , for all ] , [ 1 + ∈ i i t τ τ , where 0 > σ is defined by the equation q − = − 1 ) 2 exp( σ . Using the fact that r i t i ) 1 ( 1 + ≤ ≤ + τ , we obtain the inequality ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ≤ − t r i σ σ σ exp ) exp( ) exp( , for all 0 ≥ i and ] , [ 1 + ∈ i i t τ τ . Consequently, by distinguishing again the cases +∞ < i τ sup and +∞ = i τ sup , we have 0 1 2 0 ) exp( exp ) ; , ( x K M K t r d x t x σ σ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ≤ , for all 0 ≥ t , which implies that n ℜ ∈ 0 is Robustly Globally Exponentially Stable for (1.1). The proof is complete. < Remark 2.4: The reader should notice that the converse of Proposition 2.3 holds, i.e., if n ℜ ∈ 0 is URGAS for (1.1) then for every positive definite and radially unbounded function ) ; ( 0 + ℜ ℜ ∈ n C V , there exist a function ∞ ∈ K a and a locally bounded function ) , 0 ( } 0 { \ : +∞ → ℜ n T such that for each } 0 { \ 0 n x ℜ ∈ , D M d ∈ the solution of (1.1) ) ; , ( 0 d x t x with initial condition 0 0 ) ; , 0 ( x d x x = corresponding to D M d ∈existsn x ℜ ∈ 0 , D M d ∈ the solution of (1.1) ) ; , ( 0 d x t x with initial condition 0 0 ) ; , 0 ( x d x x = corresponding to D M d ∈ satisfies: ( ) t x d x t x , ) ; , ( 0 0 σ ≤ , 0 ≥ ∀t (2.8) Without loss of generality, we may assume that for each 0 > s the mapping ) , ( t s t σ → is strictly decreasing (if not replace ) , ( t s σ by ) exp( ) , ( t s t s − + σ ). Since ) ; ( 0 + ℜ ℜ ∈ n C V is positive definite and radially unbounded, there exist functions ∞ ∈ K a a 2 1 , such that ( ) ( ) x a x V x a 2 1 ) ( ≤ ≤ , n x ℜ ∈ ∀ (2.9) Combining (2.8) and (2.9) we obtain: ( ) ( ) ( ) ( ) t x V a a d x t x V , ) ( ) ; , ( 0 1 1 2 0 − ≤ σ , 0 ≥ ∀t (2.10) Let ) 1 , 0 ( ∈ q and let 0 ) ( > s t be the solution of the equation ( ) ( ) ( ) s q s t s a a ) 1 ( ) ( , 1 1 2 − = − σ , for each 0 > s . It can be shown by contradiction that the mapping ) ( ) , 0 ( s t s → ∋ +∞ is bounded on every compact set ) , 0 ( +∞ ⊂ S . Therefore, by virtue of (2.10), we conclude that inequalities (2.1) and (2.2) hold with ( ) ( ) ( ) 0 , : ) ( 1 1 2 s a a s a − = σ , s q s q = : ) ( and 1 )) ( ( : ) ( + = x V t x T . < The following proposition is less demanding in terms of the inequalities that guarantee URGAS. However, in contrast to Proposition 2.3, we have to assume that system (1.1) is forward complete and that hypothesis (H4) holds. We say that system (1.1) is forward complete if for every n x ℜ ∈ 0 , D M d ∈ the solution of (1.1) ) ; , ( 0 d x t x with initial condition 0 0 ) ; , 0 ( x d x x = corresponding to D M d ∈ is defined for all 0 ≥ t . Proposition 2.5: Consider system (1.1) under hypotheses (H1), (H2), (H4) and assume that system (1.1) is forward complete. Furthermore, suppose that there exist a positive definite and radially unbounded ) ; ( 0 + ℜ ℜ ∈ n C V , a positive definite function ) ; ( 0 + + ℜ ℜ ∈ C q and a locally bounded function ) , 0 ( : +∞ → ℜ n T such that for each } 0 { \ 0 n x ℜ ∈ , D M d ∈ the solution of (1.1) ) ; , ( 0 d x t x with initial condition 0 0 ) ; , 0 ( x d x x = corresponding to D M d ∈ exists on )] ( , 0 [ 0 x T and satisfies inequality (2.2). Then n ℜ ∈ 0 is URGAS for (1.1). The reader should notice that an additional difference between Proposition 2.5 and Proposition 2.3 is the fact that Proposition 2.3 demands the function ) , 0 ( } 0 { \ : +∞ → ℜ n T to( ) 0 0 ) ( ) ; , ( x t d x t x ζ μ ≤ (2.11) Without loss of generality, we may assume that + ∈ K μ is non-decreasing. Since ) ; ( 0 + ℜ ℜ ∈ n C V is positive definite and radially unbounded, there exist functions ∞ ∈ K a a 2 1 , such that inequality (2.9) holds. Combining (2.9) and (2.11) we obtain that for each } 0 { \ 0 n x ℜ ∈ , D M d ∈ the solution ) ; , ( 0 d x t x of (1.1) satisfies the following inequality: ( ) ( ) 0 0 2 0 )) ( ( )) ; , ( ( x x T a d x t x V ζ μ ≤ , )] ( , 0 [ 0 x T t ∈ ∀ (2.12) Define ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ≤ ) ( ) ( sup : ) ( 2 s x T a s p s x ζ μ for all 0 ≥ s . Since ) , 0 ( : +∞ → ℜ n T is locally bounded, it follows that ) (s p is well-defined for all 0 ≥ s and is a non-decreasing function. Moreover, it holds that 0 ) 0 ( ) ( lim 0 = = + → p s p s . Define ∫ + = s s d p s s s a 2 ) ( 1 : ) ( ξ ξ for 0 > s and 0 : ) 0 ( = a . The function ã is of class ∞ K and satisfies ) ( ) ( s p s a ≥ for all 0 ≥ s . Consequently, using (2.12) we obtain that for each } 0 { \ 0 n x ℜ ∈ , D M d ∈ the solution ) ; , ( 0 d x t x of (1.1) satisfies the following inequality: ( ) 0 0) ) ; , ( ( x a d x t x V ≤ , )] ( , 0 [ 0 x T t ∈ ∀ (2.13) Using (2.9) and (2.13), it follows that inequality (2.1) holds with ( ) ) ( : ) ( 1 1 s a a s a − = . The proof is complete. < Main Results Next, the main result of the present work is stated. ) ; ( 1 + ℜ ℜ ∈ n C V , a family of functions ) ; ( 1 ℜ ℜ ∈ n i C W with 0 ) 0 ( = i W , ) ; ( 0 + + ℜ ℜ ∈ C b i ( k i ,..., 0 = ), ∞ ∈ K g c c γ ρ , , , , 2 1 , with ) ( ) ( ) ( 2 1 s c s c s ≥ > ρ for all 0 > s , K ∈ λ with s s < ) ( λ for all 0 > s , a locally bounded function ) , 0 ( : +∞ → ℜ + r and a 1 C function ℜ → ℜ + : μ with 0 ) 0 ( = μ for which the function ) ( ) ( : ) ( 1 s s c s μ κ + = is non-decreasing, such that the following inequalities hold: ) ( )) ( ( ) , ( ) ( max 0 x W x V x d f x V D d + − ≤ ∇ ∈ ρ , for all n x ℜ ∈ (3.1) ) ( ) , ( ) ( max 1 x W x d f x W i i D d + ∈ ≤ ∇ , for all 1 ,..., 0 − = k i and for all n x ℜ ∈ with )) ( ( ) ( 2 0 x V c x W ≥ (3.2) )) ( ( ) ( x V b x W i i ≤ , for all k i ,..., 0 = and for all n x ℜ ∈ with )) ( ( ) ( 2 0 x V c x W ≥ (3.3) )) ( ( ) , ( ) ( max x V g x d f x W k D d − ≤ ∇ ∈ , for all n x ℜ ∈ with )) ( ( ) ( 2 0 x V c x W ≥ (3.4) 0 ) , ( ) ( )) ( ( max ) , ( ) ( max 0 ≤ ∇ ′ + ∇ ∈ ∈ x d f x V x V x d f x W D d D d μ , for all n x ℜ ∈ with )) ( ( ) ( )) ( ( 2 0 1 x V c x W x V c ≥ ≥ (3.5) ( ) ( ) ( ) )) ( ( )) ( ( ) ( 2 1 s s c s s c γ μ γ λ μ λ + > + , for all 0 > s (3.6) ( ) ( ) ( ) ∑ = + > + + k i i i k s b i s r k s r s g s c 0 1 2 ! ) ( )! 1 ( ) ( ) ( ) ( λ λ , for all 0 > s (3.7) ( ) ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + − ≥ + = ∈ − ∑ )! 1 ( ) ( ) ( ! max , max ) ( 1 0 )] ( , 0 [ 1 k s g s b i s s k k i i i s r τ λ τ ρ γ τ , for all 0 > s (3.8) +∞ < − ∫ + → ) ( ) ( 1 0 ) ( ) ( sup lim s s s c d γ λ τ τ ρ τ (3.9) Then n ℜ ∈ 0 is URGAS for (1.1). Remark 3.2: A sufficient condition for the existence of a locally bounded function ) , 0 ( : +∞ → ℜ + r that satisfies (3.7) is the set of inequalities ( ) +∞ < + → ) ( ) ( sup lim 0 s g s b i s λ , for k i ,..., 1 = and ( ) +∞ < − + → ) ( )) ( ( ) ( sup lim 2 0 0 s g s c s b s λ λ .( ) ( ) ( ) ( ) ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − + + + = − + = + i k i k i k s g s b k i k s g s c s b k k s r 1 1 ,..., 1 1 1 2 0 ) ( ) 1 ( ! )! 1 ( max , ) ( )) ( ( )( 1 ( )! 1 ( max 1 : ) ( λ λ λ for 0 > s and 1 : ) 0 ( = r , is a locally bounded function ) , 0 ( : +∞ → ℜ + r that satisfies (3.7). Remark 3.3: A sufficient condition for (3.9) is the existence of a constant ) 1 , 0 ( ∈ K such that: The proof of Theorem 3.1 is heavily based on the following lemma. Its proof can be found in the Appendix. s K s c s + ≥ ) ( ) ( 1 ρ and ) ( ) ( 1 s K s λ γ − ≤ ,) ; ( 1 + ℜ ℜ ∈ n C V , a family of functions ) ; ( 1 ℜ ℜ ∈ n i C W with 0 ) 0 ( = i W , ) ; ( 0 + + ℜ ℜ ∈ C b i ( k i ,..., 0 = ) , ∞ ∈ K g c c γ ρ , , , , 2 1 , with ) ( ) ( ) ( 2 1 s c s c s ≥ > ρ for all 0 > s , K ∈ λ with s s < ) ( λ for all 0 > s , a locally bounded function ) , 0 ( : +∞ → ℜ + r and a 1 C function ℜ → ℜ + : μ with 0 ) 0 ( = μ for which the function ) ( ) ( : ) ( 1 s s c s μ κ + = is non-decreasing, such that inequalities hold (3.1)-(3.8) hold. Then system (1.1) is forward complete. We are now ready to provide the proof of Theorem 3.1. 2.2) hold for each } 0 { \ 0 n x ℜ ∈ , D M d ∈ for the function ) ; ( 1 + ℜ ℜ ∈ n C V with )) ( ( : ) ( x V p x T = (3.10) ∫ − + = ) ( ) ( 1 ) ( ) ( ) ( : ) ( s s c d s r s p γ λ τ τ ρ τ , for 0 > s and 1 : ) 0 ( = p (3.11) ) ( : ) ( s s s q λ − = (3.12) The reader should notice that condition (3.9) and the fact that ) , 0 ( : +∞ → ℜ + r is a locally bounded function guarantee that the function ) , 0 ( : +∞ → ℜ + p as defined by (3.11) is locally bounded. Let } 0 { \ 0 n x ℜ ∈ , D M d ∈ (arbitrary). We next show by contradiction that there exists )] ( , 0 [ 0 x T t ∈ such that )) ( ( )) ; , ( ( 0 0 x V d x t x V λ ≤ . Then definition (3.12) automatically guarantees that inequality (2.2) holds. Assume next that )) ( ( )) ; , ( ( 0 0 x V d x t x V λ > for all )] ( , 0 [ 0 x T t ∈ . We show that this cannot happen. We first start by stating the following fact. Its proof can be found in the Appendix. FACT I: There exists ))] ( ( , 0 [ 0 x V r t ∈ with ))) ; , ( ( ( )) ; , ( ( 0 2 0 0 d x t x V c d x t x W < . Define: { } ))) ; , ( ( ( )) ; , ( ( : ))] ( ( , 0 [ inf : 0 2 0 0 0 1 d x t x V c d x t x W x V r t t < ∈ = (3.13) We next continue with the following fact. Its proof can be found in the Appendix. FACT II: The following inequality holds: )) ( ( )) ; , ( ( 0 0 1 x V d x t x V γ ≤ (3.14) We next distinguish the following cases: CASE 1: ))) ; , ( ( ( )) ; , ( ( 0 1 0 0 d x t x V c d x t x W ≤ for all )] ( , [ 0 1 x T t t ∈ . Define ) ( ) ( : ) ( 1 s c s s c − = ρ , which is a positive definite, continuous function. In this case inequality (3.1) implies that )) ( ( ) ( t V c t V − ≤ & , for )] ( , [ 0 1 x T t t ∈ , a.e., where )) ; , ( ( : ) ( 0 d x t x V t V = . Consequently, we obtain ) ) ( ( )) ( ( ) ( ) ( 1 0 ) ( )) ( ( ) ( 0 1 0 1 t x T dt t V c t V s c ds x T t x T V t V − − ≤ = ∫ ∫ & . Combining, the previous inequality with the fact that ))] ( ( , 0 [ 0 1 x V r t ∈ and definition (3.11) we get ∫ ∫ ≥ )) ( ( )) ( ( ) ( )) ( ( 0 0 1 0 ) ( ) ( x V x V t V x T V s c ds s c ds γ λ . Since (3.14) holds, the previous inequality gives )) ( ( )) ( ( 0 0 x V x T V λ ≤ , a contradiction. CASE 2: There exists )] ( , [ 0 1 x T t t ∈ with ))) ; , ( ( ( )) ; , ( ( 0 1 0 0 d x t x V c d x t x W > . In this case, continuity of mapping )) ; , ( ( )) ; , ( ( 0 0 0 d x t x V d x t x W t → guarantees the existence of times 3 2 t t < with ) ( 0 3 2 1 x T t t t ≤ < ≤ and such that: ))) ; , ( ( ( )) ; , ( ( 0 1 0 0 d x t x V c d x t x W ≤ , for all ] , [ 3 1 t t t ∈ (3.15) ))) ; , ( ( ( )) ; , ( ( 0 2 2 0 2 0 d x t x V c d x t x W = , ))) ; , ( ( ( )) ; , ( ( 0 3 1 0 3 0 d x t x V c d x t x W = (3.16) ))) ; , ( ( ( )) ; , ( ( 0 2 0 0 d x t x V c d x t x W ≥ , for all ] , [ 3 2 t t t ∈ (3.17) Inequality (3.15) in conjunction with inequality (3.1) guarantees that ( ) ∫ − − ≤ t t d V c V t V t V 2 )) ( ( )) ( ( ) ( ) ( 1 2 τ τ τ ρ , for all ] , [ 3 2 t t t ∈ (3.18) ) ( ) ( 1 t V t V ≤ , for all ] , [ 3 1 t t t ∈ (3.19) Inequalities (3.15), (3.17) and (3.5) imply that )) ( ( ) ( )) ( ( ) ( 2 2 0 3 3 0 t V t W t V t W μ μ + ≤ + (3.20) It follows from (3.16) and (3.20) that: )) ( ( )) ( ( )) ( ( )) ( ( 2 2 2 3 3 1 t V t V c t V t V c μ μ + ≤ + (3.21) If ( ) )) ( ( ) ( 2 1 3 t V t V − ≤ γ λ then using (3.14), (3.19) we obtain )) ( ( ) ( 0 3 x V t V λ ≤ , a contradiction. Thus we are left with the case ( ) )) ( ( ) ( 2 1 3 t V t V − > γ λ . In this case inequality (3.21) and the fact that the function ) ( ) ( : ) ( 1 s s c s μ κ + = is non-decreasing, give ( ) ( ) ( ) ( ) )) ( ( )) ( ( )) ( ( )) ( ( 2 2 2 2 1 2 1 1 t V t V c t V t V c μ γ λ μ γ λ + ≤ + − − The above inequality contradicts inequality (3.6) for )) ( ( 2 1 t V s − = γ . The proof is complete. < (3.5), (3.6), (3.7), (3.8), (3.9) hold as well as the following inequalities: 1 ℜ ℜ ∈ n i C W with 0 ) 0 ( = i W , ) ; ( 0 + + ℜ ℜ ∈ C b i ( k i ,..., 0 = ), ∞ ∈ K g c c γ ρ ,, , , 2 1 ,) ( ) ( )) ( ( ) ( ) , ( ) ( max 0 x W x x V x x d f x V D d φ ρ φ + − ≤ ∇ ∈ , for all n x ℜ ∈ (3.22) ) ( ) ( ) , ( ) ( max 1 x W x x d f x W i i D d + ∈ ≤ ∇ φ , for all 1 ,..., 0 − = k i and for all n x ℜ ∈ with )) ( ( ) ( 2 0 x V c x W ≥ (3.23) )) ( ( ) ( ) , ( ) ( max x V g x x d f x W k D d φ − ≤ ∇ ∈ , for all n x ℜ ∈ with )) ( ( ) ( 2 0 x V c x W ≥ (3.24) Then n ℜ ∈ 0 is URGAS for (1.1). Proof: Simply consider the dynamical system: D d x x d f x x n ∈ ℜ ∈ = , ), ( ) (+ ℜ → ℜ n q : such that ) ( ) , ( ) ( max x q x d f x V D d − ≤ ∇ ∈ for all n x ℜ ∈ , then one can construct a locally Lipschitz function ) , 0 ( : The following theorem provides stability criteria under minimal regularity requirements for system (1.1). Here we do not assume the local Lipschitz assumption (H4). +∞ → ℜ n φ and a function ∞ ∈ K ρ such that )) ( ( ) ( ) , ( ) ( max x V x x d f x V D d ρ φ − ≤ ∇ ∈ for all n x ℜ ∈ . Theorem 3.7: Consider system (1.1) under hypotheses (H1), (H2), (H3) and suppose that there exist a positive definite and radially unbounded function ) ; ( 1 + ℜ ℜ ∈ n C V , a family of functions ) ; ( 1 ℜ ℜ ∈ n i C W with 0 ) 0 ( = i W , constants 0 ≥ i b ( k i ,..., 0 = ) with ρ ≥ 0 b , 0 , ,, , , 2 1> r g c c γ ρ , with 0 2 1 > ≥ > c c ρ , ) 1 , 0 ( ∈ λ and 1 c − ≥ μ such that the following inequalities hold: ) ( ) ( ) , ( ) ( max 0 x W x V x d f x V D d + − ≤ ∇ ∈ ρ , for all n x ℜ ∈ (3.26) ) ( ) , ( ) ( max 1 x W x d f x W i i D d + ∈ ≤ ∇ , for all 1 ,..., 0 − = k i and for all n x ℜ ∈ with ) ( ) ( 2 0 x V c x W ≥ (3.27) ) ( ) ( x V b x W i i ≤ , for all k i ,..., 0 = and for all n x ℜ ∈ with ) ( ) ( 2 0 x V c x W ≥ (3.28) ) ( ) , ( ) ( max x V g x d f x W k D d − ≤ ∇ ∈ , for all n x ℜ ∈ with ) ( ) ( 2 0 x V c x W ≥ (3.29) 0 ) , ( ) ( max ) , ( ) ( max 0 ≤ ∇ + ∇ ∈ ∈ x d f x V x d f x W D d D d μ , for all n x ℜ ∈ with ) ( ) ( ) ( 2 0 1 x V c x W x V c ≥ ≥ (3.30) γ μ λ μ ) ( ) ( 2 1 + > + c c (3.31) ∑ = + > + + k i i i k b i r k r g c 0 1 2 ! )! 1 ( λ λ (3.32) ( ) ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + − − ≥ + = ∈ ∑ )! 1 ( ! 1 max ; ) ( exp min 1 0 ] , 0 [ 0 k g b i r b k k i i i r τ λ τ ρ ρ γ τ(ρ ≥ 0 b imply that ( ) ) ( ) ( exp )) ; , ( ( 0 0 0 x V t b d x t x V ρ − ≤ , for all 0 ≥ t (3.34) Indeed, inequality (3.26) implies 0 ) , ( ) ( max ≤ ∇ ∈ x d f x V D d , when ) ( ) ( 2 0 x V c x W ≤ . Moreover, inequalities (3.26), (3.28) (for ) 0 = i imply that ) ( ) ( ) , ( ) ( max 0 x V b x d f x V D d ρ − ≤ ∇ ∈ , when ) ( ) ( 2 0 x V c x W ≥ . Since ρ ≥ 0 b , we conclude that ) ( ) ( ) , ( ) ( max 0 x V b x d f x V D d ρ − ≤ ∇ ∈ for all n x ℜ ∈ . Inequality (3.34) follows directly from the previous differential inequality. The proof is exactly the same with the proof of 2) Inequality (3.14) is obtained by a combined use of the proof of Fact II in the Appendix and inequality (3.34). Details are left to the reader. < Corollary 3.8: Consider system (1.1) under hypotheses (H1), (H2), (H3) and suppose that there exist a positive definite and radially unbounded function ) ; ( 1 + ℜ ℜ ∈ n C V , a locally Lipschitz function ) , 0 ( : +∞ → ℜ n φ , a family of functions ) ; ( 1 ℜ ℜ ∈ n i C W with 0 ) 0 ( = i W , constants 0 ≥ i b ( k i ,..., 0 = ) with ρ ≥ 0 b , 0 , ,, , , 2 1> r g c c γ ρ , with 0 2 1 > ≥ > c c ρ , ) 1 , 0 ( ∈ λ and 1 c − ≥ μ such that inequalities (3.28), (3.30), (3.31), (3.32), (3.33) as well as the following inequalities hold: ) ( ) ( ) ( ) ( ) , ( ) ( max 0 x W x x V x x d f x V D d φ ρφ + − ≤ ∇ ∈ , for all n x ℜ ∈ (3.35) ) ( ) ( ) , ( ) ( max 1 x W x x d f x W i i D d + ∈ ≤ ∇ φ , for all 1 ,..., 0 − = k i and for all n x ℜ ∈ with ) ( ) ( 2 0 x V c x W ≥ (3.36) ) ( ) ( ) , ( ) ( max x V x g x d f x W k D d φ − ≤ ∇ ∈ , for all n x ℜ ∈ with ) ( ) ( 2 0 x V c x W ≥ (3.37) Then n ℜ ∈ 0 is URGAS for (1.1). Moreover, if there exist constants 2 1 0 K K < < with 2 2 2 1 ) ( x K x V x K ≤ ≤ and 1 ) ( K x ≥ φ for all n x ℜ ∈ then n ℜ ∈ 0 is Robustly Globally Exponentially Stable for (1.1). Proof: Again the proof of Corollary 3.8 is made with the help of Theorem 3.7 and system (3.25). Exponential stability follows directly from the fact that for every 1) can be useful. The following theorem uses an additional differential inequality, which can be used to replace inequality (3.8) by a less demanding inequality. Corollary 3.9 : Consider system (1.1) under hypotheses (H1), (H2), (H4) and suppose that there exist a positive definite and radially unbounded function ) ; ( 1 + ℜ ℜ ∈ n C V , a locally Lipschitz function ) , 0 ( : +∞ → ℜ n φ , a family of (3.6), (3.9) hold as well as the following inequalities: functions ) ; ( 1 ℜ ℜ ∈ n i C W with 0 ) 0 ( = i W , ) ; ( 0 + + ℜ ℜ ∈ C b i ( k i ,..., 0 = ) , ∞ ∈ K g c c , ,)) ( ( ) ( ) , ( ) ( max x V g x x d f x W k D d φ − ≤ ∇ ∈ , for all n x ℜ ∈ with )) ( ( ) ( 2 0 x V c x W ≥ and 0 ) ( max ,..., 1 ≥ = x W i k i (3.38) ( ) ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + − ≥ + = = ∈ − ∑ )! 1 ( ) ( ) ( ! max , min ) ( 1 0 ) ( )], ( , 0 [ 1 k s g x W i s s k k i i i s x V s r τ λ τ ρ γ τ , for all 0 > s (3.39) Then n ℜ ∈ 0 is URGAS for (1.1). The reader should notice that in general the function ∞ ∈ K g involved in (3.38) will be greater than the function ∞ ∈ K g involved in (3.24). Therefore, (3.39) is a less demanding inequality than (3.8). Proof: It suffices to show that the result holds for the special case 1 ) ( ≡ x φ . Then a similar argument to the one used in the proof of Corollary 3.5 can show the validity of the result to the general case. Therefore, we assume that inequalities (3.1), (3.2), (3.4) and (3.38 ) with 1 ) ( ≡ x φ hold. The reader should notice that inequality (3.8) in the proofs of Theorem 3.1 and Lemma 3.4 is used only for the derivation of inequality )) ( ( ) ( 0 1 t V t V γ ≤ , where 1 0 t t < are times with )) ( ( 0 0 1 t V r t t ≤ − (3.40) ))) ; , ( ( ( )) ; , ( ( 0 2 0 0 d x t x V c d x t x W ≥ for all ] , [ 1 0 t t t ∈ (3.41) )) ( ( ) ( 0 t V t V λ > for all ] , [ 1 0 t t t ∈(≤ = t W i k i and ) ( ) ( 0 0 T W t W ≤ for all ] , [ 1 t T t ∈ . We next distinguish the following cases: Case 1: 0 ) ( max ,..., 1 > = t W i k i for all ] , [ 1 0 t t t ∈ . In this case, inequality (3.38) implies that inequalities (A9), (A10) hold with ∞ ∈ K g replaced by ∞ ∈ K g for all ] , [ 1 0 t t t ∈ (inequality (A9) holds for ] , [ 1 0 t t t ∈ a.e.). Consequently, by virtue of (3.40) we get: ( ) ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + − ≤ + = ∈ ∈ ∑ )! 1 ( )) ( ( ) ( ! max )) ; , ( ( max 1 0 0 0 ))] ( ( , 0 [ 0 0 ] , [ 0 1 0 k t V g t W i d x t x W k k i i i t V r t t t τ λ τ τ (3.T W t W ≤ for all ] , [ 1 t T t ∈ , it follows that )) ; , ( ( max )) ; , ( ( max 0 0 ] , [ 0 0 ] , [ 0 1 0 d x t x W d x t x W T t t t t t ∈ ∈ = . We conclude that (3.43) holds in this case as well. ( ) ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + − + − ≤ + = ∈ ∑ )! 1 ( )) ( ( ) ( ! max )) ( ( ) ( 1 0 0 0 ))] ( ( , 0 [ 0 k t V g t W i t V t V k k i i i t V r τ λ τ ρ τ & (3.44) Differential inequality (3.44) directly implies that: ( ) ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + − ≤ + = ∈ − ∑ )! 1 ( )) ( ( ) ( ! max , ) ( max ) ( 1 0 0 0 ))] ( ( , 0 [ 1 0 0 k t V g t W i t V t V k k i i i t V r τ λ τ ρ τ , for all ] , [ 1 0 t t t ∈ The above inequality in conjunction with inequality (3.39) implies that inequality )) ( ( ) ( 0 1 t V t V γ ≤ holds. It should be noticed that inequality )) ( ( ) ( 0 1 t V t V γ ≤ holds as well for the case 0 1 t t = (since (3.39) implies that )) ( ( ) ( 0 0 t V t V γ ≤ ). The proof is complete. < Similarly with Corollary 3.9, we obtain the following result. Corollary 3.10: Consider system (1.1) under hypotheses (H1), (H2), (H3) and suppose that there exist a positive definite and radially unbounded function ) ; ( 1 + ℜ ℜ ∈ n C V , a locally Lipschitz function ) , 0 ( : +∞ → ℜ n φ , a family of functions ) ; ( 1 ℜ ℜ ∈ n i C W with 0 ) 0 ( = i W , constants 0 ≥ i b ( k i ,..., 0 = ) with ρ ≥ 0 b , 0 , ,, , 2 1) ( ) ( ) , ( ) ( max x V x g x d f x W k D d φ − ≤ ∇ ∈ , for all n x ℜ ∈ with ) ( ) ( 2 0 x V c x W ≥ and 0 ) ( max ,..., 1 ≥ = x W i k i (3.45) ( ) ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + − − ≥ + = ℜ ∈ ∈ ∑ )! 1 ( ) ( ) ( ! 1 sup , ) ( exp min 1 0 ], , 0 [ 0 k g x V x W i r b k k i i i x r n τ λ τ ρ ρ γ τ (3.46) Then n ℜ ∈ 0 is URGAS for (1.1). Moreover, if there exist constants 2 1 0 K K < < with 2 2 2 1 ) ( x K x V x K ≤ ≤ and 1 ) ( K x ≥ φ for all n x ℜ ∈ then n ℜ ∈ 0 is Robustly Globally Exponentially Stable for (1.1). Examples p p d x x x x x d x x x − ∈ ℜ ∈ ′ = − = − = β & & (4.1) where 0 ≥ p is a constant parameter, ℜ → ℜ : β is a locally Lipschitz mapping with 0 ) 0 ( = β and the Lyapunov function 2 2 2 1 ) ( x x x V + = (4.2) The reader should notice that the derivative of the Lyapunov function defined by (4.2) is not necessarily sign definite and classical Lyapunov analysis does not help. Of course there are Lyapunov functions that can be used directly for classical Lyapunov analysis (e.g., dy y y p x x x V x ∫ + + = 1 0 2 2 2 2 2 1 ) ( ) ( β ). Here, for illustration purposes, we apply the result of Theorem 3.1 and we show that for every 0 ≥ p , 2 0 ℜ ∈ is URGAS for system (4.1). We have: ) ( ) ( ) ( 2 ) ( 2 ) ( 1 2 2 2 1 x p x V x x d x V x V β β + − ≤ + − = & , 2 ℜ ∈ ∀x (4.3) Let ℜ → ℜ : β be an odd 1 C mapping such that its restriction on + ℜ is a convex ∞ K function and satisfies ( ) 1 1) ( x x β β ≤ for all ℜ ∈ 1 x . Inequality + + ∞ ℜ ℜ ∩ ∈ C K b such that inequality (3.3) for 0 = i holds for all 2 ℜ ∈ x . Without loss of generality we may assume that s s b ≥ ) ( 0 for all 0 ≥ s . Furthermore, since ℜ → ℜ : β is a 1 C mapping, we have: 1 1 1 2 0 ) ( ) ( 2 ) ( x x x p x W β β ′ − = & , 2 ℜ ∈ ∀x (4.4) If 0 1 ≥ x , then since the restriction of ℜ → ℜ : β on + ℜ is a convex ∞ K function we get 1 1 1 ) ( ) ( x x x β β ′ ≤ . Therefore, (4.4) implies for all 0 1 ≥ x : ) ( 2 ) ( 0 0 x W x W − ≤ & (4.5) Since ℜ → ℜ : β is an odd mapping, we have ) ( ) ( 1 1 x x − = ′ β β for all 0 1 < x . Therefore, if 0 1 < x , by virtue of convexity we get 1 1 1 ) ( ) ( x x x β β ′ − ≤ − . Thus (4.5) holds for 0 1 < x as well. + + ∞ ℜ ℜ ∩ ∈ C K b ). Finally, notice that inequality (3.6) holds with ) ( : ) ( p D d x x x x x d x x x = ∈ ℜ ∈ ′ = − + − = = & & (4.6) where 0 ≥ p is3 ) ( 3 3 4 2 max ) , ( ) ( max x p x V x x p x x dx x d f x V D d D d + + − = − ≤ − − = ∇ ∈ ∈ (4.7) where ) 2 ) 1 ( , ( : ) , ( 2 1 2 ′ − + − = x x d x x d f . Inequality (4.7) shows that inequality (3.26) holds with 3 := ρ and ( ) 2 1 2 0 3 : ) ( x p x W + = . We also have: The reader should notice that inequalities (3.28) hold with ( ) 2 1 2 0 3 2 ) , ( ) ( max x x p x d f x W D d + = ∇ ∈ (4.2 3 2 2 ) 1 ( max 3 2 ) , ( ) ( max x x x x p x x x d x p x d f x W D d D d − − + ≤ − + − + = ∇ ∈ ∈2 1 0 3 p b b + = = .3 3 2 ) , ( ) ( max ) , ( ) ( max x x p x x p x d f x V x d f x W D d D d μ μ μ − + + ≤ ∇ + ∇ ∈ ∈ (4.14) The reader can verify that ) ( ) ( ) ( 1 0 2 x V c x W x V c ≤ ≤ is equivalent to the sector condition: On the other hand, the right hand side of inequality (4.14) is non-positive provided that However, here we have used a completely inappropriate Lyapunov function, which has positive derivative in certain regions of the state space. The example simply shows that stability analysis is possible even with completely inappropriate Lyapunov functions. < + + ≤ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − + + = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + + ≤ ≤ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − + + ≤ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + − ℜ ∈ ℜ ∈ ℜ ∈ ∈ + = ℜ ∈ ∈ ∑ λ λ λ λ λ λτ τ τ λ τ ρ τ τ p x V x x V x p x V x x x V x p x V x x x V x p k g x V x W i n Concluding Remarks Novel criteria for global asymptotic stability are presented. The results (Theorem 3.1, Corollary 3.5, Theorem 3.7, Corollaries 3.8, 3.9 and 3.10) are developed for the autonomous uncertain case and are obtained by a combination of: • suitable generalizations of the "discretization approach" (Proposition 2.3 and Proposition 2.5), which are necessary and sufficient conditions for uniform robust global asymptotic stability, • the idea contained in the proof of the original Matrosov's result concerning the division of the state space into two regions: the "good region", where the derivative of the Lyapunov function has a negative upper bound and the "bad region" where the derivative of the Lyapunov function can be positive. a.e.: ) ( ) ( 1 t W t W i i + ≤ & , for 1 ,..., 0 − = k i (A1) ( ) )) ( ( ) ( 0 x V g t W k λ − ≤ & (A2) where )) ; , ( ( : ) ( 0 d x t x W t W i i = ( k i ,..., 0 = ). Inequalities (A1) and (A2) imply that the following inequality holds for all ))] ( ( , 0 [ 0 x V r t ∈ ( ) )! 1 ( )) ( ( ) ( ! ) ( 1 0 0 0 0 + − ≤ + = ∑ k t x V g x W i t t W k k i i i λ (A3) Our assumption that ))) ; , ( ( ( )) ; , ( ( 0 2 0 0 d x t x V c d x t x W ≥ for all ))] ( ( , 0 [ 0 x V r t ∈ in conjunction with the fact that )) ( ( )) ; , ( ( 0 0 x V d x t x V λ > for all ))] ( ( , 0 [ 0 x V r t ∈ and inequality (A3) gives: ( ) ( ) )! 1 ( )) ( ( ) ( ! )) ( ( 1 0 0 0 0 2 + − ≤ + = ∑ k t x V g x W i t x V c k k i i i λ λ , for all ))] ( ( , 0 [ 0 x V r t ∈ (A4) Inequality (A4) (for )) ( ( 0 x V r t = ) in conjunction with inequalities (3.3) (which give )) ( ( ) ( 0 0 x V b x W i i ≤ for k i ,..., 0 = ) implies that the following inequality must hold: ( ) ( ) ( ) )! 1 ( )) ( ( )) ( ( ) ( ! )) ( ( )) ( ( 0 1 0 0 0 0 0 2 + − ≤ + = ∑ k x V r x V g x V b i x V r x V c k k i i i λ λ which contradicts inequality (3.7) with ) ( 0 x V s = . The proof is complete. < Proof of FACT II in the proof of Theorem 3.1: Suppose first that 0 0 > t . By virtue of definition (3.13), it follows that ))) ; , ( ( ( )) ; , ( ( 0 2 0 0 d x t x V c d x t x W ≥ for all ] , 0 [ 0 t t ∈ . Using inequalities (3.2), (3.4) and the fact that )) ( ( )) ; , ( ( 0 0 x V d x t x V λ > for all ))] ( ( , 0 [ 0 x V r t ∈ ,x V b x W i i ≤ for k i ,..., 0 = ), we get from (A3): ( ) )! 1 ( )) ( ( )) ( ( ! ) ( 1 0 0 0 0 + − ≤ + = ∑ k t x V g x V b i t t W k k i i i λ (A5) Inequality (A5) and the fact that ))] ( ( , 0 [ 0 0 x V r t ∈ imply that: ( ) ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + − ≤ + = ∈ ∈ ∑ )! 1 ( )) ( ( )) ( ( ! max )) ; , ( ( max 1 0 0 0 ))] ( ( , 0 [ 0 0 ] , 0 [ 0 0 k t x V g x V b i t d x t x W k k i i i x V r t t t λ(k x V g x V b i x V t V k k i i i x V r τ λ τ ρ τ , for all ] , 0 [ 0 t t ∈ The above inequality in conjunction with inequality (3.8) implies that inequality (3.14) holds. It should be noticed that inequality (3.14) holds as well for the case 0 0 = t (since (3.8) implies that )) ( ( ) ( 0 0 x V x V γ ≤ ). The proof is complete. < Proof of Lemma 3.4: We will prove that system (1. and cannot be further continued. Standard results on the continuation of the solutions of ordinary differential equations imply that +∞ = − → ) ( lim max t V t t , where )) ; , ( ( : ) ( 0 d x t x V t V = . We next prove the following claims. } { i i t with )) ( ( ) ( 1 i i t V t V λ ≤ + , )) ( ( ) ( 0 ) ( t V t V i i λ ≤ ,d x t x V c d x t x W < . Proof of Claim 2: Suppose on the contrary that ))) ; , ( ( ( )) ; , ( ( 0 2 0 0 d x t x V c d x t x W ≥ for all ) , [ max 0 t t t ∈ . Using inequalities (3.2), (3.4) and the fact that )) ( ( )) ; , ( ( 0 0 t V d x t x V λ > for all ) , [ max 0 t t t ∈ , we obtain for ) , [ max 0 t t t ∈ a.e.: ) ( ) ( 1 t W t W i i + ≤ & , for 1 ,..., 0 − = k i (A8) ( ) )) ( ( ) ( 0 t V g t W k λ − ≤ & (A9) where )) ; , ( ( : ) ( 0 d x t x W t W i i = ( k i ,..., 0 = ). Inequalities (A8) and (A9) imply that the following inequality holds for all ) , Our assumption that ))) ; , ( ( ( )) ; , ( ( 0 2 0 0 d x t x V c d x t x W ≥ for all ) ,t V t W t V t W μ μ + ≤ + (A21) It follows from (A17) and (A21) that: )) ( ( )) ( ( )) ( ( )) ( ( 2 2 2 3 3 1 t V t V c t V t V c μ μ + ≤ + (A22) If ( ) )) ( ( ) ( t V t V c t V t V c μ γ λ μ γ λ + ≤ + − − The above inequality contradicts inequality (3.6) for )) ( ( 2 1 t V s − = γ . The proof is complete. < 3 of the paper the main results are stated and proved. Illustrative examples of the proposed approach are provided in Section 4: the examples show how we can use very simple positive definite functions (e.which do not have a sign-definite derivative. Finally, some concluding remarks are provided in Section 5. The Appendix contains the proofs of some technical steps needed in the proof of Theorem 3.1. Theorem 3. 1 : 1Consider system (1.1) under hypotheses (H1), (H2), (H4) and suppose that there exist a positive definite and radially unbounded function Lemma 3. 4 : 4Consider system (1.1) under hypotheses (H1), (H2), (H4) and suppose that there exist a positive definite and radially unbounded function Proof of Theorem 3.1: By virtue of Proposition 2.5 and Lemma 3.4, it suffices to show that inequalities (2.1), ( Corollary 3. 5 : 5Consider system (1.1) under hypotheses (H1), (H2), (H4) and suppose that there exist a positive definite and radially unbounded function . Since the estimation of the function ∞ ∈ K γ is crucial for the verification of inequalities (3.6), (3.8) and (3.9) of Theorem 3.1 and Corollary 3.5, less conservative estimates of the solution of system (1. inequalities (3.35), (3.36), (3.37), (3.28), (3.30) and (3.32) hold. Moreover, suppose that there exist constants 0 , > γ g such that inequality (3.31) holds as well as the following inequalities: section is devoted to the presentation of two illustrative examples. Both examples can be handled easily by classical Lyapunov analysis (i.e., it is easy to find a continuously differentiable, positive definite and radially unbounded function with negative definite derivative). However, here the issue is to show how we can prove robust global asymptotic (or exponential) stability by using a positive definite function with non sign definite derivative. In both examples, the simplest continuously differentiable, positive definite function 2 ) ( x x V = is used; this function fails to satisfy the requirements of Lyapunov's direct method. are left with the verification of inequalities (3.31) and (3.46). By virtue of (4.13) and previous definitions the following inequalities hold for every . In this case inequality (A22) and the fact that the function ) we denote the class of functions (takingvalues in m ℜ ⊆ Ω ) that have continuous derivatives of order j on A . * For every scalar continuously differentiable function ℜ → ℜ n V : Proof of Proposition 2.5:The key idea of the proof is to show that forward completeness + hypothesis (H4) +be locally bounded while Proposition 2.5 demands the function ) , 0 ( : +∞ → ℜ n T to be locally bounded. Since the value ) 0 ( T plays no role, it is clear that the extra assumption required for Proposition 2.5 can be replaced by the condition +∞ < → ) ( sup lim 0 x T x . +∞ < → ) ( sup lim 0 x T x imply the existence of a function ∞ ∈ K a such that inequality (2.1) holds as well. Then Proposition 2.3 guarantees that n ℜ ∈ 0 is URGAS for (1.1). Indeed, since (1.1) is forward complete and since hypothesis (H4) holds, Proposition 5.1 in [5] guarantees that system (1.1) is Robustly Forward Complete (RFC, see [5]). Lemma 2.3 in [5] guarantees the existence of functions + ∈ K μ and ∞ ∈ K ζ such that the following inequality holds for all n x ℜ ∈ 0 , D M d ∈ and 0 ≥ t : Moreover, all requirements of Theorem 3.1 are fulfilled and consequently1 φ & (3.25) Since ) , 0 ( : +∞ → ℜ n φ is locally Lipschitz, it follows that system (3.25) satisfies hypotheses (H1), (H2) and (H4). n ℜ ∈ 0 is URGAS for (3.25). Classical Lyapunov theory implies that n ℜ ∈ 0 is URGAS for (1.1). The proof is complete. < Remark 3.6: Here it should be noticed that Lyapunov's direct method is a special case of Corollary 3.5. Indeed, if there exists a positive definite continuous function It should be noticed that the result is very conservative. Indeed, by following classical Lyapunov analysis the readern n n x x x r k k i i i x r Consequently, inequality (3.46) will hold with 2 2 ) 1 ( 12 3 : + + = λ λ γ p . On the other hand, previous definitions imply that inequality (3.31) is equivalent to the following inequality: ( ) ( ) 2 1 2 1 1 2 1 2 1 1 1 2 2 3 ) 3 ( 2 3 ) 3 ( 2 3 ) 1 ( 12 p c p c c c c p c c c > − − + + − − + + − + λ λ for arbitrary constants 3 2 2 4 3 1 2 2 < < < − + c c p and ) 1 , 0 ( ∈ λ . Consequently, the maximum allowable value of 0 ≥ p must satisfy 2 6 9 − < p and the following inequality ( ) ( ) 2 1 2 1 1 2 1 2 1 ) 3 ( 2 3 3 ) ( 3 p c p c c c c c c > − + + − − − (4.18) for certain constants 3 2 2 4 3 1 2 2 < < < − + c c p . Numerical calculations show that the maximum value is greater than 236643 . 0 5 7 5 1 ≈ ; the reader can verify that inequality (4.18) holds with 5 7 5 1 = p , 6094 . 2 2 = c and 8594 . 2 1 = c . can verify that much higher values for 0 ≥ p than 236643 . 0 5 7 5 1 ≈ can be allowed. For example, the quadratic Lyapunov function ( ) 2 1 2 2 1 2 1 4 1 ) ( x x x x V σ + + = with 2 2 1+ = σ has negative definite derivative for 1 < p . The results can be used for the proof of global asymptotic stability by using continuously differentiable, positive definite functions which do not have a negative semi-definite derivative. Illustrating examples are provided, which show how we can use very simple positive definite functions (e.g. Future work can address the issue of the extension of the obtained results to the local case or the time-varying case. V r t ∈2 ) ( x x V = ), which do not have a sign-definite derivative. 1) is forward complete by contradiction. Suppose that there exists} 0 { \ 0 n x ℜ ∈ , D M d ∈ the solution ) ; , ( 0 d x t x of (1.1) is defined on ) , 0 [ max t , where ) , 0 ( max +∞ ∈ t . Consequently, we can construct an increasing sequenceCLAIM 1: There exists ) , 0 [ max 0 t t ∈ such that )) ( ( ) ( 0 t V t V λ > , for all ) , [ max 0 t t t ∈ . Proof of Claim 1: If Claim 1 were not true then for every ) , 0 [ max t t i ∈ , there would exist ) , ( max 1 t t t i i ∈ + with )) ( ( ) ( 1 i i t V t V λ ≤ + ∞ =0 , a contradiction with the fact that +∞ = ≥ , which contradicts the fact that +∞ =where ) ( : ) ( ) ( s s times i i ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = 43 42 1 o K o o λ λ λ λ for 1 ≥ i and s s = : ) ( ) 0 ( λ . A standard contradiction argument shows that max sup : t t T t i i ≤ = → and 0 ) ( → i t V . If max t T = then we obtain 0 ) ( inf lim max = − → t V t t − → ) ( lim max t V t t . If max t T < then we must have 0 ) ( inf lim = − → t V T t . Since the mapping ) (t V t → is continuous, we must have ) ( ) ( lim ) ( inf lim T V t V t V T t T t = = → → − and this implies 0 ) ( = T V . Consequently, we must have 0 ) ; , ( 0 = d x T x . Uniqueness of solutions for system ( Σ ) implies that 0 ) ; , ( 0 = d x t x , for all T t − → ) ( lim max t V t t . CLAIM 2: There exists ) , [ max 0 t t t ∈ with ))) ; , ( ( ( )) ; , ( ( 0 2 0 0 Acknowledgments:The author would like to thank Professor A. R. Teel and Professor J. Tsinias for their useful comments and suggestions.AppendixProof of FACT I in the proof of Theorem 3.1: Suppose on the contrary that ))) ; , ( ( ( )) ; , ( (. Using inequalities(3.2),(3.4)and the fact that )) ( ( )) ; ,, we obtain for ))] (. Then definition (A11) implies that ))) ; ,. Using inequalities (3.2),(3.4)and the fact that )) ( ( )) ; ,a.e.. Inequalities (A8) and (A9) imply that inequality (A10) holds for all ] , [), we get from the above inequality:contradicts inequality (3.7).CLAIM 4: The following inequality holds:Proof of Claim 4: Suppose first that. Using inequalities (3.2), (3.4) and the fact that )) ( ( )) ; ,, we obtain that inequalities (A8), (A9By virtue of (3.1) and (A14) we obtain for ] , [ 1 0 t t t ∈ a.e.:( )Differential inequality (A15) directly implies that:The above inequality in conjunction with inequality (3.8) implies that inequality (A12) holds. It should be noticed that inequality (A12) holds as well for the case).Proof of Claim 5: Suppose the contrary, that ))) ; , ( ( ( )) ; , ( (. Then inequality (3.1) and the fact thata.e.. Thus we obtain, a contradiction with the fact thatWe are now ready to finish the proof. 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[ "Optimized auxiliary oscillators for the simulation of general open quantum systems", "Optimized auxiliary oscillators for the simulation of general open quantum systems" ]	[ "F Mascherpa \nInstitut für Theoretische Physik\nUniversität Ulm\nAlbert-Einstein-Allee 11D-89069UlmGermany\n", "A Smirne \nInstitut für Theoretische Physik\nUniversität Ulm\nAlbert-Einstein-Allee 11D-89069UlmGermany\n", "D Tamascelli \nInstitut für Theoretische Physik\nUniversität Ulm\nAlbert-Einstein-Allee 11D-89069UlmGermany\n\nDipartimento di Fisica\nUniversità degli Studi di Milano\nVia Celoria 16I-20133MilanoItaly\n", "P Fernandez Acebal \nInstitut für Theoretische Physik\nUniversität Ulm\nAlbert-Einstein-Allee 11D-89069UlmGermany\n", "S Donadi \nInstitut für Theoretische Physik\nUniversität Ulm\nAlbert-Einstein-Allee 11D-89069UlmGermany\n", "S F Huelga \nInstitut für Theoretische Physik\nUniversität Ulm\nAlbert-Einstein-Allee 11D-89069UlmGermany\n", "M B Plenio \nInstitut für Theoretische Physik\nUniversität Ulm\nAlbert-Einstein-Allee 11D-89069UlmGermany\n" ]	[ "Institut für Theoretische Physik\nUniversität Ulm\nAlbert-Einstein-Allee 11D-89069UlmGermany", "Institut für Theoretische Physik\nUniversität Ulm\nAlbert-Einstein-Allee 11D-89069UlmGermany", "Institut für Theoretische Physik\nUniversität Ulm\nAlbert-Einstein-Allee 11D-89069UlmGermany", "Dipartimento di Fisica\nUniversità degli Studi di Milano\nVia Celoria 16I-20133MilanoItaly", "Institut für Theoretische Physik\nUniversität Ulm\nAlbert-Einstein-Allee 11D-89069UlmGermany", "Institut für Theoretische Physik\nUniversität Ulm\nAlbert-Einstein-Allee 11D-89069UlmGermany", "Institut für Theoretische Physik\nUniversität Ulm\nAlbert-Einstein-Allee 11D-89069UlmGermany", "Institut für Theoretische Physik\nUniversität Ulm\nAlbert-Einstein-Allee 11D-89069UlmGermany" ]	[]	A method for the systematic construction of few-body damped harmonic oscillator networks accurately reproducing the effect of general bosonic environments in open quantum systems is presented. Under the sole assumptions of a Gaussian environment and regardless of the system coupled to it, an algorithm to determine the parameters of an equivalent set of interacting damped oscillators obeying a Markovian quantum master equation is introduced. By choosing a suitable coupling to the system and minimizing an appropriate distance between the two-time correlation function of this effective bath and that of the target environment, the error induced in the reduced dynamics of the system is brought under rigorous control. The interactions among the effective modes provide remarkable flexibility in replicating non-Markovian effects on the system even with a small number of oscillators, and the resulting Lindblad equation may therefore be integrated at a very reasonable computational cost using standard methods for Markovian problems, even in strongly non-perturbative coupling regimes and at arbitrary temperatures including zero. We apply the method to an exactly solvable problem in order to demonstrate its accuracy, and present a study based on current research in the context of coherent transport in biological aggregates as a more realistic example of its use; performance and versatility are highlighted, and theoretical and numerical advantages over existing methods, as well as possible future improvements, are discussed.	10.1103/physreva.101.052108	[ "https://arxiv.org/pdf/1904.04822v1.pdf" ]	104,291,901	1904.04822	03618ee56f9f42c2162ca716fa1dccc329a697dc	Optimized auxiliary oscillators for the simulation of general open quantum systems F Mascherpa Institut für Theoretische Physik Universität Ulm Albert-Einstein-Allee 11D-89069UlmGermany A Smirne Institut für Theoretische Physik Universität Ulm Albert-Einstein-Allee 11D-89069UlmGermany D Tamascelli Institut für Theoretische Physik Universität Ulm Albert-Einstein-Allee 11D-89069UlmGermany Dipartimento di Fisica Università degli Studi di Milano Via Celoria 16I-20133MilanoItaly P Fernandez Acebal Institut für Theoretische Physik Universität Ulm Albert-Einstein-Allee 11D-89069UlmGermany S Donadi Institut für Theoretische Physik Universität Ulm Albert-Einstein-Allee 11D-89069UlmGermany S F Huelga Institut für Theoretische Physik Universität Ulm Albert-Einstein-Allee 11D-89069UlmGermany M B Plenio Institut für Theoretische Physik Universität Ulm Albert-Einstein-Allee 11D-89069UlmGermany Optimized auxiliary oscillators for the simulation of general open quantum systems (Dated: April 10, 2019) A method for the systematic construction of few-body damped harmonic oscillator networks accurately reproducing the effect of general bosonic environments in open quantum systems is presented. Under the sole assumptions of a Gaussian environment and regardless of the system coupled to it, an algorithm to determine the parameters of an equivalent set of interacting damped oscillators obeying a Markovian quantum master equation is introduced. By choosing a suitable coupling to the system and minimizing an appropriate distance between the two-time correlation function of this effective bath and that of the target environment, the error induced in the reduced dynamics of the system is brought under rigorous control. The interactions among the effective modes provide remarkable flexibility in replicating non-Markovian effects on the system even with a small number of oscillators, and the resulting Lindblad equation may therefore be integrated at a very reasonable computational cost using standard methods for Markovian problems, even in strongly non-perturbative coupling regimes and at arbitrary temperatures including zero. We apply the method to an exactly solvable problem in order to demonstrate its accuracy, and present a study based on current research in the context of coherent transport in biological aggregates as a more realistic example of its use; performance and versatility are highlighted, and theoretical and numerical advantages over existing methods, as well as possible future improvements, are discussed. A method for the systematic construction of few-body damped harmonic oscillator networks accurately reproducing the effect of general bosonic environments in open quantum systems is presented. Under the sole assumptions of a Gaussian environment and regardless of the system coupled to it, an algorithm to determine the parameters of an equivalent set of interacting damped oscillators obeying a Markovian quantum master equation is introduced. By choosing a suitable coupling to the system and minimizing an appropriate distance between the two-time correlation function of this effective bath and that of the target environment, the error induced in the reduced dynamics of the system is brought under rigorous control. The interactions among the effective modes provide remarkable flexibility in replicating non-Markovian effects on the system even with a small number of oscillators, and the resulting Lindblad equation may therefore be integrated at a very reasonable computational cost using standard methods for Markovian problems, even in strongly non-perturbative coupling regimes and at arbitrary temperatures including zero. We apply the method to an exactly solvable problem in order to demonstrate its accuracy, and present a study based on current research in the context of coherent transport in biological aggregates as a more realistic example of its use; performance and versatility are highlighted, and theoretical and numerical advantages over existing methods, as well as possible future improvements, are discussed. I. INTRODUCTION Any physical system in nature may be studied theoretically in complete isolation from its surroundings. However, since interactions with uncontrolled environmental degrees of freedom are unavoidable in practice, this condition is never actually realized. The effects of said degrees of freedom on the dynamics and general properties of a system are especially important in quantum mechanics, where the time and energy scales involved are likely to make interactions between the system and the surrounding environment a key actor in their own right in the physics at play. The goal of the theory of open quantum systems is to determine the behavior and investigate the physical properties of systems both in and out of equilibrium by properly accounting for environmental effects and other external influences (e.g. driving forces) using appropriate analytical or numerical methods [1][2][3]. The starting point of such methods may be either a microscopic model for the system and the environment, such as the spin-boson [1,4], Caldeira-Leggett [5] or more complex models, or an effective description of the system alone with the effects of the environment implicitly taken into account via a quantum master equation [6][7][8][9][10][11][12]. The former setup leads to a wide variety of potentially more complete and general treatments, but this greater range of attainable results and predictions comes at moderate to high computational costs [13][14][15][16][17][18][19]; the latter construction * [email protected] † [email protected] ‡ [email protected] is typically less expensive but applies to a constrained class of physical settings, since it either delivers accurate results only in a few well-defined limiting cases [7,20,21] or relies on equations which are difficult to derive for general systems [9-12, 22, 23]. Provided that the necessary assumptions on the system-environment interaction are satisfied, however, efficient methods for the solution of the master equation are widely available [24][25][26]. Much theoretical research in recent decades has focused on the study of complex non-Markovian environments [27][28][29][30], for which analytical results are hard to obtain except for specific models, and numerical simulation may become very challenging depending on the physical regime of interest. For thermal bosonic environments, the most commonly studied category by far, numerical methods developed for a general treatment of non-Markovian problems include e.g. Hierarchical Equations of Motion (HEOM) [14,31], Quasi-Adiabatic Path Integrals (QUAPI) [15,[32][33][34], Nonequilibrium Green's Function (NEGF) techniques [13,35], Non-Markovian Quantum State Diffusion (NMQSD) [18,19,[36][37][38], or simulated evolution of the state using the time-adaptive Density Matrix Renormalization Group (t-DMRG) [39][40][41] in combination with convenient exact mappings of the environment e.g. into one-dimensional oscillator chains well suited for such numerical methods, as in the Time-Evolving Density with Orthogonal Polynomials Algorithm (TEDOPA) [16,17,42], to name a few. These methods are often referred to as numerically exact, in the sense that they are designed to address problems from the bottom up, requiring only numerical approximations (e.g. Hilbert space truncation, discretized integrals or finite expansions of relevant functions) in order to keep the costs manageable, but otherwise posing no physical restrictions on the models themselves; these numerical errors can sometimes be bounded rigorously, e.g. for TEDOPA [43,44] or HEOM [45]. Finite bosonic environments [46] can also be used as an approximate treatment for simulation times short enough to prevent recurrence in the dynamics. An alternative route for the numerical study of such nontrivial open-system problems is to model environmental effects on a system by splitting them into coherent, information-preserving contributions and purely dissipative Markovian damping. Then one can devise effective models in which the system of interest is coupled explicitly to a finite auxiliary system acting as the non-Markovian core of the environment, and dissipation is accounted for through Markovian damping of these auxiliary degrees of freedom. This is the idea underlying approaches such as the pseudomode method [47][48][49][50], the reactioncoordinate method [51,52] or other techniques based on the same concept but differing in the ansatz used to create the effective environment and the techniques to solve for the dynamics [53][54][55][56]. Such remappings of open-system problems can be very convenient numerically, but are not always grounded in a mathematically rigorous and physically sound relation between the original and effective environments, making assessment of their accuracy challenging. In this paper, we present a new approach to general open quantum systems interacting with Gaussian bosonic environments. Our method combines the simplicity and efficiency of simulating a small set of effective degrees of freedom with analytical equivalence relations between the structure and parameters of this auxiliary system and the exact properties of the microscopic environment. Even in cases in which no exact equivalence holds, the physical error from replacing a unitary environment by a dissipative one is kept to a bare minimum and under rigorous control. Our scheme is based on a quantitatively certified recipe to construct networks of interacting, damped harmonic oscillators specifically designed to mimic any given target environment as specified by its spectral density and temperature. The reduced dynamics is then computed by solving a time-homogeneous quantum master equation of the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) type [6][7][8] for the system coupled to these effective harmonic modes and tracing them out at the end. The theoretical foundation underlying this construction lies in a recently proved equivalence theorem between unitary and non-unitary Gaussian environments in open quantum systems [57], which states that the reduced dynamics of a system coupled to an environment of either type is identical if the single-time averages and two-time correlation functions of the environment operators relevant to the interaction are the same. Exploiting this notion, we introduce a systematic procedure by which the effective environment is tailored to reproduce the correlation function of the target environment with an accuracy quantitatively controlled through known error bounds for Gaussian environments [45]. The advantages of the method proposed are the simple yet versatile structure of the effective environments, which can emulate a broad range of nontrivial unitary environments using small numbers of auxiliary modes, the small, controlled error in the resulting effective dynamics, a high flexibility in the physical regimes which can be studied at comparably low costs, such as high and low temperature and strong as well as weak coupling, and numerical simplicity, since the simulations only require solving a Lindblad equation. We have organized the presentation of our results as follows: in Section II we will outline the theoretical background and state the equivalence theorem from [57] lying at the core of our method; Section III details the procedure by which an effective environment corresponding to a nontrivial microscopic one may be constructed, and includes an analysis of the theoretical implications and approximations involved; a demonstration of our scheme on the spin-boson model as an exactly solvable test system, with accuracy and performance reports as well as a profile of the numerical advantages and disadvantages of the method in different physical regimes, is given in Section IV; Section V contains an application of the method to a system in a structured environment relevant to current research investigating coherent effects in biomolecular aggregates; in Section VI we discuss the current state of the method, focusing on its scope and applicability, numerical and conceptual strengths and limitations as well as some possible improvements; finally, Section VII summarizes our conclusions and future prospects. II. THEORETICAL FOUNDATIONS AND SCOPE OF THE METHOD The non-perturbative method we are going to introduce relies on the equivalence theorem between unitary and dissipative environments stated and proved in [57]; in order to set the stage for discussing our work, we will now introduce the relevant notation, outline the physical context in which the theorem applies and state it explicitly for reference within the paper. A. Gaussian unitary environments A wide array of open quantum system (OQS) problems, ranging e.g. from quantum Brownian motion [1,5] to dissipative cavity and circuit electrodynamics [58,59] or the study of charge and energy transfer in noisy natural or artificial aggregates [60,61], can be modeled microscopically by coupling the system of interest to an infinite collection of harmonic oscillators: H := H S ⊗ I E + I S ⊗ H E + H I(1) with the free Hamiltonian of the environment H E := ∞ 0 dω ωa † ω a ωH I := k A Sk ⊗ G Ek . The global state ρ of the system and the environment at time t is determined by the Liouville-Von Neumann equation d dt ρ(t) = − i [H, ρ(t)](2) and the initial state ρ 0 := ρ(0); the reduced state ρ S of the system at time t is obtained by taking the partial trace over the environmental degrees of freedom: ρ S (t) = Tr E [ρ(t)].(3) We are interested in the reduced dynamics of systems interacting with Gaussian environments, i.e. with H I linear in a ω and a † ω and factorizing initial conditions ρ 0 = ρ 0S ⊗ ρ 0E with ρ 0E a Gaussian state. Then the oscillators can be traced out exactly, and the reduced dynamics of the system only depends on the single-and two-time environmental averages G Ek (t) E and C E kk (t + τ, τ ) := G Ek (t + τ )G Ek (τ ) E as given by the evolution of the oscillators with no coupling to the system: G Ek (t) E = Tr E [U † E (t)G Ek U E (t)ρ 0E ](4)C E kk (t + τ, τ ) = Tr E [U † E (t + τ )G Ek U E (t)G Ek U E (τ )ρ 0E ](5) with U E (t) := e −iH E t/ . B. Gaussian dissipative environments Considering infinite environments evolving unitarily in the absence of a coupled system is one way to bring about dissipation and decoherence in the evolution of the latter when the coupling is nonzero. Alternatively, one may consider finite environments which evolve non-unitarily according to a quantum master equation (QME). In this case, one may start from a Hamiltonian H := H S ⊗ I R + I S ⊗ H R + H I(6) and a QME describing the evolution of the environment when decoupled from the system: d dt ρ R (t) = L R [ρ R (t)](7) where the new quantum Liouville superoperator L R [ρ R ] := − i [H R , ρ R ] + D R [ρ R ] includes a dissipator D R [ρ R ] := m i,j=1 Λ ij L Ri ρ R L † Rj − 1 2 L † Rj L Ri , ρ R with a positive semidefinite rate matrix Λ ij , which makes the dynamics non-unitary but ensures a completely positive and trace-preserving evolution at all positive times. The rate matrix Λ ij , which we take to be constant in time, can always be brought into diagonal form by changing the basis of operators L Ri [1]; this is the quantum dynamical semigroup master equation for Markovian open systems first derived by Gorini, Kossakowski, Sudarshan and Lindblad [6][7][8]. We will refer to this master equation simply as the Lindblad equation throughout this paper. The full state of the system and a non-unitary environment evolves according to the QME d dt ρ(t) = L[ρ(t)](8) where L[ρ] := − i [H , ρ] + D[ρ] is the complete quantum Liouvillian for the system and the environment and D := I ⊗ D R embeds the dissipator D R into the full Hilbert space of the problem. For harmonic environments coupled linearly to the system, i.e. for H R := n ω n b † n b n with [b m , b † n ] = δ mn and H I := l A Sl ⊗ F Rl with F Rl linear in the creation and annihilation operators, if one also takes the Lindblad operators L Ri linear in b n and b † n and initial conditions ρ 0 = ρ 0S ⊗ ρ 0R with a Gaussian ρ 0R then the reduced dynamics of the system will only depend on the environment through F Rl (t) R and C R ll (t + τ, τ ) := F Rl (t + τ )F Rl (τ ) R , again considering the free dynamics of the environment with no system attached, like in the unitary case: F Rl (t) R = Tr R [F Rl e L R t [ρ 0R ]](9)C R ll (t + τ, τ ) = Tr R [F Rl e L R t [F Rl e L R τ [ρ 0R ]]].(10) Note that the two-time correlation function (10) has the form one would obtain by applying the quantum regression hypothesis [62], which must be handled with some care in general but is true by construction for the Lindbladdamped environments relevant to our work. No approximation is required or implied at this stage [57]. C. Equivalence between unitary and non-unitary environments While it is clear that if two unitary Gaussian environments share the same averages G Ek (t) E and correlation functions C E kk (t + τ, τ ) at all times they will give rise to the same reduced dynamics if coupled to a system, this is not obvious if one or both environments are not unitary. In [57] it was shown, using the unitary dilation formalism for Lindblad equations [2], that this still holds for non-unitary environments under the same conditions. We restate this result here for reference. Define the reduced dynamics ρ U S (t) := Tr E [ρ(t)](11) for some system S coupled to a unitary environment and evolving according to Eq. (2) from factorizing initial conditions with the environment starting in a Gaussian state, and the reduced dynamics ρ L S (t) := Tr R [ρ(t)](12) for the same system coupled to a non-unitary environment and evolving according to Eq. (8) from factorizing initial conditions with the environment starting in a Gaussian state. Both environments are taken to be harmonic and coupled to the system through the same set of A Sk operators in H I and H I , with the corresponding G Ek and F Rk as well as the Lindblad operators L Ri of the non-unitary environment linear in the relevant creation and annihilation operators. Theorem 1 [57] Under the above assumptions, if F Rk (t) R = G Ek (t) E ∀k, t and C R kk (t + τ, τ ) = C E kk (t + τ, τ ) ∀k, k , t, τ, then ρ L S (t) = ρ U S (t) ∀t. This theorem is the cornerstone of our method; for the sake of clarity and an easier understanding of the rest of this paper, some remarks are in order. First of all, it is important to stress that Gaussianity is a key ingredient of Theorem 1, because in principle all correlation functions up to infinite order would have to be equal for two environments to have the same effect on a system, but for Gaussian environments the single-and two-time functions generate all the others. This restricts the initial state of the environment to the Gaussian family; in this work, we will only consider system-environment product states which are Gaussian in the environmental variables as initial states, leaving the free Hamiltonian, interaction operators and initial density matrix of the system arbitrary. Furthermore, since no equivalence theorem analogous to Theorem 1 for fermionic or other types of environments is known at the time of writing, we restrict our study to systems coupled to bosonic baths. Finally, for physical reasons discussed in Section VI and thoroughly analyzed in [63], in general a finite network of damped harmonic oscillators does not yield a two-time correlation function exactly equal to that of an infinite bath, so we will apply the theorem in approximate form by looking for effective parameters such that C R kk (t + τ, τ ) ≈ C E kk (t+τ, τ ) and hence ρ L S (t) ≈ ρ U S (t) (single-time expectation values of coupling operators are typically zero or can be set to zero and will no longer be dealt with in this work), relying on the fact that the error in the former approximate relation rigorously bounds that in the latter, as established in previous work [45]. Other than these caveats, no further problems arise in terms of applicability; in particular, temperature and coupling strength between system and environment pose no theoretical or computational limits in principle. In the next sections we will show how one may exploit the theorem to systematically construct simple networks of damped harmonic oscillators, which can stand in for complex, highly non-Markovian thermal baths at any temperature, by comparing the associated correlation functions (10) and (5). This procedure is independent of the system and the effective environments obtained through it can then be coupled arbitrarily strongly to any system of interest, and standard simulation methods for Lindblad equations may be used to obtain the reduced dynamics at potentially very low computational costs. III. SYSTEMATIC CONSTRUCTION OF EFFECTIVE ENVIRONMENTS From now on, we will consider unitary environments with Gaussian stationary states, such as thermal baths, and assume them to be initialized in such states, so that C E kk (t + τ, τ ) = C E kk (t, 0), which will be denoted by C E kk (t) in the following. Any harmonic oscillator network obeying a Lindblad equation of the form (7), with L R quadratic in b n and b † n , must also start from a stationary ρ 0R in order to give a time-homogeneous correlation function matrix C R kk (t) := C R kk (t + τ, τ ) = C R kk (t, 0) for operators of the form F Rk = n (c nk b n + c * nk b † n ).(13) The condition for ρ 0R to be a stationary state of Eq. (7) is L R [ρ 0R ] = 0.(14) For the initial state of our effective environments, we will therefore need a Gaussian ρ 0R satisfying this property. A. Ansatz and correlation function structure The correlation functions C R kk (t) of the auxiliary environment depend on all parameters appearing in L R , ρ 0R and the operators F Rk : unrestricted geometries and initial states allow for more generality at the expense of keeping potentially redundant parameters in the model and restricting the range of properties that can be easily calculated; to strike a balance between simplicity and versatility, we will now take an ansatz for the configuration and initial density matrix of the surrogate oscillator network such that the quantities of interest have a simple expression with little loss of generality; for a more extensive discussion of the technical details, we refer the reader to Appendix B. We choose a free Hamiltonian H R corresponding to a chain of N oscillators with a hopping interaction between nearest neighbors: H R := N n=1 Ω n b † n b n + N −1 n=1 g n b n b † n+1 + b † n b n+1 ,(15) where the couplings g n , as well as one of the coefficients c nk in the interaction operators F Rk appearing in C R kk (t), can be assumed real without loss of generality if the F Rk are nonlocal, i.e. acting on all effective modes (see Appendix B). We complete the QME by adding local thermal dissipators at zero temperature acting on each oscillator: d dt ρ R (t) = − i [H R , ρ R (t)] + N n=1 Γ n b n ρ R (t)b † n − 1 2 b † n b n , ρ R (t)(16) so that the stationary initial state satisfying Eq. (14) is just the overall vacuum state ρ 0R = N n=1 \|0 0\| n .(17) Note that a zero-temperature master equation for the effective environment does not restrict the temperature of the target environments it can simulate; the effect of a nonzero temperature in the target bath will simply be encoded in the parameters of the oscillator network, as is done in other approaches [42,[64][65][66][67]. The QME (16) and initial condition (17) lead to two decoupled sets of linear equations for b n (t) R and b † n (t) R related by Hermitian conjugation. For b n (t) R one has d dt b n (t) R = N m=1 M nm b m (t) R ,(18) where M nm := − Γ n 2 − iΩ n δ nm −i(g m δ n m+1 +g m−1 δ n m−1 ).(19) The two-time correlation function b n (t)b † m (0) R also evolves according to Eq. (18) as a direct consequence of the quantum regression hypothesis, which holds by construction in this context and states that correlation functions A(t + τ )B(t) obey the same equations of motion as the single-time expectation values A(t+τ ) [2,68]. This is equivalent to the statement that they can be written in the explicit form given in Eq. (10). Integrating Eq. (18) for b n (t)b † m (0) R and plugging the result (as well as its conjugate b † n (t)b m (0) R , which is identically zero for our initial state (17)) into the expression of C R kk (t) in terms of the operators F Rk as given in Eq. (13), one finds C R kk (t) = N n=1 (w n ) kk e λnt(20) where λ n are the eigenvalues of the matrix M defined in Eq. (19), which we assume to be non-degenerate for simplicity (see Appendix B for further discussions), and (w n ) kk = N l,m=1 c lk c * mk u n l v n m(21) are complex coefficients obtained from the definition of the operators F Rk and the left and right eigenvectors u n and v n corresponding to each λ n , normalized in such a way that N l=1 v m l u n l = δ mn as discussed in Appendix B. This exponential structure is a consequence of the Lindblad dynamics of the effective environment, which is a requirement of Theorem 1. B. Transformation to Surrogate Oscillators Consider an OQS problem described by a microscopic model of the form (1); for simplicity we will now assume a single interaction term, to which there corresponds a correlation function C E (t). Our goal is to find the matrix elements M mn and operator coefficients c n of some operator F R as given in Eq. (13) such that the resulting effective correlation function C R (t) is as close as possible to C E (t). The form of C R (t) in terms of M mn and c n is given by Eq. (20), where the eigenvalues λ n and weights w n can be thought of as functions of the free parameters Ω k , g k , Γ k and-only the w n -c k with n, k = 1, . . . , N , where N is the number of oscillators making up the effective bath. In order to determine the values of these free parameters such that C R (t) ≈ C E (t), we proceed in two steps. First, we perform a nonlinear fit on C E (t) using N damped exponentials with complex coefficients C E (t) −→C E (t) = N n=1w n eλ n t ,(22) for instance using Prony analysis [69]. Then we solve the problem of matching or getting as close as possible to theλ n andw n with the λ n (Ω k , g k , Γ k ) and w n (Ω k , g k , Γ k , c k ) from the effective environment. This is in general a highly nontrivial inversion problem involving an underdetermined, non-convex system of nonlinear equations of mixed degrees, and can be hard to solve: there is a trade-off between this complexity and the accuracy of the initial fit, with an optimum at small numbers (N 5 in all our applications) of interacting oscillators. Neither existence nor uniqueness of solutions are guaranteed for this inversion problem and physical requirements such as positivity of the rates Γ n need to be taken into account as well, so it is typically necessary to minimize some distance between C R (t) and C E (t) instead of exactly matching the best fitC E (t); this change in the correlation function is the only error introduced into the problem by the use of an effective environment. In some cases with N 3, it is possible to invert the equations exactly and obtain valid effective bath parameters; we have listed a few explicit solutions in Appendix C and will put some of them to use in our example applications. For general N , we devised a variational recipe to carry out our Transformation to Surrogate Oscillators (TSO) systematically. This is also described in detail in Appendix B and can be summarized as follows: • Sample random points in a suitably sized open set (0, g max ) N −1 , to be used as coupling constants. • Substitute each (N − 1)-tuple (g 1 , . . . , g N −1 ) into the equations relating the complex eigenvalues λ n to theλ n : this will give rates Γ n and frequencies Ω n such that the eigenvalues match; keep only the solutions with all Γ n > 0. • Compute the left and right eigenvectors of the matrices M corresponding to each solution found, plug them into Eq. (21) and minimize a distance (e.g. the Manhattan distance d Man (w,w) := N n=1 \|w n − w n \|) between w andw by varying the c n . • Assess overall accuracy of the solutions found and rank the corresponding C R (t) according to a meaningful figure of merit, such as the integral [45]. Effective environments obtained through this procedure can then be used to simulate the reduced dynamics of any model in which the interaction with the bath is mediated by the same bath operator F R : the TSO is carried out once and for all irrespective of the system coupled to the environment given, and the effective environment can be used in any problem involving the same correlation function C E (t). We also wish to remark that for composite systems with multiple local environments, the procedure applies to each independent correlation function individually and yields local effective environments to be coupled to the corresponding parts of the system in the same way as the original ones, with no further complications arising: this feature is sketched in Fig. 1. t 0 dt t 0 dt \|C R (t − t ) − C E (t − t )\| from By the same token, complex correlation functions requiring many exponentials for an accurate fit can be treated by breaking down the effective environment into smaller clusters of interacting modes, with each cluster accounting for a different component of C E (t)-or equivalently, the underlying spectral density J(ω) of the unitary environment-as shown in Fig. 2. Note that decoupling all oscillators from each other, i.e. taking all g n = 0 (which corresponds to requiring allw n to be real and positive in Eq. (22)), one recovers the noninteracting pseudomodes of [48] as a limiting case. C. Working example: Ohmic spectral density To better illustrate the technique explained in the previous subsection, let us now demonstrate how our transformation works with an explicit example. Consider an arbitrary quantum system coupled to an infinite environment through the position operator of each oscillator (for a more succinct notation, we will leave the tensor products implicit and use natural units = 1, k B = 1 from now on): H = H S + ∞ 0 dω ωa † ω a ω + A S ∞ 0 dω g(ω)(a ω + a † ω ). (23) This type of coupling for microscopic models is one of the most common in the OQS literature [1,[3][4][5]29]. The correlation function of the interaction operator G E = ∞ 0 dω g(ω)(a ω + a † ω ) for a thermal initial state at inverse temperature β = 1/T is C E β (t) = G E (t)G E (0) βE = ∞ 0 dω π J(ω) coth βω 2 cos(ωt) − i sin(ωt)(24) where the spectral density J(ω) is related to the frequencydependent coupling strength g(ω) through J(ω) = πg 2 (ω) and typically given as a starting point for studying the problem. Spectral densities are real and positive by definition, and are often categorized according to the power of ω best approximating their behavior near the origin, where they are always zero; a J(ω) ∝ ω s is called Ohmic if s = 1, and super-(sub-)Ohmic if s > 1 (s < 1). The spectral density and temperature uniquely determine C E β (t) and, consequently, the effect of the environment on the system. Note that for unitary environments the correlation function is Hermitian in time, i.e. its real part is even and its imaginary part is odd, as can be seen clearly from Eq. (24). This implies that its Fourier transform where θ(ω) is the Heaviside step function, is always real; at temperature T = 0, it is just 2J(ω)θ(ω). In fact, C E β (ω)/2 may itself be regarded as a spectral density defined over a new environment, which encompasses both positiveand negative-frequency modes and gives the correlation function C E β (t) if initialized in the vacuum [64]: this allows one to effectively rephrase arbitrary-temperature OQS problems as zero-temperature ones if it is convenient to do so, a possibility exploited by thermofield-based and other numerical methods [42,[65][66][67]. C E β (ω) = ∞ −∞ dt C E β (t)e iωt = 1 + coth βω 2 (J(ω)θ(ω) − J(−ω)θ(−ω)),(25) For non-unitary environments, in which time evolution is not an invertible map, correlation functions C R (t) are only defined at positive times; we extend the definition to negative times by imposing the same symmetry C R (−t) := C R * (t) in order to be able to compare exact and effective correlation functions in the frequency domain instead of inspecting their real and imaginary parts separately. Consider an Ohmic J(ω) with an exponential cutoff: J(ω) = πωe − ω Ωc .(26) Ohmic spectral densities define a very important class of environments entering the study of many systems, such as a particle undergoing quantum Brownian motion, or microscopic models leading to a Lindblad equation for a harmonic oscillator in a weakly coupled high-temperature environment [1,5,70]. The thermal correlation function C E β (t) corresponding to the spectral density (26) can be determined analytically as (27)). All parameters have dimensions of frequency and are given in units of Ωc; the last cn is real. C E β (t) = Ω 2 c (1 + iΩ c t) 2 + 1 β 2 ψ 1 + 1 + iΩ c t βΩ c + ψ 1 + 1 − iΩ c t βΩ c(27) where ψ (z) := 1 Γ(z) dΓ(z) dz is the polygamma function of order one. Performing our TSO on this correlation function at temperature T = Ω c according to the recipe described in the previous subsection, for N = 4 we determined the parameters given in Table I; Fig. 3 shows the Fouriertransformed effective correlation function C R β (ω) obtained using these parameters and the target C E β (ω) for comparison. As can be seen from the plot, four interacting oscillators were enough to obtain a very accurate C R β (t), with a peak in the error around ω = 0 reaching about 2% of the function value (see the inset of Fig. 3). This error affects the correlation function at very long times compared to its decay time, so we expect it to have a minor impact on the transient reduced dynamics of the system and become potentially more important at very long times. In all our tests, a small region around the origin was consistently found to be the part of the frequency domain where a general C E β (ω) is hardest to match: this is because any C R β (ω) is analytical around zero by construction, whereas C E β (ω) has discontinuous derivatives, as can be checked from Eq. (24). We stress again that the nonzero temperature is encoded in the effective parameters and not in the initial state, allowing us to treat very different temperature regimes at comparable costs, as will be made clearer in the next sections. IV. A TEST CASE: THE SPIN-BOSON MODEL We now turn to the second part of our approach: computing the reduced dynamics of a system by coupling it to the effective environment and solving the relevant Lindblad master equation (8). In order to demonstrate and quantitatively validate the method, we will show here the results we obtained for a system for which an analytical solution is known: the purely dephasing spin-boson model [1,4,71]. The Hamiltonian for this system is H = ω 0 2 σ z + ∞ 0 dω ωa † ω a ω + k 2 σ z ∞ 0 dω g(ω)(a ω + a † ω )(28) and we consider again the Ohmic spectral density defined in Eq. (26). In this model, the system and interaction Hamiltonians commute and are both diagonal in the sys-tem basis, so the populations p 0 := ρ 00 and p 1 := ρ 11 are conserved by the evolution. Any coherence in this basis present in the initial state, on the other hand, is erased according to the law (see [1] for a derivation) ρ 01 (t) = ρ * 10 (t) = e −iω0t+k 2 Γ(t) ρ 01 (0) (29) with Γ(t) := ∞ 0 dω π J(ω) coth βω 2 cos(ωt) − 1 ω 2 . Using the cutoff frequency of the environment Ω c as our energy scale, we set the parameter values ω 0 = 4Ω c and k = 1, corresponding to a strong-coupling regime. Comparable coupling strengths appear e.g. in the study of superconducting quantum transmission lines [72]. The system is initialized in the pure state ρ 0S = \|+ +\|, with \|+ := 1 √ 2 (\|0 + \|1 ) in terms of the eigenstates of σ z , and we simulated the reduced dynamics at three different temperatures T = 0, T = Ω c and T = 5 2 Ω c . Recall that the effective bath is always at zero temperature; different temperatures of the original environment require different surrogate baths. We found accurate effective correlation functions with N = 4 for the first two cases, and with N = 5 for the high-temperature regime; the parameters are given in Appendix A, and the errors of the two correlation functions at nonzero temperatures are similar to the zero-temperature case already discussed. A. Results, accuracy and performance We solved the effective Lindblad equations for all three cases using the QME integrator provided by the Python OQS package QuTiP [73,74], which implements a twelfthorder Adams-Moulton discrete integration algorithm. From the results shown in Fig. 4, we see that our simulations with effective correlation functions give quantitatively good results for the coherence ρ 01 (t) at all times and temperatures (the populations p 0 and p 1 are both equal to 1/2 throughout the evolution), as the overlap between the numerical (solid lines) and exact (dashed lines) solutions shows. The pure quantum decoherence at T = 0 induces an algebraic decay asymptotically proportional to t −k 2 , while at T > 0 the damping becomes exponential; a stronger effective coupling regime, which is determined both by k and the strength of thermal effects, induces faster relaxation in the system dynamics. The plots in the insets show the error figure E f (t) := \|f (t) − f Num (t)\| \|f (t)\| + \|f Num (t)\| ,(30) which is identical for f = [ρ 01 ] (solid line) and f = [ρ 01 ] (dashed line). This is a better estimator for the accuracy of the simulations than e.g. the absolute difference \|f (t) − f Num (t)\| because it removes the bias coming from changes in the relaxation time due to temperature, allowing us to compare all regimes on an equal footing. The error, as measured by E f (t), remains of the order of a few percent until the system has almost reached equilibrium and is comparable for the three regimes probed, mirroring the similar relative errors we had in all three effective correlation functions. The latter observation can be understood as follows: as higher temperatures or larger coupling constants increase the effects of the bath on the system, the error being carried from the correlation function into the reduced dynamics is magnified accordingly; on the other hand, these stronger effective regimes shorten the relaxation time of the system, so the cumulative effect of the error over time is not as severe as when the coupling is weaker or the temperature is lower. From these results, we conclude that the method is quite reliable and stable provided that the effective correlation functions used are reasonably accurate, and that this accuracy does not command significantly greater effort or complexity in the TSO at higher temperatures and is completely independent of the system and the coupling strength. Furthermore, it is worth noting that any method based on approximating the environment alters its correlation function and is therefore prone to the same kind of error as ours, but we use a rigorously motivated and physically meaningful quantifier to optimize our correlation functions and keep it under control. The computational cost of the simulations depends on the local dimension at which each effective mode is truncated and on the spread between the total evolution time and any faster timescales in the problem at hand, though the memory requirements scale faster with the complexity of the problem than the computation times do; to obtain our converged results for this system, which required ∼ 4 levels for most oscillators and a maximum of 7 for one mode in each simulation, all running times were below 10 minutes on a laptop. This cost grows rapidly with the number of effective modes, the local dimensions needed for convergence and the size of the system itself; on the other hand, temperature and coupling strength have a limited impact on these factors: a very strong coupling or high temperature will require higher local dimensions but also cause very rapid relaxation to equilibrium, making long simulation times unnecessary. Moreover, when the coupling is stronger and more levels are needed for convergence, this typically affects one particular mode much more than the others, leading to an effective polynomial rather than exponential scaling in the coupling strength and temperature. V. A PHYSICALLY RELEVANT APPLICATION Few-body systems non-perturbatively coupled to environments with structured spectra are ubiquitous in many fields ranging from biological physics and chemistry [61,75] to condensed matter [59,76] and thermodynamics [77,78], nanomaterial science and sensing [79] and quantum metrology [80][81][82], and have prompted much research in theoretical models and numerical methods for non-Markovian OQS. As a first example of an application of our simulation method to a system relevant for current research, we will now show some results for experimentally measurable optical properties in a model inspired by research in coherent charge and energy transfer in biological molecular aggregates [60]. A. The model We consider a simple dimer model with system parameters in the range of those found in biomolecular aggregates participating in excitation energy transfer [83,84], coupled to an environment with a realistic spectral density derived from models common in the literature [85]. We simulated the system and computed its absorption spectrum at temperatures ranging from T = 0 K to T = 300 K, comparing the results for two different versions of the environmental spectral density in order to identify the optical signatures setting them apart: in particular, we sought to determine the differences between absorption spectra obtained in the presence or absence of a strongly coupled vibrational mode in addition to a broad background spectral density. Following [84], we considered a free dimer Hamiltonian H S = E g \|g g\|+ 2 n=1 E n \|E n E n \|+(E 1 +E 2 ) \|E 12 E 12 \| + J(\|E 1 E 2 \| + \|E 2 E 1 \|),(31) where the two monomers have on-site energies E 1 = E g +12 328 cm −1 and E 2 = E g +12 472 cm −1 and interact through a hopping coefficient J = 70.7 cm −1 , and \|E 12 is the state with both monomers excited. We assume our system to be probed by a low-intensity laser source, as it is in most absorption experiments [64], and will therefore focus only on optical transitions between the ground state and the single-excitation manifold spanned by the states \|E n , ignoring the doubly excited state \|E 12 in the following since its contribution is typically negligible. Then, setting E g = 0 as our reference energy, we are left with an effective two-level Hamiltonian for the single-excitation manifold H 1ex = 2 n=1 E n \|E n E n \| + J(\|E 1 E 2 \| + \|E 2 E 1 \|),(32) whose eigenstates have an energy gap of ∆ = 201.8 cm −1 . The local excited states \|E n interact with separate environments, which account for the molecular vibrations (both within the system and in the protein scaffold around it) and the presence of a solvent. We model these degrees of freedom by coupling the monomers to independent thermal baths with the same spectral density and temperature; the physical model is sketched in the left panel of Fig. 5. We first studied the problem for a spectral density in the super-Ohmic form introduced by Adolphs and Renger [85] J AR (ω) := π 2 · 9! 2 a=1 ρ a ω 5 Ω 4 ARa e − √ ω/Ω ARa ,(33) where the two cutoff frequencies are (Ω AR1 , Ω AR2 ) = (0.557, 1.936) cm −1 and the weights of the two terms are (ρ 1 , ρ 2 ) = 288 5 ( 8 13 , 5 13 ). This spectral density accounts for environmental noise across a broad frequency range and will be referred to as J 0 (ω) in the following. Then we repeated the analysis for a spectral density featuring a strongly coupled vibrational mode in addition to the broad background: the mode was represented by adding an antisymmetrized Lorentzian peak J AL (Ω, Γ, S; ω) := S 8ΓΩ(4Ω 2 + Γ 2 )ω (4(ω − Ω) 2 + Γ 2 )(4(ω + Ω) 2 + Γ 2 ) ,(34) to the Adolphs-Renger background, and we chose a frequency Ω = 200 cm −1 , near resonance with the system, a width Γ = 10 cm −1 corresponding to a decay time (Γ/2) −1 ∼ 1 ps, and a Huang-Rhys factor S = 0.25 placing it in a strong-coupling regime with the system. We will refer to the spectral density with the strongly coupled mode as J 1 (ω). The reorganization energies corresponding to the two environments are λ 0 = ∞ 0 dω π J 0 (ω) ω = 2 a=1 ρ a Ω ARa = 19.93 cm −1 λ 1 = ∞ 0 dω π J 1 (ω) ω = λ 0 + SΩ = 69.93 cm −1 and a plot of both spectral densities is given in Fig. 6. The total Hamiltonian of our problem H tot = H 1ex + 2 n=1 ∞ 0 dω n ω n a † ωn a ωn + g(ω n ) \|E n E n \| (a ωn + a † ωn ) ,(35) can be rewritten in terms of the 'common-mode' and 'relative' creation and annihilation operators parametrized by a single frequency A ( †) ω = a ( †) ω 1 +a ( †) ω 2 √ 2 and a ( †) ω = a ( †) ω 1 −a ( †) ω 2 √ 2 . The common-mode terms then decouple from the system and the Hamiltonian reduces to H = H 1ex + ∞ 0 dω ωa † ω a ω + 1 √ 2 (\|E 1 E 1 \| − \|E 2 E 2 \|) ∞ 0 dω g(ω)(a ω + a † ω ) (36) in terms of the 'relative' modes only. Note that the ground state is decoupled from the single-excitation subspace, since no off-diagonal terms involving it appear in the Hamiltonian (36). The dynamics thus factorizes between the two subspaces unless coherences between them are present in the initial state. A sketch of the model after this rearrangement of the environmental modes and the TSO is given in the right panel of Fig. 5. Absorption experiments probe the linear response of the system; the light from the laser source can be described as interacting with the local dipole moment operators µ n := d n \|E n g\|, where d n is the classical dipole moment of the n-th site, in a perturbative manner [64,68]. Then the spectrum is obtained from the one-sided Fourier transform of the correlation function of the total dipole operator µ := 2 n=1 µ n over the initial stationary state ρ 0Abs := \|g g\| ρ β ,(37) where the bath is in a thermal state at inverse temperature β and the system is in the electronic ground state, which does not couple to the environment, since excited-state populations at equilibrium are negligible due to the very low intensity of the laser in such a setup [64,86]. Specifically, the correlation function of interest is given by the scalar product of the dipole operator µ, applied at times t 0 = 0 and t: in terms of the overall unitary evolution, one has C µ (t) := Tr U † (t)( µ † 1 + µ † 2 )U (t) · ( µ 1 + µ 2 )ρ 0Abs .(38) Note that this is formally a two-time object: we can compute it using an effective environment because the first operator acts on the system at equilibrium, so the hypotheses of Theorem 1 are not violated. The unitary dynamics acts onρ 0 := (\|E 1 + \|E 2 ) g\| ρ 0Abs , which is still a factorized object with the environment in a thermal state, so the equivalence with a suitable effective Lindblad dynamics remains well defined. We set the ansatz d 1 = d 2 = d for the geometry of the dimer in order to simplify the form of the dipole correlation function. Normalizing its value to \| d\| 2 , C µ (t) becomes C µ (t) = Tr U † (t) \|g ( E 1 \| + E 2 \|) U (t) (\|E 1 + \|E 2 ) g\| ρ 0Abs ] . (39) The absorption spectrum is then given by S Abs (ω) := ω lim tmax→∞ tmax 0 dt iC µ (t)e iωt .(40) B. TSO, results and simulation metrics In order to calculate C µ (t) and hence S Abs (ω) for our model, we determined effective parameters for the two spectral densities J 0 (ω) and J 1 (ω) at temperatures T = 0 K, T = 77 K (53.5 cm −1 ) and T = 300 K (208.5 cm −1 ), performing the TSO separately on the Adolphs-Renger background (33) and the antisymmetrized Lorentzian peak (34). This corresponds to assigning a separate effective environment to each additive part of the spectral density J 1 (ω) and can be a convenient strategy to break down structured spectra, as mentioned in an earlier section and shown in Fig. 2. The effective parameters for the two environments at each temperature are then given by a common set accounting for the background and the extra Here a strongly coupled dissipative vibrational mode has been added to the background spectral density. Note the additional spectral lines appearing in the spectrum at frequencies consistent with combined excitations of the system and the mode. parameters reproducing the peak; the Adolphs-Renger correlation function required N = 4 oscillators at all three temperatures, and the Lorentzian mode was replaced by one effective oscillator at T = 0 and two interacting ones at T > 0 using the exact methods for N = 1, 2 described in Appendix C. All parameters of the effective environments are given in Appendix A. The dipole correlation function can be calculated by solving the effective Lindblad equation with the initial pseudo-stateρ 0 = (\|E 1 + \|E 2 ) g\| ρ 0Abs . Since the memory required for a direct integration of the equation would become too large for the system coupled to six effective modes with the local dimensions needed for convergence, we carried out the simulations using the quantum jump or Monte Carlo Wave Function (MCWF) method for pure states [25,87,88] instead (the memory cost of MCWF scales linearly with the total Hilbert space dimension N for sparse Lindblad superoperators such as ours, while a master equation integrator requires at least O(N 2 )). For correlation functions, a convenient decomposition taking our initial conditionρ 0 into account is described in [88]: it can be shown that C µ (t) = 1 4 4 k=1 e i kπ 2 Tr [ 3 \|g ( E 1 \| + E 2 \|) \|φ k (t) φ k (t)\| ] ,(41) where for our simulations \|φ k (0) := \|g + e −i kπ 2 (\|E 1 + \|E 2 ) √ 3 \|0 R and the trace is taken over the system and the surrogate modes. The factor 3 appearing in the trace cancels the two normalizations 1/ √ 3 in the initial states \|φ k (0) φ k (0)\|. The effective Lindblad equation for the system and the modes was solved using another QuTiP code, since the package also provides MCWF routines. Our averages converged after as few as 250 Monte Carlo trajectories for each \|φ k (t) φ k (t)\|, i.e. 1000 trajectories in total, due to the jumps occurring in the modes but not in the system (thus partly canceling in the reduced dynamics) and the Fourier transform smearing out the remaining noise over all frequencies; we computed twice as many trajectories as a check but found no visible differences. The results for J 0 (ω) and J 1 (ω) are shown in Figs. 7 and 8, respectively. For each temperature, the system was simulated until the correlation function C µ (t) had decayed almost to zero, so that the limit in Eq. (40) could be considered legitimate: the final times we considered in our simulations are reported in Table II. The absorption spectra obtained via a discrete Fourier transform show the expected features: the result for J 0 (ω) displays absorption lines corresponding to the singleexcitation eigenstates \|ε 1,2 of H 1ex and appearing at the corresponding energy values redshifted by the bath reorganization energy; the line corresponding to the higher eigenstate is broadened due to the decay channels of that state, which couples to the environment and the lower excited state, whereas the latter gives a very narrow zero-temperature peak since it is not coupled to any lower-lying state it could decay to. At room temperature, the contribution from the upper level becomes larger than the lower one, but the energies associated with the-now markedly broadened-spectral lines no longer represent energy eigenstates of the system, since the system-environment energy eigenbasis is very different from a tensor product basis in this regime, as can be deduced from the fact that the environmental reorganization energy corresponding to the thermalized spectral density C E β (ω)/2 is comparable to J. Adding the strong peak to the spectral density, the expected sidebands associated with excitations of both the dimer and the coupled vibrational mode appear for both energy levels; a very small two-phonon sideband of the higher excited state is barely visible to the right of the main spectral curves at lower temperatures but disappeares under the strong broadening of the hightemperature spectrum. It should be noted that the timescale at which the reduced dynamics of the system reaches a steady state is of the order of a few picoseconds for both J 0 (ω) and J 1 (ω) (most of the dissipation is due to the broad Adolphs-Renger background, since the Lorentzian mode has a long lifetime); the decay times of the dipole correlation functions, on the other hand, were found to be very long, especially at low temperatures and if one wants C µ (t) to reach values small enough to avoid visible spurious effects from an incomplete decay in the Fourier transform (see again Table II). Such calculations thus require either numerical efficiency up to long simulation times or conditions, such as static disorder, which average out these long-time effects. Our computational costs scale linearly in the simulation time, making the method suitable for these tasks. As an extra analysis, we computed the reduced dynamics in the single-excitation subspace for an initial state ρ 0S = \|+ + \|, where \|+ := \|ε 1 + \|ε 2 √ 2 , for a spectral density J 1 (ω) combining the background with an attenuated antisymmetrized Lorentzian peak whose parameters are given in Table X in Appendix A. The environmental correlation function at T = 77 K and the effective one from the TSO are plotted in Fig. 9, and we computed the reduced dynamics at short times using both our method and the numerically exact TEDOPA [17,42,89,90]. The results are shown in Fig. 10 and confirm the accuracy of our effective Lindblad equation; no comparison was possible for times longer that about 1.3 ps due to the rapidly increasing cost of the TEDOPA simulation at later times. All simulations required under 3 GB of memory per thread and could therefore have been carried out on a desktop or laptop computer. To achieve higher parallelization of the work, however, we used the JUSTUS cluster at Ulm University: on a 16-cluster node, the simulations with J 1 (ω) took from 27 minutes (T = 0 K, t max = 26.54 ps) to 460 minutes (T = 77 K, t max = 13.27 ps) per 1000 trajectories; for comparison, analogous calculations (unrelated to this work) for a similar system using TEDOPA take about 60 minutes to reach 0.4 ps and about 90 to reach 0.5 ps on the same hardware, already showing strong nonlinear scaling. VI. DISCUSSION After introducing and demonstrating our new simulation method, let us now recapitulate its main theoretical and technical points, discussing its strengths, limitations and error sources in order to give a clear and concise summary of its current state and possible future improvements. A. Theoretical basis and general remarks Our method is part of a category of hybrid approaches based on rephrasing microscopic OQS models as effective Markovian problems, in which the memory of the environment is accounted for by an ad hoc auxiliary system. Although this divide-and-conquer strategy between Markovian and non-Markovian effects is a shared feature of several existing methods, the flexibility and quantitative control allowed for by the rigorous theoretical groundwork underlying our construction [45,57] are, as far as we know, unprecedented for an approach of this type. The transformation procedure we described in Section III exploits the generality of the class of mathematically admissible effective environments to tailor them in a systematic way to fit the microscopic ones given: isolating the correlation function as the single property of the environment which needs to be replicated as accurately as possible, we take advantage of the added versatility from using interacting effective modes to make this fitting procedure more accurate while keeping the number of effective degrees of freedom lower than would be possible in chain-or star-configuration schemes. It should be noted that we took one particular ansatz for the effective environment because we found it the most convenient for our needs, but many other choices (non-hopping linear couplings, interactions beyond nearest neighbors, different damping and initial stationary state, etc.) are possible. Another relevant feature is that the system on which the environment acts does not enter in this part of the procedure at all. Therefore, once the effective parameters corresponding to a given unitary bath are determined, they can be used in all problems which share the same environment: this can be convenient in any field in which standard spectral densities appear in many different situations. The second part of the method is the simulation of the system coupled to the effective environment. Here any of the analytical or numerical techniques for Lindblad master equations already developed in the literature can be used; the system part of the problem is completely unrestricted, so different strategies can be adopted depending on the problem at hand and available computational resources. We demonstrated the method on two-to three-level systems interacting with small sets of up to six effective modes, for which simple and clear solution methods like direct integration of the master equation or MCWF are still suitable. B. Numerical complexity and costs As we have shown in the examples given in the preceding sections, our method is very versatile and applies in principle to any non-Markovian, non-perturbative OQS problem involving a Gaussian bosonic bath, at any temperature and coupled arbitrarily strongly to a system. Clearly, some problem classes and physical regimes are more suited than others to this type of treatment: we will now summarize the key elements determining the computational effort for any given application. Concerning the simulation of systems, the most important variable to look at is the Hilbert space dimension, which includes both the system size and the number and local dimensions of the surrogate modes; temperature and coupling strength, on the other hand, pose less significant challenges. The number of effective oscillators and their parameters come from the TSO and depend on the spectral density and temperature of the unitary environment. The spectral density has a prominent role in determining the number of oscillators required; higher temperatures can contribute too but typically result in an effectively stronger coupling to the system instead: this does cause the oscillators to become more populated, making higher local dimensions necessary for the reduced dynamics to converge, but the added computational cost is usually less than that entailed by adding a new mode. For example, in our application described in the previous section, changing the temperature from 77 K to 300 K required higher local dimensions for three out of six modes, raising them from (6,5,6) to (10,8,8): the increase in the overall dimension is a factor of about 3.55, i.e. less than that from adding an oscillator truncated at local dimension 4. When simulating more complicated systems or environments, the Hilbert space dimensions involved are such that memory, rather than time, typically becomes the main computational concern: such situations are a topic of current relevance in the study of large systems of numerical differential equations [24,91,92] and may benefit from ongoing research in the field. For the simulations reported in this paper, with Hilbert space dimensions ranging up to ∼ 3 · 10 5 , the memory required never exceeded 3 GB per thread and we did not need to introduce any Hilbert space optimization beyond truncating the oscillators, but different dimensional reduction schemes such as matrix product operator-based methods [93][94][95] are a viable alternative when dealing with a large number of degrees of freedom between the system and the oscillators. As for the simulation time, the cost of the method scales linearly in the total evolution time considered whenever the full Hilbert space dimension can be fixed upfront, making it well suited to the study of long-time dynamics and relaxation to equilibrium. When the total effective Hilbert space of a problem is too large and one needs to resort to time-adaptive truncation schemes (such as the aforementioned tensor network-based propagators) in order to compress the memory requirements, this generally results in a nonlinear scaling in time; however, novel methods which can exploit damping in the simulated dynamics to bound the maximum effective dimensions, thus reducing these nonlinear additional costs significantly, are being developed [95] and could greatly boost the computational potential of our approach in problems involving larger systems or many surrogate modes. As to determining the parameters of the surrogate modes in the first place, the inversion problem from the target correlation function is, in general, a mathematically difficult task. Our TSO algorithm uses a randomized parametrization as a variational method to reduce the number of variables in the problem and unlock a part of the solution, which is then fed back into the inversion problem to determine the values still missing; the solution found is the best possible for the random initial values given, and a minimization on the sample according to a suitable figure of merit is carried out a posteriori. This rather involved procedure gives satisfactory results but scales poorly with the number of modes; for more than five interacting oscillators, it is already very expensive. This, however, is not a major setback for several reasons. First of all, complex environmental correlation functions typically originate from spectral densities comprising several simple terms, which can be addressedand recycled for other problems if needed-individually, as demonstrated in our example application; secondly, keeping the number of effective oscillators as low as possible is also a priority for simulation purposes and does not put significant constraints on accessible coupling or temperature regimes; finally, the cost of the TSO is not fixed but depends on the form chosen for the effective environment, so the complexity of our particular algorithm is not universal. Possible future improvements to the variational algorithm could come from employing different methods such as simulated annealing or importance sampling in the parameter search or machine-learning techniques to minimize the distance between original and effective correlation functions with respect to the parameters; finding a way to work with the map from effective environment to correlation function in the direct rather than reverse direction, if possible, would be a major simplification. C. Accuracy and error sources To complete the discussion of our method, we must now turn to the sources of error affecting the reduced dynamics, and the control we have over them. The most important error to be addressed is of physical origin and comes from the TSO. This is the error in the correlation function, whose impact on the reduced dynamics and operator expectation values at any time is rigorously bounded [45] (the paper focuses on the spinboson model in particular, but similar bounds can be derived for other finite systems following the same steps). Though under control, this error is worth a more careful analysis because it is actually a sum of two errors, one from a fundamental feature of our method, the other from a technical constraint. The former source is the very form of any correlation function defined as in Eq. (10): it has been shown that no infinite, unitary thermal environment can have a correlation function of the form obtained via the quantum regression formula for a finite, Lindblad-damped auxiliary bath, because the fluctuation-dissipation theorem [1,96], which holds for continuous, unitary thermal environments, is incompatible, strictly speaking, with the regression hypothesis [63,97]. This is reflected quantitatively in the fact that no zero-temperature correlation function obtained through the latter is exactly zero on the whole negative frequency domain; however, the violation of the fluctuation-dissipation theorem can always be reduced by adding effective modes, until the unavoidable residual error is comparable to other errors in the model at hand (except in pathological cases, such as the weakly coupled spin-boson model with pure dephasing: this system is only sensitive to the limit of the spectral density at zero frequency, where the analytical differences between unitary and effective environments emerge most clearly, as discussed in Section III). The second source contributing to the correlation function error are the constraints on the parameters in the master equation: for a fixed number of modes, not every linear combination of complex exponentials can be derived from a valid set of effective bath parameters via Eqs. (20) and (21) (for example, values obtained by setting at least one of the master equation rates Γ n to a negative value are out of physical scope). Therefore, the closest physically possible correlation function to the one given is generally not the best unconstrained fit with complex exponentials; however, it is more accurate than the closest fit with the same number of independent pseudomodes, thanks to the added flexibility from the extra coupling parameters. It should also be mentioned at this point that spectral densities of microscopic models are ultimately derived from experimental results in many applications of current interest [75,79,85,98], so any error in our TSO resulting in a correlation function still compatible with the data is immaterial in practice: any reduced dynamics within the error bounds from [45] given by these differences should be regarded as equally valid. Finally, the last error source in our method is strictly numerical and comes from the integration schemes used to solve the Lindblad equation, which necessarily involve some truncation of the Hilbert space. This error is not under rigorous control, but requires method-dependent convergence checks like any other numerical solution technique. D. Impact Finally, let us sum up the salient features of our simulation method and highlight its distinguishing qualities among existing schemes for general OQS. First of all, we wish to emphasize that our aim in proposing this approach is to offer the level of accuracy and reliability of a fully microscopic simulation while retaining the benefits of working with two-tiered effective environments, particularly their simple mathematical structure and efficient numerical simulation. Our scheme fills the gap between exact and simplified effective methods by providing the auxiliary oscillators standing in for any unitary environment with a quantitatively certified link to it, and enables very efficient simulation of nontrivial environments by keeping the number of modes much lower-thanks to the interactions among them-than any similar techniques we are aware of. For example, noninteracting pseudomodes are a special case of our surrogate oscillators, but we found that in order to reproduce an environment such as the one we discussed in our example application in Section V, we would have needed at least 20 pseudomodes to attain the same accuracy given by our TSO with 6 oscillators: while each of our interacting effective modes contributes a term C R n (t) = w n e λnt with a complex w n , or C R n (ω) = −2 [w n ] [λ n ] + [w n ](ω + [λ n ]) [λ n ] 2 + (ω + [λ n ]) 2 , in the frequency domain, to the correlation function, for independent pseudomodes all w n are real and positive and each C R n (ω) thus becomes just a Lorentzian, which is strictly positive and far less flexible for fitting purposes. An accurate simulation of our example system with noninteracting pseudomodes would thus have been much more expensive. We have also compared our simulations with calculations performed using microscopic methods and found that our accuracy is on par with numerically exact results, e.g. from TEDOPA, at least for the relatively simple applications considered in this paper: long-time dynamics are much easier to compute by solving our effective Lindblad equation in all coupling and temperature regimes due to the nonlinear scaling of TEDOPA in the evolution time; on the other hand, spectral densities with complicated shapes requiring a large number of effective oscillators are hard with our method (though the solution of our effective Lindblad equation can be optimized by techniques such as the one presented in [95]) while TEDOPA is much less sensitive to the shape of the spectral density. In our example application, performance was found to be better than TEDOPA at all temperatures for medium to long times (at zero temperature even for very short times) and comparable for short times at nonzero temperature, for a moderately structured spectral density; the scaling in system size and complexity is very similar for the two schemes. The HEOM method [14,31] is more akin to our approach in spirit, since it is also based on exponential fitting of C E (t). Much like our number of surrogate modes, the number of exponentials needed for an adequate fit is the main factor determining complexity of HEOM simulations; although no comparative study was carried out, this number should be the same for the two methods if one requires the same accuracy and uses efficient expansion techniques for the correlation function [99,100] (these overcome the well-known problem of the more traditional Matsubara-frequency expansion [101,102], which at low temperatures needs a very large number of exponential terms in order to converge). HEOM scales with the complexity of the spectral density in a similar manner as our method and can likewise account implicitly for temperature through the approximate correlation function. Long evolution times are also not problematic for most regimes; however, they can be in the presence of certain environmental features, e.g. narrow peaks with high Huang-Rhys factors corresponding to strongly coupled environmental modes, which make the hierarchy of equations converge very slowly, significantly increasing the simulation cost. We have shown in our example that including the effect of such terms in our effective baths is not particularly expensive; even higher coupling strengths than we used in that model can be dealt with efficiently. A more quantitative comparison between our effective method and HEOM will be undertaken in another paper. VII. CONCLUSIONS AND OUTLOOK We have introduced and demonstrated a new nonperturbative approach for the description and simulation of arbitrary open quantum systems in Gaussian bosonic environments, which puts no restrictions on temperature, non-Markovianity, system-environment coupling strength or system structure. Our method is based on the construction of an effective environment of an extremely versatile class, built according to a systematic recipe to capture the effects of any given unitary bath, using as few degrees of freedom as possible and with a clear measure of the error involved. This procedure is grounded in rigorous theoretical results, specifically the equivalence conditions between unitary and dissipative Gaussian environments proved in [57] and the relation between changes in the bath correlation function and in the reduced dynamics and single-time averages of the system given in [45]. The result is a mapping of the problem at hand onto a Lindblad master equation for the system coupled to one or more small networks of interacting effective modes, which can be simulated using standard numerical methods for quantum master equations. The effective modes are always at zero temperature regardless of the temperature of the original environment, giving the Lindblad equation a simple structure, and the interactions among them make a smaller number of modes necessary to account for the specific effect of any given environment than would be the case if they were all independent, with clear computational advantages. Not all modes need to be interacting; environmental spectral densities consisting of several terms may be reproduced using separate clusters of oscillators for each term, simplifying the calculation of their parameters while still exploiting the versatility of interacting oscillators in the rendering of each individual contribution. As a first example of realistic use, we applied the method to a non-perturbative problem of a kind relevant for current research, obtaining accurate predictions for experimentally significant quantities across the temperature range from absolute zero to room temperature with desktop-level computational resources. In addition, it was shown that by mapping non-Markovian problems to Markovian ones-provided that the dynamically relevant part of the effective Hilbert space is not too large-the method can deal with short as well as long evolution times at comparable costs, making it a suitable tool for the simulation of long-lived dynamical features and relaxation to equilibrium. Future work on this project will be aimed mainly at improving the recipe for determining effective environment parameters, enhancing simulation efficiency and carrying out more case studies to better assess accuracy and performance, as well as applications to more systems of theoretical or experimental interest; we also plan to undertake deeper theoretical studies on the properties of mappings between unitary and dissipative environments, in order to determine optimized figures of merit for the effective parameters, gain some insight into system-dependent sensitivity to the error in the correlation function and exploit this knowledge to improve accuracy of the reduced dynamics. We wish to thank A. Mattioni, F. Caycedo-Soler, J. Lim, A. Somoza, R. Puebla and M. Paternostro for useful discussions, suggestions and feedback about the work presented in this paper. We further acknowledge support by the State of Baden-Württemberg through bwHPC for the use of the BwUniCluster and the German Research Foundation (DFG) through grant no INST 40/467-1 FUGG for the use of the JUSTUS cluster. This work was supported by the ERC Synergy Grant BioQ, the EU Projects HY-PERDIAMOND and AsteriQs, and the BMBF projects DiaPol and NanoSpin. Appendix A: Effective parameters We list here several sets of effective parameters used in the simulations discussed in the main text, along with the local dimensions of each mode in each set at convergence. The corresponding spectral densities are defined in Eq. (26) In Section III of the main text, we defined the general form of the effective environments used in our method, and sketched the transformation algorithm by which we obtain their parameters given a target correlation function. Here we will go through the procedure in detail, in order to give a clearer view of its more technical aspects. Effective correlation function The Hamiltonian for our effective oscillators in a chain configuration with hopping interactions is (with = 1) H R := N n=1 Ω n b † n b n + N −1 n=1 g n b n b † n+1 + g * n b † n b n+1 (B1) and we consider a zero-temperature Lindblad dissipator D R [ρ R ] := N n=1 Γ n b n ρ R b † n − 1 2 b † n b n , ρ R (B2) acting locally on each mode. The interaction term with the system has the form H I := m k=1 A Sk ⊗ F Rk ,(B3) where the interaction operators F Rk of the environment are linear in the creation and annihilation operators: F Rk := N n=1 c nk b n + c * nk b † n .(B4) Assuming factorizing initial conditions ρ 0 = ρ 0S ⊗ ρ 0R with ρ 0R = N n=1 \|0 0\| n , which is Gaussian and stationary under this dynamics, meaning it satisfies L R [ρ 0R ] := −i[H R , ρ 0R ] + D R [ρ 0R ] = 0, the correlation function C R kk (t + τ, τ ) := F Rk (t + τ )F Rk (τ ) R (B5) is independent of the first evolution time τ . We will drop the τ time argument from now on and also restrict our analysis to a single interaction operator (m = 1 in Eq. (B3)), so in the following the correlation function (B5) will be denoted by C R (t). Writing it out explicitly in terms of the expression of F R , we get C R (t) = N m,n=1 c m c * n b m (t)b † n (0) R ,(B6) since terms with two creation or annihilation operators and contributions proportional to b † m (t)b n (0) R are zero for our initial vacuum state. It is easy to show that the hopping coupling constants g n can be assumed real and positive without loss of generality in C R (t): define the canonical transformation b n → e iδn b n (B7) for arbitrary real δ n . The creation operators b † n will transform with the opposite phase, preserving the canonical commutation relations. The free term in the Hamiltonian (B1) and the dissipator (B2) are invariant under this transformation; the hopping term in Eq. (B1) and the interaction operator F R defined as in (B4) are not: c n a n + c * n b † n → c n e iδn b n + c * n e −iδn b † n g n b n b † n+1 + g * n b † n b n+1 → g n e i(δn−δn+1) b n b † n+1 + g * n e −i(δn−δn+1) b † n b n+1 . Taking δ n such that g n e i(δn−δn+1) = \|g n \|, we may absorb the phase of the couplings in the still undetermined c n , without restricting the physical picture in any way. Note that this leaves one of the δ n still free as an overall phase in all the operator coefficients, which may be set e.g. so that c 1 or c N is real. The free dynamics of the oscillators with no coupling to the system is given by the Lindblad equation d dt ρ R (t) = −i[H R , ρ R (t)] + D R [ρ R (t)].(B8) Acting with the operator b n from the left on both sides and taking the trace, we get d dt b n (t) R = N m=1 M nm b m (t) R ,(B9) with Assuming that none of the eigenvalues are degenerate, which is always the case in numerical applications since the Λ matrices with equal diagonal elements are a zeromeasure set, the evolution of the expectation value b n (t) R is thus M nm := α n δ nm − i(g m δ n m+1 + g m−1 δ n m−1 ) =          α 1 −ig 1 0 . . . 0 −ig 1 α 2 . . . . . . 0 . . . . . . 0 . . . α N −1 −ig N −1 0 . . . 0 −ig N −1 α N          ,(B10)b n (t) R = N m=1 (Se Λt S −1 ) nm b m (0) R = N k,m=1 e λ k t u k n v k m b m (0) R ,(B11) and extends to the correlation functions b n (t)b † m (0) R by the quantum regression hypothesis, which is true by construction in the context of Theorem 1 of the main text [57] ; since b n (0)b † m (0) R = δ nm on our initial state, one has b n (t)b † m (0) R = N k=1 e λ k t u k n v k m ,(B12) which can now be substituted into (B6) to give the expression found in Eqs. (20) and (21) main text: C R (t) = N n=1   N k,l=1 c k c * l u n k v n l   e λnt ,(B13)c k c * l u n k v n l .(B15) If degenerate eigenvalues λ d k are present, the time evolution in the corresponding subspace will be driven by e λ d k t times growing powers of t; we did not consider this case for the sake of simplicity, but it may be useful to keep in mind that a mixed algebraic and exponential time dependence of correlation functions is not entirely ruled out by considering a Lindblad dynamics. If one wishes to explore this possibility in the TSO method, equality of two or more eigenvalues should be enforced at the level of the initial fit of the original correlation function C E (t) (see the next paragraph), since its spontaneous occurrence in the numerical procedure is virtually impossible. Inversion problem from a given correlation function To construct an effective environment whose C R (t) is as similar as possible to the C E (t) of a given unitary environment, we first fit C E (t) with a linear combination of N complex exponentials eλ nt weighted by complex coefficientsw n , with N large enough to give an accurate fit, and then work backwards from Eq. (B14) to find the parameters that give the best approximation of the target function. Since the real parameters in C R (t) are 4N − 1 (taking into account the fact that N n=1 w n = N n=1 \|c n \| 2 is real and positive by construction) and it takes 5N − 2 real parameters (N frequencies, N damping rates, N − 1 couplings and N complex coefficients c n minus one overall redundant phase) to identify an effective environment, this is a highly nontrivial inversion problem, because the map from effective environments to correlation functions is both nonlinear and many-to-one. This means that existence or uniqueness of a solution to our problem are not guaranteed in general; furthermore, we must require Γ n > 0 for all n in order for our effective master equation to be meaningful, which sets another important constraint. It is useful to break down the problem into two parts: first an inverse eigenvalue problem leading from theλ n to the dynamical matrix M , and then a system of equations relating the coefficientsw n to the interaction operator parameters c n . This allows us to deal with the sign constraints on the rates once and for all in the first half of the solution procedure, and to exploit the fact that the c n only appear in the second. To determine the relation between the eigenvalues and elements of the matrix M , it is not convenient to look for symbolic expressions for each eigenvalue in terms of the parameters, since these would necessarily involve highdegree roots of complex polynomials. A simpler approach is to consider the characteristic polynomial of M p M (λ) := det(λI − M ) = N n=1 (λ − λ n ), substitute the target eigenvaluesλ n on the right-hand side and equate the coefficients of like powers of λ, which are geometrical invariants of any operator. The result is a system of equations of degrees 1 through N stating the invariance of the sums of principal minors order by order (the trace and the determinant appearing in the first and last equation being the simplest such invariants). Now, Eq. (B16) can be regarded as a parametric system of equations in the couplings g n . With the g n fixed, it becomes an algebraic nonlinear system of N equations in N unknowns which can be solved numerically to give multiple sets of α n -i.e. frequencies Ω n and rates Γ n whose sign can be checked directly-and therefore the entire dynamical matrix M .                              Given a dynamical matrix M obtained by choosing some set of g n and solving Eq. (B16), its eigenvectors u n and v n can be substituted into the w n as defined in Eq. (B15), which then become functions of the c n only and can be equated with the target valuesw n                    N m,n=1 c * m v 1 m u 1 n c n =w 1 . . . N m,n=1 c * m v N m u N n c n =w N (B17) to solve the second half of the problem. These N complex equations are equivalent to 2N − 1 equations in 2N − 1 real unknowns, since the overall phase of all c n drops out of the left-hand side while on the right-hand side N n=1w n = C E (0) has no imaginary part. A set of c n solving Eq. (B17) does not always exist, so here we numerically minimize the Manhattan distance between the w n on the left-hand side and thew n instead. At this point, we have converted an arbitrary (N − 1)tuple of coupling constants into a trial effective correlation function C R trial (t) = N n=1 w n (g m , α m , c m )e λn(gm,αm)t which can be compared to the target C E (t) according to some figure of merit. We used the integral I 1 (t max ) := tmax 0 dt t 0 dt \|C R trial (t − t ) − C E (t − t )\| (B18) up to some final time t max such that C E (t max ) 1 in all cases where we had a closed expression for it, and I 2 (t max ) := ∆t Nmax n=1 \|C R trial (n∆t) − C E (n∆t)\| (B19) for some number of points N max and timestep ∆t = t max /N max when C E (t) was only known in integral form and needed to be evaluated for each value of the time argument. This whole procedure can be carried out for many values of the couplings in the physical parameter region, ranking the corresponding trial correlation functions by their values of the figure of merit in search of an optimum, in the spirit of the error bounds in [45] which relate the absolute difference between correlation functions to the changes in the reduced dynamics. To summarize the steps described above, in order to find an effective environment corresponding to some correlation function C E (t), we first fit it with complex exponentials, and then overcome the mismatch between the number of variables from this fit and the number of parameters in the effective environment by setting up a variational problem in the g n couplings between neighboring surrogate modes. We sample multiple (N − 1)-tuples (g 1 , . . . , g N −1 ) in a suitably sized open set (0, g max ) N −1 , solve Eq. (B16) for each of them and then plug the eigenvectors of all physically acceptable matrices M found into Eq. (B17) to determine the c n . The trial correlation functions C R trial (t) constructed from each set of parameters are ranked according to the estimators (B18) or (B19), depending on the original C E (t), and we search for the minimum of the figure of merit in the space of the g n . This variational problem is not convex in general: both the shape of the region in g n -space leading to physically admissible solutions and the dependence of the cost functions defined in Eqs. (B18) and (B19) on the couplings can be highly nontrivial, with trenches, pointed features, local minima and gaps without any solutions at all appearing at unpredictable locations. We have also found no obvious patterns giving any hints as to the existence of a region of physically acceptable values of g n (for any N ) such that there are exact solutions, or the form that such a region may have, based on the target parameters. All these mathematical features are very model-dependent; Fig. 11 shows some examples of parameter space shapes and cost function behavior for different correlation functions approximated using N = 3 surrogate oscillators. Though none of the examples shown gave a C R (t) of sufficient accuracy for practical use due to the small number of effective modes, they nonetheless give a clear qualitative idea of the variety of possible outcomes. In general, sampling the parameter space efficiently is difficult, and we are looking for ways to improve this part of the algorithm. Appendix C: Special cases with exact solutions We will now show some examples, both general and related to the specific systems treated in the main text, of analytical solutions of the inversion problem in specific cases. One and two oscillators The simplest possible effective environment is a single damped oscillator (N = 1 in Eq. (B1), with interaction operator F R = c(b + b † ) since the phase of c can be set to zero). This yields a correlation function C R (t) = c 2 e − Γ 2 \|t\|−iΩt where the time dependence at t < 0 is defined by C R (−t) := C R * (t) because the Lindblad equation only gives C R (t) for positive times, as discussed in Section III of the main text. The Fourier transform of this function is a Lorentzian of width Γ centered in Ω: C R (ω) = c 2 Γ (Γ/2) 2 + (ω − Ω) 2 . A single sharp peak at zero temperature in the target spectral density can be mapped to a mode like this by simple nonlinear fitting of C E (t) with a complex exponential, as we did for the dimer simulations in Section V of the main text: in that case, the peaks were antisymmetrized Lorentzians so the frequency and damping rate of the effective mode matched those from the original spectral density almost exactly. A less trivial, still exactly solvable case is given by two interacting oscillators and was already introduced by Garraway in [48] to show that not only sums but also differences of Lorentzians can be modeled by pseudomodes. In that paper, only one of the modes is coupled to the system (i.e. c 2 = 0); here we lift this assumption to show a more general result. The general correlation function for N > 1 has the form C R (t) = (C2) thus adding a linear frequency dependence in the numerator of the term associated with each mode. In phenomenological approaches where there is no intention of accurately simulating a specific correlation function, one may use ad-hoc combinations of weights and exponents to cancel terms in the numerator of the full C R (ω) written as a single fractional polynomial and achieve a steeper fall-off in frequency than is possible with individual Lorentzians with positive coefficients (again, one example is given in [48]). Such strategies hardly generalize beyond specific applications but can be helpful to mitigate the error associated with the behavior of C R (ω) near the origin, which does not comply with the fluctuation-dissipation theorem in general, as discussed in the main text. In the case N = 2, the eigenvalues λ 1,2 and weights w 1,2 depend on the effective environment parameters through the relations λ 1,2 (α 1,2 , g) = α 1 + α 2 2 ± α 1 − α 2 2 2 − g 2 w 1,2 (α 1,2 , g, c 1,2 ) = \|c 1 \| 2 + c 2 2 2 ± (\|c 1 \| 2 − c 2 2 )(α 1 − α 2 ) − 4ig [c 1 ]c 2 2 (α 1 − α 2 ) 2 − 4g 2 ,(C3) where c 2 is taken to be real by fixing the overall phase mentioned in the preceding section. The first equations Figure 12: Left to right: overlapping plots of C E β (ω) and the effective C R β (ω) with N = 2 for an antisymmetrized Lorentzian with peak frequency Ω AL = 2.15u and width Γ AL = 0.1u in units u = 100 cm −1 , at temperature T = 77 K; a plot of the difference C E β (ω) − C R β (ω); a plot of the minimum Manhattan distance N n=1 \|wn − wn\| as a function of g, with a very small region around g = 2.14u (shown in the inset) in which thewn can only just be matched exactly, before the Γn in the solutions change sign and higher values of g no longer give acceptable solutions. Notice both the very steep descent of the error and the abrupt end of the physically admissible region on either side of this spot, as an example of how minima in our figures of merit quickly become hard to find in a more coarse-grained sampling in higher dimensions. is readily inverted parametrically in g: α 1,2 =λ 1 +λ 2 2 ± λ 1 −λ 2 2 2 + g 2 .(C4) The domain of physically admissible solutions is the set of all g such that Γ 1,2 = −2 [α 1,2 ] > 0 and can be found by using the formula for the square root of a complex number z = z R + iz I √ z = \|z\| + z R 2 + i sgn(z I ) \|z\| − z R 2 . Using Eq. (C4) in the equation for the weights, this becomes parametric in g as well: w 1,2 = \|c 1 \| 2 + c 2 2 2 ∓ (\|c 1 \| 2 − c 2 2 ) λ 1 −λ 2 2 + 4g 2 − 4ig [c 1 ]c 2 2(λ 1 −λ 2 ) .(C5) This equation, which relates the four real quantities g, [c 1 ], [c 1 ] and c 2 to the three real numbers determining w 1,2 , may or may not have a solution depending on the g chosen, as discussed earlier: if the solution exists only for g outside the physical region in which Γ 1,2 = −2 [α 1,2 ] > 0, then it is necessary to operate variationally and minimize the distance 2 n=1 \|w n − w n (α 1,2 , g, c 1,2 )\|. If one assumes c 2 = 0 in (C3), as is done in [48], then the whole system can be inverted explicitly:                            \|c 1 \| 2 =w 1 +w 2 α 1 =w 1λ1 +w 2λ2 w 1 +w 2 α 2 =w 1λ2 +w 2λ1 w 1 +w 2 g 2 = λ 1 −λ 2 2 2 w 1 −w 2 w 1 −w 2 2 − 1(C6) but the solution only exists if theλ 1,2 ,w 1,2 given are such that the expression on the right-hand side of the last equation has a vanishing imaginary part. This is because now the phase of c 1 has also decoupled from the problem, removing a second real degree of freedom and making the system overdetermined: the balance between equations and unknowns is thus restored by this real constraint appearing on theλ 1,2 ,w 1,2 . In our applications, we used pairs of effective modes to reproduce narrow antisymmetrized Lorentzians at nonzero temperature: we found no exact solution and had to minimize 2 n=1 \|w n − w n (α 1,2 , g, c 1,2 )\| in most cases, but e.g. for an antisymmetrized Lorentzian with peak frequency Ω AL = 215 cm −1 and width Γ AL = 10 cm −1 at temperature 77 K the system can be solved exactly for 213.0 cm −1 < g < 213.8 cm −1 (Fig. 12). Note that in all cases we considered, the best fit of such thermalized peaks with two modes was always obtained by mode frequencies close to zero and a strong coupling g between the two; fitting the same function with two noninteracting modes at the positive and negative peak frequencies was consistently found to be a less accurate choice even for such seemingly obvious target functions. Three oscillators Adding a third oscillator, we found exact solutions for c 2 = 0, which we did not use in any of the simulations discussed in the main paper but can be useful in general. For N = 3, the system of eigenvalue equations is          α 1 + α 2 + α 3 =λ 1 +λ 2 +λ 3 α 1 α 2 + α 2 α 3 + α 3 α 1 + g 2 1 + g 2 2 =λ 1λ2 +λ 2λ3 +λ 3λ1 α 1 α 2 α 3 + g 2 1 α 1 + g 2 2 α 3 =λ 1λ2λ3 (C7) and one may remove α 2 from the last two equations by using the first, so that α 1 and α 3 can be regarded as effective functions of the real parameters g 1 and g 2 . With c 2 set to zero, the whole inversion problem is determined, since the equations for thew n will determine the values of g 1 and g 2 instead. Setting the overall phase so that c 3 is real, the equations can be written as                                \|c 1 \| 2 + c 2 3 =w 1 +w 2 +w 3 \|c 1 \| 2 α 1 + c 2 3 α 3 =w 1λ1 +w 2λ2 +w 3λ3 2 [c 1 ]c 3 g 1 g 2 = −w 3 (λ 1 −λ 3 )(λ 2 −λ 3 ) − (α 1 −λ 1 )(α 1 −λ 2 )(α 3 −λ 3 ) (α 3 − α 1 )(α 1 −λ 3 ) \|c 1 \| 2 − (α 3 −λ 1 )(α 3 −λ 2 )(α 1 −λ 3 ) (α 3 − α 1 )(α 3 −λ 3 ) c 2 3 (C8) where the last line again features a real expression on the left-hand side and a complex one whose imaginary part must be zero on the right-hand side. Since the first equation is real by construction, there are five real equations in the five real variables g 1 , g 2 , [c 1 ], [c 1 ], c 3 in Eq. (C8), so the existence of solutions is only subject to the constraint Γ n = −2 [α n ] > 0. If c 3 is also set to zero, then the system (C8) becomes                      c 2 1 =w 1 +w 2 +w 3 c 2 1 α 1 =w 1λ1 +w 2λ2 +w 3λ3 0 = −w 3 (λ 1 −λ 3 )(λ 2 −λ 3 ) − (α 1 −λ 1 )(α 1 −λ 2 )(α 3 −λ 3 ) (α 3 − α 1 )(α 1 −λ 3 ) c 2 1 .(C9) and can be inverted explicitly, giving c 2 1 , α 1 and α 3 . But now the system (C7) is overdetermined: the trace gives α 2 , and the last two complex equations can give g 1,2 only if theλ n andw n happen to satisfy two real relations among themselves (one because c 3 was removed from the problem, another because the phase of c 1 is now irrelevant). In particular, the expressions whose imaginary part must vanish now appear on the right-hand side of the last two lines of the full solution                                                        c 2 1 =w 1 +w 2 +w 3 α 1 =w 1λ1 +w 2λ2 +w 3λ3 w 1 +w 2 +w 3 α 2 = (w 2 +w 3 )λ 1 + (w 3 +w 1 )λ 2 + (w 1 +w 2 )λ 3 w 1 +w 2 +w 3 −w 2w3 (λ 2 −λ 3 ) 2λ 1 +w 3w1 (λ 3 −λ 1 ) 2λ 2 +w 1w2 (λ 1 −λ 2 ) 2λ 3 w 2w3 (λ 2 −λ 3 ) 2 +w 3w1 (λ 3 −λ 1 ) 2 +w 1w2 (λ 1 −λ 2 ) 2 α 3 =w 2w3 (λ 2 −λ 3 ) 2λ 1 +w 3w1 (λ 3 −λ 1 ) 2λ 2 +w 1w2 (λ 1 −λ 2 ) 2λ 3 w 2w3 (λ 2 −λ 3 ) 2 +w 3w1 (λ 3 −λ 1 ) 2 +w 1w2 (λ 1 −λ 2 ) 2 g 2 1 = −w 2w3 (λ 2 −λ 3 ) 2 +w 3w1 (λ 3 −λ 1 ) 2 +w 1w2 (λ 1 −λ 2 ) 2 (w 1 +w 2 +w 3 ) 2 g 2 2 = −w 1w2w3 (λ 2 −λ 3 ) 2 (λ 3 −λ 1 ) 2 (λ 1 −λ 2 ) 2 (w 1 +w 2 +w 3 ) w 2w3 (λ 2 −λ 3 ) 2 +w 3w1 (λ 3 −λ 1 ) 2 +w 1w2 (λ 1 −λ 2 ) 2 2 . (C10) Figure 1 :Figure 2 :Figure 3 : 123Simulation of composite systems. When applying the transformation to surrogate oscillators (TSO) to interacting systems coupled to local environments, each environment is replaced by the corresponding effective one, regardless of the properties of the system attached to it. Our procedure leads to modular structures which do not require a rederivation of the effective parameters when couplings among separate open systems are introduced. Simulation of structured environments. A system coupled to an environment with spectral density J(ω) = J (ω) + J (ω), with J (ω) a broad background and J (ω) a sharp resonance as shown in the small plots, mapped to two distinct effective environments, one with N = 4 and another with N = 2 oscillators. We will encounter a similar structure in our example application in Section VFourier transform of the Ohmic correlation function C E β (t) with βΩc = 1 (solid orange line) and of the corresponding C R β (t) from the TSO with parameters given inTable I (dashed blue line). The inset shows the difference as a function of frequency. Figure 4 : 4Spin-boson results. Time evolution of the density matrix of a qubit starting from the state ρ0S = \|+ +\|, in an Ohmic environment at three different temperatures. The solid lines show the real and imaginary parts of the coherence ρ01(t) obtained by our simulation of the equivalent Lindblad equations, the dashed lines show the analytical solution and the insets show the error as defined in Eq. (30). Both populations are identically 1/2 throughout the evolution and are not shown. Figure 5 : 5The dimer model. Left: the physical picture with the 0-, 1-and 2-excitation subspaces (spanned by the ground state \|g , the local excited states \|E1 and \|E2 of the two sites and the doubly excited state \|E12 , respectively) and independent local environments interacting with each site. Right: the equivalent model after the TSO, with no doubly excited state and a single effective environment corresponding to the relative coordinates as in Eq.(36) and coupled to both local excited states. The wavy lines represent the laser probing a sample in optical experiments. Figure 6 : 6The spectral densities J0(ω) and J1(ω) of the environments we used in our dimer model. Figure 7 :Figure 8 : 78Absorption spectrum of the dimer with bath spectral density J0(ω). Only the Adolphs-Renger background is present. The absorption maxima appear at the eigenenergies of the system Hamiltonian minus the reorganization energy λ0; the upper eigenstate gives a broader peak, since it can decay to the lower one or lose energy to the bath. Absorption spectrum of the dimer with bath spectral density J1(ω). Figure 9 :Figure 10 : 910Exact (solid orange line) and effective (dahsed blue line) correlation function for J 1 (ω) at T = 77 K; four modes were used for the background and two for the peak. The inset shows the TSO error. Note the shape related to the spectral density by Eq.(25), in particular the super-Ohmic dip at frequencies near zero and the mirrored local maximaShort-time reduced dynamics in the singleexcitation subspace of our dimer model with initial state ρ0S = \|+ + \| and spectral density J 1 (ω) (defined in the text) at T = 77 K, as simulated by our effective Lindblad equation (solid lines, colors as in legend) and TEDOPA (dashed lines). The inset shows the difference between the results. X: Antisymmetrized Lorentzian spectral density with Ω = 227.5 cm −1 , Γ = 20 cm −1 , S = 0.0379 from J 1 (ω): parameters in units u = 100 cm −1 and local dimensions. where we have introduced the shorthand α n := − Γn 2 −iΩ n . Eq. (B9) can be solved formally by diagonalizing the tridiagonal matrix M . Since M is not Hermitian, one hasM = SΛS −1 ,with Λ := diag(λ 1 , . . . , λ N ) the diagonal matrix containing the eigenvalues, S := (u 1 , . . . , u N ) a matrix made of arbitrarily normalized right eigenvectors u n and S −1 := (v 1 , . . . , v N ) T its inverse, whose rows (v n ) T are left eigenvectors. Since M is a symmetric matrix, left and right eigenvectors are the same, so S −1 is just the transpose of S up to normalization of the rows in Figure 11 : 11Shape of the space of gn such that all Γn are positive, with N = 3 and three different thermal correlation functions C E β (t) corresponding to J(ω) = ωe −ω/Ωc , J(ω) = (ω 2 /Ωc)e −ω/Ωc and J(ω) = (ω 5 /Ω 4 c )e −ω/Ωc , all at βΩc = 0.85. The axes show the values of g1 and g2 and the color denotes accuracy of the trial correlation function as estimated by I1(tmax) with Ωctmax = 25 and normalized to the maximum accuracy obtained for each case, with blue areas representing smaller errors and yellow and orange ones indicating very vague resemblance. N n=1 w n=1n e λnt (C1) with [λ n ] < 0, where the fact that the w n are complex changes the form in the frequency domain from a simple linear combination of Lorentzians toC R (ω) = −2 N n=1 [w n ] [λ n ] + [w n ](ω + [λ n ]) [λ n ] 2 + (ω + [λ n ]) 2 , Table I : IEffective parameters for N = 4 surrogate modes corresponding to the correlation function of an Ohmic bath at temperature T = Ωc (Eq. Table II : IITotal simulation times for the dipole correlation function Cµ(t) of the dimer and absolute values at the final time for J0(ω) and J1(ω), respectively. , Eq. (33) and Eq. (34) of the main text, respectively.Ohmic spectral density Mode 1 Mode 2 Mode 3 Mode 4 Ωn 2.70796 2.13014 1.15884 0.310906 gn 3.38195 1.43514 0.491546 Γn 11.9298 0.573494 0.0317143 0.000795693 cn −0.0333215 0.319 0.760716 0.579218 −0.0121362i +0.0811955i +0.0175762i d loc 3 4 5 7 Table III : IIIOhmic spectral density with cutoff frequency Ωc, temperature T = 0: parameters in units Ωc and local dimensions.Mode 1 Mode 2 Mode 3 Mode 4 Ωn 0.512683 2.53779 4.53293 0.151433 gn 1.82454 3.20774 1.60194 Γn 0.056336 4.42709 15.7371 0.110104 cn −0.962917 −0.227707 0.231179 0.818093 +0.819128i +0.0701249i −0.137866i d loc 5 4 4 7 Table IV : IVOhmic spectral density with cutoff frequency Ωc, temperature T = Ωc: parameters in units Ωc and local dimensions.Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Ωn 0.306859 0.361308 0.167597 0.0297981 0.00236395 gn 4.17718 2.1243 0.673391 0.166947 Γn 16.0093 2.76375 0.00358704 0.0949691 0.0517414 cn −0.166675 0.21927 1.61933 0.187388 1.1553 −0.0342019i +0.103791i −0.00703994i −1.07416i d loc 3 3 4 4 6 Table V : VOhmic spectral density with cutoff frequency Ωc, temperature T = 5 2 Ωc: parameters in units Ωc and local dimensions.Adolphs-Renger spectral density Mode 1 Mode 2 Mode 3 Mode 4 Ωn 0.718918 3.06543 2.96082 0.667101 gn 2.10958 3.91248 1.56527 Γn 0.00554063 15.4881 0.00291091 0.294244 cn −0.57271 −0.0147923 0.725729 0.409762 +0.06491i +0.0820348i +0.0119678i d loc 6 4 4 4 Table VI: Adolphs-Renger spectral density, temperature T = 0: parameters in units u = 100 cm −1 and local dimensions. Mode 1 Mode 2 Mode 3 Mode 4 Ωn 3.05106 2.74196 0.00670418 0.00780109 gn 2.74161 2.01796 0.33975 Γn 0.0284151 11.6481 0.00549033 0.0184315 cn −0.910465 −0.135049 0.524001 0.114767 −0.0164266i −0.0104797i +0.317767i d loc 5 4 6 8 Table VII : VIIAdolphs-Renger spectral density, temperature T = 77 K: parameters in units u = 100 cm −1 and local dimensions.Mode 1 Mode 2 Mode 3 Mode 4 Ωn 0.788783 0.414407 −0.0300357 −0.034035 gn 3.10576 0.978945 0.294823 Γn 10.4575 0.0934767 0.00983292 0.0167273 cn 0.189405 1.23326 0.0221509 0.365249 +0.0639657i +0.451035i +0.962709i d loc 4 5 10 8 Table VIII : VIIIAdolphs-Renger spectral density, temperature T = 300 K: parameters in units u = 100 cm −1 and local dimensions.Antisymmetrized Lorentzian spectral densities T = 0 K T = 77 K T = 300 K Mode 1 Mode 2 Mode 1 Mode 2 Ωn 2.00 −0.318699 0.316331 −0.00048954 0.000480821 gn − 1.976 2.00052 Γn 0.098296 0.045988 0.138442 0.00953908 0.190362 cn 0.992322 0.764199 0.676024 1.45733 0.343374 +0.000002i +0.000003i d loc 6 5 6 8 8 Table IX : IXAntisymmetrized Lorentzian spectral density with Ω = 200 cm −1 , Γ = 10 cm −1 , S = 0.25 from J1(ω): parameters in units u = 100 cm −1 and local dimensions.T = 0 K T = 77 K T = 300 K Mode 1 Mode 2 Mode 1 Mode 2 Ωn 2.275 0.662126 −0.667153 −0.00139464 0.0013106 gn − 2.1788 2.2772 Γn 0.197195 0.264596 0.0788813 0.00326568 0.396252 cn 0.440408 0.333222 0.296358 0.578109 0.169995 −0.000005i −0.176482i d loc 5 4 4 4 4 Table Appendix B: Transformation to SurrogateOscillators in detail The theory of open quantum systems. 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[ "(Pseudo)goldstinos, SUSY fluids, Dirac gravitino and gauginos", "(Pseudo)goldstinos, SUSY fluids, Dirac gravitino and gauginos" ]	[ "Karim Benakli \nUMR 7589\nSorbonne Universités\nUPMC Univ Paris 06\nLPTHE\nF-75005ParisFrance\n\nUMR 7589\nCNRS\nLPTHE\nF-75005ParisFrance\n" ]	[ "UMR 7589\nSorbonne Universités\nUPMC Univ Paris 06\nLPTHE\nF-75005ParisFrance", "UMR 7589\nCNRS\nLPTHE\nF-75005ParisFrance" ]	[]	We review the emergence and fate of goldstinos in different frameworks. First, we consider a super-Higgs mechanism when supersymmetry breaking is induced by neither an F-term nor a D-term but related to a more general stress energy-momentum tensor. This allows us to build a novel Lagrangian that describes the propagation of a spin-3 2 state in a fluid. Then we briefly review the ubiquitous pseudo-goldstinos when breaking supersymmetry in an extra dimension. We remind that the fermion (gravitino or gaugino) soft masses can be tuned to be of Dirac-type. Finally, we briefly connect the latter to the study of models with Dirac-type gaugino masses and stress the advantage of having both an F and a D-term sizable contributions for the hierarchies of soft-terms as well as for minimizing R-symmetry breaking.	10.1051/epjconf/20147100012	[ "https://www.epj-conferences.org/articles/epjconf/pdf/2014/08/epjconf_icnfp2013_00012.pdf" ]	118,457,124	1402.4286	1ea6dc3f79aeb5095e42659142bcaabe0c4cff32	(Pseudo)goldstinos, SUSY fluids, Dirac gravitino and gauginos Karim Benakli UMR 7589 Sorbonne Universités UPMC Univ Paris 06 LPTHE F-75005ParisFrance UMR 7589 CNRS LPTHE F-75005ParisFrance (Pseudo)goldstinos, SUSY fluids, Dirac gravitino and gauginos 10.1051/epjconf/20147100012 We review the emergence and fate of goldstinos in different frameworks. First, we consider a super-Higgs mechanism when supersymmetry breaking is induced by neither an F-term nor a D-term but related to a more general stress energy-momentum tensor. This allows us to build a novel Lagrangian that describes the propagation of a spin-3 2 state in a fluid. Then we briefly review the ubiquitous pseudo-goldstinos when breaking supersymmetry in an extra dimension. We remind that the fermion (gravitino or gaugino) soft masses can be tuned to be of Dirac-type. Finally, we briefly connect the latter to the study of models with Dirac-type gaugino masses and stress the advantage of having both an F and a D-term sizable contributions for the hierarchies of soft-terms as well as for minimizing R-symmetry breaking. Introduction Interest in supersymmetry (SUSY) can be motivated through different arguments among which its role as an essential ingredient of the fundamental theory unifying gravity with all the other interactions in a consistent quantum theory. At experimentally probed energies, supersymmetry is not manifest and we would like to think of that as a consequence of its spontaneous breaking at a higher energy scale. The spontaneous breaking gives rise in the global limit to a massless Goldstone fermion, the goldstino [1,2]. Once gravitational interactions are taken into account, this state is absorbed by the gravitino to become the spin-1 2 component [3,4] of the massive spin- 3 2 particle. The corresponding dynamics is described by the Rarita-Schwinger Lagrangian [5] which appears supplemented with appropriate constraints. We shall review here a few aspects when departing from the minimal set-up. In the first part, we shall construct a Lagrangian that allows to describe the propagation of a spin- 3 2 state in a fluid [6].. This is obtained as the result of super-Higgs mechanism when supersymmetry is broken by a non-vanishing energy-momentum tensor. The modification to the Rarita-Schwinger Lagrangian appears as a deformation of the quadratic mass term. This allows to describe different propagation velocities of the different gravitino helicities. The second part reviews the omnipresence of pseudo-goldstinos in models where supersymmetry is broken in different sectors of models with one extra dimension [7]. We restrict to the simplest case of at most two branes while the more general case with an arbitrary number of supersymmetry breaking sectors and more dimensions can be found in [8]. Finally, the last part discusses models Dirac gaugino masses (see [9] for a review) where the soft terms are induced by gauge mediation. In particular, we focus on the advantages of having sizable a e-mail: [email protected] F and D-term supersymmetry breaking and point out how this can help to keep R-symmetry unbroken by generating a Dirac gravitino mass. A Lagrangian for a spin-3 propagating in a fluid We start by a brief review of the well known non-linear realization of supersymmetry and super-Higgs effect. The Rarita-Schwinger Lagrangian from the super-Higgs mechanism The Volkov-Akulov Lagrangian We consider a global supersymmetric theory in flat space time. Supersymmetry is broken spontaneously when the vacuum has non-zero energy which, as we will see below, is not the case in local supersymmetric theory. If one insists on preserving Lorentz invariance, this is accomplished for N = 1 supersymmetry by giving a vev to an auxiliary field in a chiral multiplet (F-term) or in a vector multiplet (D-terms). Without loss of generality, we shall chose to focus on the F-term case in this section. As a consequence of Goldstone theorem, the low energy spectrum contains a fermionic massless mode, known as the goldstino for each broken supersymmetry. The goldstino is a spin 1 2 field (G α ,Ḡα) in the ( 1 2 , 0) ⊕ (0, 1 2 ) representation of the Lorentz group 1 . It has a mass dimension of 3 2 . Supersymmetry is non-linearly realized on the field G through ( Q +¯ Q ) × G α (x) = √ 2F α − i 1 √ 2F G(x)σ μ¯ − σ μḠ (x) ∂ μ G α (x).(1) where the F-term F is taken to be real and has a mass dimension 2, and plays the role of the order parameter of supersymmetry breaking. The invariant (up to a divergence) non-linear Lagrangian for G is given by the Volkov-Akulov Lagrangian [2] L A−V = −F 2 det δ ν μ + i 1 F 2Ḡσ ν ∂ μ G (2) = −F 2 − iḠσ μ ∂ μ G + · · · ,(3) where the dots refer to higher order terms that we do not discuss here. This canonically normalized goldstino field satisfies the Dirac equation σ μ ∂ μ G = 0, σ μ ∂ μḠ = 0 .(4) The Rarita-Schwinger Lagrangian for a massless gravitino We are interested by theories with N = 1 local supersymmetry. The supersymmetric partner of the graviton is a gravitino field (ψ μα ,ψ μα ) of spin 3 2 and mass dimension 3 2 . Following Fierz and Pauli, the irreducible spin 3 2 representation is obtained from ψ μα in the ( 1 2 , 1 2 ) ⊗ ( 1 2 , 0) = (1, 1 2 ) ⊕ (0, 1 2 ) 1 We will work in 4 dimensions and we use Wess and Bagger [10] notations. η μν = diag(−, +, +, +), 12 = − 21 = 1. ζ α is a left Weyl spinor in the ( 1 2 , 0) representation.ζα is a right Weyl spinor in the (0, 1 2 ) representation. Complex conjugation exchanges S U(2) L and S U (2) representation, andψ μα in the ( 1 2 , 1 2 ) ⊗ (0, 1 2 ) = ( 1 2 , 1) ⊕ ( 1 2 , 0) representation by imposing constraints that project out the additional spin 1 2 components. The (0, 1 2 ) and ( 1 2 , 0) parts in the decomposition of (ψ μα ,ψ μα ) are removed by imposinḡ σ μ ψ μ = 0, σ μψ μ = 0 .(5) The representations (1, 1 2 ) and ( 1 2 , 1) have dimension six each. In order to reduce the number of degrees of freedom to four we impose ∂ μ ψ μα = 0, ∂ μψ μα = 0 .(6) One can get this structure of equations and constraints from a Lagrangian. The massless gravitino Rarita-Schwinger Lagrangian is: L ψ = μνρσψ μσν ∂ ρ ψ σ .(7) The field equations are μνρσσ ν ∂ ρ ψ σ = 0, μνρσ σ ν ∂ ρψσ = 0 . By imposing on this equation the condition (5) we get σ ρ ∂ ρ ψ σ = 0, σ ρ ∂ ρψσ = 0 .(9) It is easy to see that (9) and (5) imply (6). The super-Higgs mechanism and the massive gravitino We would like to promote the non-linear realization of supersymmetry above from a global to a local realization. The graviton degrees of freedom are described by the vierbein fields e a μ where a is a tangent space index. We define e ≡ det(e a μ ). The spontaneous F-term supersymmetry breaking is associated with a stress-energy tensor with a vev T μν S = −F 2 η μν , where F is the vev of the auxiliary field as defined above. Promoting to a local parameter (x), at leading order, the supersymmetry transformation read δe a μ = − 1 M P i¯ σ a ψ μ − i σ aψ μ , δψ μ = −M p 2∂ μ , δG = √ 2F , δψ μ = −M p 2∂ μ¯ , δḠ = √ 2F¯ .(10) and, up to a divergence, the supersymmetric Lagrangian is: L = − 1 2M 2 P eR − μνρσψ μσν ∂ ρ ψ σ − F 2 e − iḠσ μ ∂ μ G − i F √ 2M p ψ μ σ νḠ +ψ μσν G + · · ·(11) where one sees that the term F 4 represents now a cosmological constant. This is problematic as we wish to work in a flat background. This issue is solved by adding a combination of a canceling contribution to the cosmological constant and a gravitino mass term (where the an anti-symmetric structure σ μν is necessary to avoid the appearance of a pathological fermionic term of the form (∂G) 2 ) : ΔL = F 2 e − m 3 2 ψ μ σ μν ψ ν − m * 3 2ψ μ σ μνψ ν − m 3 2 GG − m * 3 2ḠḠ .(12) ICNFP 2013 00012-p.3 and the total Lagrangian is invariant under supersymmetry variation δe a μ = − 1 M P i¯ σ a ψ μ − i σ aψ μ , δψ μ = −M p 2∂ μ − im * 3 2 σ μ¯ , δG = √ 2F , δψ μ = −M p 2∂ μ¯ + im 3 2σ μ , δḠ = √ 2F¯ .(13) only if: m 3 2 = F √ 3M p .(14) Finally, we can go to the unitary gauge by performing the transformation ψ μα → ψ μα + √ 2M P F ∂ μ G α + i 1 √ 6 σ μααḠα .(15) and derive the Rarita-Schwinger Lagrangian for a massive gravitino L g = μνρσψ μσν ∂ ρ ψ σ − m 3 2 ψ μ σ μν ψ ν − m * 3 2ψ μ σ μνψ ν .(16) Modified Rarita-Schwinger Lagrangian from the super-Higgs mechanism in fluids In this section we will derive a Generalized Rarita-Schwinger Lagrangian from the study the super-Higgs mechanism in fluids. More precisely, we generalize the previous section as now supersymmetry is broken spontaneously by the vacuum expectation value of the stress-energy tensor T μν which also, in general, breaks spontaneously the Lorentz symmetry. The goldstino in supersymmetric fluids: the phonino Consider for simplicity a supersymmetric field theory in thermal equilibrium described by a background stress-energy tensor taken to be a perfect fluid: T μν = diag (ε, p, p, p) .(17) where p is the pressure and ε is the energy density. This expectation value of the stress-energy tensor (17) breaks spontaneously supersymmetry and Lorentz symmetry but keeps rotational invariance. Based on the study of the supersymmetric Ward-Takahashi identity, it was argued that the associated spontaneous breaking of supersymmetry implies a massless fermionic field in the spectrum, the goldstino called here a phonino [11]. In fact, the supersymmetric Ward-Takahashi identity for the supercurrent two-point function: ∂ μ T {S μ (x)S ν (y)} ∼ δ (4) (x − y) T νρ σ ρ .(18) shows that the correlator has to have a singularity when k → 0 when going to momentum space and assuming a constant energy-momentum tensor the correlator. Note that without Lorentz invariance it is possible to have a singularity without having a massless particle. While this is known to happen for instance in a free theory, in a generic interacting system it is expected that the massless mode is present (see for example [12] ). Here, we will consider such a situation. The field equations of the phonino take the form T μνσ μ ∂ ν G = 0, T μν σ μ ∂ νḠ = 0 .(19) which can be obtained from the Lagrangian L G = − i T 4 T μνḠσ μ ∂ ν G ,(20) where T = \| Tr T μν \| 1 4 has dimension of mass. Note that for T μν = −F 2 η μν the Lagrangian (20) reduces to the previous section and the propagator of the phonino becomes that of the usual goldstino. Note that the gravitino and the goldstino remain massless in a CFT fluid. Modified Rarita-Schwinger Lagrangian In the following we will be working at the quadratic order, dropping in particular the four-fermion interaction in supergravity, and keep the lowest order of an expansion in powers of the dimensionless parameter T M p . The goldstino variation needs to modified : δG α (x) = √ 2 1 T 2 g μν T μν α + · · · ,(21) as well as the Lagrangian L = −T 4 − i T μν T 4Ḡσ μ ∂ ν G + · · · ,(22) As in the usual case, promoting the supersymmerty transformation to local ones requires dealing with the contribution of the goldstino energy density to the energy-momentum stress energy tensor. This requires adding a canceling cosmological constant term and a gravitino quadratic "mass" term. However, as the dispersion relation for the phonino is no more Lorentz invariant, we need to allow these quadratic terms to be non-Lorentz invariant. It is straightforward to see that the the Lagrangian: L = μνρσψ μσν ∂ ρ ψ σ + i 4 μνρσ n σγψμσρ σ γψ ν − i 4 μνρσ n σγ ψ μ σ ρσ γ ψ ν − i √ 2 T 2 M P T μν T 4 (ψ μσν G + ψ μ σ νḠ ) + i T μν T 4Ḡσ μ ∂ ν G + 1 4 T μν n μν T 4 GG + 1 4 T μν n μν T 4ḠḠ . is invariant under the modified supersymmetry transformations with Lorentz violating coefficients: δG α = √ 2T 2 ε α , δψ μα = −M P (2∂ μ ε α + in μν σ ν ααεα ) ,(23)δψ μα = −M P (2∂ μεα − in * μν ε α σ ν αα ) . and leads to the equation of motion for the goldstino. For n μν real, the requirement of supersymmetry in flat space implies that it should satisfy − 1 2 μνσρ λγκ ρ n νλ n σγ = T μκ M 2 P .(24) ICNFP 2013 00012-p.5 The unitary gauge is obtained by making a supersymmetry transformation to set G = 0: ψ μα → ψ μα + √ 2M P T 2 ∂ μ G α + i M P √ 2T 2 n μν σ ν ααḠα .(25) As a result we obtain the Generalized Rarita-Schwinger Lagrangian L = μνρσψ μσν ∂ ρ ψ σ − i 2 μνρσ n γ σψμσργψν + i 2 μνρσ n γ σ ψ μ σ ργ ψ ν .(26) and the corresponding equation of motion is μνρσσ ν ∂ ρ ψ σ − i 2 μνρσ n σγσρ σ γψ ν = 0 .(27) As for the usual Rarita-Schwinger case, two constraints are necessary in order to reduce the number of degrees of freedom of ψ μ to the four that describe a massive gravitino. The first is obtained by acting on the equation of motion by n μλ σ λ which gives − i 2 μνρσ n μλ n σγ σ λσ ρ σ γψ ν = 0 .(28) Using the symmetry of n μλ , this can be put in the form: T μν σ μψν = 0 ,(29) This replaces the standard F-term breaking constraintσ μ ψ μ = 0 of the gravitino. A procedure similar to the case of curved space-time [13], allows to get a second constraint by taking the component μ = 0 of (27). We shall illustrate how to apply these constraint in the next subsection. Explicit formulae for a perfect fluid Here will show how the gravitino mass can be expressed as a function of the fluid variables. We will consider relativistic ideal fluids with stress-energy tensor T μν = ( + p)u μ u ν + pη μν ,(30) where u μ is the fluid four-velocity u μ u μ = −1. In order to solve (24) we parametrize the solution n μν as n μν = (n T − n L )u μ u ν + n T η μν .(31) Plugging n μν and T μν and solving for n T and n L we get n 2 T = ε 3M 2 P , −n T (n T + 2n L ) = p M 2 P ,(32) hence n L = −n T ε + 3p 2ε .(33) Note that for F-term breaking, ε = −p, n L = n T . For constant energy density and pressure, in the fluid rest frame n ν μ = diag(n L , n T , n T , n T ). We introduce the notation / D = σ μ ∂ μ , / ∂ = σ i ∂ i ,(34) and Ψ =σ μ ψ μ ,ψ 1 2 =σ i ψ i , ψ1 2 = σ iψ i .(35) The constraint (29) can be used to solve for one of the components ψ 0 = −v σ 0 ψ 1 2 ,(36) where v = p is the phonino velocity. The component μ = 0 of equation of motion gives the constraint / ∂ψ 1 2 − in T ψ 1 2 + ∂ · ψ = 0 .(37) Putting all the constraints together leads to (σ 0 ∂ 0 + v/ ∂)ψ 1 2 − imψ 1 2 = 0 .(38) This is the Dirac equation satisfied by the longitudinal spin-1/2 mode with masŝ m = n L + n T 2 = n T 4 \|(1 − 3v)\| = √ 3 4M P p − ε 3 √ ε .(39) where as the eqs. (32) determine n L , n T only up to a sign, we have used this freedom in the last equation to have a positive massm. The projector on the transverse part of the spinor is ψ j = ψ T j − 1 2 σ j − k j / k 2k 2 ψ 1 2 + 3k j 2k 2 + 1 2 σ j / k k 2 k · ψ .(40) and the transverse part satisfies then the decoupled equation (σ 0 ∂ 0 +/ ∂)ψ T j + imψ T j = 0 .(41) In the fluid, the gravitino has two distinct propagating modes, the longitudinal and the transverse, with the same mass but different dispersion relations. Pseudo-goldstinos and Dirac gravitinos from extra dimensions In the previous section, we have discussed the presence of a goldstino associated breaking supersymmetry in the global limit. We then proceeded to coupling it to gravity in order to obtain the massive gravitino Lagrangian. We shall now discuss a different case where the supersymmetry breaking is intimately related with the gravity sector: the goldstino is given by a gravitino component along an internal dimension. Only in the limit of comparable sizes of supersymmetry breaking and compactification scales are the extra dimensions relevant and we shall therefore focus on such case. For simplicity, we will illustrate these in the very simple case of one single extra-dimension. We will also mention the possibility to tune the parameters to give Dirac type mass for the gravitino. ICNFP 2013 00012-p.7 Minimal supergravity in five dimension For the sake of establishing our notations and to illustrate the main ideas, we shall start the discussion with the simplest case of a five-dimensional space parametrized by coordinates (x μ , x 5 ) with μ = 0, · · · , 3 and x 5 ≡ y parametrizing the interval S 1 /Z 2 . The latter can be constructed as an orbifold from the circle of length 2πR (y ∼ y + 2πR) through the identification y ∼ −y. We take the theory in the bulk to be the five-dimensional supergravity with the minimal on-shell content: the fünfbein e A M , the gravitino Ψ MI and the graviphoton B M . We focus on the Lagrangian part involving the gravitino and drop the terms involving B M as well as the spin connection as we will consider for simplicity only the case of a flat extra-dimension. The on-shell Lagrangian is given by 2 [14]: L S UGRA = e 5 − 1 2 R (ω) + i 2Ψ I M Γ MNP ∂ N Ψ PI + · · ·(42) and the on-shell supersymmetry transformations are : δe A M = iΞ I Γ A Ψ MI δΨ MI = 2∂ M Ξ I + · · ·(43) where Ξ is the supersymmetry transformation parameter. The five-dimensional spinors Ψ MI and Ξ I are symplectic Majorana spinors. The five-dimensional gravitino Ψ MI will be written using the twocomponent Weyl spinors ψ MI as: Ψ M1 = ψ M1 ψ M2 , Ψ M2 = −ψ M2 ψ M1(44) and the supersymmetry transformation parameter as: Ξ 1 = −Ξ 2 = 1 2 , Ξ 2 = Ξ 1 = − 2 1 Ξ 1 = −Ξ 2 = − 1 , 2 ,Ξ 2 =Ξ 1 = 2 , 1 .(45) The on-shell supersymmetry transformations in two-component spinor notation are given by: δe a M = i 1 σ a ψ M1 + 2 σ a ψ M2 + h.c. δeˆ5 M = 2 ψ M1 − 1 ψ M2 + h.c. δψ 1M = 2∂ M 1 + · · · δψ 2M = 2∂ M 2 + · · ·(46) The fermionic part of the bulk Lagrangian expressed in two-component spinor notation reads now: L Fermi = e 5 1 2 μνρλ ψ 1μ σ ν ∂ ρ ψ 1λ + ψ 2μ σ ν ∂ ρ ψ 2λ + e 5 5 ψ 1μ σ μν ∂ 5 ψ 2ν − ψ μ2 σ μν ∂ 5 ψ 1ν −e 5 5 ψ 15 σ μν ∂ μ ψ ν2 − ψ 25 σ μν ∂ μ ψ 1ν + ψ 1μ σ μν ∂ ν ψ 25 − ψ 2μ σ μν ∂ ν ψ 15 + h.c. + · · · (47) where the five-dimensional covariant derivatives expressed have been replaced by partial derivatives as we work in a flat metric. SUSY breaking through twisted boundary conditions We will perform our study in the simplest case with no branes in the bulk other than the boundary ones at y = 0 and y = πR, as it contains all the qualitative features. The twisted boundary conditions fields basis Every generic field ϕ has a well defined Z 2 transformation: Z 2 : ϕ(y) → P 0 ϕ(−y)(48) that allows us to define the orbifold S 1 /Z 2 from the original five-dimensional compactification on S 1 . Here P 0 is the parity of the field ϕ which obeys P 2 0 = 1. The Lagrangian (42) and supersymmetry transformations (43) must be invariant under the action of the mapping (48). A possible choice of parity assignments is ψ 1μ (−y) = +ψ 1μ (y). At the point y = 0, the other fields parity transformations are determined from invariance of supersymmetry transformations under the mapping (48). We must assign a parity P π for each generic field ϕ at the point y = πR which keeps the Lagrangian and the supersymmetry transformations invariant ϕ(πR + y) = P π ϕ(πR − y). We also need to impose periodicity condition, we choose to be: ψ M1 (y + 2πR) ψ M2 (y + 2πR) = cos(2πω) sin(2πω) − sin(2πω) cos(2πω) ψ M1 (y) ψ M2 (y)(51) which correspond for ω 0 to implement a Scherk-Schwarz supersymmetry breaking in the bulk [15]. Then, invariance of the supersymmetry transformations under the Z 2 mapping (50) determines the parities of all fields. The result is given in table 1, Table 1. Parity assignments for bulk fields at y = 0 and y = πR. ψ μ+ = cos(πω)ψ μ1 − sin(πω)ψ μ2 ψ μ− = sin(πω)ψ μ1 + cos(πω)ψ μ2 ψ 5+ = sin(πω)ψ 51 + cos(πω)ψ 52 ψ 5− = cos(πω)ψ 51 − sin(πω)ψ 52 + = cos(πω) 1 − sin(πω) 2 − = sin(πω) 1 + cos(πω) 2 (52) ICNFP 2013 00012-p.9 The Periodic fields basis It is often useful to work in a basis of periodic fieldsψ MI ( ie.ψ MI (x, y + 2πR) =ψ MI (x, y)) in contrast to the multi-valued ψ MI used up to now. These are related by the rotation: ψ M1 ψ M2 = cos[ f (y)] sin[ f (y)] − sin[ f (y)] cos[ f (y)] ψ M1 ψ M2 , f (y) = ω R y(53) The supersymmetry breaking mass terms for the gravitinos is then manifest as we perform this fields transformation in the kinetic terms of the Lagrangian to give the action: S Kinetic = 2πR 0 dy d 4 x 1 2 e 5 1 2 μνρλ ψ μ1 σ ν ∂ ρψλ1 +ψ μ2 σ ν ∂ ρψλ2 +e 5 5 ψ μ1 σ μν ∂ 5ψν2 −ψ μ2 σ μν ∂ 5ψν1 − 2e 5 5 ψ 51 σ μν ∂ μψν2 −ψ 52 σ μν ∂ μψν1 − ω R e 5 5 ψ μ1 σ μνψ ν1 +ψ μ2 σ μνψ ν2 + h.c. .(54) with the fields now being periodic. Going to the new basis requires then the following redefinition for the supersymmetry transformation parameters, 1 2 = cos[ f (y)] sin[ f (y)] − sin[ f (y)] cos[ f (y)] ˜ 1 2 ,(55) and the supersymmetry transformations take now the form: δψ μ1 = 2∂ μ˜ 1 + · · · δψ μ2 = 2∂ μ˜ 2 + + · · · δψ 51 = 2∂ 5˜ 1 + 2 d f dy˜ 2 + · · · δψ 52 = 2∂ 5˜ 2 − 2 d f dy˜ 1 + · · ·(56) where · · · stand for higher order terms and terms proportional to F MN . It is important to note that the fieldsψ 51 andψ 52 transforms non linearly under supersymmetry transformations: they are the Goldstino fields associated with the supersymmetry breaking in the bulk and the supersymmetry breaking is measured by the d f dy therefore by the transformation between the two basis or stated differently, by the change of the gravitino component preserved at each point of the extra dimension. The super-Higgs mechanism From now on we drop the˜from the fermion symbols and use the periodic basis unless stated differently. Equations (56) show that four fields ψ 15 and ψ 25 transform non linearly under supersymmetry transformations. These are the goldstinos associated with breaking of supersymmetry in the bulk that shall be a"absorbed" by the two gravitinos for the super-Higgs mechanism. In order to study further this effect we will concentrate on the bi-linear terms of the fermionic fields: ψ 1μ , ψ 2μ , ψ 15 , ψ 25 . Some field redefinition are necessary to obtain standard kinetic terms for EPJ Web of Conferences 00012-p.10 the fields ψ 5I : ψ 1μ → ψ 1μ + i √ 6 σ μ ψ 25 ψ 2μ → ψ 2μ − i √ 6 σ μ ψ 15 ψ 15 → 2 √ 6 ψ 15 ψ 25 → 2 √ 6 ψ 25 .(57) This leads to the Lagrangian density: L = 1 2 1 2 μνρλ ψ 1μ σ ν ∂ ρ ψ 1λ + ψ 2μ σ ν ∂ ρ ψ 2λ + ψ 1μ σ μν ∂ 5 ψ 2ν − ψ μ2 σ μν ∂ 5 ψ 1ν − i 2 ψ 15 σ μ ∂ μ ψ 15 + ψ 25 σ μ ∂ μ ψ 25 + ψ 15 ∂ 5 ψ 25 − ψ 25 ∂ 5 ψ 15 − ω R ψ 1μ σ μν ψ 1ν + ψ 2μ σ μν ψ 2ν + ψ 15 ψ 15 + ψ 25 ψ 25 −i √ 6 2 ∂ 5 ψ 15 σ μ ψ 1μ + ∂ 5 ψ 25 σ μ ψ 2μ + ω R ψ 25 σ μ ψ 1μ − ψ 15 σ μ ψ 2μ + h.c.(58) The first line represents the five-dimensional kinetic term for the four-dimensional gravitinos, the second line corresponds to mass terms coming from the propagation in the fifth dimension, the third line ψ 1μα → ψ 1μα + 2 3 R ω ∂ μ (ψ 25α + R ω ∂ 5 ψ 15α ) + i 1 √ 6 σ μαα (ψα 25 + R ω ∂ 5ψα 15 ) . ψ 2μα → ψ 2μα + 2 3 R ω ∂ μ (ψ 15α − R ω ∂ 5 ψ 25α ) + i 1 √ 6 σ μαα (ψα 15 − R ω ∂ 5ψα 25 ) .(59) It is straightforward to check that this gauge fixing term provides the cancellation of mixing terms between gravitino and goldstino fields, which is the aim of our gauge choice : L = 1 2 1 2 μνρλ ψ 1μ σ ν ∂ ρ ψ 1λ + ψ 2μ σ ν ∂ ρ ψ 2λ + ψ 1μ σ μν ∂ 5 ψ 2ν − ψ μ2 σ μν ∂ 5 ψ 1ν − ω R ψ 1μ σ μν ψ 1ν + ψ 2μ σ μν ψ 2ν + h.c.(60) In this gauge, the gravitino propagators have poles at their physical mass and the degrees of freedom of would-be goldstinos are eliminated, traded for the longitudinal components for the gravitinos, through the super-Higgs mechanism. The equations of motion for the gravitinos ψ μI (y) in the unitary gauge can be extracted from the Lagrangian (60): − 1 2 μνρλ σ ν ∂ ρ ψ 1λ + σ μν ∂ 5 ψ 2ν − ω R σ μν ψ 1ν = 0 − 1 2 μνρλ σ ν ∂ ρ ψ 2λ − σ μν ∂ 5 ψ 1ν − ω R σ μν ψ 2ν = 0 (61) ICNFP 2013 00012-p.11 Assuming the gravitinos have a four-dimensional mass m 3/2 : μνρλ σ ν ∂ ρ ψ λI = −2m 3/2 σ μν ψ νI (62) their equations of motion can take the form: ∂ 5 ψ 2μ + m 3/2 − ω R ψ 1μ = 0 ∂ 5 ψ 1μ − m 3/2 − ω R ψ 2μ = 0(63) A solution for the equations (63) in the interval 0 < y < πR satisfying the first condition in (71): ψ 1μ (y) = cos m 3/2 − ω R y ψ 1μ (0) ψ 2μ (y) = − sin m 3/2 − ω R y ψ 1μ (0).(64) The second condition in (71) is then used to determine the gravitino mass: m 3/2 = ω R + n R , n ∈ Z(65) Pseudo-goldstinos and brane localized gravitino mass terms Gravitino mass Matter fields live on branes localized for instance at particular points y = y n . Here, we will consider the simplest case where the branes are localized on the boundaries y b = 0, πR, as it contains all the qualitative features. The generalization can be found in . On each of these branes supersymmetry can be locally broken by the matter scalar potential and a "would-be-goldstino" appears localized. The corresponding action can be written as: S = 2πR 0 dy d 4 x 1 2 L BULK + L 0 δ(y) + L π δ(y − πR) .(66) There are four fields ψ 15 , ψ 25 , χ 0 and χ π transform non linearly under supersymmetry transformations. These are the "local would be goldstinos" associated with breaking of supersymmetry in the bulk and in the two branes respectively. As we have two gravitinos then two local would be goldstinos will be absorbed in the super-Higgs effect to give mass to the gravitino fields ψ 1μ and ψ 2μ , while two linear combination of the fields ψ 15 , ψ 25 , χ 0 and χ π remain as pseudo-goldstinos. The additional bi-linear terms of the fermionic fields: ψ μ1 , ψ 2μ , ψ 15 , ψ 25 , χ 0 and χ π take the form: ΔL = δ(y) − i 2 χ 0 σ μ ∂ μ χ 0 − M 0 ψ 1μ σ μν ψ 1ν + i √ 6 2 χ 0 + ψ 25 σ μ ψ 1μ + (χ 0 + ψ 25 ) (χ 0 + ψ 25 ) + δ(y − πR) − i 2 χ π σ μ ∂ μ χ π −M π ψ 1μ σ μν ψ 1ν + i √ 6 2 χ π + ψ 25 σ μ ψ 1μ + (χ π + ψ 25 ) (χ π + ψ 25 ) + h.c.(67) The modification necessary to fix the unitary gauge is straightforward and two would-be goldstinos are eliminated, absorbed to provide the longitudinal components for the gravitinos, through the super-Higgs mechanism while two remain in the spectrum with masses and fields content explicitly given EPJ Web of Conferences 00012-p.12 in . The gravitino equations of motion in the bulk-branes system. The equations of motion for the gravitinos ψ μI (y) are then given by: − 1 2 μνρλ σ ν ∂ ρ ψ 1λ + σ μν ∂ 5 ψ 2ν − ω R σ μν ψ 1ν = 2M 0 σ μν ψ 1ν δ(y) + 2M π σ μν ψ 1ν δ(y − πR) − 1 2 μνρλ σ ν ∂ ρ ψ 2λ − σ μν ∂ 5 ψ 1ν − ω R σ μν ψ 2ν = 0(68) Again, assuming the gravitinos have a four-dimensional mass m 3/2 : μνρλ σ ν ∂ ρ ψ λI = −2m 3/2 σ μν ψ νI(69) the equations of motion become: ∂ 5 ψ 2μ + m 3/2 − ω R ψ 1μ = 2M 0 ψ 1μ δ(y) + 2M π ψ 1μ δ(y − πR) ∂ 5 ψ 1μ − m 3/2 − ω R ψ 2μ = 0(70) Integration of the equations (70) near the points y = 0 and y = πR, taking into account the parity assumptions, leads to the following expressions for the discontinuities of the odd gravitino fields: ψ 2μ (0 + ) = M 0 ψ 1μ (0) = −ψ 2μ (0 − ) ψ 2μ (πR − ) = −M π ψ 1μ (πR) = −ψ μ2 (πR + ).(71) A solution for the equations (70) in the interval 0 < y < πR satisfying the first condition in (71): ψ 1μ (y) = cos m 3/2 − ω R y + M 0 sin m 3/2 − ω R y ψ 1μ (0) ψ 2μ (y) = M 0 cos m 3/2 − ω R y − sin m 3/2 − ω R y ψ 1μ (0).(72) and the second condition in (71) is then used to determine the gravitino mass: m 3/2 = ω R + 1 πR [arctan (M 0 ) + arctan (M π )] + n R , n ∈ Z(73) Pseudo-goldstinos We will concentrate now on the would-be goldstino fields ψ 15 (y), ψ 25 (y), χ 0 and χ π . In the unitary gauge, a stationary action (in order to derive of the equations of motion) is possible if: ∂ 5 ψ 15 + ω R ψ 25 = −2δ(y)M 0 (χ 0 + ψ 25 ) − 2δ(y − πR)M π (χ π + ψ 25 ) ∂ 5 ψ 25 − ω R ψ 15 = 0.(74) which imply that the fields ψ 5I (y), in the interval 0 < y < πR can be written as: ψ 15 (y) = 1 √ πR cos ω R y + θ χ 1 + sin ω R y + θ χ 2 ψ 25 (y) = 1 √ πR sin ω R y + θ χ 1 − cos ω R y + θ χ 2(75) ICNFP 2013 00012-p.13 where χ 1 and χ 2 are y independent 4d spinors and θ is a constant which corresponds to a choice of basis for χ 1 and χ 2 . Integrating the equations (74) near y = 0 and y = π we find: ψ 15 (0 + ) + M 0 χ 0 + ψ 25 (0) = 0 ψ 15 (πR − ) − M π χ π + ψ 25 (πR) = 0(76) which implies (for M π 0 and M 0 0): χ π = 1 √ πR − sin(ωπ + θ) + 1 M π cos(ωπ + θ) χ 1 + 1 √ πR cos(ωπ + θ) + 1 M π sin(ωπ + θ) χ 2 χ 0 = − 1 √ πR sin(θ) + 1 M 0 cos(θ) χ 1 + 1 √ πR cos(θ) − 1 M 0 sin(θ) χ 2 .(77) Here we see that there are two equations for four fermion fields, and we see how the super Higgs mechanism operates: from the original two 5d and two 4d degrees of freedom (ψ 15 (y), ψ 25 (y), χ 0 and χ π ), an infinity of Kaluza-Klein modes is absorbed to give mass to the fields ψ 1μ (y) and ψ 2μ (y) and only two degrees of freedom remain in the unitary gauge: the pseudo-goldstinos χ 1 and χ 2 . Dirac gravitino and R-symmetry We will restore the explicit dependence on the (reduced) five-dimensional Planck mass M 5 = κ −1 . It is related to the four-dimensional Planck mass M P by πRM 3 5 = M 2 P .(78) The four-dimensional gravitino mass can be read from (73): m 3/2 = ω R + 1 πR [arctan (κM 0 ) + arctan (κM π )] + n R ,(79) First consider the case with M 0 = M π = 0. It is well known [15,16,20] that the Scherk-Schwarz m mechanism described above, leads for ω = 1/2 to a tower of Dirac-type Kaluza-Klein excitation of fermions the bulk fermions. The two modes with n = 0 and n = −1 lead to two degenerate Majorana fermions with mass 1/2R. The two states correspond to the two orthogonal supersymmetry charges of N = 2 supergravity. One couples to the boundary at y = 0 and the other one couples to the one at y = πR. The original N = 2 Lagrangian is invariant under an S U(2) R R-symmetry, under which the gravitinos ψ M1 and ψ M2 transform in the representation 2 of S U(2) R . For ω = 1/2 there is a remanent U(1) R symmetry left with the Dirac gravitino charged under it. This R-symmetry corresponds to the exchange of the two boundaries y = 0 ↔ y = πR. For generic ω 0, 1/2 the R-symmetry is totally broken Consider the case where there are also contribution the gravitino mass from potential on the boundary branes. Looking at the mass formula: m 3/2 = n + ω R + 1 πR [arctan (κM 0 ) + arctan (κM π )] ,(80) As it stands the localized masses can shift the value of ω π to any desired value, it could for example be canceled or shifted to ω = 1/2 , by the appropriate choice of values of M 0 and M π . However, we EPJ Web of Conferences 00012-p.14 are mainly interested in the case when these localized masses arise from dynamics on the boundary branes and can have a four-dimensional description. Then, two remarks are in order. First, the Scherk-Schwarz twist can not compensate the effects of supersymmetry breaking due to F or D-term dynamics as explicitly shown in [7]. Second, the mass formula can be approximated by: m 3/2 n + ω R + κ πR (M 0 + M π ) .(81) and as κM b << 1 they cannot have a sizable effect on the numerical value of ω if this is not already small. For small values of ω, the lightest gravitino lies far below the Kaluza-Klein tower and a fourdimensional approximation can be used to study the system within four-dimensional supergravity. Dirac gauginos A few features of the goldstinos have been exhibited in the previous section: i) there might be more than one sector breaking supersymmetry (the bulk and all the branes located inside it or at the boundaries). ii) only a global description of the model allows to identify which of the linear combination of the would-be-goldstinos is the (true) goldstino while the rest, the pseudo-goldstinos, remain as matter fermions iii) if one starts with an extended R-symmetry due to the presence of of an extended supersymmetry in the gravitational sector, it is possible to build models where the breaking parameters can be tuned to keep a part of this R-symmetry unbroken with the gravitino having a Dirac mass. While previous sections focused mainly on the gravitational sector, we would like to discuss the important role played by similar features in the case of models with Dirac gauginos [21,22]. Let us first start with the first point: why would we need two or more would-be-goldstinos? A gaugino λ a acquire Dirac masses m D (λ a ψ a ) by coupling to other chiral fermions ψ a in the adjoint representation. This gaugino mass is soft and can be seen as originating either from U(1) a non-vanishing D-terms or F-term F b . In a gauge mediation type scenario, new states are introduced at a mass scale M m (we consider a single scale for simplicity) to serve as mediators of the breaking, and they couple to the visible and secluded sector through gauge couplings of strengths g, g mD a and g mFb , respectively (see for example [23] for discussion on gravity mediation). The exchange of loops of these messengers induces soft masses that scale parametrically as: m D = g √ 2 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ a c Da 1 2 g mDa 8π 2 D a M m + b c Fb 1 2 g mFb 16π 2 F 2 b 6M 3 m ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠(82) where the coefficients c Da 1 2 and c Fb 1 2 are calculable model dependent coefficients that take into account summation on other quantum numbers of the messengers. The chiral adjoints, pairing up with the gauginos, have scalar superpartners Σ in the adjoint representation of the gauge group. The soft induced masses are parametrized as m 2 Σ tr(Σ † Σ) + 1 2 B Σ tr(Σ 2 + (Σ † ) 2 ) and are given by m 2 Σ = a g 2 mDa c DaΣ2 1 96π 2 D 2 a M 2 m + c DaΣ1 3D a 64π 2 + b g 2 mFb c FbΣ2 1 16π 2 F 2 b M 2 m B Σ = −2 a c DaΣ3 g 2 mDa 96π 2 D 2 a M 2 m − 2 b c FbΣ3 g 2 mFb 16π 2 F 2 b M 2 m(83) As the B Σ contribution tends to make tachyonic the mass of one of the component of the adjoint scalars, the generation of viable soft mass for the adjoint scalars turns out to be not totally trivial. ICNFP 2013 00012-p.15 The simplest interactions between the DG-adjoints and the messengers, as the Yukawa couplings descending from N = 2 lead to tachyonic masses as the B Σ contribution. Historically, this was the main reason for abandoning the Dirac gaugino scenario in [21]. To avoid such a result, the required forms of the adjoint-messengers interactions have been fully classified in [24,25] (see also [26]) . The scalar partners of the chiral fermions in the visible sector get leading order contribution in D/M 2 m and F/M 2 m to their soft masses at three-loop from D-term and two-loop from the F-term: m 2f = i C if (m iD ) 2 α i π log ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ m (i) Σ P m bD ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ 2 + 2c F0 i,b C if α i 4π 2 F 2 b M 2 m(84) where C if is the quadratic Casimir of the field f under group i. To allow Dirac gaugino masses generated at the leading order in the supersymmetry breaking parameter, and sufficiently heavy selectrons, we shall consider a combination of D-and F-term breaking, with both D-and F-terms comparable. This will generate a spectrum with masses of generic order of magnitude (a) gaugino masses ∼ g mD g M m . and we expect the adjoint scalars to be the most massive states. Explicit models can be found, for example, in [25]. The requirement of using both an F-term and a D-term means that the goldstino will be a linear combination of the U(1) D gaugino λ Da and of the chiral fermion χ b associated with the F-term, G = a D a λ Da + b F b χ b a D 2 a + b F 2 b .(85) The goldstino is "absorbed" through the super-Higgs mechanism as explained in the previous sections. Its coupling to matter is easy to obtain. More interesting is the fate of the fermions orthogonal to G. The obvious possibility is that all become pseudo-goldstinos, therefore remains as one of the massive matter fermions. However another possibility is one of them will be absorbed as a true goldstino by another gravitino. As λ D and χ are not visible sector fields, there is no obstruction to make them part of an N = 2 sector that will also include the gravitinos [27]. For instance, in the case of just two would be goldstinos, associated with one F and one D-term, one can use the orthogonal combination to G to be absorbed by the the second gravitino. This can be tuned as the couplings of the two gravitinos to these fields can appear different, even with opposite sign, as seen in the previous section. In this case, the second gravitino will have the same mass as the visible sector one. The two degenerate Majorana spin-3/2 states will combine to give rise to a Dirac gravitino that preserves an R-symmetry. Of course, the Higgs sector still breaks R-symmetry as this seems necessary to obtain the right size of the Higgs mass. But the induced Majorana mass for the gauginos can be kept very small. In summary: • A generic Dirac gaugino model build for phenomenological purpose would involve at least two different supersymmetry breaking sources corresponding to non-vanishing D-term and R-preserving F-term. • Each sector of the breaking gives rise to a would-be-goldstino and therefore there are at least two of them. • The gravitino is chosen to to originate in an extended N = 2 supersymmetric structure. The two gravitinos could eat two linear combination of the would-be-goldstinos giving rise to degenerate Majorana masses that combine in an R-preserving Dirac mass. following definitions have been introduced: R . The complex conjugate of a left Weyl spinor is a right Weyl spinor. 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[ "Minding Impacting Events in a Model of Stochastic Variance", "Minding Impacting Events in a Model of Stochastic Variance" ]	[ "Duarte Queiró S Sm ", "Emf M F Curado ", "F D Nobre " ]	[]	[ "PLoS ONE" ]	We introduce a generalization of the well-known ARCH process, widely used for generating uncorrelated stochastic time series with long-term non-Gaussian distributions and long-lasting correlations in the (instantaneous) standard deviation exhibiting a clustering profile. Specifically, inspired by the fact that in a variety of systems impacting events are hardly forgot, we split the process into two different regimes: a first one for regular periods where the average volatility of the fluctuations within a certain period of time W is below a certain threshold, w, and another one when the local standard deviation outnumbers w. In the former situation we use standard rules for heteroscedastic processes whereas in the latter case the system starts recalling past values that surpassed the threshold. Our results show that for appropriate parameter values the model is able to provide fat tailed probability density functions and strong persistence of the instantaneous variance characterized by large values of the Hurst exponent (Hw0:8), which are ubiquitous features in complex systems.	10.1371/journal.pone.0018149	null	5,188,812	1102.4819	a1b7a52fd53e2086e0eef370f1ad8f0c614b9429	Minding Impacting Events in a Model of Stochastic Variance 2011 Duarte Queiró S Sm Emf M F Curado F D Nobre Minding Impacting Events in a Model of Stochastic Variance PLoS ONE 63181492011Received November 15, 2010; Accepted February 22, 2011;Editor: Funding: This work was funded by Conselho Nacional de Desensolvimento Científico e Tecnoló gico (www.cnpq.br); Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro (www.faperj.br); and Marie Curie Actions FP7-PEOPLE-2009-IEF (contract nr 250589) (http://ec.europa.eu/research/ mariecurieactions/). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Competing Interests: The authors have declared that no competing interests exist. * ¤ Current address: Istituto dei Sistemi Complessi -CNR, Roma, Italy We introduce a generalization of the well-known ARCH process, widely used for generating uncorrelated stochastic time series with long-term non-Gaussian distributions and long-lasting correlations in the (instantaneous) standard deviation exhibiting a clustering profile. Specifically, inspired by the fact that in a variety of systems impacting events are hardly forgot, we split the process into two different regimes: a first one for regular periods where the average volatility of the fluctuations within a certain period of time W is below a certain threshold, w, and another one when the local standard deviation outnumbers w. In the former situation we use standard rules for heteroscedastic processes whereas in the latter case the system starts recalling past values that surpassed the threshold. Our results show that for appropriate parameter values the model is able to provide fat tailed probability density functions and strong persistence of the instantaneous variance characterized by large values of the Hurst exponent (Hw0:8), which are ubiquitous features in complex systems. Introduction For the last years the physical community has broaden its subject goals to matters that some decades ago were too distant from the classical topics of Physics. Despite being apparently at odds with the standard motivations of Physics, this new trend has given an invaluable contribution toward a more connected way of making Science, thus leading to a better understanding of the world surrounding us [1]. Within this context, the major contribution of physicists is perhaps the quantitative procedure, reminiscent of experimental physics, in which a model is proposed after a series of studies that pave the way to a reliable theory. This path has resulted in a series of findings which have helped such diverse fields as physiology, sociology and economics, among many others [2][3][4]. Along these findings, one can mention the determination of non-Gaussian distributions and long-lasting (power-law like) correlations [5][6][7]. Actually, by changing the observable, the conjunction of the two previous empirical verifications is quite omnipresent. For this reason and regardless the realm of the problem very similar models have been applied with particular notoriety to discrete stochastic processes of timedependent variance based on autoregressive conditional heteroscedastic models [8]. That is to say, most of these models are devised taking basically into account the general features one aims at reproducing, rather than putting in elements that represent the idiosyncracies of the system one is surveying. For instance, many of the proposals cast aside the cognitive essence prevailing on many of these systems, when it is well known that in real situations this represents a key element of the process [9]. On the other hand, intending to describe long-lasting correlations, long-lasting memories are usually introduced thus neglecting the fact that we do not traditionally keep in mind every happening. As a simple example, we are skilled at remembering quotidian events for some period. However, we will discard that information as time goes by, unless the specific deed either created an impact on us or has to do with something that has really touched us somehow. In this case, it is likely that the fact will be remembered forever and called back in similar or related conditions, which many times lead to a collective memory effect [10]. In this work, we make use of the celebrated heteroscedastic model, the ARCH process [11] and modify it by pitching at accommodating cognitive traits that lead to different behavior for periods of high agitation or impact. Particularly, we want to stress on the fact that people tend to recall important periods, no matter when they took place. To that end, we introduce a measure of the local volatility, as well as a volatility threshold, so that the system changes from a normal dynamics, in which it uses the previous values of the variable to determine its next value, to a situation in which it recalls the past and compares the current state with previous states of high volatility, even if this past is far. Standard models of heteroscedasticity The Engle's formulation of an autoregressive conditional heteroscedastic (ARCH) time series [11] represents one of the simplest and effectual models in Economics and Finance, for which he was laureated the Nobel Memorial Prize in Economical Sciences in 2003 [12]. Explicitly, the ARCH corresponds to a discrete time, t, process associated with a variable, z t , z t~st v t ,ð1Þ with v t being an independent and identically distributed random variable with zero mean and standard deviation equal to one. The quantity s t represents the time-dependent standard deviation, which we will henceforth name instantaneous volatility for mere historical reasons. Traditionally, a Gaussian is assigned to the random variable v t , but other distributions, namely the truncated a-stable Lévy distribution and the q-Gaussian (Student-t) have been successfully introduced as well [13,14]. In his seminal paper, Engle suggested that the values of s 2 t could be obtained from a linear function of past squared values of z t , s 2 t~a z X s i~1 b i z 2 t{i , a,b i §0 ð Þ :ð2Þ In financial practice, viz., price fluctuations modelling, the case s~1 (b 1 :b) represents the very most studied and applied of all the ARCH s ð Þ-like processes. The model has been often applied in cases where it is assumed that the variance of the observable (or its fluctuation) is a function of the magnitudes of the previous occurrences. In a financial perspective, Engle's proposal has been associated with the relation between the market activity and the deviations from the normal level of volatility a, and the previous price fluctuations making use of the impact function [8]. Alternatively, recent studies convey the thesis that leverage can be responsible for the volatility clustering and fat tails in finance [15]. Nonetheless, the heteroscedastic ARCH-like processes has been repeatedly used as a forecasting method. In other words, one makes use of the magnitude of previous events in order to indicate (or at least to bound) the upcoming event (see e.g. [16,17]). In respect of its statistical features, although the time series is completely uncorrelated, Sz t z t' Td tt' , it can be easily verified that the covariance S z t j j z t' j jT is not proportional to d tt' . As a matter of fact, for s~1, it is provable that Sz 2 t z 2 t' T decays according to an exponential law with a characteristic time t: ln b j j {1 . This dependence does not reproduce most of the empirical evidences, particularly those bearing on price fluctuations studies. In addition, the introduction of a large value of s used to give rise to implementation problems [18]. Expressly, large values of s augment the difficulty of finding the appropriate set of parameters b i f g for the problem under study as it corresponds to the evaluation of a large number of fitting parameters. Aiming to solve this short-coming of the original ARCH 1 ð Þ process, the GARCH s,r ð Þ process was introduced [19] (where G stands for generalized), with Eq. (2) being replaced by, s 2 t~a z X s i~1 b i z 2 t{i z X r i~1 c i s 2 t{i a,b i ,c i §0 ð Þ :ð3Þ In spite of the fact that the condition, bzcv1, guarantees that the GARCH 1,1 ð Þ process exactly corresponds to an infiniteorder ARCH process, an exponential decay for Sz 2 t z 2 t' T, with t: ln bzc ð Þ j j {1 is found. Although the instantaneous volatility is time dependent, the ARCH(1) process is actually stationary with the stationary variance given by, Ss 2 T~b s 2 s 2~a 1{b , (bv1),ð4Þ (herein S . . .T represents averages over samples at a specified time and c . . . . . . denotes averages over time in a single sample). Moreover, it presents a stationary probability density function (PDF), P z ð Þ, with a kurtosis larger than the kurtosis of distribution P(v). Namely, the fourth-order moment is, Sz 4 T~a 2 Sv 4 T 1zb 1{b ð Þ 1{b 2 Ss 4 T ð Þ : This kurtosis excess is precisely the outcome of the dependence of s t on the time (through z). Correspondingly, when b~0, the process is reduced to generating a signal with the same PDF of v, but with a standard variation equal to ffiffi ffi a p . At this point, it is convenient to say that, for the time being and despite several efforts, there are only analytical expressions describing the tail behavior of P(z) or the continuous-time approximation of the ARCH(1) process with the full analytical formula still unknown [14,20]. In order to cope with the long-lasting correlations and other features such as the asymmetry of the distribution and the leverage effect, different versions of the ARCH process have been proposed [8,18]. To the best of our knowledge, every of them solve the issue of the long-lasting correlations of the volatility by way of introducing an eternal dependence on z 2 i in Eq. (2), b i :b K i ð Þ, with K : ð Þ representing a slowly decaying function [8,21]. Most of these generalizations can be encompassed within the fractionally integrated class of ARCH processes, the FIARCH [22][23][24][25]. The idea supporting the introduction of a power-law for the functional form of K : ð Þ is generally based on the assumption that the agents in the market make use of exponential functions K : ð Þ with a broad distribution of relaxation times related to different investment horizons [26,27]. This type of model has achieved a huge popularity in the replication of non-Gaussian time series in several areas, such as biomedicine, climate, engineering, and physics (a few examples can be found in [28][29][30][31][32][33][34][35]). As described above, the statistical features of the macroscopic observables are the result of the nature of the interactions between the microscopic elements of the system and the relation between microscopic as well as the macroscopic observables. In the case of the ''financial'' ARCH process, it was held that z 2 i bears upon the impact of the price fluctuations on the trading activity. On the one hand, it is understood that the impact of the price fluctuations (or trading activity) on the volatility does not merely come from recent price fluctuations and it does actually involve past price fluctuations. In finance, upgraded versions of heteroscedasticity models use multi-scaling, i.e., it is assumed that the price will evolve by modulating the volatility according to the volatility over different scales (days, weeks, months, years, etc.) [36] in order to smooth their possible misjudgement about the volatility. However, in practice, these models do not differ much from FIARCH-like proposals at the level of the results we are pointing at. Alternately, it is worthwhile to look upon the ARCH proposal as a mechanism of forecast [16,17]. In this way, the simplest approach, the ARCH(1), represents an attempt to foresee future values just taking into account recent observations, whereas models like the FIARCH bear in mind all the history weighting each past-value according to some kernel functional. Minding impacting events In our case, we want to emphasize the fact that people tend to recall periods of high volatility (i.e., impact) in the system, no matter when they took place, by changing the surrounding conditions as agent-based models suggested [37,38]. Hence, we introduce a measure of the local volatility, v t~1 W X W {1 i~0 z 2 t{i ,ð5Þ and a threshold, w, so that instead of Eq. (2), the updating of s 2 t goes as follows: [39,40]. Therefore, if we assume the financial market perspective, we are implicitly presuming that the characteristic time, t, is Dirac delta or at least narrow distributed, so that the exponential functional is a valid approximation. This approach is confirmed by recent heuristic studies in which it has been verified that the largest stake of the market capitalization is managed by a small number of companies that apply very similar strategies [41]. With the second branch equation we intend to highlight the difference in behavior of the "normal" periods of trading and the periods of significant volatility, in which the future depends on the spells of significant volatility in the past as well. The values b s 2 t~a z P t i~1 b i z 2 t{i if v t{1 vw, az P t i~1 b 0 i z 2 t{i if v t{1 §w, 8 > > > < > > > :ð6Þwhere b i~b K i ð Þ~b exp { i = t ½0 i are defined as, b 0 i~b p i H v t{i {w ½ ,ð7Þ with H . . . ½ being the Heaviside function and p i is a factor that represents a measure of the similarity (in the volatility space) between the windows of size W with upper limits at z t and z t{W z1 , respectively. Analytically, this is equivalent to mapping segments in the form z 2 i , . . . ,z 2 i{W z1 È É into vectors in R zW 0 and afterward computing a normalized internal product-like weight, p i~1 N X W j~1 z 2 t{j z 2 i{j ,ð8Þ where, for the sake of simplicity, we set aside the time dependence of p i and b 0 i in the equations, while N represents the normalization factor such that P i p i~1 for all i (with fixed t). We are therefore dealing with a model characterized by 5 parameters, namely: a (the normal level of volatility) and b (the impact of the observable in the volatility), which were both first introduced by Engle in [11]; t, put forward in exponential models; and two new parameters W (representing the volatility spell) and w that we will reduce to a single extra parameter. If we think of trading activities, our proposal introduces a key parameter, the volatility threshold, w, which signals a change in behavior of the agents in the market. At present, significant stake of the trading in financial markets is dominated by short-term positions and thus a good part of the dynamics of price fluctuations can be described by Eq. (2), or by functions with an exponential kernel. As soon as the market fluctuates excessively, i.e., the volatility soars beyond the threshold, the market changes its trading dynamics. The main forecast references are obviously the periods where the volatility has reached high levels and afterward, the periods of those which are most similar; this is the rationale described by our Eq. (8). Thence, our proposal is nothing but the use of simple mechanisms that in a coarse-grained way master a good part of our decisions. Results General results In this section we present the results obtained by the numerical implementation of the model. For comparison, we will use the results of a prior model that can be enclosed in the class of FIARCH processes [25]. There, the adjustment of the parameters comes from the delicate balance between the parameter b, which is responsible for introducing deviations of the volatility from its normal level a, and the parameter controlling the memory. On the one hand, large memory has the inconvenient effect of turning constant the instantaneous volatility, so that after a seemly number of time steps the value of s becomes constant, hence leading to a Gaussian (or close to it) distribution of the variable z, independently of how large b is. On the other hand, short memory is unable to introduce long-range correlations in the volatility, although it enhances larger values of kurtosis excess. The model we introduce herein is rather more complex. In order to deal with the change of regime, we define a parameter establishing this alteration and we need to specify W and t. Henceforth, we have assumed W~t, which is very reasonable as it imposes that the volatility and the time scale that the agents in the market use to assess the evolution of the observable are the same. In order to speed up our numerical implementation, we have imposed a cutoff of 10W in the computation of the first line in Eq. (6). This approximation turns the numerical procedure much lighter with a negligible effect because the influence of the discarded past is not much relevant in numerical terms (within standard numerical implementation error). In all of our realizations, we have used a normalized level of expected volatility, a~1, and we have defined the volatility threshold in units of a= 1{b ð Þ, following a stationary approach, as well. We have adjusted the probability distributions of z by means of the distribution, P z ð Þ~Z {1 1zBz 2n À Á 1 1{q 0 ,ð9Þ the behavior of which follows a power-law distribution for large jzj with an exponent equal to 2n q 0 {1 and where (using Ref. [42], sec. 3.194), ð z n (1zBz 2n ) 1 1{q 0 dz1 z {1 ð Þ n 1zn C 2nznz1 2n ! C 2nz nz1 ð Þ 1{q 0 2n q 0 {1 À Á 2 4 3 5 B 1zn 2n C 1 q 0 {1 ! ,ð10Þ 2n q 0 {1 w1zn , and Z represents the previous integral with n~0. The fittings for the probability density distribution (9) were obtained using non-linear and maximum log-likelihood numerical procedures and the tail exponents double-checked with the value given by the Hill estimator [43,44]. As a matter of fact, values of n different from 1 have only been perceived for large values of b and small values of w (slightly larger) or large values of w (slightly smaller). For n~1 and q 0 =1, the PDF corresponds to a q 0 -Gaussian distribution (or Student-t distribution) [45] and when q 0~1 we have either the Gaussian (n~1) or the stretched distribution (n=1). Since that in the majority of the applications one is interested in the tail behavior, we have opted for following the same approach by defining the tail index as, 2 q{1~2 n q 0 {1 uq~n {1 q 0 zn{1 :ð11Þ In spite of the fact that other functional forms could have been used, we have decided on Eq. (9) because of its statistical relevance and simplicity (in comparison with other candidates involving special functions, namely the hypergeometric). Moreover, the q-Gaussian (t-Student) is intimately associated with the long-term distribution of heteroscedastic variables since it results in the exact distribution when the volatility follows an inverse-Gamma distribution [35,[46][47][48]. Concerning the persistence of the volatility, we have settled on the Detrended Fluctuation Analysis (DFA) [49], which describes the scaling of a fluctuation function related to the average aggregated variance over segments of a time series of size ', F ' ð Þ' H ,ð12Þ where H is the Hurst exponent. Although it has been shown that Fluctuation Analysis methods can introduce meaningful errors in the Lévy regime [50], we have verified that for our case, which stands within the finite second-order moment domain, the results of DFA are so reliable as other scaling methods. Let us now present our results for b~0:5, which is able to depict the qualitative behavior of the model for small b. This case corresponds to a situation of little deviation from the Gaussian, when long-range memory is considered. In accordance, we can analyse the influence of the threshold w and W . Overall, we verify a very sparse deviation from the Gaussian. Keeping W fixed and varying w, we understand that for small values of w the distribution of z t is Gaussian and the Hurst exponent of jz t j is 1=2. It is not hard to grasp this observation if we take into account that, by using small values of w, we are basically employing almost all of the past values which limits the values of instantaneous volatility to a constant value after a transient time. As we increase the value of w, we let the dynamics be more flexible and therefore the volatility is able to fluctuate, resulting in a kurtosis excess. For small values of W , the Hurst exponent is slenderly different from 1=2 and the value of the Hurst exponent increases with W . However, because of the small value of b, the rise of W turns out the distribution of z barely undistinguishable from a Gaussian. This behavior is described in Fig. 1. We have obtained a Gaussian distribution and a Hurst exponent H~0:5 for small values of w (w~0:1) and W (W~5). When we augment the value of the threshold, w~5, the system is loose and the instantaneous volatility is able to fluctuate leading to the emergence of tails (q~1:09) and a subtle increase of the Hurst exponent (H~0:52+0:01). Hiking up both W and w (W~75 and w~2), we have achieved large values of the Hurst exponent (H~0:58+0:02), but the small value of b is not sufficient to induce relevant fluctuations, bringing on a distribution that is almost Gaussian (q~1:02). The distribution fittings were assessed by computing the critical value P Ã KS~1 {a crit from the Kolmogorov-Smirnov test [51] that are equal to 0:9634 and 0:9454, respectively. As we increase the value of b, we favor the contribution of the past values of the price dynamics, thus, for the same value of W we are capable of achieving larger values of the kurtosis excess, that we represent by means of the increase of the q index. The same occurs for the Hurst exponent. This general scenery is illustrated in Fig. 2 for the value b~0:998, where we present the dependence of q and H with w, for different choices of W . Again, the higher W , the lower the tail index q, because the extension of the memory surges a weakening of the fluctuations in the volatility. The opposite occurs with the Hurst exponent, which increases towards unit (ballistic regime) as we consider W larger, for obvious reasons. In all the cases of b,W ð Þinvestigated, we verified that both q and H augment with w. The assessment of the numerical adjustments is provided in Tab. 1 in the form of the P Ã KS critical values from the Kolmogorov-Smirnov test [51]. The only case we obtained a value 1 (within a five-digit precision) was for the pair W~10 and w~5, which results in a value quite close to the limit of finite secondorder moment (a fat-tailed distribution with q~5=3). At this point it is worth saying that we have investigated the likelihood of other well-known continuous distributions, such as the stretchedexponential, the simple t-Student, Lévy, and Gaussian. Nonetheless, the fittings carried with Eq. (9) outperformed every other analyzed distribution. Concerning the instantaneous volatility, s t , we verified that the Dirac delta distribution, p s ð Þ~d s{1 ð Þ, starts misshaping and short tails appear as we depict in Fig. 3 (upper panel) for the case b~0:998, W~75 and w~0:25. Considering this particular case, we can present relevant evidence of the effectiveness of our proposed probability distribution approach. The empirical distribution function in the upper panel of Fig. 3 may be simply approximated by p s ð Þ~f 1 2c if s=1 1{f ð Þd s{1 ð Þ otherwise 8 < : ,ð13Þ with c §0, f ƒ1, and s [ 1{c,1zc ð Þ ; when f~0 we recover the homoscedastic process distribution as a particular case. Reminding that at each time step the distribution is a Gaussian (conditioned to a time-dependent value of s) the long-term distribution is, P z ð Þ~ð 1zc 1{c p s ð Þ 1 ffiffiffiffiffi ffi 2p p s exp { z 2 2s 2 ! ds,ð14Þ which gives (Ref. [42], sec. 3.351), P z ð Þ~f 4 ffiffiffiffiffi ffi 2p p c Ei { z 2 2 1{c ð Þ 2 " # {Ei { z 2 2 1zc ð Þ 2 " # ! z 1{f ffiffiffiffiffi ffi 2p p exp { z 2 2 !ð15Þ where Ei : ½ is the Exponential Integral function (see e.g. Ref. [52]). Considering c~1=2 (which is appropriate to the case shown) and taking for the sake of simplicity f~1=2, we obtain the function presented in Fig. 4, the kurtosis of which is k~1 0854 3125 &3:47 (making use of Ref. [42], sec. 5.221). Actually, this curve is represented in the scaled variable z=s so that the standard deviation, which is originally equal to cz1 ð Þ 3 z c{1 ð Þ 3 z6c 12c , becomes equal to one, like in other depicted distributions. The accordance between this distribution and the empirical distribution is quite remarkable since it emerges from no numerical adjustment and can be further improved by tuning the values of f and c. Regardless, this kurtosis value is only 2:2% larger than our numerical adjustment (see Table 1 for the goodfness of fitting). Furthermore, comparing the distributions by means of the symmetrized Kullback-Leibler divergence KL~1 2 ð P z ð Þln P z ð Þ P 0 z ð Þ dzz ð P 0 z ð Þln P 0 z ð Þ P z ð Þ dz ! , we obtain a value of 0.00014 that is 19 times smaller than the distance between our fitting and a Gaussian. These results show that the PDF of Eq. (9) not only provides a good description of the data, but it is much more manageable as well. Cases for which the kurtosis excess is relevant (qw5=4) stem from wider distributions of s (see the lower panel of Fig. 3). Actually, it is the emergence of larger values of the instantaneous volatility that brings forth fat tails. Although we have not been successful in describing the whole distribution, we have verified that, for values of qw5=4, the distribution p s ð Þ is very well described by a type-2 Gumbel distribution, p s ð Þ!exp {bs {f Â Ã s {f{1 ,ð16Þ and after certain value of s the distribution sharply decreases according to a power-law with a large exponent. We credit this sheer fall to the threshold w, which introduces a sharp change in the dynamical regime of the volatility and thus in its statistics. In finance, such a cut-off is more than plausible as real markets do suspend trading when large price fluctuations occur. This also grants feasibility to descriptions based on truncated power-law distributions [6]. Moreover, a fall off is also presented in the quantity s e of Fig. 3 in Ref. [53]. It is known that for heteroscedastic models the tail behavior of the long-term distribution is governed by the asymptotic limit of p(s) when s tends to infinity. For the case of distribution (16), this limit is the power-law s {f{1 and therefore we can verify that the asymptotic behavior of the long-term distribution of the variable z, For p(s) following an exponential decay in the form exp {cs ½ , a similar procedure yields, P z ð Þ?G c 2 8 z 2 j { { { 0, 1 2 , 1 " # , z?? ð Þ,ð19Þ where G : ½ is the Meijer G-function [42,52]. It is worth noting that in an effort to obtain a full description of p s ð Þ we also used a function allows the appearance of a crossover from a power law to an exponential decay. Nonetheless, it did not provide better results. It is worth saying that we can reduce the number of parameters to a, b and w, i.e., apply the simple ARCH(1) process, and obtain fat tails and persistence still. such as f x ð Þ~Z exp {bx {f Â Ã 1{ A B z A B exp A m x ! {m which Comparison with a real system Following this picture, we can now look for a set of parameters that enable us to replicate a historic series such as the daily (adjusted) log-index fluctuations, r t ð Þ f g, of the SP500 stock index, S t ð Þ f g, between 3rd January 1950 and 12th April 2010 (14380 data points) with, r t ð Þ~lnS tz1 ð Þ{lnS t ð Þ:ð20Þ The adjusted values of the index take into account dividend payments and splits occurred in a particular day. Inspecting over a grid of values of b, W and w, we have noted that the values of 0:9998, 22 and 1:125, respectively, yield values of q and H for z t f g that are in good agreement with a prior analysis of r t ð Þ f g which gave q~1:48+0:02 (using a simple t-Student distribution) and q~1:51+0:02 q 0~1 :47+0:003,n~0:92+0:008 [x 2 =n0 :00003, R 2~0 :999 and P Ã KS~0 :9276](using the PDF of Eq. (9)) and persistence exponent H~0:86+0:03 (see Fig. 4). Comparing the numerical distribution of our model with the data we obtained D KS~0 :014 and a P Ã KS critical value equal to 0:991 from the twosample Kolmogorov-Smirnov test [51], while the comparison between the distribution of the numerical procedure and the adjustment of the SP500 empirical distribution function yielded P Ã KS~0 :9998. Once again we have tested other possible numerical adjustments and the only other relevant distribution was the stretched exponential with n~1:3+0:02 (q 0~1 ) which has given a P Ã KS different from 1 P Ã KS~0 :9999 À Á , but a significantly larger value of x 2 [x 2 =n~0:00009, R 2~0 :9963] (see Fig. 5). It is worthy to be mentioned that all the three values of the parameters are plausible. First, within an application context, b is traditionally a value robustly greater than 0:9. Second, W is close to the number of business days in a month and last, but not least, w is somewhat above the average level of the mean variance presented above. This provides us with a very interesting picture of the dynamics. Specifically, at a relevant approximation we can describe this particular system as monitoring the magnitude of its past fluctuations with a characteristic scale of a month, from which it computes the level of impact resulting in an excess of volatility. Actually one month moving averages are established indicators in quantitative analyses of financial markets. When the volatility in a period of the same order of magnitude of W surpasses the value w= 1{b ð Þ, then the system recalls previous periods of time, no matter how long they happened, in which a significant level of volatility excess occurred. Those periods are then averaged in order to determine the level of instantaneous volatility s 2 t . Discussion We have studied a generalization of the well-known ARCH process born in a financial context. Our proposal differs from other generalizations, since it adds to heteroscedastic dynamics the ability to reproduce systems where cognitive traits exist or systems showing typical cut-off limiting values. In the former case, when present circumstances are close to extreme and impacting events, the dynamics switches to the memory of abnormal events. By poring over the set of parameters of the problem, namely the impact of past values, b, the memory scale, W , and the volatility threshold, w, we have verified that we are able to obtain times series showing fat tails for the probability density function and strong persistence for the magnitudes of the stochastic variable (directly related to the instantaneous volatility), as it happens in several processes studied within the context of complexity. In order to describe the usefulness of our model we have applied it to mimic the fluctuations of the stock index SP500, we verified that the best values reproducing the features of its time series are W close to one business month and w greater that the mean variance of the process which is much larger than the normal level of volatility for which trading is not taken into account. Concerning the volatility, we have noticed that for the problems of interest (i.e., fat tails and strong persistence), the distributions are very well described by a type-2 Gumbel distribution in large part of the domain, which explains the emergence of the tails. Materials and Methods Our results have been obtained from numerical simulation using code written in fortran language and run on the 64-bit ssolarII cluster (http://mesonpi.cat.cbpf.br/ssolar/). Figure 1 . 1Probability density functions P(z) vs z in a log-linear scale on the left column; On the right column the fluctuation functions F(') vs ' for jzj in a log-log scale. The values of the model parameters are: w~0:1,W~5 yielding q~1 and H~0:5+0:01 (upper panels); w~5,W~5 yielding q~1:09+0:01 and H~0:52+0:01 (middle panels); w~2,W~75 yielding q~1:02+0:01 and H~0:58+0:02 (lower panels). The results have been obtained from series of 4\|10 5 elements and the numerical adjustment of P(z) gave values of x 2 =n never greater than 0.00003, with R never smaller than 0.998. doi:10.1371/journal.pone.0018149.g001 Figure 2 . 2Value of the tail index q vs parameter w for several values of W and b~0:998 according to the adjustment procedures mentioned in the text in the upper panel. In the lower panel Hurst exponent H vs w. The results have been obtained from series of 4\|10 5 elements and the numerical adjustment of P(z) gave values of x 2 =n never greater than 0.00003 with R 2 never smaller than 0.998. Regarding the values of the Hurst exponent, the absolute error has never been greater that 0:015 and a linear coefficient Rw0:999. doi:10.1371/journal.pone.0018149.g002 Figure 3 . 3Probability density function of the instantaneous volatility p(s) vs s for two different b~0:998. In the upper panel: b~0:998, w~0:25 and W~75 which leads to a sharply peaked distribution around s~1 and to a P(z) tail index q~1:1. In the lower panel: b~0:998, w~2:5 and W~25 that results in a broader distribution largely described by a type-2 Gumbel distribution with b~0:421+0:002 and f~2:323+0:006 (x 2 =n~0:00011 and R20 :9982). For s5, p(s) changes its behavior to a faster decay with an exponent equal to 8:4+0:2 represented by the gray symbols. The ANOVA test of the type-2 Gumbel adjustment (up to s5) have yielded a sum of squares of 0:03553 (323 degrees of freedom) and 20:3684 (2 degrees of freedom) for the error and the model, respectively. The uncorrected value of the sum of squares is 20:4039 (325 degrees of freedom) and the corrected total is 12:5941 (324 degrees of freedom). The empirical distribution function has been obtained from series of 4\|10 5 elements. doi:10.1371/journal.pone.0018149.g003 Figure 4 . 4Probability density function P(z) vs z. The points represent the empirical distribution function for b~0:998, w~0:25 and W~75; the dashed red line is our adjustment with Eq. (9) with q~1:1+0:01, n~1 and B~(q{1)=(5{3q) [ x 2 =n~0:00003 and R 2~0 :9986]; the green line is PDF (15) with f~c~1=2 and the dotted cyan line is the Normal distribution. doi:10.1371/journal.pone.0018149.g004 Figure 5 . 5Probability density function P(z) vs z. In the upper panels and on the left side we have b~0:9998, W~22 and w~1:125 (full line) [q~1:49+0:01 with x 2 =n~0:000025 and R 2~0 :9984] and the SP500 daily log-index fluctuations (symbols) [q~1:48+0:02 with x 2 =n~0:00004 and R 2~0 :996] in the log-linear scale and on the right side the complementary cumulative distribution function D(xwz) vs z for case shown on the left. Lower panel: Fluctuation function F (') vs ' for the same parameters above [H~0:85+0:02, with R~0:998] (red circles) and for the SP500 daily logindex fluctuations [H~0:86+0:03, with R~0:997] (black squares) in a log-log scale. doi:10.1371/journal.pone.0018149.g005 Table 1 . 1Critical values P Ã KS~1 {a crit from the Kolmogorov-Smirnov test for typical pairs (W ,w) used for adjustments. PLoS ONE \| www.plosone.org March 2011 \| Volume 6 \| Issue 3 \| e18149 Acknowledgments SMDQ thanks the warm hospitality of the CBPF and its staff during his visits to the institution. The paper benefited from the comments of the referees to whom the authors are grateful.Author Contributions M Gell-Mann, The Quark and the Jaguar: Adventures in the Simple and the Complex. New York -NYAbacusGell-Mann M (1996) The Quark and the Jaguar: Adventures in the Simple and the Complex. New York -NY: Abacus. The Computational Beauty of Nature: Computer Explorations of Fractals, Chaos, Complex Systems and Adaptation. 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[ "Consequences of Symmetries on the Analysis and Construction of Turbulence Models", "Consequences of Symmetries on the Analysis and Construction of Turbulence Models" ]	[ "Integrability Symmetry ", "Geometry " ]	[]	[ "Methods and Applications" ]	Since they represent fundamental physical properties in turbulence (conservation laws, wall laws, Kolmogorov energy spectrum, . . . ), symmetries are used to analyse common turbulence models. A class of symmetry preserving turbulence models is proposed. This class is refined such that the models respect the second law of thermodynamics. Finally, an example of model belonging to the class is numerically tested.	10.3842/sigma.2006.052	[ "https://arxiv.org/pdf/physics/0605106v1.pdf" ]	7,591,142	physics/0605106	ed9fbd9515abbe57471626ed94148a5504df7c2d	Consequences of Symmetries on the Analysis and Construction of Turbulence Models 2006 Integrability Symmetry Geometry Consequences of Symmetries on the Analysis and Construction of Turbulence Models Methods and Applications 22006Received October 28, 2005, in final form May 02, 2006;arXiv:physics/0605106v1 [physics.flu-dyn] Original article is available atturbulencelarge-eddy simulationLie symmetriesNoether's theoremthermo- dynamics 2000 Mathematics Subject Classification: 22E7035A3035Q3076D0576F6576M60 Since they represent fundamental physical properties in turbulence (conservation laws, wall laws, Kolmogorov energy spectrum, . . . ), symmetries are used to analyse common turbulence models. A class of symmetry preserving turbulence models is proposed. This class is refined such that the models respect the second law of thermodynamics. Finally, an example of model belonging to the class is numerically tested. Introduction Turbulence is one of the most interesting research fields in mechanics. But with the current performance of computers, a direct simulation of a turbulent flow remains difficult, even impossible in many cases, due to the high computational cost that it requires. To reduce this computational cost, use of turbulence models is necessary. At the present time, many turbulence models exist (see [22]). However, derivation of a very large majority of them does not take into account the symmetry group of the basic equations, the Navier-Stokes equations. In turbulence, symmetries play a fundamental role in the description of the physics of the flow. They reflect existence of conservation laws, via Noether's theorem. Notice that, even if Navier-Stokes equations are not directly derived from a Lagrangian, Noether's theorem can be applied and conservation laws can be deduced. Indeed, there exists a Lagrangian (which will be called "bi-Lagrangian" here) from which Navier-Stokes equations, associated to their "adjoint" equations, can be derived. A way in which this bi-Lagrangian can be calculated is described by Atherton and Homsy in [1]. The expression of this bi-Lagrangian and the infinitesimal generators of the associated Euler-Lagrange equations are given in Appendix A. However, the conservation laws are not studied in this paper. The importance of symmetries in turbulence is not limited to the derivation of conservation laws.Ünal also used a symmetry approach to show that Navier-Stokes equations may have solutions which have the Kolmogorov form of the energy spectrum [24]. Next, symmetries enabled Oberlack to derive some scaling laws for the velocity and the two point correlations [19]. Some of these scaling laws was reused by Lindgren et al in [15] and are proved to be in good agreement with experimental data. Next, symmetries allowed Fushchych and Popowych to obtain analytical solutions of Navier-Stokes equations [8]. The study of self-similar solutions gives also an information on the behaviour of the flow at a large time [4]. Lastly, we mention that use of discretisation schemes which are compatible with the symmetries of an equation reduces the numerical errors [20,13]. Introduction of a turbulence model in Navier-Stokes equations may destroy symmetry properties of the equations. In this case, physical properties (conservation laws, scaling laws, spectral properties, large-time behaviour, . . . ) may be lost. In order to represent the flow correctly, turbulence models should then preserve the symmetries of Navier-Stokes equations. The first aim of this paper is to show that most of the commonly used subgrid turbulence models do not have this property. The second goal is to present a new way of deriving models which are compatible with the symmetries of Navier-Stokes equations and which, unlike many existing models, conform to the second law of thermodynamics. As it will be shown in appendix, conformity with this law leads to stability of the model, in the sense of L 2 . The paper will be structured as follows. In Section 2, the principle of turbulence modelling, using the large-eddy simulation approach, will be concisely presented, as well as some common models. These models will be analysed in Section 3 under the symmetry consideration. In Section 4, a class of symmetry preserving and thermodynamically consistent models are derived. One example of a model of the class is numerically tested in Section 5. Some conclusions will be drawn in Section 6. In Appendix A, it will be shown that Navier-Stokes equations can be derived from a bi-Lagrangian. At last, in Appendix B, stability of thermodynamically consistent models is proved. Large-eddy simulation Consider a three-dimensional incompressible Newtonian fluid, with density ρ and kinematic viscosity ν. The motion of this fluid is governed by Navier-Stokes equations: ∂u ∂t + div(u ⊗ u) + 1 ρ ∇p = div T, div u = 0,(1) where u = (u i ) i=1,2,3 and p are respectively velocity and pressure fields and t the time variable. T is a tensor such that ρT is the viscous constraint tensor. T can be linked to the strain rate tensor S = (∇u + T ∇u)/2 according to the relation: T = ∂ψ ∂S , ψ being a positive and convex "potential" defined by: ψ = ν tr S 2 . Since a direct numerical simulation of a realistic fluid flow requires very significant computational cost, (1) is not directly resolved. To circumvent the problem, some methods exist. The most promising one is the large-eddy simulation. It consists in representing only the large scales of the flow. Small scales are dropped from the simulation; however, their effects on the large scales are taken into account. This enables to take a much coarser grid. Mathematically, dropping small scales means applying a low-pass filter. The large or resolved scales φ of a quantity φ are defined by the convolution: φ = G δ * φ, where G δ is the filter kernel with a width δ, and the small scales φ ′ are defined by φ ′ = φ − φ. It is required that the integral of G δ over R 3 is equal to 1, such that a constant remains unchanged when the filter is applied. In practice, (u, p) is directly used as an approximation of (u, p). To obtain (u, p), the filter is applied to (1). If the filter is assumed to commute with the derivative operators (that is not always the case in a bounded domain), this leads to: ∂u ∂t + div(u ⊗ u) + 1 ρ ∇p = div(T + T s ), div u = 0,(2) where T s is the subgrid stress tensor defined by T s = u ⊗ u − u ⊗ u which must be modelled (expressed by a function of the resolved quantities) to close the equations. Currently, an important number of models exists. Some of the most common ones will be reminded here. They will be classified in four categories: turbulent viscosity, gradient-type, similarity-type and Lund-Novikov-type models. Turbulent viscosity models Turbulent viscosity models are models which can be written in the following form: T d s = ν s S,(3) where ν s is the turbulent viscosity. The superscript ( d ) represents the deviatoric part of a tensor: Q → Q d = Q − 1 3 (tr Q)I d , where I d is the identity operator. The deviatoric part has been introduced in order to have the equality of the traces in (3). In what follows, some examples of turbulent viscosity models are presented. • Smagorinsky model (see [22]) is one of the most widely used models. It uses the local equilibrium hypothesis for the calculation of the turbulent viscosity. It has the following expression: T d s = (C S δ) 2 \|S\|S, where C S ≃ 0.148 is the Smagorinsky constant, δ the filter width and \|S\| = 2 tr S 2 . • In order to reduce the modelling error of Smagorinsky model, Lilly [14] proposes a dynamic evaluation of the constant C S by a least-square approach. This leads to the so-called dynamic model defined by: T d s = C d δ 2 \|S\|S, with C d = tr(LM) tr M 2 .(4) In these terms, L = u ⊗ u − u ⊗ u, M = δ 2 \|S\|S − δ 2 \| S\| S, the tilde represents a test filter whose width is δ, with δ > δ. The last turbulent viscosity model which will be considered is the structure function model. • Metais and Lesieur [16] make the hypothesis that the turbulent viscosity depends on the energy at the cutoff. Knowing its relation with the energy density in Fourier space, they use the second order structure function and propose the structure function model: T d s = C SF δ F 2 (δ) S,(5) where F 2 is the spatial average of the filtered structure function: r → F 2 (r) = z =r u(x) − u(x + z) 2 dz dx. The next category of models, which will be reminded, consists of the gradient-type models. Gradient-type models To establish the gradient-type models, the subgrid stress tensor is decomposed as follows: T s = u ⊗ u − ( u ⊗ u + u ⊗ u ′ + u ′ ⊗ u + u ′ ⊗ u ′ ). Next, each term between the brackets are written in Fourier space. Then, the Fourier transform of the filter, which is assumed to be Gaussian, is approximated by an appropriate function. Finally, the inverse Fourier transform is computed. The models in this category differ by the way in which the Fourier transform of the filter is approximated. • If a second order Taylor series expansions according to the filter width δ is used in the approximation, one has: T s = − δ 2 12 ∇u T ∇u.(6) • The gradient model is not dissipative enough and not numerically stable [26,11]. Thus, it is generally combined to Smagorinsky model. This gives Taylor model: T s = − δ 2 12 ∇u T ∇u + Cδ 2 \|S\|S. • The Taylor approximation of the Fourier transform of the filter tends to accentuate the small frequencies rather than attenuating them. Instead, a rational approximation can be used [12,2]. This gives the following expression of the model: T s = − δ 2 12 I d − δ 2 24 ∇ 2 −1 [∇u T ∇u] + Cδ 2 \|S\|S. To avoid the inversion of the operator I d − δ 2 24 ∇ 2 , T s is approximated by: T s = − δ 2 12 G δ * [∇u T ∇u] + Cδ 2 \|S\|S.(7) G δ is the kernel of the Gaussian filter. The convolution is done numerically. The model (7) is called the rational model. Similarity-type models Models of this category are based on the hypothesis that the statistic structure of the small scales are similar to the statistic structure of the smallest resolved scales. Separation of the resolved scales is done using a test filter (symbolized by ). The largest resolved scales are then represented by u and the smallest ones by u− u. From this hypothesis, we deduce the similarity model: T s = u ⊗ u − u ⊗ u.(8) From this expression, many other models can be obtained by multiplying by a coefficient, by filtering again the whole expression or by mixing with a Smagorinsky-type model. The last models that we will consider are Lund-Novikov-type models. Lund-Novikov-type models • Lund and Novikov include the filtered vorticity tensor W = (∇u − T ∇u) in the expression of the subgrid model. Cayley-Hamilton theorem gives then the Lund-Novikov model (see [22]): −T d s = C 1 δ 2 \|S\|S + C 2 δ 2 (S 2 ) d + C 3 δ 2 W 2 d + C 4 δ 2 (S W − W S) + C 5 δ 2 1 \|S\| S 2 W − S W 2 ,(9) where the coefficients C i depend on the invariants obtained from S and W. The expression of these coefficients are so complex that they are considered as constants and evaluated with statistic techniques. • To reduce the computation cost of the previous model, Kosovic brings a simplification and proposes the following model: −T d s = (Cδ) 2 2\|S\|S + C 1 S 2 d + C 2 (S W − W S) ,(10) where the constants C, C 1 and C 2 are calculated using the theory of homogeneous and isotropic turbulence. The derivation of these models was done using different hypothesis but did not take into consideration the symmetries of Navier-Stokes equations which may then be destroyed. So, in the next section, these models will be analysed by a symmetry approach. Model analysis The (classical) symmetry groups of Navier-Stokes equations have been investigated for some decades (see for example [7,3]). They are generated by the following transformations: • The time translations: (t, x, u, p) → (t + a, x, u, p), • the pressure translations: (t, x, u, p) → (t, x, u, p + ζ(t)), • the rotations: (t, x, u, p) → (t, Rx, Ru, p), • the generalized Galilean transformations: (t, x, u, p) → (t, x + α(t), u +α(t), p − ρ x ·α(t)), • and the first scaling transformations: (t, x, u, p) → (e 2a t, e a x, e −a u, e −2a p). In these expressions, a is a scalar, ζ (respectively α) a scalar (resp. vectorial) arbitrary function of t and R a rotation matrix, i.e. R T R = I d and det R = 1. The central dot (·) stands for R 3 scalar product. If it is considered that ν can change during the transformation (which is then an equivalence transformation [9]), one has the second scaling transformations: (t, x, u, p, ν) → (t, e a x, e a u, e 2a p, e 2a ν), where a is the parameter. Navier-Stokes equations admit other known symmetries which do not constitute a oneparameter symmetry group. They are • the reflections: (t, x, u, p) → (t, Λx, Λu, p), which are discrete symmetries, Λ being a diagonal matrix Λ = diag(ι 1 , ι 2 , ι 3 ) with ι i = ±1, i = 1, 2, 3, • and the material indifference: (t, x, u, p) → (t, x, u, p), in the limit of a 2D flow in a simply connected domain [5], with x = R(t) x, u = R(t) u +Ṙ(t) x, p = p − 3ωϕ + 1 2 ω 2 x 2 , where R(t) is a 2D rotation matrix with angle ωt, ω an arbitrary real constant, ϕ the usual 2D stream function defined by: We wish to analyse which of the models cited above is compatible with these symmetries. The set of solutions (u, p) of Navier-Stokes equations (1) is preserved by each of the symmetries. We then require that the set of solutions (u, p) of the filtered equations (2) is also preserved by all of these transformations, since (u, p) is expected to be a good approximation of (u, p). More clearly, if a transformation u = curl(ϕe 3 ),T : (t, x, u, p) → ( t, x, u, p) is a symmetry of (1), we require that the model is such that the same transformation, applied to the filtered quantities: T : (t, x, u, p) → ( t, x, u, p), is a symmetry of the filtered equations (2). When this condition holds, the model will be said invariant under the relevant symmetry. The filtered equations (2) may have other symmetries but with the above requirement, we may expect to preserve certain properties of Navier-Stokes equations (conservation laws, wall laws, exact solutions, spectra properties, . . . ) when approximating (u, p) by (u, p). We will use the hypothesis that test filters do not destroy symmetry properties, i.e. φ = φ for any quantity φ. For the analysis, the symmetries of (1) will be grouped into four categories: The aim is to search which models are invariant under the symmetries within the considered category. - Invariance under translations Since almost all existing models are autonomous in time and pressure, the filtered equations (2) remain unchanged when a time or pressure translation is applied. Almost all models are then invariant under the time and the pressure translations. The generalized Galilean transformations, applied to the filtered variables, have the following form: (t, x, u, p) → ( t, x, u, p) = (t, x + α(t), u +α(t), p − ρ x ·α(t)) . All models in Section 2, in which x and u are present only through ∇u are invariant since ∇ u = ∇ u = ∇u, where ∇ = (∂/∂ x 1 , ∂/∂ x 2 , ∂/∂ x 3 ). The remaining models, i.e. the dynamic and the similarity models are also invariant because u ⊗ u − u ⊗ u = (u +α) ⊗ (u +α) − (u +α) ⊗ (u +α) = u ⊗ u − u ⊗ u. Invariance under rotations and reflections The rotations and the reflections can be put together in a transformation: (t, x, u, p) → (t, Υx, Υu, p) where Υ is a constant rotation or reflection matrix. This transformation, when applied to the filtered variables, is a symmetry of (2) if and only if T s = Υ T s T Υ.(11) Let us check if the models respect this condition. • For Smagorinsky model, we have: ∇ u = [∇( u)] T Υ = [∇(Υu)] T Υ = Υ[∇u] T Υ.(12) This leads to the objectivity of S: S = Υ S T Υ. And since \| S\| = \|S\|, (11) is verified. Smagorinsky model is then invariant. • For similarity model (8), one has: u ⊗ u = (Υu) ⊗ (Υu) = Υ(u ⊗ u) T Υ. By means of these relations, invariance can easily been deduced. • The same relations are sufficient to prove invariance of the dynamic model since the trace remains invariant under a change of orthonormal basis. • The structure function model (5) is invariant because the function F 2 is not altered under a rotation or a reflection. • Relations (12) can be used again to prove invariance of each of the gradient-type models. • Finally, since W = Υ W T Υ, Lund-Novikov-type models are also invariant. Any model of Section 2 is then invariant under the rotations and the reflections. Invariance under scaling transformations The two scaling transformations can be gathered in a two-parameter transformation which, when applied to the filtered variables, have the following expression: (t, x, u, p, ν) → e 2a t, e ab x, e b−a u, e 2b−2a p, e 2b ν . where a and b are the parameters. The first scaling transformations corresponds to the case b = 0 and the second ones to the case a = 0. It can be checked that the filtered equations (2) Since S = e −2a S, this condition is equivalent to: ν s = e 2b ν s(14) for a turbulent viscosity model. • For Smagorinsky model, we have: ν s = C S δ 2 \| S\| = e −2a C S δ 2 \|S\| = e −2a ν s . Condition (14) is violated. The model is invariant neither under the first nor under the second scaling transformations. Note that the filter width δ does not vary since it is an external scale length and has no functional dependence on the variables of the flow. • The dynamic procedure used in (4) restores the scaling invariance. Indeed, it can be shown that: C d = e 2b+2a C d , that implies: ν s = C d δ 2 \| S\| = e 2b C d δ 2 \|S\| = e 2b ν s . The dynamic model is then invariant under the two scaling transformation. • For the structure function model, we have: F 2 = e b−a F 2 and then ν s = e b−a ν sm , that proves that the model is not invariant. • Since ∇ u = e 2a ∇u the gradient model (6) violates (13), T s varying in the following way: T s = e 4a T s . This also implies that none of the gradient-type models is invariant. • It is straight forward to prove that the similarity model (8) verifies (13) and is invariant. • At last, Lund-Novikov-type models are not invariant because they comprise a term similar to Smagorinsky model. In fact, none of the models where the external length scale δ appears explicitly is invariant under the scaling transformations. Note that the dynamic model, which is invariant under these transformations, can be written in the following form: T d s = tr(LN) tr(N 2 ) \|S\|S, where N = \|S\|S − ( δ/δ) 2 \| S\| S. It is then the ratio δ/δ which is present in the model but neither δ alone nor δ alone. In summary, the dynamic and the similarity models are the only invariant models under the scaling transformations. Though, scaling transformations have a particular importance because it is with these symmetries that Oberlack [18] derived scaling laws and thatÜnal [24] proved the existence of solutions of Navier-Stokes equations having Kolmogorov spectrum. The last symmetry property of Navier-Stokes equations is the material indifference, in the limit of 2D flow, in a simply connected domain. Material indifference The material indifference corresponds to a time-dependent plane rotation, with a compensation in the pressure term. We will not write explicitly the dependence on time of the rotation matrix R. • The objectivity of S (see Section 3.2) directly leads to invariance of Smagorinsky model. T s = RT s T R + R( u ⊗ x − u ⊗ x) T R +Ṙ( x ⊗ u − x ⊗ u) T R +Ṙ( x ⊗ x − x ⊗ x) TṘ . Consequently, if the test filter is such that ). ( u ⊗ x − u ⊗ x) = 0, ( x ⊗ u − x ⊗ u) = 0, ( x ⊗ x − x ⊗ x) = 0,(15) • Under the same conditions (15) on the test filter, the dynamic model is also invariant. • The structure function model is invariant if and only if F 2 = F 2 .(16) Let us calculate F 2 . Let u z be the function x → u(x + z). Then F 2 = z =δ (Ru +Ṙx) − (Ru z +Ṙx +Ṙz) 2 dz = z =δ u − u z − T RṘz dz. Knowing that T RṘz = ωe 3 × z, we get: F 2 = F 2 + 2πω 2 δ 3 − 2ω z =δ (u − u z ) · (e 3 × z) dz. Condition (16) is violated. So, the structure function model is not invariant under the material indifference. • For the gradient model, we have: ∇ u = R ∇u T R +Ṙ T R, T ∇ u = R T ∇u T R + R TṘ .(17) Let J be the matrix such thatṘ T R = −ωJ = −R TṘ or, in a component form: J = 0 1 −1 0 . Then, (∇u T ∇u) = R(∇u T ∇u) T R + ωR∇u T RJ − ωJR T ∇u T R + ω 2 I d . The commutativity between J and R finally leads to: T s = R T s T R + ωR(∇uJ − J T ∇u) + ω 2 I d . This proves that the gradient model is not invariant. • The other gradient-type models inherit the lack of invariance of the gradient model. • It remains the Lund-Novikov-type models. We will begin with Kosovic model (10) since it is simpler. The first two terms of (10) are unchanged under the transformation. For the filtered vorticity tensor W, it follows from (17) that: W = R W T R − ωJ. Thus, S W − W S = R(S W − W S) T R − ωR(SJ − JS) T R using again the commutativity between J and R. As for them, S and J are not commutative. In fact, using properties of S, it can be shown that SJ = −JS. This implies that S W − W S = R(S W − W S) T R − 2ωRSJ T R.(18) This shows that Kosovic model is not invariant. • Lastly, consider Lund-Novikov model (9). We have: W 2 = R W 2 T R − ωR(JW + WJ) T R − ω 2 I d . Since W is anti-symmetric and the flow is 2D, W is in the form: W = 0 w −w 0 . A direct calculation leads then to JW = WJ = −wI d and W 2 = R W 2 T R − (2w − ω)ωI d . Let us see now how each term of the model (9) containing W varies under the transformation. From the last equation, we deduce the objectivity of (W 2 ) d : (W 2 ) d = R (W 2 ) d T R. For the fourth term of (9), we already have (18). And for the last term, S 2 W − S W 2 = R(S 2 W − S W 2 ) T R − ωRS 2 T RJ − (2w − ω)ωRS T R. Putting these results together, we have: T d s = RT d s T R − ωδ 2 R 2C 4 SJ − C 5 1 \|S\| S 2 J − (2w − ω)ωS T R. We conclude that Lund-Novikov model is not invariant the material indifference. This ends the analysis. Table 1 summarizes the results of the above analysis. "Y" means that the model is invariant under all the symmetries of the category, "N" the opposite and "Y * " that the model is invariant if the conditions (15) on the test filter is verified. It can be seen on this table that only two models among the nine, the dynamic and the similarity models, are invariant under the symmetry group of Navier-Stokes equations. The scaling transformations, which are of a particular importance (scaling laws, Kolmogorov spectrum, . . . ) are violated by almost all models. Smagorinsky Y Y N Y Dynamic Y Y Y Y * Structure function Y Y N N Gradient Y Y N N Taylor Y Y N N Rational Y Y N N Similarity Y Y Y Y * Lund Y Y N N Kosovic Y Y N N The dynamic and the similarity models have an inconvenience that they necessitate use of a test filter. Rather constraining conditions, (15), are then needed for these models to preserve the material indifference. In addition, the dynamic model does not conform to the second law of thermodynamics since it may induce a negative dissipation. Indeed, ν + ν s can take a negative value. To avoid it, an a posteriori forcing is generally done. It consists of assigning to ν s a value slightly higher than −ν: ν s = −ν (1 − ε), where ε is a positive real number, small against 1. Non-conformity to the second law of thermodynamics may be detrimental for a model because, as it will be shown in Appendix B, consistence with this law leads to stability of the model. Considering this lack of invariance of existing models and to non-conformity with thermodynamical principles, we propose in the next section a new way of deriving models which, on one hand, possess the symmetry group of Navier-Stokes equations and, on the other hand, are compatible with the second law of thermodynamics. Invariant and thermodynamically consistent models First, we will build a class of models which possess the symmetries of Navier-Stokes equations and next refine this class such that the models also satisfy the thermodynamics requirement. Invariance under the symmetries Suppose that S = 0. Let T s be an analytic function of S: T s = A(S).(19) By this way, invariance under the time, pressure and generalised Galilean translations and under the reflections is guaranteed. From (19), Cayley-Hamilton theorem and invariance under the rotations lead to: T d s = A(χ, ζ) S + B(χ, ζ) Adj d S,(20) where χ = tr S 2 and ζ = det S are the invariants of S (the third invariant, tr S, vanishes), Adj stands for the operator defined by (Adj S)S = (det S)I d , (Adj S is simply the comatrix of S ) and A and B are arbitrary scalar functions. Contrarily to Lund-Novikov model, these coefficient functions will not be taken constant. Next, a necessary and sufficient condition for T s defined by (20) to be invariant under the second scale transformations is that ν can be factorized: T d s = νA 0 (χ, ζ) S + νB 0 (χ, ζ) Adj d S. Lastly, T s is invariant under the first scaling transformations if T s = e −2a T s . Rewritten for A 0 and B 0 , this condition becomes: A 0 (e −4a χ, e −6a ζ) = A 0 (χ, ζ), B 0 (e −4a χ, e −6a ζ) = e 2a B 0 (χ, ζ). After differentiating according to a and taking a = 0, it follows: −4χ ∂A 0 ∂χ − 6ζ ∂A 0 ∂ζ = 0, −4χ ∂B 0 ∂χ − 6ζ ∂B 0 ∂ζ = 2B 0 . To satisfy these equalities, one can take A 0 (χ, ζ) = A 1 ζ χ 3/2 , B 0 (χ, ζ) = 1 √ χ B 1 ζ χ 3/2 . Finally, if v = ζ χ 3/2 then T d s = νA 1 (v) S + ν 1 √ χ B 1 (v) Adj d S.(21) A subgrid-scale model of class (21) remains then invariant under the symmetry transformations of Navier-Stokes equations. In fact several authors were interested in building invariant models for a long time. But because they did not use Lie theory, they did not consider some symmetries such as the scaling transformations which are particularly important. Three of the few authors who considered all the above symmetries in the modeling of turbulence areÜnal [25] and Saveliev and Gorokhovski [23]. The present manner to build invariant models generalises theÜnal's one in the sense that it introduces ν and the invariants of S into the models. In addition,Ünal used the Reynolds averaging approach (RANS) instead of the large-eddy simulation approach (LES) for the turbulence modelling. Saveliev and Gorokhovski in [23] used the LES approach but derive their model in a different way than in the present article. Let us now return to considerations which are more specific to large eddy simulation. We know that T s represents the energy exchange between the resolved and the subgrid scales. Then, it generates certain dissipation. To account for the second law of thermodynamics, we must ensure that the total dissipation remains positive that is not always verified by models in the literature. In order to satisfy this condition, we refine class (21). Consequences of the second law of thermodynamics At molecular scale, the viscous constraint is: T = ∂ψ ∂S . The potential ψ = ν tr S 2 is convex and positive that ensures that the molecular dissipation is positive: Φ = tr(TS) ≥ 0. The tensor T s can be considered as a subgrid constraint, generating a dissipation Φ s = tr(T s S). To preserve compatibility with the Navier-Stokes equations, we assume that T s has the same form as T: T s = ∂ψ s ∂S .(22) where ψ s is a potential depending on the invariants χ and ζ of S. This hypothesis refines class (21) in the following way. Since tr S = 0, one deduces from (22): T d s = 2 ∂ψ s ∂χ S + ∂ψ s ∂ζ Adj d S. Comparing it with (21), one gets: 1 2 νA 1 (v) = ∂ψ s ∂χ , ν 1 √ χ B 1 (v) = ∂ψ s ∂ζ . This leads to: ∂ ∂ζ 1 2 A 1 (v) = ∂ ∂χ 1 √ χ B 1 (v) . If g is a primitive of B 1 , a solution of this equation is A 1 (v) = 2g(v) − 3vg ′ (v) and B 1 (v) = g ′ (v).(23) Then, the hypothesis (22) involves existence of a function g such that: T d s = ν 2g(v) − 3vg ′ (v) S + ν 1 √ χ g ′ (v) Adj d S.(24) Now, let Φ T be the total dissipation. We have: Φ T = tr[(T + T s )S]. Using (21) et (23), one can show that Φ T ≥ 0 ⇐⇒ 1 + A 1 (v) + 3vB 1 (v) ≥ 0 ⇐⇒ 1 + g(v) ≥ 0.(25) In summary, a model belonging to class (24) with a continuous function g verifying (26) out of this interval. Another important property of such a model is its stability, in the sense that the L 2 -norm of the filtered velocity remains bounded. In fact, all models which are consistent with the second law of thermodynamics are stable. This will be proved in Appendix B. 1 + g ≥ 0,(26) In the next section, we show that our approach can lead to numerically efficient results. A very simple model of class (24) is then chosen and compared to the two most popular models which are Smagorinsky and the dynamic models (see [22,14]). Numerical test We choose a simple linear function for g: g(v) = Cv. where the constant C can depend on the filter width and other parameters. Let d be the ratio: d = δ ℓ , where ℓ is a length scale related to the size of the domain. The introduction of this ratio is also useful to have the right dimensions. We now take: C = (C s d) 2 , where C s is a pure constant, set to be equal to Smagorinsky constant, i.e. C s ≃ 0.16. Doing so, condition (25) is verified and one has: In the present paper, this model will be called "invariant model". We use this model to simulate a flow within a ventilated room (Nielsen's cavity [17]) which interests us particularly for applications in building field. The results will then be compared to those provided by the Smagorinsky model and the dynamic model. T d s = ν(C s d) 2 − det S \|\|S\|\| 3 S + 1 \|\|S\|\| Adj d S . The geometry of the room is presented on Fig. 1. For this configuration, we take ℓ = 1m. The code used for the resolution was developed by Chen et al and is described in [6]. The spatial discretization is performed by a finite difference scheme. Fig. 2 compares the velocity profiles given by Smagorinsky, the dynamic and the invariant models with experimental data at x 1 /L = 2/3 and x 3 /W = 0.5. It can be observed on it that the invariant model gives a better result than Smagorinsky and dynamic models, without need of a test filtering. The result is in good agreement with experiments, except near the floor. Notice that no wall model was used. Conclusion In this article, we presented a new class of physically compatible subgrid turbulence models. The main ingredient used is the symmetry group of the Navier-Stokes equations which contains a fundamental information on the properties of the flow. The second principle of thermodynamics was also introduced. From a practical point of view, conformity with this principle ensures stability of the model. A simple model of the class was tested and encouraging results was obtained. However, the aim of this test was not to present a complete analysis of the model but to check that the symmetry approach can lead to good numerical results. Further studies will be done in future works on the choice of the parameters of the model and on analysis of the numerical results. The way presented here for deriving symmetry compatible models is a general way. It can be applied to other equations (non-isothermal fluid, . . . ). Other parameters can also be included. For example, dependence of the model on the viscosity ν can be replaced by dependence on the dissipation rate. A Noether's theorem to Navier-Stokes equations Noether's theorem can be applied to evolution equations which can be derived from a Lagrangian, i.e. evolution equations which can be expressed in an Euler-Lagrange form: ∂L(r) ∂r − Div ∂L(r) ∂ṙ = 0,(27) where r is the dependent variable, r = r(y), y = (y i ) i the independent variable, L the Lagrangian and Div the operator: f → Div f = i df dy i . From the infinitesimal generators of (27), conservation laws are deduced. Navier-Stokes equations cannot be directly written in the form (27). However, thanks to an approach of Atherton and Homsy [1], see also [10], which consists in extending the Lagrangian notion, it will be shown in this appendix that Noether's theorem can be applied to Navier-Stokes equations. We will say that an evolution equation ∂u ∂t · v − u · ∂v ∂t + q − 1 2 u · v − p div v + ν tr ( T ∇u · ∇v) . where v = (v i ) i and q are the adjoint variables. The corresponding adjoint equations are − ∂v ∂t + (v · ∇u T − u · ∇v) = ∇q + ν∆v, div v = 0. Noether's theorem can then be applied. The infinitesimal generators of the couple of equations (Navier-Stokes equations and their adjoint equations) are: X 0 = ∂ ∂t , Y 0 = ζ(t) ∂ ∂p , B Stability of thermodynamically consistent models After an eventual change of variables such that u vanishes along the boundary Γ of the domain Ω, the filtered equations can be written in the following form: ∂u ∂t + div(u ⊗ u) + 1 ρ ∇p − div(T − T s ) = F , div u = 0 (29) associated to the conditions u = 0 sur Γ, u(0, x) = γ(x) on Ω, Ω p(t, x) dx = 0 ∀ t ∈ [0, t f ]. F is an appropriate function of t and x and t f is the final observation time. Proposition. Let (u, p) be a regular solution of (29) where T s is symmetric and verif ies the condition: tr[(T − T s )S] ≥ 0. Then: u(t, x) L 2 (Ω) ≤ γ(x) L 2 (Ω) + t f 0 F (τ, x) L 2 (Ω) dτ . This proposition ensures a finite energy when the model conforms to the second law of thermodynamics. Proof . Let (•, •) denote the scalar product of L 2 (Ω) and (u, p) a regular solution of (29). From the first equation of (29) and the boundary condition, we have: ∂u ∂t , u + b(u, u, u) − 1 ρ (p, div u) + (T − T s , ∇u) = (F , u), where b is defined by the trilinear form b(u 1 , u 2 , u 3 ) = div(u 1 ⊗ u 2 ), u 3 . From integrals by parts, the boundary condition and the divergence free condition, it can be shown that b(u, u, u) = 0 and (p, div u) = 0. Since (T − T s ) is symmetric, it follows that ∂u ∂t , u + (T − T s , S) = (F , u). Consequently, u L 2 (Ω) d dt u L 2 (Ω) ≤ (F , u) ≤ F L 2 (Ω) u L 2 (Ω) . After simplifying by the L 2 -norm of u and integrating over the time, it follows: u(t, x) L 2 (Ω) ≤ γ(x) L 2 (Ω) + t 0 F (τ, x) L 2 (Ω) dτ ≤ γ(x) L 2 (Ω) + t f 0 F (τ, x) L 2 (Ω) dτ . This ends the proof of the proposition. e 3 3the unit vector perpendicular to the plane of the flow and • the Euclidean norm. translations, containing time translations, pressure translations and the generalized Gali- are invariant under the two scaling transformations if and only if T s = e 2b−2a T s . Figure 1 .Figure 2 . 12Geometry of the ventilated room. Mean velocity profiles at x/L = 2/3. F (from a "bi-Lagrangian" if there exists an (non necessarily unique) applicationL : (r, s) → L(r, s) ∈ R such that (28) is equivalent to ∂L(r, s) ∂s − Div ∂L(r, s) ∂ṡ = 0.s is called the adjoint variable and the equation∂L(r, s) ∂r − Div ∂L(r, s) ∂ṙ = 0is called the adjoint equation of (28). The Noether theorem can then be applied since the evolution equation, associated to his adjoint, can be written in an Euler-Lagrangian form∂L(w) ∂w − Div ∂L(w) ∂ẇ = 0,where w = (r, s). Navier-Stokes equations are derived from a bi-Lagrangian L((u, p), (v, q)) = 1 2 , u) + (T − T s , S) = (F , u). Now, using the main hypothesis, we have: (T − T s , S) = Ω tr[(T − T s )S] , u) ≤ (F , u). Table 1 . 1Results of the model analysis. Y=invariant, N=not invariant, Y * =invariant if (15) is verified. where ζ, the α i 's, η and σ are arbitrary scalar functions. Conservation laws for Navier-Stokes equations can be deduced from these infinitesimal generators. However, that requires non-trivial calculations and is not done in this paper.In the last section, we will prove that a model which is consistent with the second law of thermodynamics, i.e. such that the total dissipation remains positive, is stable. 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[ "CRITICAL POINTS BETWEEN VARIETIES GENERATED BY SUBSPACE LATTICES OF VECTOR SPACES", "CRITICAL POINTS BETWEEN VARIETIES GENERATED BY SUBSPACE LATTICES OF VECTOR SPACES" ]	[ "Pierre Gillibert " ]	[]	[]	We denote by Conc A the semilattice of all compact congruences of an algebra A. Given a variety V of algebras, we denote by Conc V the class of all semilattices isomorphic to Conc A for some A ∈ V. Given varieties V and W of algebras, the critical point of V under W is defined as crit(V; W) = min{card D \| D ∈ Conc V − Conc W}. Given a finitely generated variety V of modular lattices, we obtain an integer ℓ, depending on V, such that crit(V; Var(Sub F n )) ≥ ℵ 2 for any n ≥ ℓ and any field F .In a second part, using tools introduced in [5], we prove that:for any finite field F and any ordinal n such that 2+card F ≤ n ≤ ω. Similarly crit(Var(Sub F 3 ); Var(Sub K 3 )) = ℵ 2 , for all finite fields F and K such that card F > card K. A congruence-lifting of a (∨, 0)-semilattice S is an algebra A such that Con c A ∼ = S. Given a variety V of algebras, the compact congruence class of V, denoted by Con c V, is the class of all (∨, 0)-semilattices isomorphic to Con c A for some A ∈ V. As illustrated by[12], even the compact congruence classes of small varieties of lattices are complicated objects. For example, in case V is the variety of all lattices, Con c V contains all distributive (∨, 0)-semilattices of cardinality at most ℵ 1 , but not all distributive (∨, 0)-semilattices (cf.[15]).Given varieties V and W of algebras, the critical point of V and W, denoted by crit(V; W), is the smallest cardinality of a (∨, 0)-semilattice in Con c (V) − Con c (W) if it exists, or ∞, otherwise (i.e., if Con c V ⊆ Con c W).	10.1016/j.jpaa.2009.10.013	[ "https://arxiv.org/pdf/0809.4323v3.pdf" ]	16,240,914	0809.4323	8b7f4135637f8cb318d65f193e6be0db3cd74c25	CRITICAL POINTS BETWEEN VARIETIES GENERATED BY SUBSPACE LATTICES OF VECTOR SPACES 28 May 2010 Pierre Gillibert CRITICAL POINTS BETWEEN VARIETIES GENERATED BY SUBSPACE LATTICES OF VECTOR SPACES 28 May 2010 We denote by Conc A the semilattice of all compact congruences of an algebra A. Given a variety V of algebras, we denote by Conc V the class of all semilattices isomorphic to Conc A for some A ∈ V. Given varieties V and W of algebras, the critical point of V under W is defined as crit(V; W) = min{card D \| D ∈ Conc V − Conc W}. Given a finitely generated variety V of modular lattices, we obtain an integer ℓ, depending on V, such that crit(V; Var(Sub F n )) ≥ ℵ 2 for any n ≥ ℓ and any field F .In a second part, using tools introduced in [5], we prove that:for any finite field F and any ordinal n such that 2+card F ≤ n ≤ ω. Similarly crit(Var(Sub F 3 ); Var(Sub K 3 )) = ℵ 2 , for all finite fields F and K such that card F > card K. A congruence-lifting of a (∨, 0)-semilattice S is an algebra A such that Con c A ∼ = S. Given a variety V of algebras, the compact congruence class of V, denoted by Con c V, is the class of all (∨, 0)-semilattices isomorphic to Con c A for some A ∈ V. As illustrated by[12], even the compact congruence classes of small varieties of lattices are complicated objects. For example, in case V is the variety of all lattices, Con c V contains all distributive (∨, 0)-semilattices of cardinality at most ℵ 1 , but not all distributive (∨, 0)-semilattices (cf.[15]).Given varieties V and W of algebras, the critical point of V and W, denoted by crit(V; W), is the smallest cardinality of a (∨, 0)-semilattice in Con c (V) − Con c (W) if it exists, or ∞, otherwise (i.e., if Con c V ⊆ Con c W). Introduction We denote by Con A (resp., Con c A) the lattice (resp., (∨, 0)-semilattice) of all congruences (resp., compact congruences) of an algebra A. For a homomorphism f : A → B of algebras, we denote by Con f the map from Con A to Con B defined by the rule (Con f )(α) = congruence of B generated by {(f (x), f (y)) \| (x, y) ∈ α}, for every α ∈ Con A, and we also denote by Con c f the restriction of Con f from Con c A to Con c B. Let I be a poset. A direct system indexed by I is a family (A i , f i,j ) i≤j in I such that A i is an algebra, f i,j : A i → A j is a morphism of algebras, f i,i = id Ai , and f i,k = f j,k • f i,j , for all i ≤ j ≤ k in I. Denote by Sub V the subspace lattice of a vector space V , and by M n the variety of lattices generated by the lattice M n of length two with n atoms, for 3 ≤ n ≤ ω. Using the theory of the dimension monoid of a lattice, introduced by F. Wehrung in [13], together with some von Neumann regular ring theory, we prove in Section 3 that if V is a finitely generated variety of modular lattices with all subdirectly irreducible members of length less or equal to n, then crit(V; Var(Sub F n )) ≥ ℵ 2 for any field F . As an immediate application, crit(M n ; M 3 ) ≥ ℵ 2 for every n with 3 ≤ n ≤ ω (cf. Corollary 3.12). Thus, by using the result of M. Ploščica in [10], we obtain the equality crit(M m ; M n ) = ℵ 2 for all m, n with 3 ≤ n < m ≤ ω. Our proof does not rely on the approach used by Ploščica in [11] to prove the inequality crit(M 0,1 m ; M 0,1 n ) ≥ ℵ 2 , and it extends that result to the unbounded case. We also obtain a new proof of that result in Section 4, that does not even rely on the approach used by Ploščica in [10] to prove the inequality crit(M m ; M n ) ≤ ℵ 2 . Let V be a variety of lattices, let D be a diagram of (∨, 0)-semilattices and (∨, 0)homomorphisms. A congruence-lifting of D in V is a diagram L of V such that the composite Con c • L is naturally equivalent to D. In Section 4, we give a diagram of finite (∨, 0)-semilattices that is congruenceliftable in M n , but not congruence-liftable in Var(Sub F 3 ), for any finite field F and any n such that 2+card F ≤ n ≤ ω. As the diagram of (∨, 0)-semilattices is indexed by some "good" lattice, we obtain, using results of [5], that crit(M n ; Var(Sub F 3 )) = ℵ 2 . This implies immediately that crit(M 4 ; M 3,3 ) = ℵ 2 . Let F and K be finite fields such that card F > card K, we also obtain crit(Var(Sub F 3 ); Var(Sub K 3 )) = ℵ 2 . In a similar way, we prove that crit(M ω ; V) = ℵ 2 , for every finitely generated variety of lattices V such that M 3 ∈ V. Basic concepts We denote by dom f the domain of any function f . A poset is a partially ordered set. Given a poset P , we put Q ↓ X = {p ∈ Q \| (∃x ∈ X)(p ≤ x)}, Q ↑ X = {p ∈ Q \| (∃x ∈ X)(p ≥ x)}, for any X, Q ⊆ P , and we will write ↓X (resp., ↑X) instead of P ↓ X (resp., P ↑ X) in case P is understood. We shall also write ↓p instead of ↓{p}, and so on, for p ∈ P . A poset P is lower finite if P ↓ p is finite for all p ∈ P . For p, q ∈ P let p ≺ q hold, if p < q and there is no r ∈ P with p < r < q, in this case p is called a lower cover of q. We denote by P = the set of all non-maximal elements in a poset P . We denote by M(L) the set of all completely meet-irreducible elements of a lattice L. A 2-ladder is a lower finite lattice in which every element has at most two lower covers. S. Z. Ditor constructs in [1] a 2-ladder of cardinality ℵ 1 . For a set X and a cardinal κ, we denote by: [X] κ = {Y ⊆ X \| card Y = κ}, [X] ≤κ = {Y ⊆ X \| card Y ≤ κ}, [X] <κ = {Y ⊆ X \| card Y < κ}. Denote by P the category with objects the ordered pairs (G, u) where G is a pre-ordered abelian group and u is an order-unit of G (i.e., for each x ∈ G, there exists an integer n with −nu ≤ x ≤ nu), and morphisms f : (G, u) → (H, v) where f : G → H is an order-preserving group homomorphism and f (u) = v. We denote by Dim the functor that maps a lattice to its dimension monoid, introduced by F. Wehrung in [13], we also denote by ∆(a, b) for a ≤ b in L the canonical generators of Dim L. We denote by K ℓ 0 the functor that maps a lattice to the pre-ordered abelian universal group (also called Grothendieck group) of its dimension monoid. If L is a bounded lattice then (the canonical image in K ℓ 0 (L) of) ∆(0 L , 1 L ) is an order-unit of K ℓ 0 (L). If f : L → L ′ is a 0, 1-preserving homomor- phism of bounded lattices, then K ℓ 0 (f ) : (K ℓ 0 (L), ∆(0 L , 1 L )) → (K ℓ 0 (L ′ ), ∆(0 L ′ , 1 L ′ ) ) preserves the order-unit. All our rings are associative but not necessarily unital. Micol's thesis [9,Theorem 1.4] for the unital case). Hence L is a functor from the category of regular rings to the category of 0-lattices with 0-lattice homomorphisms. • We denote by V the functor from the category of unital rings with morphisms preserving units to the category of commutative monoids, that maps a unital ring R to the commutative monoid of all isomorphism classes of finitely generated projective right R-modules and any homomorphism f : R → S of unital rings to the monoid homomorphism V (f ) : V (R) → V (S), i e i R → i f (e i )S. We denote by Id R (resp., Id c R) the lattice of all two-sided ideals (resp., finitely generated two-sided ideals) of any ring R. We denote by Sub E the subspace lattice of a vector space E. We denote by M n (F ) the F -algebra of n × n matrices with entries from F , for every field F and every positive integer n. A matricial F -algebra is an F -algebra of the form M k1 (F ) × · · · × M kn (F ), for positive integers k 1 , . . . , k n . For a finitely generated projective right module P over a unital ring R, we denote by [P ] the corresponding element in K 0 (R), that is, the stable isomorphism class of P . We refer to [7,Section 15] for the required notions about the K 0 functor. A K 0 -lifting of a pre-ordered abelian group with order-unit (G, u) is a regular ring R such that (K 0 (R), [R]) ∼ = (G, u). A K 0 -lifting of a diagram G : I → P is a diagram R : I → P such that (K 0 (−), [−]) • R ∼ = G. We denote by ∇ the functor that sends a monoid to it maximal semilattice quotient, that is, ∇(M ) = M/≍ where ≍ is the smallest congruence of M such that M/≍ is a semilattice. We denote by ∇ the functor that maps a partially preordered abelian group G to ∇(G + ) where G + is the monoid of all positive elements of G. We denote by Var(L) (resp., Var 0 (L), resp., Var 0,1 (L)) the variety of lattices (resp., lattices with 0, resp., bounded lattices) generated by a lattice L. A lattice K is a congruence-preserving extension of a lattice L, if L is a sublattice of K and Con c i : Con L → Con K is an isomorphism, where i : L → K is the inclusion map. We denote by M n and M n,m the lattices represented in Figure 1, for 3 ≤ m, n ≤ ω, and by M n and M n,m , respectively, the lattice varieties that they generate. We also denote by M 0 n the variety of lattices with 0 generated by M n , and so on. a∧b ≤ c∨d implies either a ≤ c ∨ d or b ≤ c ∨ d or a ∧ b ≤ c or a ∧ b ≤ d. The lattice M n satisfies Whitman's condition for all n ≥ 3. Lower bounds for some critical points The following proposition is proved in [13,Proposition 5.5]. Then there exists an isomorphism π : Dim L → (Z + ) (P ) such that π(∆(a, b)) = (\|a, b\| ξ \| ξ ∈ P ) for all a ≤ b in L. This makes it possible to prove the following lemma, which gives an explicit description of K ℓ 0 (L) for every modular lattice L of finite length (in such a case the set P is finite). Lemma 3.2. Let L be a modular lattice of finite length, set X = M(Con L). Then there exists an isomorphism π ′ : K ℓ 0 (L) → Z X such that π ′ (∆(a, b)) = (lh([a/θ, b/θ]) \| θ ∈ X), for all a ≤ b in L. In particular (K ℓ 0 (L), ∆(0, 1)) is isomorphic to (Z X , (lh(L/θ)) θ∈X ). Proof. Denote by P be the set of all projectivity classes of prime intervals of L. For any ξ ∈ P denote by θ ξ the largest congruence of L that does not collapse any prime intervals in ξ. As L is modular of finite length, the congruences of L are in one-toone correspondence with subsets of P (cf. [6, Chapter III]), and so the assignment ξ → θ ξ defines a bijection from P onto X. Moreover any prime interval not in ξ is collapsed by θ ξ , for any ξ ∈ P . Let a ≤ b in L, let ξ ∈ P . Let a 0 ≺ a 1 ≺ · · · ≺ a n in L such that a 0 = a and a n = b. Let 0 ≤ r 1 < r 2 < · · · < r s < n be all the integers such that [a r k , a r k +1 ] ∈ ξ for all 1 ≤ k ≤ s. Thus \|a, b\| ξ = s. Set r s+1 = n. As [a r k , a r k +1 ] ∈ ξ and [a r k +t , a r k +t+1 ] ∈ ξ for all 1 ≤ t ≤ r k+1 − r k − 1, we obtain that a r k /θ ξ ≺ a r k +1 /θ ξ = a r k +2 /θ ξ = · · · = a r k+1 /θ ξ , for all 1 ≤ k ≤ s. Thus the following covering relations hold: a/θ ξ = a r1 /θ ξ ≺ a r2 /θ ξ ≺ · · · ≺ a rs /θ ξ ≺ a rs+1 /θ ξ = b/θ ξ . So lh([a/θ ξ , b/θ ξ ]) = s = \|a, b\| ξ . We conclude the proof by using Proposition 3.1. We shall always apply this result to unital regular rings R such that V (R) is cancellative (i.e., R is unit-regular), so K 0 (R) + = V (R), and to lattices L such that Dim L is cancellative, so K ℓ 0 (L) + ∼ = Dim L. Here G + denotes the positive cone of G, for any partially pre-ordered abelian group G. (i) ∇ • Dim ∼ = Con c on lattices (ii) ∇ • V ∼ = Con c •L on The following theorem is proved in [7,Theorem 15.23]. Theorem 3.4. Let F be a field, let R be a matricial F -algebra, and let S be a unit-regular F -algebra. (1) Given any morphism f : (K 0 (R), [R]) → (K 0 (S), [S]) in P, the category of pre-ordered abelian groups with order-unit (cf. Section 2), there exists an F -algebra homomorphism φ : R → S such that K 0 (φ) = f . (2) If φ, ψ : R → S are F -algebra homomorphisms, then K 0 (φ) = K 0 (ψ) if and only if there exists an inner automorphism θ of S such that φ = θ • ψ. The following lemma is folklore. Lemma 3.5. Let F be a field, let u = (u k ) 1≤k≤n be a family of positive integers, let R = n k=1 M u k (F ). Then (K 0 (R), [R]) ∼ = (Z n , u). Lemma 3.6. Let F be a field. Let I be a 2-ladder, let G i = (Z ni , u i = (u i k ) 1≤k≤ni ) such that u i is an order-unit, let R i = ni k=1 M u i k (F ) for all i ∈ I. Let f i,j : G i → G j for all i ≤ j in I such that G = (G i , f i,j ) i≤j in I is a direct system in P. Then there exists a direct system (R i , φ i,j ) i≤j in I of matricial F -algebra which is a K 0 -lifting of (G i , f i,j ) i≤j in I . Proof. By Lemma 3.5 there exists an isomorphism τ i : (K 0 (R i ), [R i ]) → G i = (Z ni , u i ) in P, for all i ∈ I. Let g i,j = τ −1 j • f i,j • τ i , for all i ≤ j in I. For i = j = 0 (the smallest element of I), we put φ 0,0 = id R0 . Let i ∈ I with a lower cover i ′ . It follows from Theorem 3.4(1) that there exists ψ i ′ ,i : R i ′ → R i such that K 0 (ψ i ′ ,i ) = g i ′ ,i . If i has only i ′ as lower cover, assume that we have a direct system (R j , φ j,k ) j≤k≤i ′ lifting (G j , f j,k ) j≤k≤i ′ . Set φ j,i = ψ i ′ ,i • φ j,i ′ for all j < i, and φ i,i = id Ri . It is easy to see that (R i , φ j,k ) j≤k≤i is a direct system lifting (G j , f j,k ) j≤k≤i . Let i has two distinct lower covers i ′ and i ′′ , and set ℓ = i ′ ∧ i ′′ . Assume that we have direct system (R j , φ j,k ) j≤k≤i ′ and (R j , φ j,k ) j≤k≤i ′′ lifting (G j , f j,k ) j≤k≤i ′ and (G j , f j,k ) j≤k≤i ′′ respectively. The following equalities hold K 0 (ψ i ′ ,i • φ ℓ,i ′ ) = K 0 (ψ i ′ ,i ) • K 0 (φ ℓ,i ′ ) = g i ′ ,i • g ℓ,i ′ = g ℓ,i Similarly K 0 (ψ i ′′ ,i • φ ℓ,i ′′ ) = g ℓ,i = K 0 (ψ i ′ ,i • φ ℓ,i ′ ), thus, by Theorem 3.4(2), there exists an inner automorphism θ of R i such that θ • ψ i ′′ ,i • φ ℓ,i ′′ = ψ i ′ ,i • φ ℓ,i ′ . Put φ i ′ ,i = ψ i ′ ,i and φ i ′′ ,i = θ • ψ i ′′ ,i . Thus φ i ′ ,i • φ i ′ ∧i ′′ ,i ′ = φ i ′′ ,i • φ i ′ ∧i ′′ ,i ′′ , so we can construct a direct system (R j , φ j,k ) j≤k≤i . Hence, by induction, we obtain a direct system (R i , φ i,j ) i≤j in I of matricial F - algebras, such that K 0 (φ i,j ) = g i,j for all i ≤ j in I as required. Lemma 3.7. Let F be a field. Let L be a bounded modular lattice such that all finitely generated sublattices of L have finite length. Assume that card L ≤ ℵ 1 . Then there exists a locally matricial ring R such that Con L ∼ = Con L(R) and L(R) ∈ Var 0,1 (Sub F n \| n < ω). Moreover if there exists n < ω such that n ≥ lh(K) for each simple lattice K ∈ Var(L) of finite length, then there exists a locally matricial ring R such that Con L ∼ = Con L(R) and L(R) ∈ Var 0,1 (Sub F n ). Proof. Let I be a 2-ladder of cardinality ℵ 1 . Pick a surjection ρ : I ։ L and denote by L i the sublattice of L generated by ρ(I ↓ i) ∪ {0, 1}, for each i ∈ I. Furthermore, denote by f i,j : L i → L j the inclusion map, for all i ≤ j in I. Then L = (L i , f i,j ) i≤j in I is a direct system of modular lattices of finite length and 0, 1-lattice embeddings. Assume that there exists n < ω such that n ≥ lh(K) for each simple lattice K ∈ Var(L) of finite length. Let G = K ℓ 0 • L, set X i = M(Con L i ) for all i ∈ I, and set r i x = lh(L i /x) for each x ∈ X i . The congruence lattice of any modular lattice of finite length is Boolean (cf. [6, Chapter III]), in particular, every subdirectly irreducible modular lattice of finite length is simple. This applies to the subdirectly irreducible lattice L i /x, which is therefore simple. Thus r i x ≤ n, for all i ∈ I and all x ∈ X i . By Lemma 3.2, G i ∼ = (Z Xi , (r i x ) x∈Xi ) for all i ∈ I. Set R i = x∈Xi M r i x (F ). By Lemma 3.5, (K 0 (R i ), [R i ]) ∼ = (Z Xi , (r i x ) x∈X ) ∼ = G i . By Lemma 3.6, there exists a direct system R = (R i , φ i,j ) i≤j in I with morphisms preserving units, such that: K 0 • R ∼ = G = K ℓ 0 • L. (3.1) Moreover: L(R i ) ∼ = L x∈Xi M r i x (F ) ∼ = x∈Xi L(M r i x (F )) ∼ = x∈Xi Sub F r i x ∈ Var 0,1 (Sub F n ). Let R = lim − → R. As L preserves direct limits, L(R) ∼ = lim − → (L • R), but L • R is a diagram of Var 0,1 (Sub F n ), so L(R) ∈ Var 0,1 (Sub F n ) . Moreover the following isomorphisms hold: Con c L(R) ∼ = ∇(K 0 (R)) by Proposition 3.3 ∼ = ∇(K 0 (lim − → R)) ∼ = ∇(lim − → (K 0 • R)) as K 0 preserves direct limits ∼ = ∇(lim − → (K ℓ 0 • L)) by (3.1) ∼ = ∇(K ℓ 0 (lim − → L)) as K ℓ 0 preserves direct limits ∼ = ∇(K ℓ 0 (L)) ∼ = Con c L by Proposition 3.3. The other case, without restriction on finite lengths of simple lattices, is similar. Lemma 3.7 works for bounded lattices, however any lattice can be embedded into a bounded lattice. In the rest of this section, using this result, we extend Lemma 3.7 to unbounded lattices. Lemma 3.8. Let L be a lattice, let L ′ = L⊔{0, 1} such that 0 is the smallest element of L ′ and 1 is the largest. Let f : L ֒→ L ′ be the inclusion map. Then Con c f is a injective (∨, 0)-homomorphism and (Con c f )(Con c L) is an ideal of Con c L ′ . Proof. Let θ ∈ Con c L, let L ′ θ = (L/θ) ⊔ {0, 1} such that 0 is the smallest element of L ′ θ and 1 is its largest element. The following map g : L ′ → L ′ θ x →      0 if x = 0 1 if x = 1 x/θ if x ∈ L is a lattice homomorphism, and ker g = θ ∪ {(0, 0), (1, 1)}, so the latter is a congruence of L ′ . It follows that (Con c f )(θ) = θ ∪ {(0, 0), (1, 1)}. Thus Con c f is an embedding. Let β = n i=1 Θ L ′ (x i , y i ) ∈ Con c L ′ , such that β ⊆ (Con c f )(θ). We can assume that x i = y i for all 1 ≤ i ≤ n. Thus, as (x i , y i ) ∈ θ ∪ {(0, 0), (1, 1)}, (x i , y i ) ∈ θ for all 1 ≤ i ≤ n. Let α = n i=1 Θ L (x i , y i ), then (Con c f )(α) = β. Thus (Con c f )(Con c L) is an ideal of Con c L ′ . F. Wehrung proves the following proposition in [14,Corollary 4.4]; the result also applies to the non-unital case, with a similar proof. Proposition 3.9. For any regular ring R, Con c L(R) is isomorphic to Id c R. Lemma 3.10. Let R be a regular ring, and let I be a two-sided ideal of R. Then the following assertions hold (1) The set I is a regular subring of R. (2) Any right (resp., left ) ideal of I is a right (resp., left ) ideal of R. Let J be a right ideal of I, let a ∈ J, let x ∈ R. As I is regular there exists y ∈ I such that a = aya, so ax = ayax, but a ∈ I, so yax ∈ I, moreover J is a right ideal of I, so ax = ayax ∈ J. Thus J is a right ideal of R. Similarly any left ideal of I is a left ideal of R. Thus Id(I) = Id(R) ↓ I. Let a ∈ R idempotent. If aR ⊆ I, then a ∈ I, so aI ⊆ aR = aaR ⊆ aI, and so aI = aR, thus aR ∈ L(I). So L(I) = L(R) ↓ I. Theorem 3.11. Let F be a field. Let V be a variety of modular lattices (resp., a variety of bounded modular lattices). Assume that all finitely generated lattices of V have finite length. Then crit(V; Var 0 (Sub F n \| n ∈ ω)) ≥ ℵ 2 (resp., crit(V; Var 0,1 (Sub F n \| n ∈ ω)) ≥ ℵ 2 ). Moreover for L ∈ V of cardinality at most ℵ 1 , there exists a regular ring A such that Con L ∼ = Con L(A) and L(A) ∈ Var 0 (Sub F n \| n ∈ ω) (resp., L(A) ∈ Var 0,1 (Sub F n \| n ∈ ω)). If there exists n < ω such that lh(K) ≤ n for each simple lattice K ∈ V of finite length, then: crit(V; Var 0 (Sub F n )) ≥ ℵ 2 (resp., crit(V; Var 0,1 (Sub F n )) ≥ ℵ 2 ). Moreover for L ∈ V of cardinality at most ℵ 1 , there exists a regular ring A such that Con L ∼ = Con L(A) and L(A) ∈ Var 0 (Sub F n ) (resp., L(A) ∈ Var 0,1 (Sub F n )). Observe that L(A) is, in addition, relatively complemented; in particular, it is congruence-permutable. Proof. The bounded case is an immediate application of Lemma 3.7. Let V be a variety of modular lattices in which finitely generated lattices have finite length. Let L ∈ V such that card L ≤ ℵ 1 , let L ′ = L ⊔ {0, 1} as in Lemma 3.8 and let D be the ideal of Con c L ′ corresponding to Con c L. By Chapter I, Section 4, Exercise 14 in [6] we have L ′ ∈ V, thus, by Lemma 3.7, there exists a regular ring R such that L(R) ∈ Var 0 (Sub F n ), and Con c L(R) ∼ = Con c L ′ . By Proposition 3.9, Con c L(R) ∼ = Id c R. Let I be the ideal of R corresponding to D. Then Con L ∼ = Id D ∼ = Id R ↓ I ∼ = Id I ∼ = Con L(I). Moreover L(I) = L(R) ↓ I belongs to W. We obtain the following generalization of M. Ploščica's results in [11]. In Section 4 we shall give another (∨, 0)-semilattice of cardinality ℵ 2 , congruenceliftable in M m , but not congruence-liftable in M n . An upper bound of some critical points Using the results of [5], we first prove that if a simple lattice of a variety of lattices V has larger length than all simple lattices of a finitely generated variety of lattices W, then crit(V; W) ≤ ℵ 0 . Remark 4.1. Let x ≺ y in a lattice L. Let (α i ) i∈I be a family of congruences of L, if (x, y) ∈ i∈I α i , then (x, y) ∈ α i for some i ∈ I. In particular there exists a largest congruence separating x and y. Such a congruence is completely meet-irreducible, and in a lattice of finite height all completely meet-irreducible congruences are of this form. Proof. If L has no finite maximal chain then lh(L) ≥ n is immediate. Assume that C is a finite maximal chain of L. Denotes by 0 = x 0 ≺ x 1 ≺ · · · ≺ x m = 1 the elements of C. Denote by f : C → L the inclusion map. Let k ∈ {0, . . . , m − 1}. We have x k ≺ x k+1 , hence Θ L (x k , x k+1 ) is joinirreducible in Con c L. As Con c L is Boolean, Θ L (x k , x k+1 ) is an atom of Con c L. Let θ be an atom of Con c L, we have: θ ≤ Θ L (0, 1) = m−1 k=0 Θ L (x k , x k+1 ) So there exists k ∈ {0, . . . , m − 1} such that θ ≤ Θ L (x k , x k+1 ). As Θ L (x k , x k+1 ) is an atom of Con c L, we have θ = Θ L (x k , x k+1 ). It follows that Con c f is surjective, so m ≥ n and so lh(L) ≥ n. Proof. As D is finite, taking a sublattice, we can assume that card K ≤ ℵ 0 . Let n be the greatest length of a lifting of D in W. As lh(K) > n, there exists a chain C of K of length n+1 (resp., we can assume that C has 0 and 1). Let f : C → K be the inclusion map. Assume that there exists a lifting g : C ′ → K ′ of Con c f in W. As f is an embedding, g is also an embedding. As Con c K ′ ∼ = Con c K ∼ = D, lh(K ′ ) ≤ n. Moreover Con c C ′ ∼ = Con c C ∼ = 2 n+1 , thus, by Lemma 4.2, lh(C ′ ) = n + 1. So n ≥ lh(K ′ ) ≥ lh(C ′ ) = n + 1; a contradiction. Therefore Con c f has no lifting in W. So, as card K ≤ ℵ 0 and by [5, Corollary 7.6], crit(V; W) ≤ ℵ 0 (in the bounded case f preserves bounds, thus the result of [5] also applies). Moreover if V is a finitely generated variety of modular lattices, then the finite (∨, 0)-semilattices with congruence-lifting in V are the finite Boolean lattices. Finite Boolean lattices are also liftable in W. Hence crit(V; W) = ℵ 0 . The following corollary is an immediate application of Theorem 4.3 and Theorem 3.11. It shows that the critical point between a finitely generated variety of modular lattices and a variety generated by a lattice of subspaces of a finite vector space, cannot be ℵ 1 . Corollary 4.4. Let V be a finitely generated variety of modular lattices, let F be a finite field, let n ≥ 1 be an integer. If there exists a simple lattice in K ∈ V such that lh(K) > n, then crit(V; Var(Sub F n )) = ℵ 0 , else crit(V; Var(Sub F n )) ≥ ℵ 2 . We shall now give a diagram of (∨, 0)-semilattices S, congruence-liftable in M n , such that for every finitely generated variety V, generated by lattices of length at most three, the diagram S is congruence-liftable in V if and only if M n ∈ V. Let n ≥ 3 be an integer. Set n = {0, 1, . . . , n − 1}, and set: I n = {P ∈ P(n) \| either card(P ) ≤ 2 or P = n}. Denote by a 0 , . . . , a n−1 the atoms of M n . Set A P = {a x \| x ∈ P } ∪ {0, 1}, for all P ∈ I n . Let f P,Q : A P → A Q be the inclusion map for all P ⊆ Q in I n . Then A = (A P , f P,Q ) P ⊆Q in In is a direct system in M 0,1 n . The diagram S is defined as Con c • A. Lemma 4.5. Let B = (B P , g P,Q ) P ⊆Q in In be a congruence-lifting of Con c • A by lattices, with all the maps g P,Q inclusion maps, for all P ⊆ Q in I n . Let u < v in B ∅ . Let P ∈ I n then: Θ BP (u, v) = B P × B P , the largest congruence of B P . Let ξ = (ξ P ) P ∈In : Con c • A → Con c • B be a natural equivalence. Let x, y ∈ n distinct. Let b x ∈ [u, v] B {x} and b y ∈ [u, v] B {y} . Set P = {x, y}. Let c ∈ {0, 1}. Then the following assertions hold: (1) If Θ B {x} (u, b x ) = ξ {x} (Θ A {x} (c, a x )), then Θ BP (u, b x ) = ξ P (Θ AP (c, a x )). (2) If Θ B {z} (u, b z ) = ξ {z} (Θ A {z} (c, a z )) for all z ∈ {x, y}, then b x ∧ b y = u. (3) If Θ B {x} (b x , v) = ξ {x} (Θ A {x} (c, a x )), then Θ BP (b x , v) = ξ P (Θ AP (c, a x )). (4) If Θ B {z} (b z , v) = ξ {z} (Θ A {z} (c, a z )) for all z ∈ {x, y}, then b x ∨ b y = v. (5) If Θ B {x} (u, b x ) = ξ {x} (Θ A {x} (c, a x )) and Θ B {y} (b y , v) = ξ {y} (Θ A {y} (c, a y )), then b x ≤ b y . Proof. As f P,Q preserves bounds, Con c f P,Q preserves bounds, thus Con c g P,Q preserves bounds, for all P ⊆ Q in I n . Let u < v in B ∅ . As B ∅ is simple, Θ B ∅ (u, v) is the largest congruence of B ∅ . Moreover, Con c g ∅,P preserves bounds, for all P ∈ I n . Hence: Θ BP (u, v) = B P × B P , the largest congruence of B P . (1) The following equalities hold: c, a x )). Θ BP (u, b x ) = (Con c g {x},P )(Θ B {x} (u, b x )) = (Con c g {x},P )(ξ {x} (Θ A {x} (c, a x ))) by assumption = ξ P • (Con c f {x},P )(Θ A {x} (c, a x )) = ξ P (Θ AP ( (2) The following containments hold: (1). (4) Similar to (2). (5) The following containments hold: Θ BP (u, b x ∧ b y ) ⊆ Θ BP (u, b x ) ∩ Θ BP (u, b y ) = ξ P (Θ AP (c, a x )) ∩ ξ P (Θ AP (c, a y )) by (1) = ξ P (Θ AP (c, a x ) ∩ Θ AP (c, a y )) = ξ P (id AP ) = id BP . so u = b x ∧ b y . (3) Similar toΘ BP (b y , b x ∨ b y ) ⊆ Θ BP (u, b x ) ∩ Θ BP (b y , v) = ξ P (Θ AP (c, a x )) ∩ ξ P (Θ AP (c, a y )) by (1) and (3) = ξ P (Θ AP (c, a x ) ∩ Θ AP (c, a y )) = ξ P (id AP ) = id BP . so b y = b x ∨ b y , thus b x ≤ b y . The following lemma shows that if we have some "small" enough congruencelifting of Con c • A in a variety, then M n belongs to this variety. Proof. Let ξ = (ξ P ) P ∈In : Con c • A → Con c • B be a natural equivalence. As f P,Q is an embedding, Con c f P,Q separates 0, so Con c g P,Q separates 0, hence g P,Q is an embedding, thus we can assume that g P,Q is the inclusion map from B P into B Q , for all P ⊆ Q in I n . Let u < v in B ∅ . By Lemma 4.5, Θ B {x} (u, v) is the largest congruence of B {x} . Moreover B {x} is the 3-element chain, so u is the smallest element of B {x} while v is its largest element. Denote by b x the middle element of B {x} . The congruence ξ {x} (Θ A {x} (0, a x )) is join-irreducible, thus it is either equal to Θ B {x} (u, b x ) or to Θ B {x} (b x , v). Set: X ′ = {x ∈ n \| ξ {x} (Θ A {x} (0, a x )) = Θ B {x} (u, b x )}, X ′′ = {x ∈ n \| ξ {x} (Θ A {x} (0, a x )) = Θ B {x} (b x , v)}. As Θ A {x} (0, a x ) is the complement of Θ A {x} (a x , 1) and Θ B {x} (u, b x ) is the comple- ment of Θ B {x} (b x , v) , we also get that: X ′ = {x ∈ n \| ξ {x} (Θ A {x} (a x , 1)) = Θ B {x} (b x , v)} X ′′ = {x ∈ n \| ξ {x} (Θ A {x} (a x , 1)) = Θ B {x} (u, b x )}. Moreover n = X ′ ∪ X ′′ . As card n ≥ 3, either card X ′ ≥ 2 or card X ′′ ≥ 2. Assume that card X ′ ≥ 2. Let x, y in X ′ distinct. By Lemma 4.5 (2) , b x ∧ b y = u. By Lemma 4.5(4), b x ∨ b y = v. Now assume that X ′′ = ∅. Let z ∈ X ′′ . As ξ {x} (Θ A {x} (0, a x )) = Θ B {x} (u, b x ) and ξ {z} (Θ A {z} (0, a z )) = Θ B {z} (b z , v), it follows from Lemma 4.5(5) that b x ≤ b z . Sim- ilarly, as ξ {z} (Θ A {z} (a z , 1)) = Θ B {z} (u, b z ) and ξ {y} (Θ A {y} (a y , 1)) = Θ B {y} (b y , v), it follows from Lemma 4.5(5) that b z ≤ b y . Thus b x ≤ b y . So u = b x ∧ b y = b x > u, a contradiction. Thus X ′′ = ∅, so X ′ = n, and so {u, b 0 , b 1 , . . . , b n , v} is a sublattice of B n isomorphic to M n . The case card X ′′ ≥ 2 is similar. We shall now use a tool introduced in [5] to prove that having a congruence-lifting of Con c • A is equivalent to having a congruence-lifting of some (∨, 0)-semilattice of cardinality ℵ 2 . This requires the following infinite combinatorial property, proved by A. Hajnal and A. Máté in [8], see also [3,Theorem 46.2]. This property is also used by M. Ploščica in [10]. Definition 4.8. A finite subset V of a poset U is a kernel, if for every u ∈ U , there exists a largest element v ∈ V such that v ≤ u. We denote this element by V · u. We say that U is supported, if every finite subset of U is contained in a kernel of U . We denote by V · u the largest element of V ∩ u, for every kernel V of U and every ideal u of U . As an immediate application of the finiteness of kernels, we obtain that any intersection of a nonempty set of kernels of a poset U is a kernel of U . A sharp ideal of (U, \|·\|) is an ideal u of U such that {\|v\| \| v ∈ u} has a largest element. For example, for every u ∈ U , the principal ideal U ↓ u is sharp; we shall often identify u and U ↓ u. We denote this element by \|u\|. We denote by Id s (U, \|·\|) the set of all sharp ideals of (U, \|·\|), partially ordered by inclusion. A sharp ideal u of (U, \|·\|) is extreme, if there is no sharp ideal v with v > u and \|v\| = \|u\|. We denote by Id e (U, \|·\|) the set of all extreme ideals of (U, \|·\|). Let κ be a cardinal number. We say that (U, \|·\|) is κ-compatible, if for every order-preserving map F : Id e (U, \|·\|) → P(U ) such that card F (u) < κ for all u ∈ Id e (U, \|·\|) = , there exists an order-preserving map σ : I → Id e (U, \|·\|) such that: (1) The equality \|σ(i)\| = i holds for all i ∈ I. (2) The containment F (σ(i)) ∩ σ(j) ⊆ σ(i) holds for all i ≤ j in I. Lemma 4.10. Let X be a set, let (A x ) x∈X be a family of sets, let: U = P ∈[X] <ω x∈P A x . We view the elements of U as (partial ) functions and "to be greater" means "to extend". Then U is a supported poset. Proof. Let V be a finite subset of U . Let Y x = {u x \| u ∈ V and x ∈ dom u} for all x ∈ X. Let D = u∈V dom u. Let: W = {u ∈ U \| dom u ⊆ D and (∀x ∈ dom u)(u x ∈ Y x )} the set D, and the sets Y x for x ∈ X are all finite, so W is finite. Let u ∈ U , let P = {x ∈ dom u \| x ∈ D and u x ∈ Y x }. Then u ↾ P ∈ W . Moreover let w ∈ W such that w ≤ u. Let x ∈ dom w, then x ∈ D, and u x = w x ∈ Y x , thus dom w ⊆ P , so w ≤ u ↾ P . Therefore u ↾ P is the largest element of W ↓ u. Using Lemma 4.10 and Proposition 4.7 we can construct a ℵ α -compatible lower finite norm-covering of I n , the poset constructed earlier. Lemma 4.11. Let α be an ordinal. Let U = P ∈P(n) ℵ P α+2 , partially ordered by inclusion. Let \|·\| : U → I n u → \|u\| = dom u if card(dom u) ≤ 2 n otherwise. Then (U, \|·\|) is a ℵ α -compatible lower finite norm-covering of I n . Moreover card U = ℵ α+2 . Proof. By Lemma 4.10, the set U is supported. Moreover \|·\| preserves order, so (U, \|·\|) is a norm-covering of I n . The poset U is lower finite. Extreme ideals are of the form ↓u, where u ∈ U and dom u ∈ I n , so we identify the corresponding extreme ideal with u. Thus Id e (U, \|·\|) = {u ∈ U \| dom u ∈ I n }. Let F : Id e (U, \|·\|) → P(U ) be an order-preserving map such that card F (u) < ℵ α for all u ∈ Id e (U, \|·\|) = , let G : [ℵ α+2 ] 2 → [ℵ α+2 ] <ℵα s →    im v \| u ∈ P ∈In−{n} s P and v ∈ F (u)    . By Proposition 4.7, there exists A ⊂ ℵ α+2 such that card A = n and a ∈ G({b, c}) for all distinct a, b, c ∈ A. Let u : n → A be a one-to-one map. Let φ : I n → Id e (U, \|·\|), P → u ↾ P . Then \|φ(P )\| = P . Let P Q in I n , let v ∈ F (u ↾ P ) ↓ (u ↾ Q). Let x ∈ dom v − P . As P ∈ I n , and P = n, card P ≤ 2. Let P ′ = {y, z} ⊆ n, such that y, z are distinct, P ⊆ P ′ , and x ∈ P ′ . Let s = {u y , u z }, then u ↾ P ′ ∈ s P ′ , as v ∈ F (u ↾ P ) ⊆ F (u ↾ P ′ ), v x ∈ G(s). Moreover u x , u y , u z ∈ A are distinct, thus u x ∈ G({u y , u z }) = G(s), so v x = u x in contradiction with v ≤ u, so dom v ⊆ P , and so v ≤ u ↾ P . Using the results of [5] together with Lemma 4.11, we obtain the following result. Lemma 4.12. Let V be a variety of algebras with a countable similarity type, let W be a finitely generated congruence-distributive variety such that crit(V; W) > ℵ 2 . Let D : I n → S be a diagram of finite (∨, 0)-semilattices. If D is congruence-liftable in V, then D is congruence-liftable in W. Proof. In this proof we use, but do not give, many definitions of [5]. By Lemma 4.11 there exists (U, \|·\|) a ℵ 0 -compatible lower finite norm-covering of I n such that card U = ℵ 2 . Let J be a one-element ordered set. By [5,Lemma 3.9], W is (Id e (U, \|·\|) = , J, ℵ 0 )-Löwenheim-Skolem. Let A = (A P , f P,Q ) P ⊆Q in In be a congruence-lifting of D in V. As Con c A P is finite, using [5,Lemma 3.6], taking sublattices we can assume that A P is countable for all P ∈ I n . By [5,Lemma 6.7], there exists an U -quasi-lifting (τ, Cond( A, U )) of D in V. Moreover: card Cond( A, U ) ≤ V ∈[U] <ω card u∈V A \|u\| ≤ V ∈[U] <ω ℵ 0 ≤ ℵ 2 As crit(V; W) > ℵ 2 , there are B ∈ W and an isomorphism ξ : Con c Cond( A, U ) → Con c B. So (τ •ξ −1 , B) is an U -quasi-lifting of D. Moreover W is (Id e (U, \|·\|) = , J, ℵ 0 )-Löwenheim-Skolem, hence, by [5,Theorem 6.9], with I = I n , there exists a congruencelifting of D in W. A similar proof, using Lemma 3.6, Lemma 3.7, Lemma 6.7, and Theorem 6.9 in [5] together with Lemma 4.11, yields the following generalization of Lemma 4.12. Lemma 4.13. Let α ≥ 1 be an ordinal. Let V and W be varieties of algebras, with similarity types of cardinality < ℵ α . Let D = (D P , ϕ P,Q ) P ⊆Q in In be a direct system of (∨, 0)-semilattices. Assume that the following conditions hold: (1) crit(V; W) > ℵ α+2 . (2) card(D P ) < ℵ α , for all P ∈ I n − {n}. (3) card(D n ) ≤ ℵ α+2 . (4) D is congruence-liftable in V. Then D is congruence-liftable in W. The following lemma implies, in particular, that a modular lattice of length three is a congruence-preserving extension of one of its subchains. Lemma 4.14. Let L be a lattice of length at most three, let u, v in L such that Θ L (u, v) = L × L. If Con c L ∼ = 2 2 , then there exists x ∈ L with u < x < v such that L is a congruence-preserving extension of the chain C = {u, x, v}. Proof. As Con c L ∼ = 2 2 , lh([u, v]) ≥ 2. If lh([u, v]) = 2, then let C = {u, x, v}, where x is any element such that u < x < v. Let i : C → L the inclusion map. The morphism Con c i : Con c C → Con c L is onto, moreover Con c C ∼ = 2 2 ∼ = Con c L, so Con c i is an isomorphism. Now assume that [u, v] has length three. As lh(L) ≤ 3, lh(L) = 3, u is the smallest element of L, and v is the largest element. Assume that L has a sublattice isomorphic to N 5 , as labeled in Figure 2. Then C = {u, y, z, v} is a maximal chain of L. Let i : C → L be the inclusion map. By Lemma 4.2, Con c i is surjective. Thus, as Con L ∼ = 2 2 , and Θ L (u, y), Θ L (y, z), and Θ L (z, v) are all the atoms of Con L, Θ L (y, z) ⊆ Θ L (u, y) ∩ Θ L (y, z) ∩ Θ L (z, v) = id L , a contradiction. Thus L does not contain any lattice isomorphic to N 5 , that is, L is modular. Proof. By Lemma 4.15, crit(V; W) ≤ ℵ 2 . Assume that lh(K) ≤ 2 for each simple K ∈ V and M 3 ∈ W. As Sub F 2 2 ∼ = M 3 ∈ W, it follows from Theorem 3.11 that crit(V; W) ≥ ℵ 2 . Assume that lh(K) ≤ 3 for each simple K ∈ V and Sub F 3 ∈ W for some field F , it follows from Theorem 3.11 that crit(V; W) ≥ ℵ 2 . Similarly we obtain the following critical points. As an immediate consequence we obtain: Proof. If F is infinite then the result is obvious. So we can assume that F is finite. If n ≤ 1+card F , then M n is a sublattice of M 1+card F ∼ = Sub F 2 ∈ Var(Sub F 3 ), thus M n ∈ Var(Sub F 3 ). Now assume that M n ∈ Var(Sub F 3 ). By Jónsson's Lemma, M n is a homomorphic image of a sublattice of Sub F 3 . As M n satisfies Whitman's condition, it follows from the Davey-Sands Theorem [2, Theorem 1] that M n is projective in the class of all finite lattices. Therefore, as Sub F 3 is finite, M n is a sublattice of Sub F 3 . Thus there exist distinct subspaces A, B, V 1 , V 2 , . . . , V n of F 3 such that V i ∩ V j = A and V i + V j = B, for all 1 ≤ i < j ≤ n. Let i, j, k distinct. Then: dim V i + dim V j = dim B + dim A = dim V i + dim V k . Thus dim V j = dim V k . But dim A < dim V 1 < dim B ≤ dimdim B = dim(V 1 ⊕ V 2 ) = dim V 1 + dim V 2 = 2 · dim V 1 . Thus dim B is even, moreover dim B ≤ 3, hence dim B = 2. In both cases M n is a sublattice of Sub E for some F -vector space E of dimension two. But Sub E ∼ = M 1+card F , thus n ≤ 1 + card F . As each simple lattice of M n is of length at most two, it follows from Theorem 3.11 that crit(M n ; Var 0 (Sub F n )) ≥ ℵ 2 , and crit(M 0,1 n ; Var 0,1 (Sub F n )) ≥ ℵ 2 . (4.5) All the other desired equalities are consequences of (4.4), (4.5). As each simple lattice of Var(Sub F 3 ) is of length at most three, it follows from Theorem 3.11 that: crit(Var(Sub F 3 ); Var 0 (Sub K n )) ≥ ℵ 2 ,(4.7) crit(Var 0,1 (Sub F 3 ); Var 0,1 (Sub K n )) ≥ ℵ 2 . (4.8) All the other desired equalities are consequences of (4.6), (4.7), (4.8). Proof. Set n = 2m − 1 ≥ 3. Let A = (A P , f P,Q ) P ⊆Q in In be the direct system of M 0,1 n introduced just before Lemma 4.5. Assume that crit(M 0,1 n ; V) > ℵ 2 . By Lemma 4.12, there exists a congruence-lifting B = (B P , g P,Q ) P ⊆Q in In of Con c • A in V. Let ξ = (ξ P ) P ∈In : Con c • A → Con c • B be a natural equivalence. Taking a sublattice of B ∅ , we can assume that B ∅ is a chain u < v. Moreover, as the map f P,Q is an inclusion map, we can assume that g P,Q is an inclusion map, for all P ⊆ Q in I n . Let x ∈ n. By Lemma 4.5, Θ B {x} (u, v) is the largest congruence of B {x} . Thus: Θ B {x} (u, v) = ξ {x} (Θ A {x} (0, a x )) ∨ ξ {x} (Θ A {x} (a x , 1)) Therefore there exist t x 0 = u < t x 1 < · · · < t x r+1 = v in B {x} such that, for all 0 ≤ i ≤ r: 1))}. By symmetry we can assume that card X ′ ≥ card X ′′ (we can replace the diagram A by its dual if required). As n = X ′ ∪ X ′′ and card n = n = 2m − 1, card X ′ ≥ m. Let x, y ∈ X ′ distinct, it follows from Lemma 4.5(2) that b x ∧ b y = u. So we obtain a family of elements (b x ) x∈X ′ greater than u such that b x ∧ b y = u (resp., b x ∧ b y = u = 0) for all x = y in X ′ , a contradiction. either (t x i , t x i+1 ) ∈ ξ {x} (Θ A {x} (0, a x )) or (t x i , t x i+1 ) ∈ ξ {x} (Θ A {x} (a x , 1)) Set b x = t x 1 . Put: X ′ = {x ∈ n \| Θ B {x} (u, b x ) = ξ {x} (Θ A {x} (0, a x ))} X ′′ = {x ∈ n \| Θ B {x} (u, b x ) = ξ {x} (Θ A {x} (a x , With a similar proof using Lemma 4.13 instead of Lemma 4.12 we obtain the following lemma. crit(M 0,1 ω ; V) = ℵ 2 . Proof. Let V be a finitely generated variety of lattices, let m be the maximal cardinality of a simple lattice of V. Thus the assumptions of Lemma 4.22 are satisfied, so a fortiori crit(M 0,1 2m−1 ; V) ≤ ℵ 2 , and so crit(M 0,1 ω ; V) ≤ ℵ 2 . Denote by F 2 the two-element field. Let V be a variety of lattices with 0 (resp., with 0 and 1), such that M 3 ∈ V. The variety M ω is locally finite, thus all finitely generated lattices of M ω are of finite length. Moreover all simple lattices of M ω have length at most two. Thus, by Theorem 3.11: crit(M ω ; Var 0 (Sub F 2 2 )) ≥ ℵ 2 (resp., crit(M 0,1 ω ; Var 0,1 (Sub F 2 2 )) ≥ ℵ 2 ). Moreover Sub F 2 2 ∼ = M 3 , so crit(M ω ; V) ≥ ℵ 2 . Acknowledgment I thank Friedrich Wehrung whose help, advise, and careful reading of the paper, led to several improvements both in form and substance. I also thank Miroslav Ploščica for the stimulating discussions that led to the diagram A introduced in Section 4. Partially supported by the institutional grant MSM 0021620839. • We denote by L(R) the poset of principal right ideals of every regular ring R. The results of Fryer and Halperin in [4, Section 3.2], imply that, L(R) is a 0-lattice, and for any homomorphism f : R → S of regular rings, the map L(f ) : L(R) → L(S), I → f (I)S is a 0-lattice homomorphism (cf. Figure 1 . 1The lattices M n and M n,m . A lattice L satisfies Whitman's condition if for all a, b, c, and d in L: Proposition 3. 1 . 1Let L be a modular lattice without infinite bounded chains. Let P be the set of all projectivity classes of prime intervals of L. Given ξ ∈ P , denote by \|a, b\| ξ the number of occurrences of an interval in ξ in any maximal chain of the interval [a, b]. Proposition 3 . 3 . 33The following natural equivalences hold ( 3 ) 3In particular Id(I) = Id(R) ↓ I, and L(I) = L(R) ↓ I.Proof. The assertion (1) follows from [7, Lemma 1.3]. Corollary 3 . 12 . 312Let m, n be ordinals such that 3 ≤ n < m ≤ ω. Then the equality crit(M m ; M n ) = ℵ 2 holds. Proof. Every simple lattice of M n has length at most two. Moreover, Sub F 2 2 ∼ = M 3 ∈ M n , where F 2 is the two-element field. Thus, by Theorem 3.11, crit(M m ; M n ) ≥ ℵ 2 . Conversely, M. Ploščica proves in [10] that there exists a (∨, 0)-semilattice of cardinality ℵ 2 , congruence-liftable in M m , but not congruence-liftable in M n . So crit(M m ; M n ) ≤ ℵ 2 . Lemma 4. 2 . 2Let L be a lattice and let n ≥ 0. If Con c L ∼ = 2 n then lh(L) ≥ n. Moreover, if C is a finite maximal chain of L, then Con c f is surjective, where f : C → L is the inclusion map. Theorem 4. 3 . 3Let V be a variety of lattices (resp., a variety of bounded lattices), let W be a finitely generated variety of lattices, let D be a finite (∨, 0)-semilattice. If there exists a lifting K ∈ V of D of length greater than every lifting of D in W, then crit(V; W) ≤ ℵ 0 . Moreover if V is a finitely generated variety of modular lattices and W is not trivial, then crit(V; W) = ℵ 0 . Lemma 4 . 6 . 46Let B = (B P , g P,Q ) P ⊆Q in In be a congruence-lifting of Con c • A by lattices. Assume that B {x} is a chain of length two for all x ∈ n. Then M n can be embedded into B n . Proposition 4 . 7 . 47Let n ≥ 0 be an integer, let α be an ordinal, let κ ≥ ℵ α+2 , let f :[κ] 2 → [κ] <ℵα . Then there exists Y ∈ [κ] n such that a ∈ f ({b, c}) for all distinct a, b, c ∈ Y .Now recall the definition of supported poset and norm-covering introduced in [5,Section 4]. Definition 4.9. A norm-covering of a poset I is a pair (U, \|·\|), where U is a supported poset and \|·\| : U → I, u → \|u\| is an order-preserving map. Figure 2 . 2The lattice N 5 . Corollary 4 . 17 . 417The following equalities hold crit(M n ; M m,m ) = ℵ 2 ; crit(M 0,1 n ; M m,m ) = ℵ 2 ; crit(M 0,1 n ; M 0,1 m,m ) = ℵ 2 ; crit(M n ; M 0 m,m ) = ℵ 2 ; crit(M n ; M 0 m ) = ℵ 2 , for all n, m with 3 ≤ m < n ≤ ω. Proof. Let n ′ ≤ n be an integer such that m < n ′ < ω. As M n ′ ∈ M m,m , it follows from Lemma 4.15 that crit(M 0,1 n ′ ; M m,m ) ≤ ℵ 2 , thus: crit(M 0,1 n ; M m,m ) ≤ ℵ 2 . (4.1) Moreover M 3 ∈ M m,m , simple lattices of M m,m are of length at most 3, and finitely generated lattices of M n have finite length (and are even finite). Thus, by Theorem 3.11 crit(M n ; M 0 m,m ) ≥ ℵ 2 . (4.2) Similarly: crit(M 0,1 n ; M 0,1 m,m ) ≥ ℵ 2 . (4.3) All the desired equalities are immediate consequences of (4.1), (4.2), and (4.3). Corollary 4 . 418. crit(M 4,3 ; M 3,3 ) ≤ ℵ 2 .This question was suggested by M. Ploščica. Lemma 4 . 19 . 419Let F be field. Then M n ∈ Var(Sub F 3 ) if and only if n ≤ 1 + card F . F 3 = 3. If dim A = 1, then M n is isomorphic to {A/A, V 1 /A, . . . , V n /A, B/A} which is a sublattice of Sub(B/A), with dim B/A = 2. If dim A = 0, then: Corollary 4 . 20 . 420Let F be a finite field and let n > 1 + card F . Then:crit(M n ; Var(Sub F 3 )) = ℵ 2 ; crit(M n ; Var 0 (Sub F 3 )) = ℵ 2 ; crit(M 0,1 n ; Var(Sub F 3 )) = ℵ 2 ; crit(M 0,1 n ; Var 0,1 (Sub F 3 )) = ℵ 2 . Proof. ByLemma 4.19, M n ∈ Var(Sub F 3 ), moreover simple lattices of Var(Sub F 3 ) are of length at most three. Thus, by Lemma 4.15: crit(M 0,1 n ; Var(Sub F 3 )) ≤ ℵ 2 . (4.4) Corollary 4 . 21 . 421Let F and K be finite fields. If card F > card K then:crit(Var(Sub F 3 ); Var(Sub K 3 )) = ℵ 2 ; crit(Var(Sub F 3 ); Var 0 (Sub K 3 )) = ℵ 2 ; crit(Var 0,1 (Sub F 3 ); Var(Sub K 3 )) = ℵ 2 ; crit(Var 0,1 (Sub F 3 ); Var 0,1 (Sub K 3 )) = ℵ 2 .Proof. By Lemma 4.19, M 1+card F ∈ Var(Sub K 3 ), moreover simple lattices of Var(Sub K 3 ) are of length at most three. Thus, by Lemma 4.15: crit(Var 0,1 (Sub F 3 ); Var(Sub K 3 )) ≤ ℵ 2 .(4.6) Lemma 4. 22 . 22Let V be a finitely generated variety of lattices (resp., a finitely generated variety of lattices with 0), let m ≥ 2 an integer. Assume that for eachsimple lattice K of V, there do not exist b 0 , b 1 , . . . , b m−1 > u in K such that b i ∧b j = u (resp., b 0 , b 1 , . . . , b m−1 > 0 such that b i ∧ b j = 0), for all 0 ≤ i < j ≤ m − 1.Then crit(M 0,1 2m−1 ; V) ≤ ℵ 2 . Lemma 4 . 23 . 423Let V be a variety of lattices (resp., a variety of lattices with 0), let m ≥ 2 an integer. Assume that for each simple latticeK of V, there do not exist b 0 , b 1 , . . . , b m−1 > u in K such that b i ∧ b j = u (resp., b 0 , b 1 , . . . , b m−1 > 0 such that b i ∧ b j = 0), for all 0 ≤ i < j ≤ m − 1. Then crit(M 0,1 2m−1 ; V) ≤ ℵ 3 .Theorem 4.24. Let V be either a finitely generated variety of lattices or a finitely generated variety of lattices with 0. If M 3 ∈ V then: crit(M ω ; V) = ℵ 2 ; crit(M 0 ω ; V) = ℵ 2 . Let V be a finitely generated variety of bounded lattices. If M 3 ∈ V then: As Con L ∼ = 2 2 and lh(L) = 3, L is not distributive. Hence there exists a sublattice of L isomorphic to M 3 , let a < x 1 , x 2 , x 3 < b be its elements. As L is modular, [a,By symmetry, we may assume that b < c. Observe that a = u and c = v. Set C = {u, b, v} and C 1 = {u, x 1 , b, v}.Let i : C → L and i 1 : C 1 → L be the inclusion maps. As C 1 is a maximal chain,, Con c i 1 and Con c i have the same image, thus Con c i is onto, so Con c i is an isomorphism.The result of Lemma 4.14 does not extend to length four or more. The lattice ofFigure 3is not a congruence-preserving extension of any chain with extremities u and v. Lemma 4.15. Let n ≥ 4 be an integer, let V be a finitely generated variety of lattices such that M n ∈ V. If lh(K) ≤ 3 for each simple lattice K of V, then crit(M 0,1 n ; V) ≤ ℵ 2 . Proof. We consider the diagram A introduced just before Lemma 4.5. Assume that crit(M 0,1 n ; V) > ℵ 2 . As M n ∈ M 0,1 n , A is a diagram of M 0,1 n indexed by I n . By Lemma 4.12, the diagram Con c • A has a congruence-lifting B = (B P , g P,Q ) P ⊆Q in In in V. As Con B n ∼ = 2, the lattice B n is simple, thus, by assumption on V, lh(B n ) ≤ 3, and so lh(B {x} ) ≤ 3, for all x ∈ n. The lattice B ∅ is simple, so, taking a sublattice, we can assume that B ∅ = {u, v}, with u < v. By Lemma 4.14, we can assume that B {x} is a chain of length two, for each x ∈ n. So by Lemma 4.6, M n is a sublattice of B n , and so M n ∈ V, a contradiction. Theorem 4.16. Let V be a finitely generated variety of modular lattices and W be finitely generated variety of lattices. Let n ≥ 3 be an integer such that M n ∈ V − W. If lh(K) ≤ 3 for each simple K ∈ V, then crit(V; W) ≤ ℵ 2 . Moreover if either lh(K) ≤ 2 for each simple K ∈ V and M 3 ∈ W or lh(K) ≤ 3 for each simple K ∈ V and Sub F 3 ∈ W for some field F , then crit(V; W) = ℵ 2 . Cardinality questions concerning semilattices of finite breadth. S Z Ditor, Discrete Math. 481S. Z. Ditor, Cardinality questions concerning semilattices of finite breadth, Discrete Math. 48, no. 1 (1984), 47-59. An application of Whitman's condition to lattices with no infinite chains. B A Davey, B Sands, Algebra Universalis. 72B. A. Davey and B. Sands, An application of Whitman's condition to lattices with no infinite chains, Algebra Universalis 7, no. 2 (1977), 171-178. Combinatorial Set Theory: Partition Relations for Cardinals. P Erdös, A Hajnal, A Máté, R Rado, Studies in Logic and the Foundations of Mathematics. AmsterdamNorth-Holland Publishing Co106P. Erdös, A. Hajnal, A. Máté and R. Rado, "Combinatorial Set Theory: Partition Relations for Cardinals". Studies in Logic and the Foundations of Mathematics, 106. North-Holland Publishing Co., Amsterdam, 1984. 347 p. The von Neumann coordinatization theorem for complemented modular lattices. K D Fryer, I Halperin, Acta Sci. Math. (Szeged). 17K. D. Fryer and I. Halperin, The von Neumann coordinatization theorem for complemented modular lattices, Acta Sci. Math. (Szeged) 17 (1956), 203-249. Critical points of pairs of varieties of algebras. P Gillibert, Internat. J. Algebra Comput. 191P. Gillibert, Critical points of pairs of varieties of algebras, Internat. J. Algebra Comput. 19, no. 1 (2009), 1-40. General Lattice Theory. G Grätzer, B. A. Davey, R. Freese, B. Ganter, M. Greferath, P. Jipsen, H. A. Priestley, H. Rose, E. TSecond editionG. Grätzer, "General Lattice Theory. Second edition", new appendices by the author with B. A. Davey, R. Freese, B. Ganter, M. Greferath, P. Jipsen, H. A. Priestley, H. Rose, E. T. . S E Schmidt, F Schmidt, R Wehrung, Wille, Birkhäuser VerlagBaselxx+663 pSchmidt, S. E. Schmidt, F. Wehrung, and R. Wille. Birkhäuser Verlag, Basel, 1998. xx+663 p. Von Neumann Regular Rings. K R Goodearl, Krieger Publishing Co., IncMalabar, FLSecond edition. Robert E.. xviii+412 pK. R. Goodearl, "Von Neumann Regular Rings. Second edition". Robert E. Krieger Publishing Co., Inc., Malabar, FL, 1991. xviii+412 p. Set mappings, partitions, and chromatic numbers. A Hajnal, A Máté, Studies in Logic and the Foundations of Mathematics. Bristol; North-Holland, Amsterdam80Logic Colloquium '73A. Hajnal and A. Máté, Set mappings, partitions, and chromatic numbers, Logic Colloquium '73 (Bristol, 1973), p. 347-379. Studies in Logic and the Foundations of Mathematics, Vol. 80, North-Holland, Amsterdam, 1975. On representability of * -regular rings and modular ortholattices. F , TU DarmstadtPhD thesisF. Micol, "On representability of * -regular rings and modular ortholattices". PhD thesis, TU Darmstadt, January 2003. Available online at http://elib.tu-darmstadt.de/diss/000303/diss.pdf . Separation properties in congruence lattices of lattices. M Ploščica, Colloq. Math. 83M. Ploščica, Separation properties in congruence lattices of lattices, Colloq. Math., 83 (2000), 71-84. Dual spaces of some congruence lattices. M Ploščica, Topology and its Applications. 131M. Ploščica, Dual spaces of some congruence lattices, Topology and its Applications, 131 (2003), 1-14. Congruence lattices of free lattices in nondistributive varieties. M Ploščica, J Tůma, F Wehrung, Colloq. Math. 762M. Ploščica, J. Tůma, and F. Wehrung, Congruence lattices of free lattices in nondistributive varieties, Colloq. Math. 76, no. 2 (1998), 269-278. The dimension monoid of a lattice. F Wehrung, Algebra Universalis. 403F. Wehrung, The dimension monoid of a lattice, Algebra Universalis 40, no. 3 (1998), 247- 411. A uniform refinement property for congruence lattices. F Wehrung, Proc. Amer. Math. Soc. 1272F. Wehrung, A uniform refinement property for congruence lattices, Proc. Amer. Math. Soc. 127, no. 2 (1999), 363-370. A solution to Dilworth's congruence lattice problem. F Wehrung, Adv. Math. 2162F. Wehrung, A solution to Dilworth's congruence lattice problem, Adv. Math. 216, no. 2 (2007), 610-625.	[]
[ "Realistic many-body models for Manganese Monoxide under pressure", "Realistic many-body models for Manganese Monoxide under pressure" ]	[ "Jan M Tomczak \nResearch Institute for Computational Sciences\nAIST\n305-8568TsukubaJapan\n\nCREST\nJapan Science and Technology Agency\n\n", "T Miyake \nResearch Institute for Computational Sciences\nAIST\n305-8568TsukubaJapan\n\nCREST\nJapan Science and Technology Agency\n\n", "F Aryasetiawan \nResearch Institute for Computational Sciences\nAIST\n305-8568TsukubaJapan\n\nCREST\nJapan Science and Technology Agency\n\n\nGraduate School of Advanced Integration Science\nChiba University\n263-8522ChibaJapan\n" ]	[ "Research Institute for Computational Sciences\nAIST\n305-8568TsukubaJapan", "CREST\nJapan Science and Technology Agency\n", "Research Institute for Computational Sciences\nAIST\n305-8568TsukubaJapan", "CREST\nJapan Science and Technology Agency\n", "Research Institute for Computational Sciences\nAIST\n305-8568TsukubaJapan", "CREST\nJapan Science and Technology Agency\n", "Graduate School of Advanced Integration Science\nChiba University\n263-8522ChibaJapan" ]	[]	In materials like transition metals oxides where electronic Coulomb correlations impede a description in terms of standard band-theories, the application of genuine many-body techniques is inevitable. Interfacing the realism of density-functional based methods with the virtues of Hubbardlike Hamiltonians, requires the joint ab initio construction of transfer integrals and interaction matrix elements (like the Hubbard U) in a localized basis set. In this work, we employ the scheme of maximally localized Wannier functions and the constrained random phase approximation to create effective low-energy models for Manganese monoxide, and track their evolution under external pressure. We find that in the low pressure antiferromagnetic phase, the compression results in an increase of the bare Coulomb interaction for specific orbitals. As we rationalized in recent model considerations [P hys.Rev.B79, 235133(2009)], this seemingly counter-intuitive behavior is a consequence of the delocalization of the respective Wannier functions. The change of screening processes does not alter this tendency, and thus, the screened on-site component of the interaction -the Hubbard U of the effective low-energy system -increases with pressure as well. The orbital anisotropy of the effects originates from the orientation of the orbitals vis-à-vis the deformation of the unit-cell. Within the high pressure paramagnetic phase, on the other hand, we find the significant increase of the Hubbard U is insensitive to the orbital orientation and almost exclusively owing to a substantial weakening of screening channels upon compression.	10.1103/physrevb.81.115116	[ "https://arxiv.org/pdf/1006.0565v1.pdf" ]	119,203,208	1006.0565	07254870600d684b0df6172b1bb9e8a9c0e05625	Realistic many-body models for Manganese Monoxide under pressure 3 Jun 2010 Jan M Tomczak Research Institute for Computational Sciences AIST 305-8568TsukubaJapan CREST Japan Science and Technology Agency T Miyake Research Institute for Computational Sciences AIST 305-8568TsukubaJapan CREST Japan Science and Technology Agency F Aryasetiawan Research Institute for Computational Sciences AIST 305-8568TsukubaJapan CREST Japan Science and Technology Agency Graduate School of Advanced Integration Science Chiba University 263-8522ChibaJapan Realistic many-body models for Manganese Monoxide under pressure 3 Jun 2010arXiv:1006.0565v1 [cond-mat.str-el]PACS numbers: In materials like transition metals oxides where electronic Coulomb correlations impede a description in terms of standard band-theories, the application of genuine many-body techniques is inevitable. Interfacing the realism of density-functional based methods with the virtues of Hubbardlike Hamiltonians, requires the joint ab initio construction of transfer integrals and interaction matrix elements (like the Hubbard U) in a localized basis set. In this work, we employ the scheme of maximally localized Wannier functions and the constrained random phase approximation to create effective low-energy models for Manganese monoxide, and track their evolution under external pressure. We find that in the low pressure antiferromagnetic phase, the compression results in an increase of the bare Coulomb interaction for specific orbitals. As we rationalized in recent model considerations [P hys.Rev.B79, 235133(2009)], this seemingly counter-intuitive behavior is a consequence of the delocalization of the respective Wannier functions. The change of screening processes does not alter this tendency, and thus, the screened on-site component of the interaction -the Hubbard U of the effective low-energy system -increases with pressure as well. The orbital anisotropy of the effects originates from the orientation of the orbitals vis-à-vis the deformation of the unit-cell. Within the high pressure paramagnetic phase, on the other hand, we find the significant increase of the Hubbard U is insensitive to the orbital orientation and almost exclusively owing to a substantial weakening of screening channels upon compression. In materials like transition metals oxides where electronic Coulomb correlations impede a description in terms of standard band-theories, the application of genuine many-body techniques is inevitable. Interfacing the realism of density-functional based methods with the virtues of Hubbardlike Hamiltonians, requires the joint ab initio construction of transfer integrals and interaction matrix elements (like the Hubbard U) in a localized basis set. In this work, we employ the scheme of maximally localized Wannier functions and the constrained random phase approximation to create effective low-energy models for Manganese monoxide, and track their evolution under external pressure. We find that in the low pressure antiferromagnetic phase, the compression results in an increase of the bare Coulomb interaction for specific orbitals. As we rationalized in recent model considerations [P hys.Rev.B79, 235133(2009)], this seemingly counter-intuitive behavior is a consequence of the delocalization of the respective Wannier functions. The change of screening processes does not alter this tendency, and thus, the screened on-site component of the interaction -the Hubbard U of the effective low-energy system -increases with pressure as well. The orbital anisotropy of the effects originates from the orientation of the orbitals vis-à-vis the deformation of the unit-cell. Within the high pressure paramagnetic phase, on the other hand, we find the significant increase of the Hubbard U is insensitive to the orbital orientation and almost exclusively owing to a substantial weakening of screening channels upon compression. PACS numbers: I. INTRODUCTION A. Realistic models for correlated materials While standard band structure methods, like density functional theory (DFT) in the local spin density approximation (LSDA), 1,2 in principle allow for a treatment of the realistic complexity of materials arising from orbital, spin and structural degrees of freedom, the failure of these techniques in the presence of strong electronic correlations is well documented. 3 Many-body techniques, on the other hand, while potentially treating correlation effects with accuracy, often lack the realism to account for the electronic structure of a material from a firstprinciples basis. Hence, when it comes to the ab initio description of correlated matter, the joining of these two fields is of paramount importance. Often, this combining amounts to constructing effective, yet realistic, low energy models, in the spirit of the Hubbard or Anderson model. These consist of a one-particle part, extracted from the Kohn-Sham band structure, and two-particle interaction terms. There are different paradigms for choosing the matrix elements of the latter. Sometimes, these values are treated as mere empirical parameters, adjusted to correctly reproduce some experimental finding. However, there exist first-principle methods, among which the most popular are the constrained LDA technique, 4 and the constrained random phase approximation (cRPA). 5 In this article we are chiefly concerned with the evolution of interaction matrix elements under pressure within a realistic setup. While the influence of pressure onto structures and band-structures has been widely studied on the DFT level, changes in the interaction have received much less attention. Yet, the pronounced sensitivity of correlated matter on external parameters 6 heralds the need for an accurate treatment of all ingredients to the electronic structure. For the LDA+U 7 technique, this issue has been investigated within a linear response based approach [8][9][10] , and the reproduction of crystal and electronic structures, as well as phase stabilities were indeed found to require the description of pressure induced changes in the Hubbard U . Here, we are interested in setting the stage for manybody methods that go beyond density functional methods. In this vein, a particularly versatile technique is the use of maximally localized Wannier functions 11,12 within the cRPA setup, 13,14 which is the method of choice of this article. In this formalism it is possible to turn off precisely those screening channels that are to be left to the solution of the effective model. 5,67 Moreover, the use of many-body techniques often requires the use of localized basis sets, and the Wannier construction is geared at exactly this. While the Wannier-cRPA technique has been already applied to several systems, such as 3d transition metals, 13 the perovskite SrVO 3 , 13 and oxypnictides 15,16 , the evolution of the respective Coulomb interaction matrix elements under external stimuli such as pressure have not been addressed in the realistic case. In recent model considerations, 14 we established generic behaviors of maximally localized Wannier functions and interaction matrix elements under pressure for simple setups. We rationalized that as a consequence of the delocalization of Wannier functions under compression, not only do transfer integrals increase, but also -counter-intuitively -the matrix elements of the bare Coulomb interaction augment. In this paper, we extent this work to a more complex scenario by applying the fully ab initio Wannier-cRPA formalism to manganese monoxide, MnO, under pressure. In particular, we monitor the evolution of transfer integrals, on-site matrix elements of the bare Coulomb interaction, and of the partially screened Coulomb interaction (the Hubbard U, and the Hund J), for different realistic low-energy systems. We will distinguish between pressure induced changes related to the construction of the sub-system, and changes in the screening strength as provided by the band-structure and transition matrix elements in the polarization. We find that while the former mainly determine the behavior of the interactions in the low temperature, low pressure anti-ferromagnetic phase of MnO, changes in the Hubbard U of the high pressure paramagnetic phase are dominated by screening related effects. B. Manganese monoxide Approaches to the crystal and electronic structure Concerning the failures of the LDA in the realm of correlated materials, the focus material of this work, manganese monoxide, MnO, is rather well behaved at first glance : Below 118 K the system is antiferromagnetic and due to the fact that the Mn3d orbitals are halffilled, spin-dependent band-structure calculations 17-24 do indeed yield an insulator. However, above the Néel temperature, which is much smaller than the experimental charge gap of 3.9 eV, 25 MnO is paramagnetic, and believed to be a Mott insulator. A straight forward bandstructure calculation in the paramagnetic phase, however, yields a metal. A genuine feature of correlated materials is their sensitivity to external stimuli, leading to rich phase-diagrams with regimes of different spin or orbital orders, and metalinsulator transitions. 6 MnO is no exception and both experiments and theory helped to elucidate the phases of manganese monoxide as a function of temperature versus pressure. See Fig. 1 in Ref. 26 for a graphical depiction of compiled results. The paramagnetic insulator MnO, which is of rocksalt structure (B1), contracts below the Néel temperature along the 111 direction, resulting in a rhombohedral distorted phase (dB1). MnO has been the prototype material for the discovery and description of the superexchange mechanism. [27][28][29] It was rationalized by Terakura et al. 17 that this strong manganese superexchange via the oxygen orbitals 27,28 is responsible for a spin oder that is of AF II type, [30][31][32] with alternating spins along the distortion axis. With increasing pressure, the Néel temperature rises 33,34 and reaches room temperature (300 K) at 30GPa. 26 It has been shown that beyond the volume compression, it is the rhombohedral distortion that stabilizes the AF II order under pressure with respect to other spin orientations. 18 At constant temperature (300 K), a further transition occurs at 90 GPa, 26,35,36 when the system becomes, again, a paramagnetic insulator, yet of NiAs (B8) structure. 26 Notably, the high pressure B8 structure was predicted by DFT calculations in the generalized gradient approximation (GGA). 18 At still higher pressure, the system undergoes an iso-structural Mott insulator to metal transition at 105 GPa, 26 signaled by a volume collapse of 6.6%, 26 metallic lustre, 36 an increase of reflectance, 37,38 a resistance decrease of several orders of magnitude 39 and a quenching of the magnetic moment. 26 The transition was anticipated, from theoretical considerations 40 and ab initio calculations. 19 Yet, both works forward different mechanism : In Ref. 40 a pressure induced increase in the crystal-field was invoked to explain the metalization, while in Ref. 19 , in which the rhombohedral distortion is neglected, a growing band-width was seen at the origin of the transition. We will come back to this in the discussion of our results, which favor the first point of view. On the theory side, numerous works that aim at reproducing the described phase-diagram from first principles resort to improvements within the density functional formalism. Among these are the generalized gradient approximation (GGA), 18-24 selfconsistent Hartree, 41 LDA/GGA+U, 18,[20][21][22][23]42,42 selfinteraction corrected (SIC) LSDA. 43,44 Also applied were various hybrid functional 45 based approaches 20,21,23,24 . Although relying on the coincidence of a half-filled dshell and the spin order for insulating results, it is due to the fact that (unlike for the other transition metal monoxides) the 3d electrons are not falsely itinerant, that the above methods succeed in yielding at times reasonable structures, lattice constants, bulk moduli, magnetic moments, and sometimes even values for the charge gap (see especially the SIC and hybrid functional references from above). Results have been obtained also within the GW approximation. [46][47][48] While pioneering work 49 still had to resort to numerical simplifications, the increase in computer performance nowadays even allows the application of self-consistency schemes to MnO, 50 which considerably improve on the gap value of the antiferromagnetic phase with respect to LDA calculations. Yet, despite these successes, it is evidently desirable to have a many-body technique that works in a wider context, capturing correlated metals, ordered and Mott insulators. In this vein, the probably most generally applicable approach nowadays is LDA+DMFT, 51 where a Hubbard Hamiltonian consisting of a one-particle part stemming from the LDA and an interaction part of Hubbard-Hund type is solved within dynamical mean field theory (DMFT). 52 ,68 While DFT methods and their generalizations, including LDA+U, 7 cannot address finite-temperature insulators without broken symmetries, LDA+DMFT is in principle capable to deal with all the different phases. Concerning MnO, the technique has first been used for the calculation of phonon dispersions. 53 Recently LDA+DMFT was applied by Kuneš et al. 54 to study the isostructural Mott insulator to metal transition witnessed at 105GPa (300 K). The results reproduced quite accurately all experimental findings, in particular the critical pressure and the joint collapse of volume and magnetic moment. 54 Often it is criticized that LDA+U and LDA+DMFT calculations are not fully ab initio since the interactions are treated as "parameters" that are in a sense tuneable. Indeed in the above LDA+DMFT work, 54 the Hubbard U and the Hund J were chosen to be constant as a function of pressure. Although adjustable parameters are to be preferred over artificially broken phases, it is true that especially in transition metal monoxides there are several competing energy scales 55 so that the values of the interactions have to be known with greater precision than in other compounds. Among those relevant energies are : the crystal field splitting, that measures the degree of non-degeneracy of the centers of gravity of the d orbitals, e.g. ∆ cf =ǭ eg −ǭ t2g for an octahedral coordination, the charge transfer energy ∆ ct =ǭ t2g −ǭ 2p here defined between the O2p and the Mn t 2g , the transfer integrals (hoppings) t t2g , t eg , the hybridizations t pd , and the onsite Coulomb interaction U of the d orbitals. Earlier works on the Hubbard U In the pioneering work of Anisimov et al. 7 that introduced the LDA+U formalism, the constrained LDA technique was applied to MnO at zero pressure and a value U = 6.9 eV was found. The Hubbard U is thus of the same order as the charge transfer energy ∆ ct (see below). Therewith MnO is in the intermediate regime between a charge transfer and a Mott-Hubbard insulator 55 and changes in U might considerably alter the many-body spectra and their interpretation. An empirical technique that was applied to MnO is the fitting of configuration interaction (CI) cluster calculations to photoemission experiments. Although of course depending on the details of the chosen cluster, we cite the values U = 7.5 eV, ∆ ct = 7.0 eV of Fujimori et al. 56 and U = 8.5 eV, ∆ ct = 8.8 eV of van Elp et al.. 25 Still, CI cluster calculations cannot account for delocalized bands. As a consequence, the struggle of d electrons between localized and itinerant behavior and the mixing with the more extended O2p and Mn4s orbtials is biased. Using a semi-empirical Anderson (d-) impurity Hamiltonian, Zaanen and Sawatzky 29 estimated ∆ and U to be both around 9 eV, and found accurate values also for the Néel temperature. An Anderson Hamiltonian was recently also applied to results from resonant inelastic x-ray scattering (RIXS) spectra 57 : the parameters of a single manganese d impurity model were adjusted to yield U = 7.2 eV and ∆ ct = 6.5 eV. In this approach, the delocalized character of the oxygen and Mn4s orbitals is better captured than within CI based methods. Indeed, this is a step toward LDA+DMFT calculations, albeit without obeying the self-consistency condition that re-lates the manganese impurity with the periodic solid. The Hubbard U can also be extracted from GW calculations. Indeed the RPA technique is a crucial ingredient to the GW approximation. Pioneering model GW calculations suggested a value of U = 8.0 eV 49 for the antiferromagnetic phase, albeit only empirically extracted from the energy difference between occupied and unoccupied e g and t 2g states. II. METHOD The method of using the Wannier orbital construction in conjunction with the constrained RPA has been presented in detail in Ref. 13 . Here, we shall just give a brief summary as is necessary for understanding our results. A. Band-structure In this work, we employ the LSDA in the full-potential (FP) LMTO 58 realization and use experimental structures and lattice parameters as provided by Yoo et al. 26 at 300 K 69 and variable pressure. We use up to 4096 reducible Brillouin zone points, the LMTO basis set includes orbitals up to l = 4(3) for Mn(O), and we use local orbitals 58 to incorporate semi-core and high-lying states. B. Maximally localized Wannier functions The one-particle part of the effective low energy systems is extracted from the (LMTO) band-structure calculation by the construction of a Wannier basis 59 of the desired subset of orbitals. This procedure not being unique, 59 we choose to use Wannier functions that are maximally localized. 11,12 While the initial Hilbert space is spanned by all Kohn-Sham wave functions, H = span{ψ KS kα }, for the aim of constructing effective low energy models necessitates the choosing of a subspace, H low ⊂ H, made up by only selective Kohn-Sham orbitals. The Wannier functions are then deduced from the latter. Complications occur if the bands of the orbitals that constitute the desired sub Hilbert space are entangled with high energy bands. For details on how then to construct maximally localized Wannier functions see 12 , and for a comparison with other Wannier function schemes, for instants the approach in which the on-site screened Coulomb interaction is maximized, see e.g. the Refs. 13,14,60 . C. constrained RPA The matrix elements of the bare Coulomb interaction v(r, r ′ ) = e 2 4πǫ0 1/ \|r − r ′ \| in the Wannier basis χ W * Rα (r) are setup low energy subspace H low polarization P low given by d span ψ KS kα \| α = Mn3d P d pd span ψ KS kα \| α = Mn3d, O2p P d +Pp+P pd +P dp pd+p span ψ KS kα \| α = Mn3d, O2p P dV αβα ′ β ′ R,R ′ = e 2 4πǫ 0 × (1) d 3 rd 3 r ′ χ W * Rα (r)χ W Rβ (r) 1 \|r − r ′ \| χ W * R ′ α ′ (r ′ )χ W R ′ β ′ (r ′ ) From the latter, partially screened interaction matrix elements are obtained using the cRPA technique. 5,13 Using the Kohn-Sham orbitals, the total RPA polarization can be expressed by P (r, r ′ , ω) = (2) spin occ kn unocc k ′ n ′ ψ KS * kn (r)ψ KS k ′ n ′ (r)ψ KS * k ′ n ′ (r ′ )ψ KS kn (r ′ ) × 1 ω − ǫ k ′ n ′ + ǫ kn + ı0 + − 1 ω + ǫ k ′ n ′ − ǫ kn − ı0 + Its matrix elements within the Wannier basis set are given analogous to the above bare interaction. The strength of screening channels can thus be influenced by the overlap integrals (matrix elements) of the wave functions, and by the energy difference of the Kohn-Sham excitations that appear in the denominators. From the form of the fully screened interaction, W (ω) = [1 − P (ω)v] −1 v (that also appears in the GW formalism [46][47][48] ) it can be shown, 5 that the screening contributions are actually additive. Indeed one can split the polarization, P = P low + P r , into the transitions within a low-energy orbital subspace, P low of H low , and the rest P r = P high−high + P high−low + P low−high , where the latter includes, both, transitions within the high-energy subspace, and between the two subspaces. Then the fully screened interaction can be written recursively, i.e. W = 1 − P low W low r −1 W low r where W low r = v 1 − P r v(3) is the Coulomb interaction when all screening processes that do not involve transitions within the low-energy orbital subspace have been accounted for 70 . For our compound, MnO, we will consider three different setups, that are also summarized in Tab. I. • In the "d-setup", a many body model is obtained from maximally localized Wannier functions that are constructed for the space of the Mn3d Kohn-Sham (KS) orbitals, ψ KS k,3d , and from a partially screened interaction W d r that is given by constraining the polarization to transitions that are not fully within the d-subspace 71,72 . • MnO being close to the charge transfer regime, it is sensible to include the oxygen 2p orbitals for a realistic many-body treatment. In such a "pdsetup" one constructs a Wannier basis for the space of Mn3d and O2p Kohn-Sham wavefunctions, and chooses P r = P − P d − P p − P pd − P dp . • Yet, often, a Hubbard U is only put on the most correlated orbitals, here the Mn3d ones. In that case, while still constructing a full pd Wannier basis, one should allow for oxygen screening in the RPA (i.e. P r = P − P d ), since those channels are blocked when the interactions between the p and d orbitals are omitted. Therewith, W r (r, r ′ ) of the current setup and the pd one are the same, yet their matrix-element will differ due to the Wannier different basis. As an outlook, we like to mention that in the described procedure to generate the effective many-body problem, couplings in the interaction between the low energy subspace and the other orbital degrees of freedom are discarded -an approximation that might prove insufficient if the subspace is chosen too restrictively. A framework that separates the Hilbert space, while retaining the influence of self-energy effects between subspaces has been recently proposed in Ref. 61 . D. Wannier functions and Coulomb matrix elements under pressure Before applying the above techniques to our compound of interest, we shall first discuss what circumstances will influence the Wannier functions and the interaction values. We can distinguish three conceptually different mechanisms : The choice of the orbital sub-space, the consequences of structural changes onto the localization of Wannier functions, and the changes related to the strength of screening processes. The choice of the orbital sub-space The construction of an effective low energy system depends very much upon the choice of the orbitals that it consists of. In particular, the dimension of the orbital sub-space H low has a strong impact on the localization of the Wannier functions. Indeed, from the maximally localized Wannier function point of view, a larger subspace will allow, for a given orbital, for the mixing in of more Kohn-Sham orbitals. Therewith, the variational freedom to localize the basis functions becomes larger, and the interaction elements bigger. See, for an example, Ref. 15 . We thus expect the interaction matrix elements of the pd-model to be larger than in the d-only model. Wannier functions under pressure External pressure, or structural changes in general, provide another impetus to alter not only the one-particle band-structure, but also the Wannier functions and thus the bare Coulomb interaction matrix elements. The fact that Wannier functions are not eigenfunctions even of the non-interacting problem, may result in counter-intuitive tendencies. In Ref. 14 , we established generic behaviors of maximally localized Wannier functions and interaction matrix elements under pressure. In order to put things into perspective, we shall here briefly summarize some findings of this work : As a model system, we consider a tight-binding parametrization of a one dimensional solid, in which case Wannier functions are already maximally localized if inversion symmetry is verified, 11,62 and limit the discussion to the case of a single band. Details can be found in Ref. 14 . As building blocks of the basis functions we take hydrogen-like 1s orbitals in one dimension, χ(x) = 1/ √ a 0 exp(− \|x\| /a 0 ), with the Bohr radius a 0 . The Bloch eigenfunction is given by ψ k (x) = A k R e ıkR χ(x − R), where A k assures the normalization. The maximally localized Wannier functions are then ψ R (x) = n A n χ(x − na − R), with the lattice constant a, while A n is a Fourier transform of A k and also incorporates the normalization of the Wannier function. In the limit of large atomic separation (a ≫ a 0 ), the Wannier functions will equal the atomic orbitals. Thus A n = δ n,0 for the representative site "0". When pressure is applied, and the lattice constant shrinks, overlaps of the (non-orthogonal) atomic orbitals causes contributions from neighboring sites to mix into the basis functions, and the distribution A n broadens. This leads to an increase in the (nearest neighbor) transfer integral t and a concomitant increase of the spread (as defined in 11 ). In this sense, charge carriers become more delocalized, as expected. The tails of the Wannier functions extent over several lattice constants, before the originally exponential decay sets in. However, as can be shown, 14 the coefficient A 0 in the above decomposition on the array of atomic orbitals, i.e. the strength of the on-site orbital, grows, and thus becomes larger than one. This results in a greater probability density \|ψ R (x)\| 2 around the site origins, x− R = 0, when pressure is applied. This in turn causes a larger local Coulomb interaction matrix element V , Eq. (2) . Hence, the pressure induced delocalization is accompanied by a larger local interaction, whereas, intuitively, one might have expected the opposite. In our one band toy-model, since P r = 0 (see Eq. (3)) the screened Coulomb interaction U equals the bare Coulomb interaction V . Moreover it is maximal in the maximally localized Wannier basis. 14 P [GPa] tp−t 2g ∆ct t 2 p−t 2g /∆ct tt 2g −t 2g 0 (dB1) 0. 60 Screening strength and the interaction. While the above changes in overlaps and hybridizations directly enter the calculation by modifying the Wannier functions, that influence the matrix elements of any quantity, pressure moreover enters the construction of the effective low energy system by means of the bandstructure. Indeed, according to Eq. (3), pressure induced modifications in the one-particle excitations will alter the screening strengths of possible transitions. When bandwidths, crystal fields or bonding/antibonding splittings get enhanced upon the compression of the system, the screening will become less effective and, therewith, the Hubbard U larger. Besides this energetic effect, evident from the denominators in Eq. (3), there is also an effect of matrix elements when expressing the polarization in the Wannier basis. For an example of the interplay of both effects see Ref. 14 . Contrary to the preceding section, the transition energies and the matrix elements affect not the bare Coulomb interaction, but only screened quantities. Explicitly marked are also the bands constructed from the Wannier orbitals that are used for the construction of the effective "pd" low energy system (the bands of the effective "d-only" model achieve the same precision). Using the maximally localized Wannier function formalism for entangled bands, 12 the choice of Hilbert space for the Mn3d and O2p orbitals results in a clean distinction from the Mn 4s band that crosses the d-bands. The corresponding LSDA densities of states (DOS) of the pd-model are shown in Fig. 2. As we can see, pressure induces an increase in the crystal-field splitting, ∆ cf , i.e. a larger separation between the e g and t 2g Kohn-Sham excitations (both the occupied and unoccupied), to an extent that the moving of the e g weight towards the Fermi level eventually causes the system to become metallic. 40 Concomitant with the increase in crystalfield, pressure results in a growing Mn-O hybridization t pd = p0\|H KS \|d0 , where H KS is the Kohn-Sham Hamiltonian. This effect is especially large for the e g orbitals, as is clear from the octahedral coordination. For values, see the later discussed Tab. II (occupied t 2g orbitals), and III (e g orbitals). Most notably is the fact, that the partial d-bandwidths do not increase with pressure. The origin of this is identified when comparing results for the experimental dB1 phase with the undistorted B1 (NaCl) structure of the same volume. As evident from Fig. 1 (d,e), and Fig. 2, the omission of the rhombohedral distortion causes the bandwidths to substantially increase. Indeed, in fictitious B1 MnO, the closing of the gap can be mainly attributed to the change in bandwidth, as was stated in Ref. 19,63 where the distortion was neglected. The evolution of the bandwidths can be quantified by looking at the effective Mn3d nearest neighbor transfer elements for a given spin. The latter consists of two major contributions : the direct hopping t dd = d0\|H KS \|dR (R nearest neighbor unit-cell of 0), and the oxygen mediated transfer t 2 pd /∆ ct , with the hybridization t pd , and the charge transfer energy ∆ ct . The (largest) nearest (spin) neighbor transfers of the occupied t 2g orbitals, t t2g −t2g , the corresponding hybridization with the oxygen 2p orbitals, t p−t2g , and the charge transfer energy ∆ ct =ǭ occ t2g −ǭ 2p as measured by the band centers 73 , are collected in Tab. II for ambient pressure and 100GPa, for both, the experimental dB1 and the undistorted B1 structure. As a result of the joint increase of t pd and ∆ ct , the oxygen mediated contribution to the t 2g bandwidth remains roughly of the same magnitude, irrespective of pressure and distortion. The direct t 2g -transfer for the distorted dB1 structure is also constant with pressure. However, it is substantially structure dependent. Indeed the nearest neighbor t 2g -t 2g transfer doubles when the distortion is neglected, causing the effective hopping to augment by 64%, demonstrating the critical need to include the distortion for a proper description of dB1 MnO under pressure. As to the unoccupied t 2g orbitals, the transfers and energy differences are larger, but the same argument as above holds, for the trends are the same. We conclude that within LDA (in contrast to previous works 19 (see also the recent 63 ) relying on the undistorted structure), the gap closure under pressure is propelled by the change in crystal field and not by a broadening of band-widths. The relevant quantities for the unoccupied (occupied) e g orbitals are compiled in Table III. For the occupied e g orbitals the situation is opposite to the t 2g : While the oxygen mediated contribution to the band-width, t 2 p−eg / ∆ occ ct , at 100GPa depends sizeably on the structure (B1/dB1), the direct hopping is rather insensitive in this respect. For the unoccupied e g none of the values at 100GPa depends much on the structure, as can be inferred also from the similar shapes of the density of states. Yet, overall the effective bandwidths of the dB1 phase are always a little smaller than for the undistorted phase. Thus, the rhombohedral distortion is effectively reducing all partial d-bandwidths. The Coulomb interaction a. The bare interaction. As explained in a model context in Ref. 14 , changes in the orbital overlaps and hybridizations are expected to have an impact on the bare Coulomb interaction matrix V of Eq. (2). The on-site elements of the latter for antiferromagnetic dB1 MnO are shown in Fig. 3(a), both for a pd and a d-only Wannier basis set, and for different pressures. Owing to the distortion axis, the degeneracy properties of the Mn3d orbitals do not change with pressure. Moreover, in the current case, the maximally localized Wannier functions retain the degeneracy of the LMTOs, and while the LDA wavefunctions were merely the initial guesses for the maximally localized Wannier functions, we keep the labeling of "t 2g ", "e g ", "d xy " and alike. In some cases, especially for entangled bands, the localization procedure might cause the resulting Wannier functions to acquire a different symmetry. In the case of the pd-model, the matrix elements are sensitive to pressure only for the O2p and the unoccupied e g orbitals. There, the changes in the interaction are quite significant and reach up to 10%. This is a natural consequence of our above remarks : It is precisely those orbitals whose bondings changes the most under pressure, namely the hybridizations between them increase considerably. As summarized in the Method section, and discussed in detail in Ref. 14 , this allows the respective Wannier functions to accumulate more weight at small distance, while an overall increase in spread accounts for the gain in hybridization. This is also the reason why, for the pd-model, the e g elements are larger than the t 2 g ones : the larger hybridizations of the e g orbitals with the oxygen 2p ones, allow the corresponding maximally localized Wannier functions to reduce their extension. As anticipated, the elements of the d-model are lower than for the pd case, since the Wannier functions are more extended. Having restrained the variational freedom to the d orbitals, their matrix elements acquire a larger dependence on pressure than in the pd case. This is a result of the fact that the intra-d hoppings, especially the e g ones, must make up for the dispersion caused by oxygen mediated transfers that are not present in the smaller basis. b. The partially screened interaction. While the bare Coulomb interaction was mainly a mean to analyze the pressure dependence of the Wannier functions, the quantity that is needed for the formulation of the effective models is the partially screened Coulomb interaction. In Fig. 3 (b) we show, for the same pressures as before, the Hubbard U for the three different types of models, as introduced in the Method's section : the full pd-model, the pd-model with pd-screening (for a treatment in which only the d-orbitals will be supplemented by a local interaction), and the d-only model. The magnitudes of the corresponding interactions decrease in the given order : The interactions in the pd-model are smaller when the p-screening is present, and the interactions in the d-only model are yet smaller because of the larger extension of the Wannier functions. The pressure dependence of the elements has the same tendencies as the bare Coulomb interactions. Yet, we note that the relative difference, (U 100GP a − U 0GP a )/U 0GP a , is larger for the Hubbard U than for V , the pressure influence thus stronger. This is understood from a decrease in screening strength owing to the larger separation in energy between occupied and unoccupied states as pressure increases (see e.g. Ref. 14 for a discussion), as apparent from the band-structure, Fig. 2. On an absolute scale, and for the given pressure range, the Hubbard U changes by up to 1 eV for both the Mn3d and the O2p orbitals -a non-negligible effect given the aforementioned modifications in the magnitude of the transfer integrals. For the sake of completeness and to put the values further into perspective, Table IV (upper part) As a function of pressure, both U pd and U nn dd increase. As was true for the diagonal elements of the Hubbard U, the pressure dependence of these off-diagonal interactions is a feature mostly intrinsic to the Wannier functions and not the screening strength. The Hund's coupling, J = W r αβ,βα 0,0 (ω = 0, α = β), is compiled below in Table V. While being largely anisotropic with respect to the orbital (for the Mn d orbitals, values range from 0.4 eV to 0.7 eV at ambient pressure), it does not depend significantly on pressure. B. MnO at high pressure : the paramagnetic B8 phase The band-structure The B8 phase of MnO is of NiAs (B8) structure. As discussed in Ref. 26 , the B8 and dB1 structure are rather similar, and a transition from the high pressure dB1 to the B8 phase requires a moving of oxygen atoms from "distorted octahedral" to "perfect trigonal" positions. Yet, in the pressure dependence, there is an important difference : Augmenting pressure in the dB1 phase leads to an increase of the trigonal angle, and thus to anisotropic changes. This results in an almost pressureinsensitive bandwidths, as discussed above. In the B8 phase, the c/a ratio basically remains unchanged, and it is thus only the uniform modification of the lattice constant that accounts for the change in volume. We note that the different coordination causes the d-orbitals to split into two doubly degenerate e 1,2 g and a single a g orbitals (see e.g. Ref. 42 ). Fig. 4 shows the band-structure of B8 MnO for various pressures, along with the bands in the Wannier gauge for the d-model. The corresponding DOS is displayed in Fig. 5. The lowest indicated pressure is in fact smaller than the largest value we had chosen for the dB1 phase. This is justified by the fact that both phases are coexisting over a finite pressure range, as evidenced from optics 37,38 and x-ray experiments. 26 Of course, the LDA yields a metal for all pressures. Since the Mott transition within the paramagnetic B8 phase only takes place at finite temperatures, this is congruent with experiments. However, we used again experimental structures and volumes at ambient temperature, 26 since our aim is the construction of a model for the metal-insulator transition within the B8 phase. Continuing our remarks from the Methods section, this model construction is approximate. While it is perfectly admissible that the one-particle dispersion is metallic, one might object that the pd screening that we include in the determination of the Hubbard U could be overestimated since the exact Green's function will have excitations with a larger separation because of the insulating nature of the system. In the LDA band-structure, Fig. 4, the d-bandwidth becomes slightly larger with increasing pressure. As mentioned in Ref. 54 this restricted increase is a result of the fact that the d-bandwidth is mainly constituted by the hybridization contribution of the O2p orbitals. Yet, while t pd increases with pressure, so does the charge transfer energy ∆ ct , and thus the effective hopping, t 2 pd /∆ ct , does not increase much (cf. our discussion of the dB1 phase.). The preponderant effect of pressure on the bandstructure is a moving away from the Fermi surface of the occupied O2p and the unoccupied Mn 4s, 4p and 4d orbitals, owing to greater hybridization splittings. The interaction Given the above changes in bandwidths, the bare Coulomb interaction is expected to be roughly pressure independent. As can be inferred from Fig. 6 (a), which displays this quantity for both the pd and the d-only model, this is found indeed to be the case. The absolute values are comparable in size to those of the dB1 phase. The pressure dependence is decisively different for the partially screened interaction -the Hubbard U-shown in Fig. 6, as above, again for the three different types of models. As pressure augments, the changes induced to the bands-structure lead to a decrease in screening strength, resulting in an almost isotropic increase of the effective interaction of up to about 0.1 eV/10GPa for the pd-model. The average nearest neighbor interaction elements U pd and U nn dd (for definitions see above) for the pd-model are compiled in Tab. IV. The trend with pressure is the same as for the intra-atomic elements, i.e. they grow with pressure. However, like in the dB1 phase, they are smaller by a factor of about four with respect to the intra-atomic elements. While the pressure dependence of the diagonal terms of the screened interaction could be attributed almost exclusively to changes in the screening, it is notable that the off-diagonal elements of already the bare Coulomb matrix elements change by almost 10% from the lowest to the highest pressure considered. What had been said about the Hund's coupling of the dB1 phase is also true here, as seen from Table VI. Being less susceptible to screening processes, it does not show a significant pressure dependence. However, it is, again, sensitive to the combination of orbitals involved, and values can vary by as much as 65%. Since the physics of the high spin to low spin transition 54 within the B8 phase is sensitive not to the diagonal elements of the interactions (which change considerably), but mostly to changes in J 74 (that are small), arguments made about the transition mechanism 54 may not be affected when using the pressure dependent matrix elements instead of constant ones. Yet, spectra will change on a quantitative level. IV. SUMMARY, CONCLUSIONS AND PERSPECTIVES In conclusion, we have studied the impact of external pressure onto the construction of effective low energy many-body models for the realistic example of manganese monoxide. This compound is particularly challenging for a first principle description, since energies, such as the transfer integrals, the Hubbard U and the charge transfer energies are of the same magnitude. On the level of band-structures, we concluded that the pressure induced vanishing of the gap in the antiferromagnetic phase of MnO is driven by the crystal field rather than by an increase of band-widths, the latter of which is found to be limited by the growing rhombohedral distortion. The high pressure paramagnetic phase, on the contrary, exhibits the usual pressure caused increase of hybridizations and bandwidths. Using the constraint RPA approach within setups of maximally localized Wannier functions, we further found that, for a quantitative description, the pressure dependence of the Hubbard U can not be neglected. While the pressure sensitivity in the low pressure antiferromagnetic phase of MnO is dominated by compression induced changes of the bare Coulomb interaction, the effective interaction of the high pressure paramagnetic phase is mostly determined by modifications in the band-structure that affect the screening properties of the system. Our ab initio constructed low energy models can provide a quantitative starting point for the application of sophisticated many-body methods. Fig. 1 1displays the LSDA band-structure of antiferromagnetic MnO for different pressures and structures. online) LDA band-structure of antiferromagnetic B1/dB1 MnO. Shown are the Kohn-Sham bands (solid lines) and the eigenvalues of the Wannier Hamiltonian (dashed). The vertical axis is energy (in eV) with the Fermi level at the origin. While (b) -(d) correspond to the true distorted dB1 structure at different pressures, (a) & (d) show the band-structure for the artificial B1 (NaCl) structure for the indicated pressures. FIG. 2 : 2(Color online) Density of states (DOS) (in 1/eV) of antiferromagnetic B1/dB1 MnO for increasing pressure (top to bottom). Shown are the site-summed contributions of one spin component of the pd-model. The lowest panel displays fictitious B1 MnO at 100GPa at the experimental volume. Different partial DOS are depicted as indicated in the graph at the bottom. For P = 0GPa, the DOS of the dB1 and the B1 phase do not differ significantly on the shown scale. FIG. 3 : 3(Color online) local Coulomb interaction of the antiferromagnetic dB1 phase of MnO for different pressures. (a) diagonal elements of the local, bare interaction V for the pd and d-only model, and resolved for the different orbitals α. (b) zero frequency limit of the RPA partially screened local interaction Uαα = Wr αα,αα 0,0 (ω = 0) for the pd model (with and without p screening), and the d-only model. For comparison are shown also the results of undistorted, B1 (NaCl) structured MnO. : Comparison between the direct and oxygen mediated contributions to the effective transfer elements of the unoccupied (occupied) eg orbitals for the B1/dB1 phase of Mn. ∆ct =ǭe g −ǭ2p is the charge transfer energy. Shown are the largest transfers for nearest neighbors with the same spin. All energies in eV. IV: Averaged nearest neighbor inter-atomic interactions U pd = Wr pp,dd 0,0 (ω = 0) and U nn dd = Wr d 1 d 1 ,d d 1/2 is a short hand for the d-orbitals of the 1./2. Mn atom in the cell) for the pd-model of the B1/dB1 phase (top) and the B8 phase (bottom) for different pressures. d 1/2 is a short hand for the d-orbitals of the 1./2. Mn atom in the cell) within the pd-model and averaged over the respective orbital subset. Both elements are roughly a factor of four smaller than the intra-atomic interactions. α = β) of antiferromagnetic dB1 MnO and the d-only model for the extreme pressures 0 and 100GPa. FIG . 4: (color online) LSDA band-structure of paramagnetic MnO in the B8 high pressure phase. Shown are the Kohn-Sham bands (solid) and the eigenvalues of the Wannier Hamiltonian (dashed). The vertical axis is energy (in eV) with the Fermi level at the origin. FIG. 5 : 5(Color online) LDA density of states (DOS) of the paramagnetic B8 phase of MnO for various pressures. The Fermi level corresponds to ω = 0. α = β) of paramagnetic B8 MnO for the d-only model and the pressures 96 and 160GPa. FIG. 6 : 6(Color online) local Coulomb interaction of the paramagnetic B8 phase of MnO for different pressures. (a) diagonal elements of the local, bare interaction V αα,αα 0,0 for the pd and d-only model, and resolved for the different orbitals α. (b) zero frequency limit of the RPA partially screened local interaction Uαα = Wr αα,αα 0,0 (ω = 0) for the pd model (with and without p screening), and the d-only model. TABLE I : ISummary of the low-energy setups that are used : Shown are the choices for the construction of the low energy subspace, and the polarization. See text for details. TABLE II : IIComparison between the direct and oxygen mediated contributions to the effective transfer element of the occupied t2g orbitals for the B1/dB1 phase of MnO. ∆ct = ǫ occ t 2g −ǭ2p is the charge transfer energy. Shown are the largest transfers for nearest neighbors of same spin. All energies in eV. TABLE AcknowledgmentsWe gratefully acknowledge discussions with and comments by R. Sakuma, J. Kuneš, K. Terakura and R. Co-hen. This work was in part supported by the G-COE program of MEXT(G-03) and the Next Generation Supercomputer Project, Nanoscience Program from MEXT, Japan.0,0(ω = 0) (using the conventions of Eq.(2)). For a discussion on the frequency dependence of the screened interaction see the Ref.5. 72 On a more general note, we stress, that the Wannier cRPA approach can be viewed as a first step towards the already mentioned combination of GW 46-48 with the dynamical mean field theory (DMFT) 52 "GW+DMFT", 66 in which the Hubbard U is promoted from a parameter to a quantity that obeys a self-consistency relation. 73 In this definition, it also includes the respective contributions from the exchange splitting. 74 J. Kuneš, private communication. 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First- principles approach to the electronic structure of strongly correlated systems: Combining the gw approximation and dynamical mean-field theory. Phys. Rev. Lett., 90(8):086402, Feb 2003. a d-manifold, the screening originating from transitions of other bands to and from the d-bands are eliminated. In the constrained LDA technique for the Hubbard U of e.g.. although they are not accountable for by the low energy system -a reason why the constraint LDA technique often overestimates the values of UIn the constrained LDA technique for the Hubbard U of e.g. a d-manifold, the screening originating from transi- tions of other bands to and from the d-bands are elimi- nated, although they are not accountable for by the low energy system -a reason why the constraint LDA tech- nique often overestimates the values of U. This discards correlation induced changes in the charge density. See e.g. 64 for a charge self-consistency scheme within DMFT. This discards correlation induced changes in the charge density. See e.g. 64 for a charge self-consistency scheme within DMFT. Still, both LDA and RPA work at T = 0. Still, both LDA and RPA work at T = 0. subspace are entangled, the separation P = P low + Pr is not uniquely defined. For a way to overcome this arbitrariness see the recent 65. Since in our case the entanglement is not too pronounced we do not employ this refinementIn case the bands of the "low" and "high" subspace are entangled, the separation P = P low + Pr is not uniquely defined. For a way to overcome this arbitrariness see the recent 65 . Since in our case the entanglement is not too pronounced we do not employ this refinement. While the RPA screening and thus also the effective interactions are energy dependent, so far, most many-body techniques only use static interactions. which is why we will discuss in this work only the low energy limit, i.e. for example the Hubbard U αβ = Wr ααWhile the RPA screening and thus also the effective in- teractions are energy dependent, so far, most many-body techniques only use static interactions, which is why we will discuss in this work only the low energy limit, i.e. for example the Hubbard U αβ = Wr αα,ββ	[]
[ "Probabilistic Software Modeling ", "Probabilistic Software Modeling " ]	[ "Hannes Thaller [email protected] \nInstitute for Software Systems Engineering\nJohannes Kepler University Linz\nAustria\n" ]	[ "Institute for Software Systems Engineering\nJohannes Kepler University Linz\nAustria" ]	[]	Software Engineering and the implementation of software has become a challenging task as many tools, frameworks and languages must be orchestrated into one functioning piece. This complexity increases the need for testing and analysis methodologies that aid the developers and engineers as the software grows and evolves. The amount of resources that companies budget for testing and analysis is limited, highlighting the importance of automation for economic software development. We propose Probabilistic Software Modeling, a new paradigm for software modeling that builds on the fact that software is an easy-to-monitor environment from which statistical models can be built. Probabilistic Software Modeling provides increased comprehension for engineers without changing the level of abstraction. The approach relies on the recursive decomposition principle of object-oriented programming to build hierarchies of probabilistic models that are fitted via observations collected at runtime of a software system. This leads to a network of models that mirror the static structure of the software system while modeling its dynamic runtime behavior. The resulting models can be used in applications such as test-case generation, anomaly and outlier detection, probabilistic program simulation, or state predictions. Ideally, probabilistic software modeling allows the use of the entire spectrum of statistical modeling and inference for software, enabling in-depth analysis and generative procedures for software.	null	[ "https://arxiv.org/pdf/1806.08942v2.pdf" ]	49,409,446	1806.08942	9812fe526683e202d8dabb429e7c0f573ec88673	Probabilistic Software Modeling * Hannes Thaller [email protected] Institute for Software Systems Engineering Johannes Kepler University Linz Austria Probabilistic Software Modeling * 10.4230/LIPIcs.xxx.yyy.pAdvised by Alexander Egyed ([email protected]) Co-Advised by Lukas Linsbauer ([email protected])1998 ACM Subject Classification I165 Computing methodologies Uncertainty quantificationD27 DistributionMaintenanceand Enhancement Keywords and phrases Software ModelingProbabilistic ModelingStatistical Inference Software Engineering and the implementation of software has become a challenging task as many tools, frameworks and languages must be orchestrated into one functioning piece. This complexity increases the need for testing and analysis methodologies that aid the developers and engineers as the software grows and evolves. The amount of resources that companies budget for testing and analysis is limited, highlighting the importance of automation for economic software development. We propose Probabilistic Software Modeling, a new paradigm for software modeling that builds on the fact that software is an easy-to-monitor environment from which statistical models can be built. Probabilistic Software Modeling provides increased comprehension for engineers without changing the level of abstraction. The approach relies on the recursive decomposition principle of object-oriented programming to build hierarchies of probabilistic models that are fitted via observations collected at runtime of a software system. This leads to a network of models that mirror the static structure of the software system while modeling its dynamic runtime behavior. The resulting models can be used in applications such as test-case generation, anomaly and outlier detection, probabilistic program simulation, or state predictions. Ideally, probabilistic software modeling allows the use of the entire spectrum of statistical modeling and inference for software, enabling in-depth analysis and generative procedures for software. Introduction Software and software development is an omnipresent and growing industry, and many modern advances in other domains hinge on the success in computer science. Many software systems never stop growing and evolving, consuming thousands of developer days; hence, knowledge and understanding of these systems are essential. This growth poses new requirements in software engineering, not only on an organizational level but also on a functional level. How can we understand the behavior of a system? How can we apply reuse on a broad scale? How can we guarantee a specific behavior? How can we automate development efforts? Many of these questions are already (partially) answered by existing modeling techniques while others are not. Probabilistic Software Modeling The two main modeling approaches currently are Model-Driven Engineering (MDE) [18] and Formal Methods (FM) [7], which share some commonalities but have different goals. In MDE, engineers create domain models on a higher abstraction level than source code to scale the overall development process. A typical technique is to create a Domain Specific (Visual) Language (DSL) from which code and reference implementations can be generated and reused. Another prominent MDE example is the Unified Modeling Language (UML) [17] that conceptualizes many aspects of software development. Formal Methods focuses on software specifications via logic. Resulting models are unambiguous machine interpretable specifications used to generate the application code. The resulting code needs less testing since many program states and transitions can be verified by automatic proofs. Relative to MDE, FM lowers the abstraction level as concepts such as conditions and invariants, need to be designed, elicited and foremost proven before a single line of code is executed. Many other specialized modeling techniques stem from these two approaches, e.g., model-based testing borrows techniques from formal methods. While these modeling techniques offer significant benefits in their respective areas, both lack a way to inspect software and its runtime behavior in a direct, visual and interactive way an engineer would appreciate. MDE lifts the abstraction level on which engineers reason about the best structure for reuse, while FM lowers the abstraction level down into the regime of logic where no ambiguity is allowed. The class diagram is depicting the structure of a Nutrition Advisor system. NutritionAdvisor is the main class and entry point for requests. Its advice method takes a Person entity and returns a textual advice regarding its nutrition. Person stores nutrition relevant properties that BmiService service uses to compute the Body Mass Index (BMI) on which the textual advice is based. We propose Probabilistic Software Modeling (PSM), a data-driven modeling paradigm for software that allows simulation and analysis of software systems. PSM builds a Probabilistic Model (PM) from the system under inspection by analyzing its structure, i.e., types, properties and executables, and behavior via static and dynamic code analysis. These probabilistic models are then used in typical predictive and generative tasks known from statistical inference. For example, the visualization of a property's distribution at runtime (e.g., Person.weight in Figure 1 with behavior in Figure 2), or prediction of the next most likely value (or distribution of values) of a property's active state, etc. Furthermore, inference allows the simulation of entire sub-systems through probabilistic execution of the corresponding models. PSM positions itself as a complementary software modeling approach to MDE or FM. It offers engineers and programmers to interact with the software's structure and behavior through statistical models. Forward and Lethbridge [9] showed that many issues with code-centric comprehension and analysis, i.e., fixing bugs, analyzing structure and behavior by inspecting the source code, come from the fact that developers cannot see and feel the structure and behavior of their system, especially after changes. This results in a typical behavior where programmers, even if they are confident in their code through tests, tend to inspect changes in code via debugging to see an instantiation of its runtime behavior. PSM tries to alleviate these issues by making structure and behavior explicit in models. The usage patterns of PSM are focused on increasing confidence through comprehension and white-boxing of software. PSM provides fine-grained control of variables and their interactions within a system, letting the engineers see and manipulate what happens at runtime through PMs and statistical inference. Idealistically, PSM allows companies that provide software components such as APIs, plugins, frameworks, to exchange their probabilistic models with partners. These models are then used to inspect and check the behavior, generate tests from the models, or to simulate and infer the behavior at the boundaries of internal and external components via the PMs of the partners. That is a unified communication framework via models without the need for a full running system. Again, PSM is a complementary approach to software modeling, and existing practices should benefit from it. MDE can be paired with PSM to check the compatibility and integrity of a new configuration build via MDE. FM could benefit from PSM by using it to check and generate tests from non-critical sub-systems in a less rigorous and data-driven way. In sum, PSM is a modeling paradigm for the analysis of systems on the abstraction level of code, hence, complements existing techniques in their effort for better software comprehension. Applications Probabilistic software modeling is, to the best of our knowledge, the first attempt to model the software's structure and behavior purely in terms of probabilistic models. Hence, it might not be clear what applications PSM can enable. As mentioned, PMs allow for predictive and generative tasks, however, this discrimination is purely conceptual in the context of PSM as the underlying techniques (e.g., approximate inference by sampling [15]) and their associated notations are the same. Predictive tasks use the PMs to predict probabilities, values, and distributions according to criteria in an inferential query. For instance, given the Person class and its properties in Figure 1, it is possible to query the PM how likely a specific value or value range is, e.g., P (W eight > 80.0). By conditioning on other variables, it is possible to infer relationships between variables. For example, P (W eight \| 169.0 < Height < 170.0) results in the distribution of W eight values for persons with a height of 169cm. Generative tasks sample observations from the PMs underlying distributions by either using the distribution as-is or by conditioning on other variables. For instance, the PM can be used to instantiate objects of the Person class by sampling for its properties P (Height, W eight) ∼ (69.54, 169.59) T . Similar to the predictive tasks, it is possible to sample values from conditioned distributions, effectively generating samples according to some criteria. Both cases are inferential tasks that allow a broad range of possible applications. The following non-exhaustive list are applications that are part of the future research. Predictive Applications Predictive applications seek to quantify, visualize and analyze the behavior of software. Visualization & Comprehension applications of PSM have the goal to white-box software by making the runtime explicit for engineers. The visualizations show the distribution of properties, inputs, and outputs of executables. These visualizations can be adapted by conditioning the underlying source models with specific variables. For example, what does the Body Mass Index (BMI) look like for people with a height of 160cm? Furthermore, untypical examples can be visualized and followed through other models effectively visualizing how far they ripple through the system and the effect they have on downstream components. For instance, how far can an invalid weight of −10 ripple through the application without being caught and which effect does it have? Integrity & Compatibility Evaluation quantifies the consistency between and within software components. Integrity measures the amount of divergence between models of different versions of a component. The divergence is the integrity measure aggregating the distribution of accepted and unaccepted values. This distribution can be visualized, conditioned and used for further inferential tasks, e.g., spotting regression and analyzing expected behavior changes. Compatibility evaluation measures the amount of divergence between models of different components and enables the same applications as integrity evaluation such as quantification, visualization, inferential queries. Anomaly Detection [6,11] applications can be employed during development but also in live systems. A PM is used as reference behavior to check whether the newly observed behavior lies within the bounds of its distributions. The implementation might be as simple as the inferential query P (W eight = x) < 0.1 where x is the currently observed value. The condition triggers an action if the likelihood of the current value is smaller then 0.1. Additionally, PSM can compute the ripple distance 1 and effect of the anomaly for detailed reports that summarize the incident. Generative Applications Generative applications reproduce the system's runtime behavior by generating new instances of code elements such as classes, parameters, and return values. Test-case generation [2,5] can be realized by sampling from the PMs. For example, test-suites can be organized into typical, rare, and impossible where the categories indicate the probability range the sampled values are allowed to have. A test-suite with rare invocations for the NutritionAdvisor.advice(p:Person) could be constructed by sampling values that with 0.02 < P (P erson) = P (Height, W eight) < 0.1 from the types PM. This allows to generate tests regarding the prevalence of states and values providing a better understanding of the behavior. Furthermore, test-case generation can be used to evaluate the integrity and compatibility of components partially. For integrity evaluation, the PM of the old version generates tests for the new version to find regressions. For compatibility evaluation, the call-site PM generates tests for the receiving site to find incompatibilities. Simulation, similar to test-case generation, samples values from the PMs and conditions downstream PMs with the sampled values. This effectively executes the system in a probabilistic fashion with the expected behavior according to the underlying distributions. Approach Probabilistic Software Modeling builds on the fact that software systems and programming languages are based on the recursive decomposition principle [16]. Any complex information event on one level can be described on another level by a number of components and a process specifying the relations among these. Reinterpreted for PSM and modern object-oriented software development: Instead of modeling the entire software (structure and behavior) in one complex probabilistic model, a recursive decomposition it into smaller models and their relationships yields a model with similar behavior. For that, the level of abstraction needs to be fixed. As mentioned, PSM is designed to analyze software on the same abstraction level developers are used to making, types, properties, and executables an appropriate choice. Figure 3 Example request to the Nutrition Advisor depicting its behavior. The client makes a new request which is handled via the advisor method of the NutritionAdvisor class. The method retrieves the properties of the person object and computes the BMI using the bmiService. Based on the result it returns a textual advice. Figure 1 shows an example of a nutrition advisor application. NutritionAdvisor takes a Person, extracts its height and weight and feeds it into the BmiService.bmi(.) method that computes the BMI of a person. Based on the result, it returns a textual advice regarding its nutrition. Figure 3 illustrates one request to this application in the form of a sequence diagram. In this example, the person is 168.59cm tall and 69.54kg heavy resulting in a BMI of 24.466 by which the textual advice is computed. A practical and straightforward application which exists (at least up to the advice) in various forms on the Internet. Usage Scenario PSM is essentially a two-step approach in which we (1) analyze the structure of the application via static code analysis, and then (2) the behavior via dynamic code analysis (i.e., runtime monitoring). Figure 4 illustrates this basic usage scenario. Starting with the Source Code that has some Structure , we first analyze these static structural aspects (i.e., properties, types, executables) to define a Probabilistic Model Network that mimics this structure. After that, the source code is executed to trigger a Runtime Behavior which provides the Behavior Observations via dynamic code analysis. At last, PSM approximates the behavior of each node within the probabilistic model network using the behavior observations to build the final Model . Figure 2 shows an example of such an approximated behavior by visualizing the distribution of Person.weight property in the Nutrition Advisor example. The blue histogram is the monitored data from the dynamic code analysis while the yellow line is the fitted distribution. In this running example, the data was sampled from the True Distribution (purple), which is the target quantity (usually unknown) the PM approximates. At this point, often the question arises whether it is possible to acquire a meaningful dataset. Software is developed around data, and engineers have an excellent understanding of it. That is, engineers solve problems, problems are represented by their data. Hence, engineers need to have an understanding of the data to build a solution (problem-solving process [12,19]). A shop handles articles, a video-on-demand service offers videos, and the control system of a spaceship handles the remaining ∆v, thus acquiring a meaningful dataset is often not a problem. Figure 5 shows the network of probabilistic models of the Nutrition Advisor example containing the random variables for types, properties, and executables. Properties are random variables. Types are quantified by their properties, hence, are joint models of their property's random variables. In the case where a type does not have a property, the distribution of the type itself is empty. The objects of a type that has no state are indistinguishable from each other; therefore there is no information to extract. This applies transitively as the bmiService property in the N utritionAdvisor class shows. Executables are random variables that are conditioned on their input parameters, property reads and method invocations (return values). For instance, advice(.) is conditioned on person (parameter), height, weight (property reads) and bmi(.) (method invocation). All classes that are considered by PSM are within the probabilistic modeling universe and can be used for the above-mentioned applications in Section 2. However, there will always be external types beyond the modeling universe illustrated by the Context class in Figure 5. Boundaries between external and modeled code elements are bridged by interpreting the external elements as latent variables. The effect of these variables on the PMs in the modeling universe is subject to the data and its distribution. That is, no specific variables are constructed for external elements, but their dependencies and effects to universe-internal elements are described through the runtime data itself (statistical dependencies). Modeling To summarize, PSM uses random variables to construct small PMs of types, properties, and executables that are connected by conditioning on related elements. Each PM is a generative model capable of producing the distribution of the code element according to the dynamically observed values. Inference Method Requirements The PMs in Figure 5 can be implemented with various statistical modeling techniques. However, the application context imposes specific requirements onto the modeling method. Modeling techniques have to be 1. decidable, 2. computationally efficient, 3. non-parametric, 4. universal. Decidable such that a clear exit criterion for the density approximation exists. Computationally efficient such that large amounts of data and models can be fitted with a limited budget. Non-parametric such that no assumptions about the data itself have to be made. Universal such that any given distribution can be approximated. The prototype will be based on Variational Autoencoders (VAEs) [14], and Probabilistic Programming [10] as these were deemed most promising. Limitations PSM has several limitations and constraints. One constraint is that subsystems need to be runnable. There is no limitation on the size of a subsystem as models can be built from functions which are the smallest runnable piece of code. However, the fact that it needs a runtime in itself may be a limitation. Furthermore, the need for a sensible runtime extends this limitation. PSM models reflect a program and enable probabilistic inference on them. Hence, the fitted behavior will be similar to the program's behavior at runtime. Running a program with arbitrary data reduces the PSM analysis capabilities to interactions of elements since the distributions of the elements themselves might not be representable for the real world. At last, the programming language must be reasonably structured as models are on the level of executables, i.e., the recursive decomposition principle must apply to the programming language. Whether this decomposition is enforced via classes and their objects, or via procedures is irrelevant. Furthermore, the language must allow for runtime monitoring. Related Work The Unified Modeling Language (UML) [17] is a prominent example of visual modeling of software. Its primary purpose is to aid engineers in their design efforts by defining diagrams that capture module, sequence, requirement or state concerns. Another goal of UML is communication by providing a unified visual representation of software design. UML is often associated with Model-driven Engineering (MDE) [18] as it can be used as a modeling language. MDE is a modeling approach with a wide variety of techniques targeted for reuse and synthesis of software using UML or DSLs. It achieves this by increasing the level of abstraction resulting in increased productivity, portability, maintainability and interoperability. However, increasing the abstraction by introducing a new artifact adds mapping costs between the high-level abstraction and the low-level representations. Often the (low-level) code and the (high-level) model gets desynchronized [9], thus reducing its usefulness. Also, technical issues and bug fixes cannot be solved in models. Compared to PSM, MDE helps to improve engineering tasks before the modeled code is written. On the other hand, PSM aids engineers in understanding the actual instantiation of the system by letting them interact with its behavior. The study from Forward and Lethbridge [9] suggests synergetic effects between MDE and PSM in their endeavor of software comprehension. MDE supports engineers in solving model-centric issues, i.e., design, module decomposition and reuse, while PSM offers support for code-centric issues such as behavior comprehension, regression analysis, and fixing of bugs, that are often tedious in the source code. Formal Methods is a modeling approach that focuses on software specification in logical terms. Once a system's behavior is fully specified by formal models, the entire application can be synthesized from it. This, ideally, makes testing obsolete as the specification itself can be part of proofs. However, the transition from the uncertain real world into the perfect mathematical world has several issues. It is time-consuming, needs personnel with the appropriate education, and changes the standard development process quite drastically [7,1]. Furthermore, both boundaries are limited by humans, i.e., the formal model is just as good as the requirements document. Similarly, the translated code is only as correct as the human implemented translator is. However, safety-critical systems greatly benefit from FM because of its ability to provide proof of the behavior [1]. Compared to PSM, FM moves the comprehension of the application and its requirements into the design phase when the specification models are created. PSM may mitigate some of the issues by reducing the FM part to critical systems while non-critical systems are comprehended using PSM. Symbolic Execution [13] avoids structural abstraction level changes by combining testing and formal methods. It generalizes testing by replacing real variable values, with symbolic values and keeps track of the symbolic program path. This allows the exploration of multiple program paths simultaneously giving an understanding of which program points are reachable or not. Typical applications are programs path space comprehension, test input generation, and finding bugs or vulnerabilities of the code. Challenges are for example path explosions, external environments, and loop invariants [3]. In comparison to PSM, Symbolic Execution abstracts the concrete values to path conditions while PSM models a distribution of the encountered values. Symbolic Execution finds inconsistencies in the code by systematically solving for all possible program paths. PSM, on the other hand, builds distributions of the application components and captures questionable behavior across them. Probabilistic Program Analysis (PPA) [8] is an extension of Symbolic Execution that quantifies program paths and points utilizing probabilities, instead of the binary decision satisfiable and not satisfiable. So given a program and a statement of interest, Probabilistic Program Analysis can quantify the likelihood a specific statement is executed. Regardless of the similar names, PSM and Probabilistic Program Analysis do not share many similarities. PSM models the content of a statement over multiple executions, while PPA quantifies the likelihood that a statement is executed (PPA). Research Goals and Evaluation Strategies The goal of this work is to devise and define the essential 1. characteristics, 2. methodologies, and 3. applications, that characterize probabilistic software modeling. These definitions and strategies should guide the implementation of a prototypical PSM system and the introduction of the modeling paradigm itself. Characteristics will be concerned with when and where PSM is the right choice based on the success of the implemented applications. Methodologies describe how to exploit the benefits and drawbacks of structural and temporal dependencies of the system's random variables (types, properties, executables). Applications provide means to quantify the approach empirically, and its appropriateness in the respective domain. The combination of the three research goals (characteristics, methodologies, applications) will allow the evaluation of PSM and its capabilities to solve the problem of software comprehension. The implementation of a prototypical PSM system will drive the evaluation of the research. PSM tries to estimate the densities of code elements at runtime, hence, is an unsupervised machine learning approach that is typically harder to evaluate as well-known supervised methods. Benchmarks, such as DaCapo [4], will be used to evaluate the prototype on real-world examples measuring the estimated fit of the PMs. This fit is evaluated by visual inspection (qualitative) of interesting cases (such as strings), and by metrics (quantitative) that measure the error between sampled and original dataset quantifying bias and variance of the models. On a higher level, the prototype will be evaluated by implementing concrete applications mentioned in Section 2. These will be subject to studies that compare the PSM solution to the current state-of-art. Both qualitative and quantitative evaluations of the studies mentioned above will provide evidence whether PSM improves on the current state-of-art of supporting program comprehension. That is, by either improving existing techniques (such as test-case generation) or by additional support (such as inferential queries, simulation). Conclusion We presented Probabilistic Software Modeling (PSM), a complementary software modeling approach. The goal of PSM is to support software comprehension from a code-centric perspective. The approach complements MDE in that it provides engineers a way to reason about software and its runtime after a system has been implemented. Further, this reasoning is done on the same level as the implementation concepts (types, properties, and executables). An overview of the approach was given on a conceptual level which was then compared with existing other modeling approaches to see commonalities and possible synergies. At last, we provided an outlook of the research methodology, possible applications, and future studies to evaluate PSM. Figure 1 1BmiService advice(p:Person): String Person height: float weight: float BmiService bmi(height: float, weight: float): float Figure 2 2Distribution of the W eight variable of the P erson class depicting its behavior at runtime. The histogram shows all weight values that objects of type Person had during the runtime. These values were generated using the True Distribution (purple) and approximated by the Fitted Distribution (yellow). Figure 4 4Source Code has some structural and behavioral properties that are extracted via static and dynamic code analysis. This results a network of probabilistic models and behavior observations that are combined by PSM into the final model . Figure 5 5advice(.) \| person, height, weight, bmiService, bmi(.)) Graphical model of the Nutrition Advisor's structure of probabilistic models and random variables. A Property is a model with one random variable representing the property itself. 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[ "DATA COMPLIANCE IN PHARMACEUTICAL INDUSTRY Interoperability to align Business and IT", "DATA COMPLIANCE IN PHARMACEUTICAL INDUSTRY Interoperability to align Business and IT" ]	[ "Néjib Moalla [email protected] \nPRISMa Laboratory Lyon 2\nIUT Lumière Lyon 2\nUniversité\n160, 69500BRON\n", "Abdelaziz Bouras [email protected] \nPRISMa Laboratory Lyon 2\nIUT Lumière Lyon 2\nUniversité\n160, 69500BRON\n", "Gilles Neubert [email protected] \nPRISMa Laboratory Lyon 2\nIUT Lumière Lyon 2\nUniversité\n160, 69500BRON\n", "Yacine Ouzrout [email protected] \nPRISMa Laboratory Lyon 2\nIUT Lumière Lyon 2\nUniversité\n160, 69500BRON\n" ]	[ "PRISMa Laboratory Lyon 2\nIUT Lumière Lyon 2\nUniversité\n160, 69500BRON", "PRISMa Laboratory Lyon 2\nIUT Lumière Lyon 2\nUniversité\n160, 69500BRON", "PRISMa Laboratory Lyon 2\nIUT Lumière Lyon 2\nUniversité\n160, 69500BRON", "PRISMa Laboratory Lyon 2\nIUT Lumière Lyon 2\nUniversité\n160, 69500BRON" ]	[]	The ultimate quest in the pharmaceutical sector is Product Quality. We aim with this work to guarantee conformity between production and Marketing Authorizations data (Authorizations to Make to Market: AMM in Europe). These MA detail the process for manufacturing the medicine and compliance to the requirements imposed by health organisations like Food and Drug Administration (FDA) and Committee for Medicinal Products for Human use (CHMP). The paper deals with the communication between heterogeneous information systems with different business structures and concepts. The goal is to maintain compliance of technical data in production with corresponding regulatory definition of pharmaceutical data in the MA. Our approach to modelling present an Interoperability Framework based on a multi-layer separation to identify the organisational aspects, business trades, information systems and technologies for each involved entity into communication between these two systems. We have used RM-ODP Reference Model for the characterisation of each layer. Interoperability is guaranteed if communication of product data respects these levels.In this work, we are also focus on the meaning of concepts and terms used by integrate business constraints in the data structure and their transformation into rules. This helps to identify integration rules to perform connections between data and their native information system and communication rules, as transformation and correspondence, to ensure the mapping through all product states.	10.5220/0002460300790086	[ "https://arxiv.org/pdf/1811.07693v1.pdf" ]	27,644,595	1811.07693	6a73064329505b615c2de8c3bd0b37008fb0c806	DATA COMPLIANCE IN PHARMACEUTICAL INDUSTRY Interoperability to align Business and IT Néjib Moalla [email protected] PRISMa Laboratory Lyon 2 IUT Lumière Lyon 2 Université 160, 69500BRON Abdelaziz Bouras [email protected] PRISMa Laboratory Lyon 2 IUT Lumière Lyon 2 Université 160, 69500BRON Gilles Neubert [email protected] PRISMa Laboratory Lyon 2 IUT Lumière Lyon 2 Université 160, 69500BRON Yacine Ouzrout [email protected] PRISMa Laboratory Lyon 2 IUT Lumière Lyon 2 Université 160, 69500BRON DATA COMPLIANCE IN PHARMACEUTICAL INDUSTRY Interoperability to align Business and IT Pharmaceutical SectorInformation SystemsComplianceCoupling and Integrating heterogeneous Data SourcesInteroperabilityData Semantic Aspect The ultimate quest in the pharmaceutical sector is Product Quality. We aim with this work to guarantee conformity between production and Marketing Authorizations data (Authorizations to Make to Market: AMM in Europe). These MA detail the process for manufacturing the medicine and compliance to the requirements imposed by health organisations like Food and Drug Administration (FDA) and Committee for Medicinal Products for Human use (CHMP). The paper deals with the communication between heterogeneous information systems with different business structures and concepts. The goal is to maintain compliance of technical data in production with corresponding regulatory definition of pharmaceutical data in the MA. Our approach to modelling present an Interoperability Framework based on a multi-layer separation to identify the organisational aspects, business trades, information systems and technologies for each involved entity into communication between these two systems. We have used RM-ODP Reference Model for the characterisation of each layer. Interoperability is guaranteed if communication of product data respects these levels.In this work, we are also focus on the meaning of concepts and terms used by integrate business constraints in the data structure and their transformation into rules. This helps to identify integration rules to perform connections between data and their native information system and communication rules, as transformation and correspondence, to ensure the mapping through all product states. INTRODUCTION The pharmaceutical industry is distinguished among process industries by the need to comply with regulatory constraints imposed by organizations like In this operating context, the issue of product quality is one of high priority for a company in order to maintain its credibility compared to its customers. One of the key factors of quality is the good management of product data. Product data comes in several types and formats specific to various business trades and are supported by several heterogeneous information systems. The challenge is to enable communication among these systems and the process of guaranteeing the validity and the conformity of exchanged information. This challenge is seldom addressed systematically. Indeed, considering the complexity of information systems architectures for the production, there is a general tendency to check conformance only between the AMM files and the Standard Working Instructions (SWI). Our Scope in this paper covers the problem of communicating product data between information system supporting the MA and the ERP for structuring production data. Deliver one product according to his description in the MA requires that we have the right information in the ERP; otherwise, we risk manufacturing a non compliant product, not delivering our product in time to respect customer commitments, destroy these product and lost money. The pivotal problem of medical data is the absence of machine readable structures [3]. Most healthcare data is narrative text and often not accessible. Generally, relates works [3] [4] have tendency to treat this problem by structuring drug information using XML standards to relating and searching drug information using topics Maps. Performing data mapping between regulatory and industrial product definition present a hard task that require regrouping effort from different sectors like regulatory affairs, industrial operations, information systems … Some pharmaceutical industries are specialized in biologic development of medicines. The implication of a deviation in manufacturing or subcontracting can run the gamut from very minor to catastrophic. Our challenge consists in delivering the right product data value through manufacturing states in production information system. During manufacturing process, the product passes from one state to another. For each one, we find one or several components and we have to validate their corresponding specifications by data coming from MA information system. The following Figure (Figure 1) presents a hierarchical structuring for one product in an ERP. When we have to ensure compliance for one data from MA to ERP, we must find and validate product data value for each component through different product states. Our contribution in this paper is to use a modelling approach for the communication between information systems in a pharmaceutical organization. We also propose a methodology for structuring and exchanging product data while ensuring their conformance. In the following section we present some modelling approaches and adapt them to our problem. Further, we propose a data exchange structure that ensures conformity between information systems. Finally, using our approach we present a case study at Sanofi-Pasteur, a developer and producer of vaccines for human use. MODELLING APPROACHES Characteristics of pharmaceutical industry Characteristic of the pharmaceutical industry the information for product data is compiled together from information contributed by various functional divisions that interact with each other in the creation and manufacture of the product. As we can see in figure 2, each of the following divisions contributes different types of information: directed research, and is interested in the development of mixture processes of excipients, tests and stability conditions of the final solution that can be defined as a drug. The information system is used to structure data about clinical trials and tests for validity. At this stage, we begin to define an explicit product structure. • Industrialization division: defines the industrial infrastructure which will support the production of a defined product quantity on the basis of a definition of product solution. At this stage, we start to define technical data describing the operations of product manufacture and the tools used. • • Production division: deals with planning, scheduling and follow-up of production based on the data describing industrial infrastructure and product composition. At this stage, we identify static data compared to external dynamic data like work orders and internal generated by the enterprise resources planning (ERP) like buying orders of raw material. • Distribution division: defines the conditions for handling the product for customer delivery in accordance with the description of the conditions of manufacture delivered by R&D division. At this stage, product handling information is documented. Figure 2: Information system sources in pharmaceutical industry From one stage to another, product data are recorded using a specific structure and format. Each local information system is defined in accordance with the needs which are relevant to their business trades. The definition of a product for pharmaceutical industry is not tied to physical shape except in the packaging stage. The company submits to the Health Authorities entire product specifications along with documented information. These deposed documents constitute the request of Marketing Authorization. When health authorities approve this request, they give the Marketing Authorization (MA). In the delivered documents to authorities, it is necessary to present all information whose justifies product creation process, including pre-clinical tests, clinical trials, tests of validities and the appendices such as bibliography. Only after the process reaches the Industrialisation stage that MA documents can get defined. Once approved in one country, this MA is used as reference document to manufacture product. It's considered as a contract with the authority of a given country by the company that's respects the regulatory constraints. Concerning the American market, the FDA is responsible for the checking the adequacy of the product delivered and manufacturing processes against the acquired authorization. The major quest of each pharmaceutical company is product quality. This objective is achieved only by ensuring a better degree of conformity between existing information in these MA documents and those used for the production. In the following sections, we will propose means to use the MA data, which can be deciphered only by pharmacists, to adapt them to logisticians needs. The approach used makes it possible to ensure interoperability between the information systems supports while satisfying some business constraints. Why interoperability? The IEEE standard computer dictionary defines interoperability as "the ability of two or more systems or components to exchange information and to use information that has been exchanged". Also, the EU Software Copyright Directive [5] defines interoperability between computing components generally to mean "the ability to exchange information and mutually to use the information which has been exchanged" 1 . This does not mean that each component must perform in the same way, or contain all of the same functionality, as every other one -interoperability is not a synonym for cloning. Rather, interoperability means that the components, which may differ in functionality, can share information and use that information to function in the manner in which they were designed to. The European Interoperability Framework definition [5] identifies three separate aspects: • Technical -is concerned with defining business goals, modelling business processes and bringing about the collaboration of administrations that wish to exchange information, but that may have a different internal organisation and structure for their operations. • Semantic -is concerned with ensuring that the precise meaning of exchanged information is understandable by any other application not initially developed for this purpose. Semantic interoperability enables systems to combine received information with other information resources and to process it in a meaningful manner. • Organisational -covers the technical issues of linking up computer systems and services. This includes key aspects such as open interfaces, interconnection services, data integration and middleware, data presentation and exchange, accessibility and security services When we aim to better exchange data between information systems, we have to be sure that these interoperability types are well identified and structured. Therefore, it's necessary to identify the area of our investigation and its specifications: structures, business constraints … Ways to achieve interoperability Seeking to achieve interoperability among divisions in collaborative enterprise, we face three core challenges • Need for Flexibility, due to need for change and exception handling following variation in registrations and commitments • Complexity, the richness of interdependencies with and among divisions, their activities, resources and skills. Heterogeneity, need for flexibility and complexity must be managed at different levels: • Knowledge, approaches, methods and skills needed for innovation, problem solving and work performance, the shared language and frames of reference needed for communication. • Rapid formation of VEs, allowing partners to join along the way When we try to ensure interoperability between information systems in regards of main challenges through these levels, we need to identify first different concerned entities in the company, their characteristics and link between them. To perform this task, we have to find an enterprise modelling approach bringing out all actors, processes, business constraints, work practices, etc. and identifying ties between them. Enterprise Modelling Approaches We can find in literature three main enterprise modelling approaches: Enterprise Framework and architectures; define a framework as a fundamental structure which allows defining the main sets of concepts to model and build an enterprise. We describe two types of frameworks: those for integrating enterprise modelling (such as Zachman, CIMOSA, etc.) and frameworks for integrating enterprise applications (such as ISO 15745, the missing approach, etc.) Enterprise Modelling language; can be define as the art of "externalising" enterprise knowledge by representing the enterprise in term of its organisation and operations. distributed systems, and cover such concerns as policy, naming, behaviour, dependability and communication. To these concepts, it is necessary to analyse the company according to several view points which influence the information structure. The view points included are: • The enterprise viewpoint: is concerned with the purpose, scope and policies governing the activities of the specified system within the organization of which it is a part; • The information viewpoint: is concerned with the kinds of information handled by the system and constraints on the use and interpretation of that information; • The computational viewpoint: is concerned with the functional decomposition of the system into a set of objects that interact at interfaces -enabling system distribution; • The engineering viewpoint: is concerned with the infrastructure required to support system distribution; • The technology viewpoint: is concerned with the choice of technology to support system distribution. How achieve interoperability can ensure compliance? The modelling of the interconnections between information systems using these concepts, and various view points, makes it possible to ensure a structuring in accordance with definite objectives. This structuring is defined in information system urbanization (cf. project SUSIE [8]). Figure 3 presents these various points of view as level of abstraction. Figure 3: levels of information structuring Interoperability between information systems could be defined more finely if this type of modelling is undertaken. In an industrial framework, structuring business knowledge in an information processing system does not imply facilitation of communication with another business system. Data interpretation changes according to the business and the challenge is in the ability to preserve information semantics when communicate across these levels. These problems also appear when we communicate product data coming from two different structuring perspectives. A proposition of communication architecture between two information systems is presented in figure 4. Figure 4: communication architecture between heterogeneous information systems Building similar interoperability architecture can align Business, Knowledge and IT through semantic framework to ensure compliance when exchanging data. In the following section, we will be building our decomposition to present a deployment of an architecture of communication adapted to our context. METHODOLOGY TO ENSURE CONFORMANCE In our context, the objective to establish communication between information systems is to ensure the conformity of the product data in one system in relation to each other. Based on the description of information in an MA, it is necessary to return product data values to the ERP useful for the production. Information system to communicate In our scope, we identify these two entities involved in communication: • MA information system: are generally managed by the regulatory affairs division and constitute a regrouping of information. A MA is composed of electronic documents coming from several sources and contain for example scanned documents, reports and attached papers. The semantic structuring of these authorisations provides a format and contents harmonized according to a pharmaceutical vision. It follows a specific format called Common Technical Document (CTD) [9] [10], defined by the ICH. The Figure 5 presents the model of information structuring. It is a 5 modules structure. The fives modules in the figure 2 are a) specific customer information (module1), b) summary module which contains information updated from version to another (module2) c) product quality (module 3) d) non clinical study reports (module 4) and e) clinical studies reports (module 5). The specificity of the levels of information in the MA is not significant for the retrieval and the use of the data for production division. Only pharmacists seek information from regulatory data to answer requests for product data consultation. Faced with very large number of MA documents that run into thousands of pages, makes it very difficult to answer requests for data validation. • Production information system: was structured data necessary for the production. The ERP (Enterprise Resource Planning) system manages this data and regrouping complex functionalities of "provisioning and scheduling" and generates new dynamic data based on product data definition. So, non-in conformity product data definition, invariably leads to the manufacture of a non conforming product. Each of these divisions presents a specific vision of the product with local knowledge tied to business rates. It's a big challenge to communicate these two systems due to the complexity of bringing together product definition data coming from heterogeneous information systems. To ensure this conformity at product data definition level, it is necessary to define a communication platform to include all perspectives, in particular, organisational, business, informational, and technical [11]. Our purpose is to present data exchange architecture allowing as to translate product data definition from regulatory systems to production one. Which data we need to translate? When we analyze information in these two systems, it's difficult to find a common data product structure. Between pharmaceutical and operational scope, we don't find necessary the same type or meaning of information. Thus, it's necessary to look first for defining what kind of information we need to ensure conformance. When fixed, we analyze product structure in each system to find issues of communication. The CTD format of MA presents in module 3 information about product quality. In this module, we can find a pharmaceutical description of the product and its various states of manufacture. Similar in production, we define another collection of product states. These states are not necessarily coherent among them or are significant from one product to another. The best issues that we find to communicate product information from MA to ERP repose on structuring product data by states. For example, in biologic pharmacy, we define two families of product state: biologic states and pharmaceutics states, from seeds to final product states. The product structure is defined in these two systems as a specific series of product states. Our problem was translated to ensure conformance of data values for each product states from one system to another. But there are not the same definition of product states and not the same data semantic too. For example the shelf life of an intermediate substance is 3 years at storage temperature of -70°C if it was preserved as is, but -20°C if it is lyophilised. The finished product has a 1 year shelf life at 5 °C storage temperature. We assume that the product has a fixed number of states on an information system. It's necessary to identify from production and regulatory information system, the entire specification of each state. It gathers for each stage, the value to be validated, the rules defined formally to extract data from an information system and informal business constraints helping to ensure the communication. We call "product state reference frame", the structuring of one product datum that assign for each product state, the data value, rules applied to extract data from information system, and business constraint helping to understand the choice of data value. For each product state, we need to define also same component of the bill of materials of this state. For example, when our product present at final state two bottles, we need to specify shelf life for these two entities. For other example of this product state, this information is not significant. The application of this reference frame to product data consists in seeking, for one data values, all states in accordance with rules and business constraints already identified. Figure 6 illustrates this structuring. This reference frame represents the data profile in an information system. It must be updated during a potential structural modification and can be published in the organization to ensure better comprehension and exploitation of product data. Each line of this reference frame contains a state of product, the value to be validated, the rules which allow the extraction and the transformation of the data and business constraints for the application of these rules. For the information system of the production, we also find rules allowing for the integration of the new value. Rules definition The definition of the rules is a tedious phase and requires defining rules at three levels: Information system rules in production They are rules to specify when extract or insert data into the information system. Difficulty arises when attempting to insert data. Indeed, the information systems for production are characterized by the reuse of product states information. Locking to two drugs, it is extremely probable to have the same excipients in the pharmaceutical solution. In this case, there are invariably one or more specific common production states with same coding in the system. In production, following a request for modification of product state data, it is necessary to check if this state is also used for another product. Considering the complexity of architecture on information production system and interconnections between information inside, it is tiresome to seek products by a simple indication of an intermediate state. For example, such request can take two days to return an answer that we must sort to find required information. If we schematize the product states by a tree structure with graphs, the interconnection between graphs can be possible with any node except the first. Figure 7 shows an example of these interconnections. Each product has 6 states: S1 to S6. Figure 7: overlap of product states for different products in production information system Integration rules must cover procedures of checking of the use of one state, as well as the impact of one modification if there are common states with other products. The impact of one modification or transformation on a data is sometimes governed by abstract business constraints. For example the date of manufacture of a product is calculated starting from the first valid test of stability. If we want to change the shelf life of a state, the expiry date which is equal to the date of manufacture added to the shelf life, must be revalidated. This aspect touches primarily on data transparency. This is why we planned to add informal business trade field in the reference frame for each product state to help the mapping the reference frame of the information system supporting MAs. Mapping rules They are rules for mapping between product states reference frames by establishing links between active product states. It is also a regulatory pharmaceutical responsibility that is necessary to share with production to ensure the coherence of rules. The product states are not same across information systems and certainly across reference frames. From one product to another, a state may or may not exist. We use business knowledge as a reference to create these links of communication between active states. This knowledge is indexed on both MA and production reference frames. The mapping rules allow formalizing fields of data to be inter-connected (links n .. n) as well as transformations to use all of the values of the fields in the regulatory reference frame to generate the corresponding values in the product states reference frame of production. Figure 8 illustrates examples of connection modes. One state in the first reference frame can correspond to one or more states in the second and vice versa. To generate mapping rules, we use data and rules from the two reference frames. For example, mapping rules can be the sum, the average, the min or a data which exist only in one of the information systems. Regulatory information systems rules According to pharmaceutical data structuring, the information system which manages the MA is not able to be directly interfaced to the regulatory product states reference frame. It is possible to have several MA for only one product, and conversely, one MA for several products. These characteristics are relocated on product states, which increase the complexity of the information retrieval. It is very frequent to find for example two product authorizations with various destinations (country) or presentations (packaging contain) and having a common product state but with different value. This difference is due to the history of the negotiations between the company and health authority about MA content. CASE STUDY This case study presents an illustration of a work developed with Sanofi Pasteur Company, a firm specialised on biologic development and the production of vaccines for human use. The purpose of this work is to ensure conformance, from MA to the ERP, for three data: Site of Manufacturing, Shelf Life, and Storage Condition. All MA data was structured in e-TRAC (Electronic Tracking of Registrations and Commitments) information systems. Access to these data is ensured through web interface allowing us exporting defined report from RA-Cockpit reporting module. As presented in figure 9, we can: a) export data for one product line to create report b) distribute this report by product licence number as criteria to identify different product data c) for each product data, instantiate three reference frame for regulatory product states d) apply mapping rules to generate corresponding ERP (here SAP) product states reference frame e) use same specific criteria to data structuring in SAP to validate data comparing to those coming from SAP reference frame. Validate data in SAP As mentioned before, it is extremely probable to have the same product state in different product decomposition, so, we can find the same value for the same product state in different SAP reference frame. In SAP systems, we identify each entity, called item, by one code. There is why, we instantiate a second SAP reference Frame with just SAP code and data value field. It's necessary to find the code of each product state. Due to specific information structuring in SAP in Sanofi-Pasteur Company, we can find the item code for the last state (final product) and use item code filiations to find code for all product states. Actually we have two SAP reference frames, one with data value coming from mapping rules with regulatory reference frame, and the second with data value and item code from SAP. We define here some rules of coherence: • For the same product state, we have necessary the same data value, else we notify exception • For same item codes in the second SAP reference frames we have to find the same value, else we notify exception • It's frequent to find two or more MA or registrations that differ just by product name from one country to another. For example we can define GRIPPE vaccines in Europe but when structuring product information in regulatory information systems, we have to separate product by country. When we seek to validate our three data for grippe in Europe, we have to find the same data value in regulatory reference frame for all country in Europe, else we notify exception. CONCLUSION In this paper, we presented a modelling methodology for information systems especially interested in the structuring and explaining dependences between product data in the pharmaceutical field. Our objective was to ensure data compliance between two information systems, one related to the Marketing Authorizations (MA) and the other with production, through the establishment of communication architecture. Faced with information systems, we have chosen to ensure mapping between product states information along product manufacturing life cycle. In spite of differences in the business visions, the product remains at the core of information structuring in these two systems. Applied to Grippe Line Product in Sanofi-Pasteur Company, proposed concepts provide a very interesting solution by ensuring data compliance of 94,6 % of final products for proposed data (Site of Manufacturing, Shelf Life, and Storage Condition). Some final products states in the ERP have the same definition, but not the same utility because they refer to product with different quality level. Actually we have to treat manually these specific products. Our methodology is based in analysing information coming from MA information systems to validate him in the ERP. But what about product data in the ERP not concerned by these concepts? How to find him? And which criteria are needed to cover him by defined concepts. We aim, with the next step of this work, to optimise our methodology of communication by adding more rules and more constraints, not only to extract or integrate data through reference frame, but between product states in one reference frame also. Food and Drug Administration (FDA) [1], Committee for Medicinal Products for Human uses (CHMP), the guidelines of International the Conference of Harmonisation (ICH) [2]. Further constraints are imposed by the conventions signed with national and international authorities, called Marketing Authorisation (MA) -Authorization to Make to Market (AMM) in Europe -for the manufacture of drugs. Figure 1 : 1Manufacturing product states and state components. 1 Council Directive of 14 May 1991 on the legal protection of computer programmes (91/250/EEC); Figure 5 : 5MA structure: CTD format [10] Figure 6 : 6product states reference frame Figure 8 : 8Mapping links Figure 9 9Figure 9: communication Scenario Research division: searches for new drugs or substances that can contribute to the creation of new drugs. At this stage studies conducted are reported and indexed in the form reports. • Research & Development division: conducts There are two categories of languages: those defined at high level of abstraction as Constructs for enterprise modelling (for example, EN/ISO 19440, ODP, UEML,GRAI,…) which are independent of the technology of implementation; and languages more related to a specific technology presents a methodology for the structuring of distributed services of data processing carried out in an environment of heterogeneous information systems. Our problem is to combine these model objectives in the formalization of the systems functionalities to ensure communication independently of the tools for implementation. In the RM-ODP model, the modelling concepts, that contributes to this formalization, cover:such as INTERNET technology based languages (example, ebXML, etc.). Enterprise Knowledge space; based on a new technology called Active Knowledge Models (AKM) and implement types of views from the knowledge dimensions that contain mutual and complex dependencies of domains. As combination of these modelling approaches, RM-ODP model (Open Distributed Processing - Reference Model) [6] [7] • Basic modelling concepts: the basic concepts are concerned with existence and activity: the expression of what exists, where it is and what it does. • Specification concepts: addressing notions such as type and class that are necessary for reasoning about specifications, the relations between specifications, provide general tools for design, and establish requirements on specification languages. • Structuring concepts: builds on the basic modelling concepts and the specification concepts to address recurrent structures in Table 1 : 1The resulting problem space Pharmaceutical CGMPS For The 21st Century -A Risk-Based Approach Final Report. Sixth International Conference on Harmonisation. 6 Osaka JapanU.S Food and Drug AdministrationNew Horizons and Future ChallengesU.S Food and Drug Administration, September 2004, "Pharmaceutical CGMPS For The 21st Century -A Risk-Based Approach Final Report", Department of Health and Human Services, 32 p [2] SUMMARY REPORT, 13-15 November 2003, "New Horizons and Future Challenges", Sixth International Conference on Harmonisation, ICH 6 Osaka Japan, Implementing health care systems using XML Standards. R Schweigera, M Brumhardb, S Hoelzerc, J Dudecka, International Journal of Medical Informatics. 74Schweigera R., Brumhardb M., Hoelzerc S., Dudecka J., 16 April 2004, "Implementing health care systems using XML Standards", International Journal of Medical Informatics (2005) 74, 267-277 EBXML Technical Architecture Specification V1.0.4, ebXML Technical Architecture Project Team. EBXML Technical Architecture Specification V1.0.4, ebXML Technical Architecture Project Team,16 February Second Version of State of the Art in Enterprise Modelling Techniques and Technologies to Support Enterprise Interoperability. Advanced Technologies for interoperability of Heterogeneous Enterprise Networks and their Applications). Work package -A1.1, Version 1.3.1ATHENA Project (Advanced Technologies for interoperability of Heterogeneous Enterprise Networks and their Applications), February, 2005, "Second Version of State of the Art in Enterprise Modelling Techniques and Technologies to Support Enterprise Interoperability", Work package -A1.1, Version 1.3.1 Information technology -Open Distributed Processing -Reference Model (RM-ODP): Architectural. ISO/IEC 10746-3. First EditionISO/IEC 10746-3, "Information technology -Open Distributed Processing -Reference Model (RM-ODP): Architectural", First Edition, 1996 Information technology -Open Distributed Processing -Reference Model (RM-ODP): Architectural semantics. ISO/IEC 10746-4. First EditionISO/IEC 10746-4, "Information technology -Open Distributed Processing -Reference Model (RM-ODP): Architectural semantics". First Edition 1998 PLM -de la difficulté d'intégrer et de gérer l'évolution du système d'information technique avec le progiciel. Figay Nicolas, Eads Ccr, ParisNicolas FIGAY (EADS CCR) « PLM -de la difficulté d'intégrer et de gérer l'évolution du système d'information technique avec le progiciel » MICAD 2005, Paris Organisation Of The Common Technical Document For The Registration Of Pharmaceuticals For Human Use. International Conference On Harmonisation Of Technical Requirements For Registration Of Pharmaceuticals For Human Use. 14pIch Harmonised Tripartite GuidelineIch Harmonised Tripartite Guideline, "Organisation Of The Common Technical Document For The Registration Of Pharmaceuticals For Human Use", International Conference On Harmonisation Of Technical Requirements For Registration Of Pharmaceuticals For Human Use, November 8, 2000, 14 p Application of product data management technologies for enterprise integration. X Gao, A Hayder, P G Maropoulos, W M Cheung, International Journal of Computer Integrated Manufacturing. 16Taylor & FrancisX. Gao, A. Hayder, P. G. Maropoulos and W. M. Cheung, "Application of product data management technologies for enterprise integration", International Journal of Computer Integrated Manufacturing, Taylor & Francis, Vol 16 N° 7-8, 491 -500, October-December 2003.	[]
[ "Decay of geodesic acoustic modes due to the combined action of phase mixing and Landau damping", "Decay of geodesic acoustic modes due to the combined action of phase mixing and Landau damping" ]	[ "A Biancalani \nMax-Planck Institute for Plasma Physics\n85748GarchingGermany\n", "F Palermo \nMax-Planck Institute for Plasma Physics\n85748GarchingGermany\n", "C Angioni \nMax-Planck Institute for Plasma Physics\n85748GarchingGermany\n", "A Bottino \nMax-Planck Institute for Plasma Physics\n85748GarchingGermany\n", "F Zonca \nENEA C. R. Frascati\nVia E. Fermi 4565-00044FrascatiCPItaly\n\nInst. for Fusion Theory and Simulation\nZhejiang University\n310027HangzhouP. R. China\n" ]	[ "Max-Planck Institute for Plasma Physics\n85748GarchingGermany", "Max-Planck Institute for Plasma Physics\n85748GarchingGermany", "Max-Planck Institute for Plasma Physics\n85748GarchingGermany", "Max-Planck Institute for Plasma Physics\n85748GarchingGermany", "ENEA C. R. Frascati\nVia E. Fermi 4565-00044FrascatiCPItaly", "Inst. for Fusion Theory and Simulation\nZhejiang University\n310027HangzhouP. R. China" ]	[]	Geodesic acoustic modes (GAMs) are oscillations of the electric field whose importance in tokamak plasmas is due to their role in the regulation of turbulence. The linear collisionless damping of GAMs is investigated here by means of analytical theory and numerical simulations with the global gyrokinetic particle-in-cell code ORB5. The combined effect of the phase mixing and Landau damping is found to quickly redistribute the GAM energy in phasespace, due to the synergy of the finite orbit width of the passing ions and the cascade in wave number given by the phase mixing. When plasma parameters characteristic of realistic tokamak profiles are considered, the GAM decay time is found to be an order of magnitude lower than the decay due to the Landau damping alone, and in some cases of the same order of magnitude of the characteristic GAM drive time due to the nonlinear interaction with an ITG mode. In particular, the radial mode structure evolution in time is investigated here and reproduced quantitatively by means of a dedicated initial value code and diagnostics.	10.1063/1.4967703	[ "https://arxiv.org/pdf/1608.03447v1.pdf" ]	119,176,925	1608.03447	98bfdf228044926bda31dc53c565564205f69ba7	Decay of geodesic acoustic modes due to the combined action of phase mixing and Landau damping 11 Aug 2016 A Biancalani Max-Planck Institute for Plasma Physics 85748GarchingGermany F Palermo Max-Planck Institute for Plasma Physics 85748GarchingGermany C Angioni Max-Planck Institute for Plasma Physics 85748GarchingGermany A Bottino Max-Planck Institute for Plasma Physics 85748GarchingGermany F Zonca ENEA C. R. Frascati Via E. Fermi 4565-00044FrascatiCPItaly Inst. for Fusion Theory and Simulation Zhejiang University 310027HangzhouP. R. China Decay of geodesic acoustic modes due to the combined action of phase mixing and Landau damping 11 Aug 2016contact of main author: Geodesic acoustic modes (GAMs) are oscillations of the electric field whose importance in tokamak plasmas is due to their role in the regulation of turbulence. The linear collisionless damping of GAMs is investigated here by means of analytical theory and numerical simulations with the global gyrokinetic particle-in-cell code ORB5. The combined effect of the phase mixing and Landau damping is found to quickly redistribute the GAM energy in phasespace, due to the synergy of the finite orbit width of the passing ions and the cascade in wave number given by the phase mixing. When plasma parameters characteristic of realistic tokamak profiles are considered, the GAM decay time is found to be an order of magnitude lower than the decay due to the Landau damping alone, and in some cases of the same order of magnitude of the characteristic GAM drive time due to the nonlinear interaction with an ITG mode. In particular, the radial mode structure evolution in time is investigated here and reproduced quantitatively by means of a dedicated initial value code and diagnostics. Introduction Plasmas in magnetic-confinement-fusion plasmas are turbulent, due to the presence of strong spatial gradients. Turbulence is often observed accompanied by zonal (i.e. axisymmetric) radial electric fields, generating poloidal ExB flows in tokamaks. These zonal flows are usually constituted by two components, one nearly constant in time, dubbed zero-frequency zonal flows (ZFZFs) [1,2,3,4,5,6,7], and one oscillating with a frequency of the order of the sound frequency (ω ∼ c s /R, where c s is the sound speed and R is the tokamak major radius), dubbed Geodesic Acoustic Modes (GAMs) [8,9]. The importance of both components is due to their nonlinear interaction with turbulence, for they shear and distort convective and turbulent cells leading to the turbulence saturation and consequently to a reduction of the transport. The peculiarity of GAM oscillations resides in the different shearing efficiency that the zonal flows have in relation to their oscillatory behavior. In fact, while ZFZFs suppress turbulence efficiently, oscillations make the action of the zonal flow less effective [10,11]. ZFs/GAMs are also known to transfer energy to the long-wavelength components of the turbulence [12,13]. Historically, the Landau damping has been recognized as one of the main damping mechanisms of GAMs. The Landau damping is due to the nonuniformity of the ion distribution function in velocity-space, and it is therefore an intrinsically kinetic effect. Its evaluation has been performed with increasing accuracy, where finite-orbit-width (FOW) of the passing ions were initially neglected [14,9], then included to the first order [15,16,17,9], then included to higher orders [18,19]. Finally, the effect of the flux surface shape was also included [20]. All analytical calculations of the Landau damping of GAMs have been performed so far by assuming a uniform plasma, i.e. by neglecting the effect of a gradient of the equilibrium density or temperature profiles. More recently, the phase mixing, which was previously investigated for non-ionized fluids [21], for Langmuir waves in plasmas [22], and for shear-Alfvén waves in plasmas [23,24,25], has also been investigated for GAMs analytically [9,26]. Contrarily to the Landau damping, the phase mixing can be studied also in fluid theory, whenever the local spectrum of oscillation of a wavethe continuous spectrum -varies in space (for the GAM in a tokamak, this happens due to the variation of the plasma temperature across the magnetic flux-surfaces). In this case, the spatial shape of a perturbation will be deformed in time due to the effect of the different frequency of oscillation, occurring at different positions. When performing a Fourier transform in space, the effect of the phase mixing is that of shifting the peak of the perturbation in time, towards higher values of the wavenumber. Therefore, the amplitude of the perturbation in Fourier space calculated at the initial wavenumber, decays in time, giving rise to the continuum damping. Gyrokinetic simulations of GAMs with realistic profiles of ASDEX Upgrade had already shown a peculiar behaviour at the edge, which was not been possible to explain with a description based on the Landau damping alone [27]. In particular the damping rate at the edge, where the temperature profile is steep, was measured to be much stronger than that predicted by the theory of Landau damping. More recently, the investigation of this strong damping observed in gyrokinetic simulations has started, showing a new mechanism based upon the combined action of the phase mixing and the Landau damping, and the first results have been shown in Ref. [28]. In this work, we complete the level of understanding of this problem, by describing the details of the combined effect of the nonuniformities in velocity-and in real-space, causing the cascade in the wavenumber and the effective damping at high wavenumbers, with a dedicated initial-value code and diagnostics. In fact, as previously mentioned, the phase mixing, acting because of the nonuniformity in space, damps only each single mode in Fourier space by continuum damping, but in the whole real space the energy of the electric field is conserved, so the actual total damping is zero. Therefore, although our analysis is linear, we can state that the values of the damping rate calculated as pure Landau damping and as pure phase mixing do not just linearly sum up. In fact, the Landau damping of GAMs strongly increases with the radial wave number, due to the finite orbit width of the passing ions. Consequently the effect of the phase mixing is to greatly amplify the efficiency of the Landau damping. This general effect, recently proved to be crucial (see Ref. [28]), is explored here in specific cases chosen for their practical experimental relevance, and quantitatively proved to be indispensabile for an interpretation of realistic GAM dynamics. In particular, the phase mixing and Landau damping are first studied separately and then the resulting combined effect is shown to form a cascade of the energy in wavenumber (analogously to that generated in a turbulent system) with a strong absorption at the small scales [24,25,9], which is reproduced here quantitatively with a dedicated initial value code. The investigation of the Landau damping with analytical theory and numerical simulations with ORB5 was performed already in Ref. [40,28], and recalled here for sake of completeness. The investigation of the phase mixing, on the other hand, is performed in this work with the help of an initial value code which solves the GAM dynamics by adopting the real part of the kinetic frequency, and neglects its imaginary part. This initial value code for the study of the phase mixing, is used to reproduce the evolution of the radial structure of the GAM perturbation, modified by the phase mixing alone. In tokamaks, the combined effect of Landau damping and phase mixing can play a crucial role in the dynamics and in the decay of GAM at the tokamak edge, where the temperature gradients are very large. Depending on the confinement regime, low confinement (L-mode), improved (low) confinement (I-mode) and high confinement (H-mode), different parameter values are achieved in the experiments, in particular changing the relative strength of temperature and density gradients at the edge of the plasma. The sequence of events in the transition phases between these regimes, particularly regarding the turbulence suppression, the development of the edge temperature and density gradients and the increase of the shear flow, is not completely established yet. GAMs are considered to be potential key players in the dynamics of the transition to I-and to H-modes [29], and their absence can enhance the effects of the shear flow on the turbulence [11]. GAMs are reported to be regularly observed in L-mode, and are also observed in I-mode, whereas they are not observed in H-mode. It will be shown that the proposed mechanism of GAM decay is consistent with the observed existence or non-existence of GAMs in the different confinement regimes [30]. Our investigation is carried out by means of analytical theory and numerical simulations. For the numerical simulations, we use the global particle-in-cell code ORB5 [31], which now includes all extensions made in the NEMORB project [32,33]. The ORB5 code uses a Lagrangian formulation based on the gyrokinetic Vlasov-Maxwell equations [34,35,36,37,38]. Due to the method of derivation of the GK Vlasov-Maxwell equations from a discretized Lagrangian, the symmetry properties of the starting Lagrangian are passed to the Vlasov-Maxwell equations, and the conservation theorems for the energy and momentum are automatically satisfied [39]. The code solves the full-f gyrokinetic Vlasov equation for ions and the drift-kinetic equation for electrons. Only linear collisionless electrostatic simulations are considered in this paper, with electrons treated as adiabatic, and with magnetic flux surfaces with circular concentric poloidal sections. This paper is structured as follows. In Sec. 2, the Landau damping in uniform plasmas is studied with ORB5 and compared with analytical theory. In Sec. 3, the phase mixing signatures are identified and compared with results of numerical simulations, obtained for realistic values of tokamak plasmas. Sec. 4 is devoted to a description of the combined action of the phase mixing and Landau damping. Finally, Sec. 5 discusses the application to predict the regimes where such damping overcomes the drive given by the turbulence, and the summary of the results. Landau damping in uniform plasmas In this Section, we recall the results about the Landau damping of GAMs in uniform plasmas, i.e. in the absence of gradients of the equilibrium profiles (i.e. the safety factor, density and temperature profiles are flat), shown in Ref. [28]. The GAM dynamics is independent on the radius to a good approximation, and the radial shape of the initial perturbation does not change in time, at least for the first few GAM oscillations (after which, global effects can occur). For our regimes of interest, a linear estimation of the frequency and damping rate is given by the analytical theory where first-order ion finite-orbit-width effects are retained [16,17]. These simulations serve as a verification of ORB5, extending the work previously done in Ref. [40]. In particular, we study the dependence of the damping rate on the radial wave-number k r , for different values of the safety factor q. We initialize a zonal perturbation of the radial component of the electric field, i.e. a perturbation independent of poloidal and toroidal angles, and depending only on the radius. In general in this paper, except where otherwise stated, we consider initial perturbations with a sinusoidal radial dependence, i.e. monochromatic in Fourier space. The perturbation evolves in time in a linear electrostatic collisionless simulation with adiabatic electrons. For each simulation, we investigate the evolution in time of the radial electric field, and in particular the values frequency and damping rate at each radial location, and the radial structure at each time. This type of numerical experiment is known as a Rosenbluth-Hinton test [2]. Different simulations are investigated, with wavenumber chosen within the range 0.02 ≤ k r ρ i ≤ 0.2, and safety factor within the range 1 ≤ q ≤ 3.5. The value of the plasma temperature is defined by ρ * = ρ s /a = 1/160. The ion Larmor radius is defined as ρ i = √ 2 T i0 /T e0 ρ s , with ρ s = √ T e0 m i /(eB 0 ) and with T i0 /T e0 being the ion/electron temperature calculated in the middle of the radial domain (in this Section, T e = T i = T e0 = T i0 ). The magnetic field B 0 is calculated at the magnetic axis (r=0). We choose a tokamak with an inverse aspect ratio ǫ = a/R 0 = 0.1. We consider analytical magnetic equilibria with circular flux surfaces. In this limit, the flux surface coordinate r = ψ/ψ edge is a good approximation of the radial coordinate (normalized to the minor radius a). The reference simulation has a spatial grid of (r × θ × φ) = (256 × 64 × 4) and a time step of 100 Ω −1 i , with Ω i being the ion cyclotron frequency. All the simulations have been performed with 10 8 ion markers. As a general remark, we find very good agreement between theory and simulations, consistently with Ref. [40]. In particular, our results with ORB5 are consistent with the analytical theory, showing that, at realistic values of q, the damping rate strongly increases with k r ρ i (see Ref. [16,17,28])). Phase mixing in the presence of a radial nonuniformity In this Section, we investigate the effect of a radial nonuniformity of the equilibrium, in generating the phase mixing. Ronsebluth-Hinton tests similar to those described in the previous Section are performed, but with the main difference that the equilibrium radial profiles are not considered flat. This introduces a radial dependence in the GAM dynamics, for example of the GAM local frequency: ω G = ω G (r). In this case, the GAM dynamics in its simplest form (no Landau damping considered here) can be described by the vorticity equation [9,26]: ∂ ∂r ∂ 2 ∂t 2 − ω 2 G (r) ∂ ∂r φ(r, t) = 0(1) This equation has been shown to describe the phase mixing [21,22,23,24,25], i.e. the cascade of energy from bigger to smaller spatial scales. We can recover the phase mixing by integrating Eq. 1 in space once, which yields the radial component of the electric field E(r, t) = E 0 exp(−iω G (r)t), and approximating for simplicity the frequency profile as linear in space: ω G (r) ≃ ω G0 + ω ′ G · (r − r 0 ) . Now it is sufficient to calculate the Fourier transform of the electric field in a radial region centered at r 0 , and with half-width ν, to obtain [28]: E(k r , t) = 2E 0 exp(−iω G0 t) exp(−ik r r 0 ) sin((ω ′ G t − k r )ν)/(ω ′ G t − k r )(2) which can be written, in the limit of radially localized perturbation, as: E(k r , t) = 2πE 0 exp(−iω G0 t) exp(−ik r r 0 )δ(k r − ω ′ G t)(3) As a first signature of the phase mixing, we observe the cascade of energy in Fourier space, in the form of a linear dependence of the wavenumber on time: k r ∝ ω ′ G t(4) As a second signature, we observe that the scalar potential dependence on time can be now estimated from E(r, t) = ik r (t)φ(r, t), obtaining for its amplitude the characteristic decay which takes the name of continuum damping: \|φ(r, t)\| ∝ (ω ′ G t) −1(5) Before proceeding further, we note that locally the phase-mixing does not dissipate the energy of the GAM, being the local total energy (ExB + thermodynamic) equal to dE(t) = ρ m \|v E (t = 0)\| 2 /2 dr, with ρ m being the mass density, and v E = cE/B 0 being the ExB drift. The investigation of the global properties of GAMs, including the radial propagation of energy, are out of the scope of this paper, and will be discussed separately. We now want to investigate the signatures of the phase mixing in the results of the numerical simulations in an equilibrium obtained from realistic values of parameters. To this purpose, we have selected a group of parameters defining the equilibrium of ASDEX Upgrade for the shot number 20787 [41], at the radial location where the GAM is detected (r ≃ 0.9 a), consistently with Ref. [27,28]. These parameters will be used for the simulations described in the rest of the paper. The magnetic equilibrium is characterized by a magnetic field on axis of B 0 = 2 T, the major and minor radii of R 0 = 1.65 m and a = 0.5 m, and a safety factor of q = 3. The plasma has a local deuterium and electron temperatures of T i = T e = 170 eV. We want to investigate the results of several simulations, with different temperature profiles with temperature gradients in the range k T a = a∇T /T = 1 − 16 at r=0.5 (density and safety factor profiles are kept flat for simplicity). The GAM characteristic radial size is estimated as λ GAM ≃ 10 cm. Starting from these values we obtain the following quantities: an ion cyclotron frequency Ω i = 0.96 · 10 8 rad/s, an ion thermal velocity v thi = 2T i /m i = 0.9 · 10 5 m/s and a Larmor radius ρ i = 1.33 · 10 −3 m (with ρ * = 1/530). We initialize an electric field with sinusoidal profile with k r a = 10π, equivalent to k r ρ i = 0.084. We note that the GAM frequency depends strongly on the plasma temperature (for the order of magnitude we have ω GAM ∼ c s /R ∝ √ T , whereas ω GAM does not depend on n and depends weakly on q in the considered regime of parameters), therefore we start by considering a simplified problem where the density and safety factor profiles are considered flat and the temperature profile has a radial dependence described as (see Fig. 1): T (r) T (r 0 ) = exp ∆ k T tanh r − r 0 ∆(6) where r 0 = 0.5 is the reference position in the simulation where the peak of the temperature gradient is located, and where the measurements are performed, k T = d ln(T )/dr is the value of the temperture gradient, and ∆ = 0.2 is the radial size of the temperature gradient peak (in this paper, electron and ion temperatures profiles are always considered equal). The results of these simulations are investigated, with a particular attention in this paper to the evolution of the radial shape of the perturbation (see Fig. 2). In order to isolate the physics of the phase mixing, we repeat the simulation with a dedicated initial value code, which evolves the electric field at each position with its local frequency: E ph−mix (r, t) = E(r, 0) exp(i ω G (r)t)(7) where the frequency is given at each position by the analytical theory, and depends on the local temperature value. No Landau damping is considered in this code. This initial value code gives the evolution in time of the radial structure of the perturbation, due to the pure phase-mixing. Effects of the Landau damping in modifying the radial structure are neglected in this code. The comparison of the results of ORB5 and the phase-mixing code are shown in Fig. 2. The first feature of the radial structure evolution, observed in both codes, is the increase of the wavenumber in time. This can be observed to occur in both ORB5 and in the phase-mixing code, with similar values. We conclude that the wavenumber calculated in gyrokinetic simulation is approximated to a good extent by the phase-mixing code. As a second remark, we note that the results of ORB5 show a damping in time at r=0.5, whereas the phase-mixing code does not show it, for construction. This damping will be investigated in the next Section. As a third remark, we note that the results of ORB5 show a slight displacement in time of the position of the peak near r=0.5. This second-order global effect due to the radial GAM propagation, which is not present in the phase-mixing code, will be investigated in a separate paper. As time goes on, we measure the wavenumbers at the center of the radial domain, for both results of ORB5 and of the phase-mixing code, and plot them in time. The result can be seen in Fig. 3, where the value of the analytical prediction given by the local approximation, Eq. 4, is also shown as a black line. One can see that the wavenumber measured in the numerical simulations is found to grow more slowly than the local analytical prediction on average. This is due to the local approximation, which does not consider the finite size of the temperature gradient peak. In fact the GAM perturbation experiences a lower temperature gradient away from the reference position r=0.5, and therefore the global effect is an average of the temperature gradient profile, rather than the value measured at r=0.5 only. In Fig. 3, the analytical prediction taking into account also this average to the first order is shown. One can see that the behaviour of the wavenumber measured with the numerical simulations is described for a longer time by the analytical prediction where the global effects are considered to the first order, and eventually higher order effects should be also taken into account. In conclusion, we observe that the phase mixing affects the numerical simulation and we can now extend this result to the combined effect of Landau damping and phase mixing. Combined effect of phase mixing and Landau damping We are now ready to investigate the combined effects of the phase mixing and Landau damping, which have been studied separately in the previous Sections. The same simulations as in Sec. 3 are considered, where the realistic conditions at the tokamak edge of AUG shot # 20787 taken in Ref. [41], are reproduced at the center of the radial domain of the simulations. As shown in Fig. 3, the radial shape of the electric field perturbation evolves in time with higher and higher wavenumber, consistently with Ref. [9,26]. In the same Figure, we can observe also that the value of the electric field at r=0.5 is lower and lower at each oscillation. In this Section, we want to investigate this damping with a proper theoretical model [28]. The values of q and k r ρ i allow us to treat the Landau damping with the analytical theory with first-order FOW effects (i.e. with a quadratic dependence on k r ρ i ). We can then write the damping rate due to the combined action of phase mixing and Landau damping (PL) as a function of time: γ P L (k T , t) = f (v T i , q, τ e ) + (k r0 + ω ′ G t) 2 g(v T i , q, τ e )(8) where f and g are the cofficients given in Ref. [16,17]. Consequently, the evolution of the electric field envelop is described at each time by: 1 E ∂E ∂t = γ P L (k T , t)(9) Equation 9 is solved with an initial-value code in order to calculate theoretically the evolution in time of the GAM electric field envelop. The frequency is also known from Ref. [16,17], and does not change in time to a good approximation. The comparison of the analytical prediction with the result of ORB5 is shown in Fig. 4. We note that the GAM decay is described to a good extent by the PL model, whereas the Landau damping alone greatly overestimates the GAM amplitude at large times. This is due to the fact that the damping rate, which is the Landau damping corresponding to the initial wavenumber at t=0, grows quadratically in time due to the evolution of the wavenumber (see Fig. 4). We also note that some discrepancy is present, due to global effects, such as the initial formation of the GAM eigenmode during the first oscillation, and the radial displacement of the GAM perturbation at later times. The analysis of these corrections due to radial propagation will be done in a separate paper. Figure 4: On the left, the damping rate as a function of time for a simulation with k T a = 10. On the right, electric field measured in the middle of the radial domain for the same simulation. The black line is measured in the gyrokinetic simulation, and it is compared with the red dashed line, given by the theory of the Landau damping alone, and with the blue line, given by the PL mechanism. We now investigate the dependence of the GAM decay on the value of the temperature gradient, with both gyrokinetic simulations and the analytical theory of the PL mechanism. In particular, the evolution in time of the electric field is measured at r=0.5 in gyrokinetic simulations with different values of k T ranging from 2 ≤ k T a ≤ 16. The time of half-decay is measured for each simulation and shown in Fig. 5, where it is compared with the analytical model of the PL mechanism. We observe that the decay time strongly decreases with the increase of the temperature gradient. The theory for the evolution of the envelop of the electric field, discussed in Sec. 4, describes well the strong damping with respect to the case k T a = 0 in which only Landau damping acts. We note that for large values of k T a, the half-decay time in the simulations is slightly larger than the damping expected by the theory, which is due to the radial propagation not investigated in this paper. Summary and conclusions. Understanding the linear damping mechanisms of GAMs is a crucial step for reaching a complete understanding of their dynamics, and in particular of their nonlinear interaction with turbulence in tokamaks. Landau damping has been historically recognized as the main linear damping mechanism of GAMs, due to their low frequency (for instance with respect to other higher frequency, less damped modes, like incompressible shear-Alfvén modes). Landau damping has been shown to depend on the radial shape of GAMs, in particular increasing with the local radial wavenumber. Moreover, the phase mixing has been predicted to affect GAMs, by means of analytical theory [9,26]. The main signature of phase mixing has been shown to be a cascade of the GAM energy from lower to higher values of the wavenumber. In previous works [27,28], the results of gyrokinetic simulations in the presence of a temperature gradient have been investigated, and in Ref. [28] the combined effect of the phase mixing and the Landau damping has been shown to generate a novel damping mechanism, responsible for a very efficient absorbption of the GAM energy by the bulk plasma [28]. This phase mixing / Landau damping (PL) mechanism, has been shown to increase up to an order of magnitude the effective damping of GAMs, for realistic tokamak conditions. In this work, we have investigated the details of the radial structure evolution in time of the GAM electric field in the presence of a temperature gradient. The cascade of the wavenumber in time has been investigated with gyrokinetic simulations and with a specific initial value code dedicated to the investigation of the phase-mixing. The radial structure of gyrokinetic simulations obtained with ORB5 has been shown to be approximated to a good extent by the one obtained with the initial value code. Second order effects, like the radial displacement of the GAM peak in time, have been neglected at this stage and have been left for a future work. Finally, consistently with Ref. [28], the combined effect of the phase-mixing and Landau damping has been investigated, and shown to decrease the half-decay time up to one order of magnitude, for realistic tokamak parameters, and using new simulations where the temperature gradient is peaked at the center of the radial domain. From these results, we deduce that the PL mechanism can play an important role in the suppression of GAM oscillations in the regions characterized by a strongly nonuniform temperature profile such as in the H and I modes. We note that simulations performed with a density gradient different from zero have given results very close to the simulations performed with a flat density profile. This is in agreement with Eq. 8 that does not depends on the density gradient. Moreover, we note that γ P L in Eq. 8 depends also on q. However, the gradient of q has a weak influence on the phase mixing and consequently on γ P L . This aspect has been verified by numerical simulations showing that the main parameter in the PL mechanism is the temperature gradient. In order to investigate and to quantify the importance of the PL damping mechanism on these regions, we compare it with the characteristic drive rate γ RD given by the nonlinear coupling with the ITG mode in Ref. [9], and discussed in Ref. [28]. The characteristic drive time can be calculated as t RD ≃ 1/γ RD , and should be compared with the PL decay time. If t P L < t RD , this means that the PL damping rate exceeds the energy transfer rate from the ITG turbulence to the GAM, and as a consequence we may expect no observation of GAMs. The drive time is found to be t RD < t s for the L-mode, t RD ∼ t s for the I-mode, and t RD ∼ 10 t s for the H-mode, with t s = 2 −1/2 R/v ti being the sound time unit. Therefore, we observe that the PL model explains the observation of GAMs for L and I modes, but a desappearance is predicted for the H-mode, for k T a > 10 (see Ref. [28] for the detailed calculation). We note that this analysis of orders of magnitude gives results which are consistent with experimental results that show the existence of GAM in I-mode regime in spite of the strong temperature gradient comparable to that of H-mode [29,30]. These results are also in agreement with the dynamics of GAM observed at the I-H transition [29]. As a future work, a more detailed description of a particular experimental shot will be done by using the micro-turbulence features observed in the experiment of AUG and comparing with linear and nonlinear gyrokinetic simulations where global effects are also investigated. typical simulations has a spatial grid of (r, θ, φ) = 1024 x 64 x 4 and a time step of 50 Ω −1 i . The number of ion markers has been chosen as N i = 10 8 . A set of poloidal harmonics going from m=-10 to m=10 has been selected by means a filter in Fourier space in the poloidal direction. Dirichlet boundary conditions on the potentials have been imposed at the outer boundary, and Neumann boundary conditions at the inner boundary. In a collisionless gyrokinetic simulation, the GAM amplitude is damped in time due to the combined effect of phase mixing and Landau damping, where the phase mixing creates a cascade in the radial wavenumber and the Landau damping effectively damps the higher wave numbers [28]. Ultimately, the energy of the GAM goes into the bulk ion microscopic kinetic energy due to the Landau damping. In this Section, this energy channel among the global GAM mode and the microscopic ion energy is investigated. In this model the rate of change of the particle kinetic energy must be equal to the power transfer from the particles to the field (power balance) (see Ref. [32]). The power balance not only gives an indication of the quality of the simulations, but also provides a measure of the instantaneous growth (decay) rate. Defining dE k /dt the rate of variation of kinetic energy and dE f /dt the power transfer, the relative error of the power balance is: δE rel = dE f /dt − dE k /dt dE f,max /dt(10) where dE f,max /dt is the maximum value of dE f /dt in the time interval of interest. In Fig. 6, the relative error of two simulations is shown, namely one with flat temperature profile, and one with k T a = 10. For this specific runs, that falls within 3% at the time of half decay. the European Enabling Research Project on "Verification and development of new algorithms for gyrokinetic codes", WP15-ER-01/IPP-01, and the European Enabling Research Project on "Micro-turbulence properties in the core of tokamak plasmas: close comparison between experimental observations and theoretical predictions", WP15-ER-01/IPP-02. Simulations were performed on the IFERC-CSC Helios supercomputer within the framework of the VERIGYRO and the ORBFAST project. Figure 1 : 1Temperature profile normalized at r=0.5 (left) and temperature gradient (right), for a simulation with k T a = 4 at r=0.5. Figure 2 : 2On the left, zoom of the electric field profile measured with ORB5 at 5 different times when the maximum at r=0.5 is reached, for a simulation with k T a = 4 at r=0.5. On the right, the same simulation performed with the phase-mixing initial value code. The blue line, labelled as "0", is the initial value of the simulations. Figure 3 : 3Evolution in time of the wavenumber measured with ORB5 (red crosses), predicted analytically in the local limit (black line) and with initial-value code with first order global effects (blue crosses). The linear fit of the effective wavenumber cascade is also shown with a blue dashed line. Figure 5 : 5Half-decay time for simulations with different temperature gradients. The results of numerical simulations are shown with blue crosses, and the theory with a red continuous line. Figure 6 : 6The relative error of the power balance, for the simulation with k T a = 10, in a time zoom corresponding to about 10 GAM oscillations. AcknowledgementsInteresting and useful discussions with G. Conway, E. Poli, P. Manz, Z. Qiu, B. Scott and AS-DEX Upgrade team are kindly acknowledged. 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[ "GLOBAL WELL-POSEDNESS OF REGULAR SOLUTIONS TO THE THREE-DIMENSIONAL ISENTROPIC COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DEGENERATE VISCOSITIES AND VACUUM", "GLOBAL WELL-POSEDNESS OF REGULAR SOLUTIONS TO THE THREE-DIMENSIONAL ISENTROPIC COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DEGENERATE VISCOSITIES AND VACUUM" ]	[ "Xin Zhouping ", "Shengguo And ", "Zhu " ]	[]	[]	In this paper, the Cauchy problem for the three-dimensional (3-D) isentropic compressible Navier-Stokes equations with degenerate viscosities is considered. By introducing some new variables and making use of the " quasi-symmetric hyperbolic"-"degenerate elliptic" coupled structure to control the behavior of the fluid velocity, we prove the global-in-time well-posedness of regular solutions with vacuum for a class of smooth initial data that are of small density but possibly large velocities. Here the initial mass density is required to decay to zero in the far field, and the spectrum of the Jacobi matrix of the initial velocity are all positive. The result here applies to a class of degenerate density-dependent viscosity coefficients, is independent of the BD-entropy, and seems to be the first on the global existence of smooth solutions which have large velocities and contain vacuum state for such degenerate system in three space dimensions.where I 3 is the 3 × 3 identity matrix, According to Liu-Xin-Yang [22], if we restrict the gas flow to be isentropic, such dependence is inherited through the laws of Boyle and Gay-Lussac:i.e., θ = AR −1 ρ γ−1 , and one finds that the viscosity coefficients are functions of the density. In this paper, we will focus on the cut-off inverse power force model whose viscosities have the forms shown in(1.4). The corresponding conclusion for the isentropic flow of the Sutherland's model will be shown in our forthcoming paper Xin-Zhu [31]. Throughout this paper, we adopt the following simplified notations, most of them are for the standard homogeneous and inhomogeneous Sobolev spaces:f (t, x) T,Ξ = ∇f (t, x) L ∞ ([0,T ]×R 3 ) + ∇ 2 f (t, x) L ∞ ([0,T ];H 2 (R 3 )) .A detailed study of homogeneous Sobolev spaces can be found in[8].	10.1016/j.aim.2021.108072	[ "https://arxiv.org/pdf/1806.02383v2.pdf" ]	102,351,812	1806.02383	9d35db7e1b86f139a88cae5c41190d82cf7df804	GLOBAL WELL-POSEDNESS OF REGULAR SOLUTIONS TO THE THREE-DIMENSIONAL ISENTROPIC COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DEGENERATE VISCOSITIES AND VACUUM 7 Apr 2019 Xin Zhouping Shengguo And Zhu GLOBAL WELL-POSEDNESS OF REGULAR SOLUTIONS TO THE THREE-DIMENSIONAL ISENTROPIC COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DEGENERATE VISCOSITIES AND VACUUM 7 Apr 2019 In this paper, the Cauchy problem for the three-dimensional (3-D) isentropic compressible Navier-Stokes equations with degenerate viscosities is considered. By introducing some new variables and making use of the " quasi-symmetric hyperbolic"-"degenerate elliptic" coupled structure to control the behavior of the fluid velocity, we prove the global-in-time well-posedness of regular solutions with vacuum for a class of smooth initial data that are of small density but possibly large velocities. Here the initial mass density is required to decay to zero in the far field, and the spectrum of the Jacobi matrix of the initial velocity are all positive. The result here applies to a class of degenerate density-dependent viscosity coefficients, is independent of the BD-entropy, and seems to be the first on the global existence of smooth solutions which have large velocities and contain vacuum state for such degenerate system in three space dimensions.where I 3 is the 3 × 3 identity matrix, According to Liu-Xin-Yang [22], if we restrict the gas flow to be isentropic, such dependence is inherited through the laws of Boyle and Gay-Lussac:i.e., θ = AR −1 ρ γ−1 , and one finds that the viscosity coefficients are functions of the density. In this paper, we will focus on the cut-off inverse power force model whose viscosities have the forms shown in(1.4). The corresponding conclusion for the isentropic flow of the Sutherland's model will be shown in our forthcoming paper Xin-Zhu [31]. Throughout this paper, we adopt the following simplified notations, most of them are for the standard homogeneous and inhomogeneous Sobolev spaces:f (t, x) T,Ξ = ∇f (t, x) L ∞ ([0,T ]×R 3 ) + ∇ 2 f (t, x) L ∞ ([0,T ];H 2 (R 3 )) .A detailed study of homogeneous Sobolev spaces can be found in[8]. Introduction The time evolution of the mass density ρ ≥ 0 and the velocity u = u (1) , u (2) , u (3) ⊤ of a general viscous isentropic compressible fluid occupying a spatial domain Ω ⊂ R 3 is governed by the following isentropic compressible Navier-Stokes equations (ICNS): ρ t + div(ρu) = 0, (ρu) t + div(ρu ⊗ u) + ∇P = divT. (1.1) Here, x = (x 1 , x 2 , x 3 ) ∈ Ω, t ≥ 0 are the space and time variables, respectively. In considering the polytropic gases, the constitutive relation, which is also called the equations of state, is given by P = Aρ γ , γ > 1, (1.2) where A > 0 is an entropy constant and γ is the adiabatic exponent. T denotes the viscous stress tensor with the form µ(ρ) = αρ δ , λ(ρ) = βρ δ , (1.4) for some constant δ ≥ 0, µ(ρ) is the shear viscosity coefficient, λ(ρ) + 2 3 µ(ρ) is the bulk viscosity coefficient, α and β are both constants satisfying α > 0, and 2α + 3β ≥ 0. (1.5) Let Ω = R 3 . We look for smooth solutions, (ρ(t, x), u(t, x)) to the Cauchy problem for (1.1)-(1.5) with the initial data and far field behavior: (ρ, u)\| t=0 = (ρ 0 (x) ≥ 0, u 0 (x)) for x ∈ R 3 , (1. 6) ρ(t, x) → 0, as \|x\| → ∞ for t ≥ 0. (1.7) In the theory of gas dynamics, the compressible Navier-Stokes equations can be derived from the Boltzmann equations through the Chapman-Enskog expansion, cf. Chapman-Cowling [3] and Li-Qin [18]. Under some proper physical assumptions, the viscosity coefficients and the heat conductivity coefficient κ are not constants but functions of the absolute temperature θ such as: µ(θ) =a 1 θ 1 2 F (θ), λ(θ) = a 2 θ 1 2 F (θ), κ(θ) = a 3 θ 1 2 F (θ) (1.8) for some constants a i (i = 1, 2, 3). Actually in [3] for the cut-off inverse power force model, if the intermolecular potential varies as r −a , where r is intermolecular distance, then in (1.8): F (θ) = θ b with b = 2 a ∈ [0, +∞). In particular, for Maxwellian molecules, a = 4 and b = 1 2 ; while for elastic spheres, a = ∞ and b = 0. As a typical model whose F is not a power function of θ, the Sutherland's model is well known where F (θ) = θ θ + s 0 , (s 0 > 0 : Sutherland's constant). (1.9) There is a lot of literature on the well-posedness of solutions to the problem (1.1)-(1.6) in multi-dimensional space. For 3-D constant viscous flow (δ = 0 in (1.4)) with inf x ρ 0 (x) > 0, it is well-known that the local existence of classical solutions has been obtained by a standard Banach fixed point argument by Nash [26], which has been extended to be a global one by Matsumura-Nishida [25] for initial data close to a nonvacuum equilibrium in some Sobolev space H s (s > 5 2 ). Hoff [11] studied the global weak solutions with strictly positive initial density and temperature for discontinuous initial data. However, these approaches do not work when inf x ρ 0 (x) = 0, which occurs when some physical requirements are imposed, such as finite total initial mass and energy in the whole space. The main breakthrough for the well-posedness of solutions with generic data and vacuum is due to Lions [27], where he established the global existence of weak solutions with finite energy to the isentropic compressible flow provided that γ > 9 5 (see also Feireisl-Novotný-Petzeltová [6] for the case γ > 3 2 ). However, the uniqueness problem of these weak solutions is widely open due to their fairly low regularities. Recently, Xin-Yan [30] proved that any classical solutions of viscous non-isentropic compressible fluids without heat conduction will blow up in finite time, if the initial data has an isolated mass group. For density-dependent viscosities (δ > 0 in (1.4)) which degenerate at vacuum, system (1.1) has received extensive attentions in recent years. In this case, the strong degeneracy of the momentum equations in (1.1) near vacuum creats serious difficulties for well-posedness of both strong and weak solutions. A mathematical entropy function was proposed by Bresch-Desjardins [1] for λ(ρ) and µ(ρ) satisfying the relation λ(ρ) = 2(µ ′ (ρ)ρ − µ(ρ)), (1.10) which offers an estimate µ ′ (ρ)∇ √ ρ ∈ L ∞ ([0, T ]; L 2 (R d )) provided that µ ′ (ρ 0 )∇ √ ρ 0 ∈ L 2 (R d ) for any d ≥ 1. This observation plays an important role in the development of the global existence of weak solutions with vacuum for system (1.1) and some related models, see Bresh-Dejardins [2], Li-Xin [17], Mellet-Vassuer [24] and some other interesting results, c.f. [14,32]. However, the regularities and uniquness of such weak solutions remain open especially in multi-dimensional cases. In this paper, we study the global well-posedness of regular solutions to the system (1.1) with density-dependent viscosities given in (1.4) and initial data such that inf x ρ 0 (x) = 0. Then the analysis of the degeneracies in momentum equations (1.1) 2 requires some special attentions. The major concerns are: (1) The degeneracy of time evolution in momentum equations (1.1) 2 . Note that the leading coefficient of u t in momentum equations vanishes at vacuum, and this leads to infinitely many ways to define velocity (if it exits) when vacuum appears. Mathematically, this degeneracy leads to a difficulty that it is hard to find a reasonable way to extend the definition of velocity into vacuum region. For constant viscosities, a remedy was suggested by Cho-Choe-Kim (for example [4]), where they imposed initially a compatibility condition −divT 0 + ∇P (ρ 0 ) = √ ρ 0 g, for some g ∈ L 2 (R 3 ), which, is roughly equivalent to the L 2 -integrability of √ ρu t (t = 0), and plays a key role in deducing that ( √ ρu t , ∇u t ) ∈ L ∞ ([0, T * ]; L 2 (R 3 )) for a short time T * > 0. Then they established successfully the local well-posedness of smooth solutions with non-negative density in R 3 , which, recently, has been shown to be a global one with small energy but large oscillations by Huang-Li-Xin [12] in R 3 and Li-Xin [16] in R 2 . (2) The strong degeneracy of the elliptic operator divT caused by vacuum for δ > 0. In [4,12] for δ = 0, the uniform ellipticity of the Lamé operator L defined by Lu = −α△u − (α + β)∇divu plays an essential role in the high order regularity estimates on u. One can use standard elliptic theory to estimate \|u\| D k+2 by the D k -norm of all other terms in momentum equations. However, for δ > 0, viscosity coefficients vanish as density function connects to vacuum continuously. This degeneracy makes it difficult to adapt the approach in the constant viscosity case in [4,12] to the current case. (3) The strong nonlinearity for the variable coefficients of the viscous term due to δ > 0. It should be pointed out here that unlike the case of constant viscosities, despite the weak regularizing effect on solutions, the elliptic part divT will also cause some troubles in the high order regularity estimates. For example, to establish some uniform a priori estimates independent of the lower bound of density in H 3 space, the key is to handle the extra nonlinear terms such as div ∇ k ρ δ S(u) for S(u) = α(∇u + (∇u) ⊤ ) + βdivuI 3 , where k = 0, 1, 2, 3. Therefore, much attentions need to be paid in order to control these strong nonlinearities, especially for establishing the global well-posedenss of classical solutions. Recently, there have some interesting works to overcome these difficulties mentioned above. In Li-Pan-Zhu [20] for the case δ = 1, the degeneracies of the time evolution and the viscosity can be transferred to the possible singularity of the special source term ∇ρ ρ · S(u). Based on this observations, via establishing a uniform a priori estimates in L 6 ∩ D 1 ∩ D 2 space for the quantity ∇ρ ρ , the existence of the unique local classical solution in 2-D space (see also Zhu [33] for 3-D case) to (1.1) has been obtained under the assumptions ρ 0 (x) → 0, as \|x\| → ∞, which also applies to the 2-D shallow water equations. However, this result only allows vacuum at the far field, and the corresponding problem with vacuum appearing in some open sets, or even at a single point is still unsolved. Later, via introducing a proper class of solutions and establishing the uniform weighted a priori estimates for the higher order term ρ δ−1 2 ∇ 4 u, the same authors [21,34] gave the existence of 3-D local classical solutions to system (1.1) for the cases 1 < δ ≤ min{3, (γ + 1)/2}. Some other interesting results can also be seen in Ding-Zhu [5] and Li-Pan-Zhu [19]. 1.1. Symmetric formulation. In order to deal with the issues mentioned above, we first need to analyze the structure of the momentum equations (1.1) 2 carefully, which can be decomposed into hyperbolic, elliptic and source parts as follows: ρ u t + u · ∇u + ∇P Hyperbolic = −ρ δ Lu Elliptic + ∇ρ δ · S(u) Source . (1.11) For smooth solutions (ρ, u) away from vacuum, these equations could be written into u t + u · ∇u + Aγ γ − 1 ∇ρ γ−1 − δ δ − 1 ∇ρ δ−1 · S(u) Principal order = −ρ δ−1 Lu Higher order . (1.12) Note that if ρ is smooth enough, one could pass to the limit as ρ → 0 on both sides of (1.12) and formally have u t + u · ∇u = 0 as ρ(t, x) = 0. (1.13) So (1.12)-(1.13) indicate that the velocity u could be governed by a nonlinear degenerate parabolic system when vacuum appears in some open sets or at the far field. Recently, in Geng-Li-Zhu [9], via introducing two new quantities: ϕ = ρ δ−1 2 , and φ = 4Aγ (γ − 1) 2 ρ γ−1 2 , system (1.1) can be rewritten into a system that consists of a transport equation for ϕ, and a "quasi-symmetric hyperbolic"-"degenerate elliptic" coupling system for U = (φ, u) ⊤ :                    ϕ t + u · ∇ϕ + δ − 1 2 ϕdivu = 0 Transport equations , U t + 3 j=1 A j (U )∂ j U Symmetric hyperbolic = −ϕ 2 L(u) Degenerate elliptic + H(ϕ) · Q(u) First order source , (1.14) where ∂ j U = ∂x j U , A j (U ) =   u (j) γ−1 2 φe j γ−1 2 φe ⊤ j u (j) I 3   , j = 1, 2, 3, L(u) = 0 Lu , H(ϕ) = 0 ∇ϕ 2 , Q(u) = 0 0 0 Q(u) , Q(u) = δ δ − 1 S(u),(1.15) and e j = (δ 1j , δ 2j , δ 3j ) (j = 1, 2, 3) is the Kronecker symbol satisfying δ ij = 1, when i = j and δ ij = 0, otherwise. Based on this reformulation, a local well-posedness for arbitrary δ > 1 in some non-homogenous Sobolev spaces has been obtained for the compressible degenerate viscous flow in [9], which could be shown as follows: Theorem 1.1. [9] Let γ > 1 and δ > 1. If initial data (ρ 0 , u 0 ) satisfy ρ 0 ≥ 0, ρ γ−1 2 0 , ρ δ−1 2 0 , u 0 ∈ H 3 ,(1.(A) ρ ≥ 0, ρ δ−1 2 ∈ C([0, T * ]; H 3 ), ρ γ−1 2 ∈ C([0, T * ]; H 3 ); (B) u ∈ C([0, T * ]; H s ′ ) ∩ L ∞ ([0, T * ]; H 3 ), ρ δ−1 2 ∇ 4 u ∈ L 2 ([0, T * ]; L 2 ), (C) u t + u · ∇u = 0 as ρ(t, x) = 0, where s ′ ∈ [2, 3) is an arbitrary constant. 1.2. Singularity formation. We consider first whether the local regular solution in Theorem 1.1 can be extended globally in time. In contrast to the classical theory for the constant viscosity case, we show the following somewhat surprising phenomenon that such an extension is impossible if the velocity field decays to zero as t → +∞ and the initial total momentum is non-zero. More pricisely, let P(t) = ρu (total momentum). Theorem 1.2. Let 1 < min{γ, δ} ≤ 2. Assume \|P(0)\| > 0. Then there is no global regular solution (ρ, u) obtained in Theorem 1.1 satisfying the following decay (ρ 0 , u 0 )(x), if V satisfies    ρ 0 (x) = 0, ∀ x ∈ V ; Sp(∇u 0 ) ∩ R − = ∅, ∀ x ∈ V, (1.18) where Sp(∇u 0 (x)) denotes the spectrum of the matrix ∇u 0 (x). By exploring the hyperbolic structure in (1.1) in vacuum region, one can confirm exactly that hyperbolic singularity set does generate singularities from local regular solutions in finite time. Theorem 1.3. Let γ > 1 and δ > 1. If the initial data (ρ 0 , u 0 )(x) have a non-empty hyperbolic singularity set V , then the regular solution (ρ, u)(t, x) on R 3 × [0, T m ] obtained in Theorem 1.1 with maximal existence time T m blows up in finite time, i.e., T m < +∞. 1.3. Global-in-time well-posedness of smooth solutions. We now come to the major task to construct global smooth solutions for the system (1.1). Due to Theorems 1.2-1.3, one needs to identify a class of initial data and a proper energy space to avoid the two singularity mechanisms shown above. Let u = ( u (1) , u (2) , u (3) ) ⊤ be the solution in [0, T ] × R 3 of the following Cauchy problem: u t + u · ∇ u = 0, u(t = 0, x) = u 0 (x) (1.19) for x ∈ R 3 . We give the definition of regular solutions considered in this paper: Definition 1.2. Let T > 0 be a finite constant. A solution (ρ, u) to the Cauchy problem (1.1)-(1.7) is called a regular solution in [0, T ] × R 3 if (ρ, u) satisfies this problem in the sense of distribution and: (A) ρ ≥ 0, ρ δ−1 2 , ρ γ−1 2 ∈ C([0, T ]; H s ′ loc ) ∩ L ∞ ([0, T ]; H 3 ); (B) u − u ∈ C([0, T ]; H s ′ loc ) ∩ L ∞ ([0, T ]; H 3 ), ρ δ−1 2 ∇ 4 u ∈ L 2 ([0, T ]; L 2 ); (C) u t + u · ∇u = 0 as ρ(t, x) = 0, where s ′ ∈ [2, 3) is an arbitrary constant. The main result on the global well-posedness of regular solutions to the three-dimensional isentropic compressible Navier-Stokes equations with degenerate viscosities and vacuum could be stated as follows. (1.20) and any one of the following conditions (P 1 )-(P 3 ): Theorem 1.4. Let parameters (γ, δ, α, β) satisfy γ > 1, δ > 1, α > 0, 2α + 3β ≥ 0,(P 1 ) 2α + 3β = 0; (P 2 ) δ ≥ 2γ − 1; (P 3 ) δ = γ. If the initial data (ρ 0 , u 0 ) satisfy (A 1 ) ρ 0 ≥ 0 and ρ γ−1 2 0 3 + ρ δ−1 2 0 3 ≤ D 0 (γ, δ, α, β, A, κ, u 0 Ξ ), (A 2 ) u 0 ∈ Ξ and there exists a constant κ > 0 such that , Dist Sp(∇u 0 (x)), R − ≥ κ for all x ∈ R 3 , where D 0 > 0 is some constant depending on (α, β, δ, A, γ, κ, u 0 Ξ ), then for any T > 0, there exists a unique regular solution (ρ, u) in [0, T ] × R 3 to the Cauchy problem (1.1)-(1.7). Particularly, when condition (P 2 ) holds, the smallness assumption on ρ δ−1 2 0 could be removed. Moreover, if 1 < min(γ, δ) ≤ 3, (ρ, u) is a classical solution to the Cauchy problem (1.1)-(1.7) in [0, T ] × R 3 . Remark 1.1. The conditions (A 1 )-(A 2 ) identify a class of admissible initial data that provides unique solvability to the Cauchy problem (1.1)-(1.7). Such initial data contain the following examples: ρ 0 (x) = ǫ 1 (1 + \|x\|) 2σ 1 , ǫ 1 g 2σ 2 (x), ǫ 1 exp{−x 2 }, ǫ 1 \|x\| (1 + \|x\|) 2σ 3 , where ǫ 1 > 0 is a sufficiently small constant, 0 ≤ g(x) ∈ C 3 c (R 3 ), and σ 1 > 3 2 max 1 δ − 1 , 1 γ − 1 , σ 2 > 3 max 1 δ − 1 , 1 γ − 1 , σ 3 > 3 2 max 1 δ − 1 , 1 γ − 1 + 1 2 ; u 0 = Ax + b + ǫ 2 f (x) , where A is a 3 × 3 constant matrix whose eigenvalues are all greater than 2κ, ǫ 2 > 0 is a sufficiently small constant, f ∈ Ξ and b ∈ R 3 is a constant vector. Remark 1.2. It is worth pointing out that, for any γ > 1, even 2α+ 3β = 0, that is to say (P 1 ) fails, we can still deal with the corresponding well-posedness problem for a relatively wide class of (α, β, δ) under the following condition: (P 0 ) 0 < M 1 = 2α+3β 2α+β < 3 2 − 1 δ and M 2 = −3δ + 1 + 1 2 (δ−1) 2 4(2α+β) + 4δ 2 (2α+β) (δ−1) 2 M 2 1 + 2M 1 δ < −1, which can be seen in §5. Next we indicate that the set of parameters (α, β, δ) satisfying (P 0 ) must be non-empty. First, for fixed δ > 1, let α = a 1 η, β = a 2 η, where η > 0, a 1 > 0 and 2a 1 + 3a 2 ≥ 0 are all constants. Thus M 1 = 2a 1 +3a 2 2a 1 +a 2 . Then one can adjust the values of a 1 and a 2 to ensure that M 1 < 3 2 − 1 δ holds. Second, let x = (δ − 1) 2 /(4(2a 1 + a 2 )η), and consider the following function: F (x) = x + 1 x δ 2 M 2 1 + 2M 1 δ − 6δ + 4, whose minimum value is F (δM 1 ) = 4M 1 δ − 6δ + 4 < 0 due to M 1 < 3 2 − 1 δ . Thus, one needs only to choose η to ensure that x belongs to a small neighbor of δM 1 and F < 0, which is equivalent to M 2 < −1. Remark 1.3. Without generality, we can assume that u 0 (0) = 0, which can be achieved by the following Galilean transformation: t ′ = t, x ′ = x + u 0 (0)t, ρ ′ (t ′ , x ′ ) = ρ(t, x), u ′ (t ′ , x ′ ) = u(t, x) − u 0 (0). The above well-posedness theory is still available for some other models such as the ones shown in the following two theorems. Theorem 1.5. Let the viscous stress tensor T in (1.1) be given by T = ρ δ (2α∇u + βdivuI 3 ). Then the same well-posedness theory as in Theorem 1.4 holds in this case. The rest of the paper is organized as follows: In §2, we show some decay estimates for the classical solutions of the multi-dimensional Burgers equations and the global wellposedenss of an ordinary differential equation, which will be used later. In §3, we prove the finite time singularity formation in Theorems 1.2-1.3. §4- §8 are devoted to establishing the global well-posedness of regular solutions stated in Theorem 1.4. We start with the reformulation of the original problem (1.1)-(1.7) as (4.1) in terms of the new variables, and establish the local-in-time well-posedness of smooth solutions to (4.1) in the case that the initial density is compactly supported in §4. Note that one cannot apply the result in [9] directly here, the initial velocity here does not have a uniform upper bound in the whole space and stays in a homogenous sobolev space. It is also worth pointing out that recently, Li-Wang-Xin [15] prove that classical solutions with finite energy to the Cauchy problem of the compressible Navier-Stokes systems with constant viscosities do not exist in general inhomogeneous Sobolev space for any short time, which indicates in particular that the homogeneous Sobolev space is crucial as studying the well-posedness (even locally in time) for the Cauchy problem of the compressible Navier-Stokes systems in the presence of vacuum. This local smooth solution to (4.1) is shown to be global in time in §5- §6 by deriving a uniform (in time) a priori estimates independent of the size of the initial density's support through energy methods based on suitable choice of time weights. Here, we have employed some arguments due to Grassin [10] and Serre [28] to deal with the nonlinear convection term u · ∇u. Next, the assumption that the initial density has compact support is removed in §7. Finally, in §8, the proof of Theorem 1.4 is completed by making use of the results obtained in §7. Furthermore, we give an appendix to list some lemmas that are used in our proof, and outline some proofs of properties shown in §2 and some inequalities used frequently in this paper. Preliminary This section will be devoted to show some decay estimates for the classical solutions of the multi-dimensional Burgers equations and the global well-posedenss of an ordinary differential equation, which will be used frequently in our proof. First, let u be the solution to the problem (1.19) in d-dimensional space. Then, along the particle path X(t; x 0 ) defined as d dt X(t; x 0 ) = u(t, X(t; x 0 )), x(0; x 0 ) = x 0 , (2.1) u is a constant in t: u(t, X(t; x 0 )) = u 0 (x 0 ) and ∇ u(t, X(t; x 0 )) = I d + t∇u 0 (x 0 ) −1 ∇u 0 (x 0 ). Based on this observation, one can have the following decay estimates of u, which play important roles in establishing the global existence of the smooth solution to the problem considered in this paper, and their proof could be found in [10] or the appendix. Proposition 2.1. [10] Let m > 1 + d 2 . Assume that ∇u 0 ∈ L ∞ (R d ), ∇ 2 u 0 ∈ H m−1 (R d ), and there exists a constant κ > 0 such that for all x ∈ R d , Dist Sp(∇u 0 (x)), R − ≥ κ, then there exists a unique global classical solution u to the problem (1.19), which satisfies (1) ∇ u(t, x) = 1 1+t I d + 1 (1+t) 2 K(t, x), for all x ∈ R d , t ≥ 0; (2) ∇ l u(·, t) L 2 (R d ) ≤ C 0,l (1 + t) d 2 −(l+1) , for 2 ≤ l ≤ m + 1; (3) ∇ 2 u(·, t) L ∞ (R d ) ≤ C 0 (1 + t) −3 ∇ 2 u 0 L ∞ (R d ) , where the matrix K(t, x) = K ij : R + × R d → M d (R d ) satisfies K L ∞ (R + ×R d ) ≤ C 0 1 + κ −d ∇u 0 d−1 L ∞ (R d ) . Here, C 0 is a constant depending only on m, d, κ and u 0 , and C 0,l are all constants depending on C 0 and l. Moreover, if u 0 (0) = 0, then it holds that \| u(t, x)\| ≤ \|∇ u\| ∞ \|x\|, for any t ≥ 0. (2.2) Secondly, we give a global well-posedness to the Cauchy problem of an ordinary differential equation: Proposition 2.2. For the constants b, C i and D i (i = 1, 2) satisfying a > 1, D 1 − (a − 1)b < −1, D 2 < −1, C i ≥ 0, for i = 1, 2, (2.3) there exists a constant Λ such that there exists a global smooth solution to the following Cauchy problem      dZ dt (t) + b 1 + t Z(t) = C 1 (1 + t) D 1 Z a (t) + C 2 (1 + t) D 2 Z, Z(x, 0) = Z 0 < Λ. (2.4) Its proof can be found in the appendix. For simplicity, we use the following notations. For matrices A 1 , A 2 , A 3 , B = (b ij ) = (b 1 , b 2 , b 3 ), a vector N = (n 1 , n 2 , n 3 ) ⊤ , set A = (A 1 , A 2 , A 3 ),              divA = 3 j=1 ∂ j A j , W · BW = 3 i,j=1 b ij w i w j , \|B\| 2 2 = B : B = 3 i,j=1 b 2 ij , N · B = n 1 b 1 + n 2 b 2 + n 3 b 3 . (2.5) Singularity formation In order to prove Theorem 1.2, we define: m(t) = ρ (total mass), E k (t) = 1 2 ρ\|u\| 2 (total kinetic energy). Then the regular solution (ρ, u)(t, x) in [0, T ] × R 3 defined in Theorem 1.1 has finite mass m(t), momentum P(t) and kinetic energy E k (t). Indeed, due to 1 < γ ≤ 2, m(t) = ρ ≤ C φ 2 γ−1 ≤ C\|φ\| 2 2 < +∞, which, together with the regularity shown in Theorem 1.1, implies that P(t) = ρu ≤ \|ρ\| 2 \|u\| 2 < +∞, E k (t) = 1 2 ρ\|u\| 2 ≤ C\|ρ\| ∞ \|u\| 2 2 < +∞. (3.1) The case for 1 < δ ≤ 2 can be verified similarly. Next it is shown that the total mass and momentum are conserved. P(t) = P(0), m(t) = m(0), for t ∈ [0, T ]. Proof. The momentum equations imply that P t = − div(ρu ⊗ u) − ∇P + divT = 0, (3.2) where one has used the fact that ρu (i) u (j) , ρ γ and ρ δ ∇u ∈ W 1,1 (R 3 ), for i, j = 1, 2, 3, due to the regularities of the solutions. Similarly, one can also get the conservation of the total mass. Now we are ready to prove Theorem 1.2. It follows from Lemma 3.1 that \|P(0)\| ≤ ρ(t, x)\|u\|(t, x) ≤ √ 2m 1 2 (t)E 1 2 k (t) = √ 2m 1 2 (0)E 1 2 k (t),(3.3) which yields that there exists a unique positive lower bound for E k (t), E k (t) ≥ \|P(0)\| 2 2m(0) > 0 for t ∈ [0, T ]. (3.4) Thus one gets that C 0 ≤ E k (t) ≤ 1 2 m(0)\|u(t)\| 2 ∞ for t ∈ [0, T ]. Obviously, that there exists a positive constant C u such that \|u(t)\| ∞ ≥ C u for t ∈ [0, T ]. Then Theorem 1.2 follows. Finally, we prove Theorem 1.3. It follows from the definition of regular solutions given in Theorem 1.2 that in the vacuum domain, the velocity satisfies u t + u · ∇u = 0, which, along with the formula ∇u(t, X(t; x 0 )) = I d + t∇u 0 (x 0 ) −1 ∇u 0 (x 0 ) and (1.18), yields the desired conclusion. Local-in-time well-posedness with compactly supported density The rest of this paper is devoted to proving Theorem 1.4. In this section, we first reformulate the original Cauchy problem (1.1)-(1.7) as (4.1) below in terms of some variables, and then establish the local well-posedness of the smooth solutions to (4.1) in the case that the initial density has compact support. Let u be the unique classical solution to (1.19) obtained in Proposition 2.1. In terms of the new variables (ϕ, W = (φ, w = u − u)) = ρ δ−1 2 , 4Aγ (γ − 1) 2 ρ γ−1 2 , u − u with w = (w (1) , w (2) , w (3) ) ⊤ , the Cauchy problem (1.1)-(1.7) can be reformulated into                        ϕ t + (w + u) · ∇ϕ + δ − 1 2 ϕdiv(w + u) = 0, W t + 3 j=1 A * j (W, u)∂ j W + ϕ 2 L(w) = H(ϕ) · Q(w + u) + G(W, ϕ, u), (ϕ, W )\| t=0 = (ϕ 0 , W 0 ) = (ϕ 0 , φ 0 , 0), x ∈ R 3 , (ϕ, W ) = (ϕ, φ, w) → (0, 0, 0) as \|x\| → ∞ for t ≥ 0, (4.1) where A * j (W, u) =   w (j) + u (j) γ−1 2 φe j γ−1 2 φe ⊤ j (w (j) + u (j) )I 3   , j = 1, 2, 3, G(W, ϕ, u) = − B(∇ u, W ) − D(ϕ 2 , ∇ 2 u), B(∇ u, W ) =   γ−1 2 φdiv u (w · ∇) u   , D(ϕ 2 , ∇ 2 u) = 0 ϕ 2 L u ,(4.2) and L, H and Q are given in (1.15). The main result in this section can be stated as follows: Theorem 4.1. Let (1.20) hold. If initial data (ϕ 0 , φ 0 , u 0 ) satisfy (A 1 ) ϕ 0 ≥ 0, φ 0 ≥ 0 and ϕ 0 , φ 0 ∈ H 3 ; (A 2 ) u 0 ∈ Ξ and there exists a constant κ > 0 such that for all x ∈ R 3 : Dist Sp(∇u 0 (x)), R − ≥ κ; (A 3 ) ϕ 0 and φ 0 are both compactly supported: supp x ϕ 0 = supp x φ 0 ⊂ B R ; where B R is the ball centered at the origin with radius R > 0, then there exist a time T * = T * (α, β, A, γ, δ, ϕ 0 , W 0 ) > 0 independent of R and a unique classical solution (ϕ, φ, w) in [0, T * ] × R 3 to (4.1) satisfying (ϕ, φ) ∈ C([0, T * ]; H 3 ), w ∈ C([0, T * ]; H s ′ ) ∩ L ∞ ([0, T * ]; H 3 ), ϕ∇ 4 w ∈ L 2 ([0, T * ]; L 2 ), for any constant s ′ ∈ [2, 3). Moreover, for t ∈ [0, T * ], ϕ(t, z(t; ξ 0 )) = φ(t, z(t; ξ 0 )) = 0, and w(t, z(t; ξ 0 )) = 0 for ξ 0 ∈ R 3 /suppϕ 0 , (4.3) where the curve z(t; ξ 0 ) is given via d dt z(t; ξ 0 ) = (w + u)(t, z(t; ξ 0 )), z(0; ξ 0 ) = ξ 0 . (4.4) The next three subsections will be devoted to prove Theorem 4.1. 4.1. Uniform a priori estimates for the linear problem. In order to show the local well-posedenss for (4.1), we will consider the following linearized approximate problem:                        ϕ t + (v + u N ) · ∇ϕ + δ − 1 2 hdiv(v + u) = 0, W t + 3 j=1 A * j (V, u N )∂ j W + (ϕ 2 + η 2 )L(w) = H(ϕ) · Q(v + u) + G(W, ϕ, u), (ϕ, W )\| t=0 = (ϕ 0 , W 0 ) = (ϕ 0 , φ 0 , 0), x ∈ R 3 , (ϕ, W ) = (ϕ, φ, w) → (0, 0, 0) as \|x\| → ∞ for t ≥ 0, (4.5) where A * j (V, u N ) =   v (j) + u N (j) γ−1 2 φe j γ−1 2 φe ⊤ j v (j) + u N (j) I 3   , j = 1, 2, 3, (4.6) with the vector u N = uF (\|x\|/N ). F (x) ∈ C ∞ c (R 3 ) is a truncation function satisfying 0 ≤ F (x) ≤ 1, and F (x) =    1 if \|x\| ≤ 1, 0 if \|x\| ≥ 2,(4.7) and N ≥ 1 is a sufficiently large constant. η > 0 is a constant, and V = (ψ, v) ⊤ . (h, ψ) are both known functions and v = (v (1) , v (2) , v (3) ) ⊤ ∈ R 3 is a known vector satisfying: (h, ψ, v)(0, x) = (ϕ 0 , φ 0 , 0), h ∈ C([0, T ]; H 3 ), ψ ∈ C([0, T ]; H 3 ), v ∈ C([0, T ]; H s ′ ) ∩ L ∞ ([0, T ]; H 3 ), h∇ 4 v ∈ L 2 ([0, T ]; L 2 ), (4.8) for any constant s ′ ∈ [2, 3). We also assume that for t ∈ [0, T * ], (4.5) can be obtained by the standard theory [4,7] at least when 0 < η < +∞ and 1 ≤ N < +∞. ψ(t, X(t; ξ 0 )) = h(t, X(t; ξ 0 )) = 0 and v(t, X(t; ξ 0 )) = 0 for ξ 0 ∈ R 3 /suppϕ 0 . (4.9) Now the following global well-posedness in [0, T ]×R 3 of a classical solution (ϕ N η , W N η ) = (ϕ N η , φ N η , w N η ) to(ϕ N η , W N η ) in [0, T ] × R 3 to (4.5) satisfying (ϕ N η , φ N η ) ∈ C([0, T ]; H 3 ), w N η ∈ C([0, T ]; H 3 ) ∩ L 2 ([0, T ]; H 4 ). (4.10) Next we give some a priori estimates for solutions (ϕ N η , φ N η , w N η ) in H 3 in the following Lemmas 4.2-4.3, which are independent of (R, N, η). For simplicity, we denote (ϕ N η , φ N η , w N η ) as (ϕ, φ, w), and W N η as W in the rest of Subsection 4.1. For this purpose, we fix a positive constant c 0 large enough such that 2 + ϕ 0 3 + φ 0 3 + u 0 Ξ ≤ c 0 , (4.11) and sup 0≤t≤T * h(t) 2 3 + ψ(t) 2 3 + v(t) 2 2 ) + ess sup 0≤t≤T * \|v(t)\| 2 D 3 + T * 0 \|h∇ 4 v\| 2 2 dt ≤ c 2 1 , (4.12) for some constant c 1 ≥ c 0 > 1 and time T * ∈ (0, T ), which will be determined later (see (4.23)) and depend only on c 0 and the fixed constants (α, β, γ, A, δ, T ). In the rest of this section, C ≥ 1 will denote a generic positive constant depending only on fixed constants (α, β, γ, A, δ, T ), but independent of (R, N, η), which may be different from line to line. It follows from the proof of Proposition 2.1 that u T,Ξ ≤ Cc 4 0 . (4.13) Based on this fact, one can establish the following estimates for ϕ. Lemma 4.2. Let (ϕ, W ) be the unique classical solution to (4.5) in [0, T ] × R 3 . Then 1 + ϕ(t) 2 3 ≤Cc 2 0 for 0 ≤ t ≤ T 1 = min(T * , c −5 1 ). Proof. Applying ∇ k (0 ≤ k ≤ 3) to (4.5) 1 , multiplying both sides by ∇ k ϕ, and integrating over R 3 , one gets 1 2 d dt \|∇ k ϕ\| 2 2 ≤ C\|div(v + u N )\| ∞ \|∇ k ϕ\| 2 2 + CΛ k 1 \|∇ k ϕ\| 2 + CΛ k 2 \|∇ k ϕ\| 2 , (4.14) where Λ k 1 =\|∇ k ((v + u N ) · ∇ϕ) − (v + u N ) · ∇ k+1 ϕ\| 2 , Λ k 2 = \|∇ k (hdiv(v + u))\| 2 . It follows from Lemma 10.1 and Hölder's inequality that \|Λ k 1 \| 2 ≤C u T,Ξ + v 3 ϕ(t) 3 , \|Λ k 2 \| 2 ≤C u T,Ξ + v 3 h(t) 3 + C\|h∇ 4 v\| 2 ,(4.15) where one has used Proposition 2.1 and the fact that for 0 ≤ t ≤ T * , \| u(t, x)\| ≤ 2N \|∇ u\| ∞ and \|∇F (\|x\|/N )\| ≤ CN −1 for N ≤ \|x\| ≤ 2N. (4.16) Then, it follows from (4.14)-(4.15), Gronwall's inequality and (4.12) that ϕ(t) 3 ≤ ϕ 0 3 + c 5 1 t + c 1 t 1 2 exp(Cc 4 1 t) ≤ Cc 0 for 0 ≤ t ≤ T 1 = min{T * , c −5 1 }. Lemma 4.3. Let (ϕ, W ) be the unique classical solution to (4.5) in [0, T ] × R 3 . Then W (t) 2 3 + t 0 \|ϕ∇ 4 u\| 2 2 ds ≤Cc 2 0 for 0 ≤ t ≤ T 2 = min{T 1 , c −10 1 }. Proof. Applying ∇ k to (4.5) 2 , multiplying both sides by ∇ k W , and integrating over R 3 by parts, we have 1 2 d dt \|∇ k W \| 2 + α\| ϕ 2 + η 2 ∇ k+1 w\| 2 2 + (α + β)\| ϕ 2 + η 2 div∇ k w\| 2 2 = (∇ k W ) ⊤ divA * j (V, u N )∇ k W − ∇ϕ 2 · ∇ k S(w) · ∇ k w − 3 j=1 ∇ k (A * j (V, u N )∂ j W − A * j (V, u N )∂ j ∇ k W · ∇ k W + − ∇ k ((ϕ 2 + η 2 )Lw) + (ϕ 2 + η 2 )L∇ k w + ∇ϕ 2 · Q(∇ k v) · ∇ k w + ∇ k (∇ϕ 2 · Q(v + u)) − ∇ϕ 2 · Q(∇ k (v + u)) · ∇ k w + ∇ϕ 2 · Q(∇ k u) · ∇ k w + ∇ k G(W, ϕ, u) · ∇ k W ≡ 8 i=1 I i . (4.17) Now consider the terms on the right-hand side of (4.17). It follows from Lemmas 10.1 and 4.2, Proposition 2.1, Hölder's and Young's inequalities, (4.13) and (4.16) that I 1 = (∇ k W ) ⊤ divA * j (V, u N )∇ k W ≤ C(\|∇V \| ∞ + \|∇ u\| ∞ )\|∇ k W \| 2 2 ≤ Cc 4 1 \|∇ k W \| 2 2 , I 2 = − ∇ϕ 2 · S(∇ k w) · ∇ k w ≤C\|∇ϕ\| ∞ \|ϕ∇ k+1 w\| 2 \|∇ k w\| 2 ≤ α 20 \|ϕ∇ k+1 w\| 2 2 + Cc 2 0 \|∇ k w\| 2 2 , I 3 = − 3 j=1 ∇ k (A * j (V, u N )∂ j W ) − A * j (V, u N )∂ j ∇ k W ∇ k W ≤C(\|∇V \| ∞ + \|∇ u\| ∞ )\|∇W \| 2 2 ≤ Cc 4 1 \|∇W \| 2 2 when k = 1, I 3 ≤C (\|∇V \| ∞ + \|∇ u\| ∞ ) ∇W 1 + (\|∇ 2 V \| 3 + \|∇ 2 u\| 3 )\|∇W \| 6 \|∇ 2 W \| 2 ≤Cc 4 1 ∇W 2 1 when k = 2, I 3 ≤C (\|∇V \| ∞ + ∇ u W 1,∞ ) ∇W 2 + (\|∇ 3 V \| 2 + \|∇ 3 u\| 2 )\|∇W \| ∞ \|∇ 3 W \| 2 + C(\|∇ 2 V \| 3 + \|∇ 2 u\| 3 )\|∇ 2 W \| 6 \|∇ 3 W \| 2 ≤ Cc 4 1 ∇W 2 2 when k = 3, I 4 = − ∇ k ((ϕ 2 + η 2 )Lw) − (ϕ 2 + η 2 )L∇ k w · ∇ k w ≤C\|ϕ∇ϕ\| ∞ \|∇ 2 w\| 2 \|∇w\| 2 ≤ Cc 2 0 ∇w 2 1 when k = 1, I 4 ≤C \|ϕ∇ϕ\| ∞ \|∇ 3 w\| 2 + \|∇ϕ · ∇ϕ\| 3 + \|ϕ∇ 2 ϕ\| 3 \|∇ 2 w\| 6 \|∇ 2 w\| 2 ≤Cc 2 0 ∇ 2 w 2 1 when k = 2, I 4 ≤C \|∇ 3 ϕ\| 2 \|ϕ∇ 3 w\| 6 \|∇ 2 w\| 3 + \|∇ϕ\| ∞ \|∇ 2 ϕ\| 3 \|∇ 2 w\| 6 \|∇ 3 w\| 2 + C \|∇ 2 ϕ\| 3 \|ϕ∇ 3 w\| 6 + \|∇ϕ\| 2 ∞ \|∇ 3 w\| 2 + \|∇ϕ\| ∞ \|ϕ∇ 4 w\| 2 \|∇ 3 w\| 2 ≤ α 20 \|ϕ∇ 4 w\| 2 2 + Cc 2 0 ∇ 2 w 2 1 + Cc 2 0 when k = 3,(4.18) and I 5 = ∇ϕ 2 · Q(∇ k v) · ∇ k w ≤C\|ϕ\| ∞ \|∇ϕ\| ∞ \|∇ k+1 v\| 2 \|∇ k w\| 2 ≤ Cc 2 0 c 1 \|∇ k w\| 2 , when k ≤ 2,I 5 = ∇ϕ 2 · Q(∇ k v) · ∇ k w ≤C \|∇ϕ\| 2 ∞ \|∇ 3 w\| 2 + \|∇ 2 ϕ\| 3 \|ϕ∇ 3 w\| 6 + \|∇ϕ\| ∞ \|ϕ∇ 4 w\| 2 \|∇ 3 v\| 2 ≤Cc 2 0 c 1 \|∇ 3 w\| 2 + Cc 2 0 c 2 1 + α 20 \|ϕ∇ 4 w\| 2 2 when k = 3, I 6 = ∇ k (∇ϕ 2 · Q(v + u)) − ∇ϕ 2 · Q(∇ k (v + u)) · ∇ k w ≤C u T,Ξ + v 3 ϕ 2 3 w 2 ≤ Cc 6 1 w 2 when k ≤ 2, I 6 ≤C u T,Ξ + v 3 ϕ 3 ϕ 3 \|∇ 3 w\| 2 + \|ϕ∇ 3 w\| 6 + I * 6 ≤ α 20 \|ϕ∇ 4 w\| 2 2 + C c 6 1 w 3 + c 10 1 + I * 6 when k = 3, I 7 = ∇ϕ 2 · Q(∇ k u) · ∇ k w ≤ C\|∇ϕ\| ∞ \|ϕ\| ∞ \|∇ k+1 u\| 2 \|∇ k w\| 2 ≤ Cc 6 1 \|∇ k w\| 2 , I 8 = − γ − 1 2 ∇ k (φdiv u)∇ k φ − ∇ k (w · ∇ u)∇ k w − ∇ k (ϕ 2 L u)∇ k w ≤C u T,Ξ W 2 3 + W 3 ϕ 2 3 + I * 8 δ 3,k ,(4.19) where integrations by parts have been used for the terms I 5 -I 6 and I 8 when k = 3. And the terms I * 6 and I * 8 δ 3,k can be estimated similarly by integration by parts as I * 6 = ϕ∇ 4 ϕ · Q(v + u)) · ∇ 3 w ≤ α 20 \|ϕ∇ 4 w\| 2 2 + Cc 6 1 w 3 + Cc 10 1 , I * 8 δ 3,k = ϕ 2 L∇ 3 u · ∇ 3 w ≤ α 20 \|ϕ∇ 4 w\| 2 2 + Cc 6 0 w 3 + Cc 10 0 . Due to (4.17)-(4.19), one gets that 1 2 d dt \|∇ k W \| 2 + 1 2 α\| ϕ 2 + η 2 ∇ 4 w\| 2 2 ≤ Cc 6 1 W 2 3 + Cc 10 1 ,(4.20) which, along with Gronwall's inequality, implies that for 0 ≤ t ≤ T 2 = min{T 1 , c −10 1 }, W (t) 2 3 + t 0 (ϕ 2 + η 2 )\|∇ 4 w\| 2 2 ds ≤ W 0 2 3 + Cc 10 1 t exp(Cc 6 1 t) ≤ Cc 2 0 . (4.21) Combining the estimates obtained in Lemmas 4.2-4.3 shows that 1 + ϕ(t) 2 3 + W (t) 2 3 + 3 k=0 t 0 (ϕ 2 + η 2 )\|∇ k+1 w\| 2 2 ds ≤Cc 2 0 ,(4.22) for 0 ≤ t ≤ min{T * , c −10 1 }. Therefore, defining the constant c 1 and time T * by c 1 = C 1 2 c 0 , T * = min{T, c −10 1 },(4.23) we then deduce that for 0 ≤ t ≤ T * , ϕ(t) 2 3 + φ(t) 2 3 + w(t) 2 3 + 3 k=0 t 0 (ϕ 2 + η 2 )\|∇ k+1 w\| 2 2 ds ≤ c 2 1 . (4.24) In other words, given fixed c 0 and T , there exist positive constant c 1 and time T * , depending solely on c 0 , T and the generic constant C such that if (4.12) holds for h and V , then (4.24) holds for the solution to (4.5) in [0, T * ] × R 3 . Here, it should be noted that the definitions of (c 1 , T * ) are all independent of the parameters (R, N, η). 4.2. Passing to the limits as η → 0 and N → +∞. Due to the a priori estimate (4.24), one can solve the following problem:                        ϕ t + (v + u) · ∇ϕ + δ − 1 2 hdiv(v + u) = 0, W t + 3 j=1 A * j (V, u)∂ j W + ϕ 2 L(w) = H(ϕ) · Q(v + u) + G(W, ϕ, u), (ϕ, W )\| t=0 = (ϕ 0 , W 0 ) = (ϕ 0 , φ 0 , 0), x ∈ R 3 , (ϕ, W ) = (ϕ, φ, w) → (0, 0, 0) as \|x\| → ∞ for t ≥ 0.(ϕ, W ) in [0, T * ] × R 3 to (4.25) such that (ϕ, φ) ∈ C([0, T * ]; H 3 ), w ∈ C([0, T * ]; H s ′ ) ∩ L ∞ ([0, T * ]; H 3 ), ϕ∇ 4 w ∈ L 2 ([0, T * ]; L 2 ), for any constant s ′ ∈ [2, 3). Moreover, (ϕ, W ) satisfies (4.24), and for t ∈ [0, T * ], ϕ(t, y(t; ξ 0 )) = φ(t, y(t; ξ 0 )) = 0 and w(t, y(t; ξ 0 )) = 0 for ξ 0 ∈ R 3 /suppϕ 0 . (4.26) Proof. Step 1: Passing to the limit as η → 0. Let N ≥ 1 be a fixed constant. Due to Lemma 4.1, for every η > 0, there exists a unique classical solution (ϕ N η , W N η ) to the linear Cauchy problem (4.5) satisfying (4.24) in [0, T * ] × R 3 , where these estimates and the life span T * > 0 are all independent of (N, R, η). Due to the estimate (4.24) and the equations in (4.5), it holds that ϕ N η t 2 + φ N η t 2 + w N η t 1 + t 0 w N η t 2 2 ds ≤ C 0 (N ), for 0 ≤ t ≤ T * ,(4.27) where the constant C 0 (N ) depends only on the generic constant C, (ϕ 0 , φ 0 , u 0 ) and N , and is independent of (R, η). By virtue of the uniform estimate (4.24) and (4.27) independent of (R, η), there exists a subsequence of solutions (still denoted by) (ϕ N η , W N η ), which converges to a limit (ϕ N , W N ) = (ϕ N , φ N , w N ) in weak or weak* sense as η → 0: (ϕ N η , W N η ) ⇀ (ϕ N , W N ) weakly* in L ∞ ([0, T * ]; H 3 (R 3 )), (ϕ N η t , φ N η t ) ⇀ (ϕ N t , φ N t ) weakly* in L ∞ ([0, T * ]; H 2 (R 3 )), w N η t ⇀ w N t weakly* in L ∞ ([0, T * ]; H 1 (R 3 )), w N η t ⇀ w N t weakly in L 2 ([0, T * ]; H 2 (R 3 )). (4.28) The lower semi-continuity of weak convergence implies that (ϕ N , W N ) also satisfies the corresponding estimates (4.24) and (4.27) except that of ϕ N ∇ 4 w N . Again by the uniform estimate (4.24) and (4.27) independent of η, and the compactness in Lemma 10.8 (see [29]), it holds that for any R 0 > 0, there exists a subsequence of solutions (still denoted by) (ϕ N η , W N η ), which converges to the same limit (ϕ N , W N ) as above in the following strong sense: (ϕ N η , W N η ) → (ϕ N , W N ) in C([0, T * ]; H 2 (B R 0 )), as η → 0. (4.29) Let β i = 1, 2, 3 for i = 1, 2, 3, 4. Because ϕ N η ∇ 4 w N η L 2 L 2 T * is uniformly bounded with respect to N and η, there exists a vector g = (g 1 , g 2 , g 3 ) ∈ L 2 ([0, T * ]; L 2 (R 3 )) such that, R 3 T * 0 ϕ N η ∂ β 1 β 2 β 3 β 4 w N η f dxdt → R 3 T * 0 gf dxdt as η → 0 (4.30) for any f ∈ L 2 ([0, T * ]; L 2 (R 3 )). For any f * ∈ C ∞ c ([0, T * ] × R 3 ) , it follows from integration by parts, (4.28) and (4.29) that lim η→0 R 3 T * 0 ϕ N η ∂ β 1 β 2 β 3 β 4 w N η f * dxdt = lim η→0 − R 3 T * 0 ∂ β 1 ϕ N η ∂ β 2 β 3 β 4 w N η f * + ϕ N η ∂ β 2 β 3 β 4 w N η ∂ β 1 f * dxdt = − R 3 T * 0 ∂ β 1 ϕ N ∂ β 2 β 3 β 4 w N f * + ϕ N ∂ β 2 β 3 β 4 w N ∂ β 1 f * dxdt = − R 3 T * 0 ∂ β 2 β 3 β 4 w N ∂ β 1 (ϕ N f * )dxdt = R 3 T * 0 ϕ N ∂ β 1 β 2 β 3 β 4 w N f * dxdt. (4.31) Because C ∞ c ([0, T * ] × R 3 ) is dense in L 2 ([0, T * ]; L 2 (R 3 )), one can get ϕ N η ∇ 4 w N η ⇀ ϕ N ∇ 4 w N weakly in L 2 ([0, T * ] × R 3 ), as η → 0. (4.32) Thus, (ϕ N , W N ) satisfies (4.24). Now, it is easy to show that (ϕ N , W N ) is a weak solution in the sense of distribution to the following problem:                      ϕ N t + (v + u N ) · ∇ϕ N + δ − 1 2 hdiv(v + u) = 0, W N t + 3 j=1 A * j (V, u N )∂ j W N + (ϕ N ) 2 L(w N ) = H(ϕ N ) · Q(v + u) + G(W N , ϕ N , u), (ϕ N , W N )\| t=0 = (ϕ 0 , W 0 ) = (ϕ 0 , φ 0 , 0), x ∈ R 3 , (ϕ N , W N ) = (ϕ N , φ N , w N ) → (0, 0, 0) as \|x\| → ∞ for t ≥ 0,(4. 33) and has the following regularities (ϕ N , φ N , w N ) ∈ L ∞ ([0, T * ]; H 3 ), (ϕ N t , φ N t ) ∈ L ∞ ([0, T * ]; H 2 ), ϕ N ∇ 4 w N ∈ L 2 ([0, T * ]; L 2 ), w N t ∈ L ∞ ([0, T * ]; H 1 ) ∩ L 2 ([0, T * ]; H 2 ). (4.34) Step 2: Passing to the limit as N → +∞. One can prove the existence, uniqueness and time continuity of the classical solutions to (4.25) as follows. Step 2.1: Existence. According to Step 1, for every N ≥ 1, there exists a solution (ϕ N , W N ) to (4.33) with the a priori estimate (4.24) independent of (R, N ). Due to (4.24) and (4.33), for any R 0 > 0, it holds that ϕ N t H 2 (B R 0 ) + φ N t H 2 (B R 0 ) + w N t H 1 (B R 0 ) + t 0 ∇ 2 w N t 2 L 2 (B R 0 ) ds ≤ C 0 (R 0 ), (4.35) where the constant C 0 (R 0 ) is independent of (R, N ). It follows from (4.24) and (4.35) that there exists a subsequence of solutions (still denoted by) (ϕ N , W N ), which converges to a limit (ϕ, W ) in weak or weak* sense as η → 0: (ϕ N , W N ) ⇀ (ϕ, W ) weakly* in L ∞ ([0, T * ]; H 3 (R 3 )), (ϕ N t , φ N t ) ⇀ (ϕ t , φ t ) weakly* in L ∞ ([0, T * ]; H 2 (B R 0 )), w N t ⇀ w t weakly* in L ∞ ([0, T * ]; H 1 (B R 0 )), w N t ⇀ w t weakly in L 2 ([0, T * ]; H 2 (B R 0 )),(4.36) for any R 0 > 0. The lower semi-continuity of weak convergence implies that (ϕ, W ) also satisfies the estimate (4.24) except that for ϕ∇ 4 w. Again by virtue of the uniform estimates (4.24) and (4.35) independent of N and the compactness in Lemma 10.8 (see [29]), for any R 0 > 0, there exists a subsequence of solutions (still denoted by) (ϕ N , W N ), which converges to the same limit (ϕ, W ) = (ϕ, φ, w) in the following strong sense: (ϕ N , W N ) → (ϕ, W ) in C([0, T * ]; H 2 (B R 0 )), as N → +∞. (4.37) Then, (4.36)-(4.37) and a similar proof as for (4.32) show that ϕ N ∇ 4 w N ⇀ ϕ∇ 4 w weakly in L 2 ([0, T * ] × R 3 ), as N → +∞. (4.38) Thus, (ϕ, W ) satisfies (4.24). It is easy to show that (ϕ, W ) is a weak solution in the sense of distribution to (4.25) with the properties that (ϕ, φ, w) ∈ L ∞ ([0, T * ]; H 3 ), (ϕ t , φ t ) ∈ L ∞ ([0, T * ]; H 2 loc ), ϕ∇ 4 w ∈ L 2 ([0, T * ]; L 2 ), w t ∈ L ∞ ([0, T * ]; H 1 loc ) ∩ L 2 ([0, T * ]; H 2 loc ). (4.39) Step 2.2: Time-continuity. (4.25) and (4.9) imply that d dt ϕ(t, X(t; ξ 0 )) = 0, d dt φ(t, X(t; ξ 0 )) = 0, d dt w(t, X(t; ξ 0 )) = −w · ∇ u, for t ∈ [0, T * ] and ξ 0 ∈ R 3 /suppϕ 0 , which yields that ϕ(t, X(t; ξ 0 )) = φ(t, X(t; ξ 0 )) = 0, w(t, X(t; ξ 0 )) = 0, for t ∈ [0, T * ] and ξ 0 ∈ R 3 /suppϕ 0 . Thus for any positive time t ∈ [0, T * ], the solution (ϕ, φ, w) is compactly supported with respect to the space variable x. Let ξ 0 ∈ ∂suppϕ 0 and \|ξ 0 \| ≤ R. It follows form Proposition 2.1 that \|X(t; ξ 0 )\| ≤ R + ∇ u L ∞ L ∞ T * t 0 \|X(τ ; ξ 0 )\|dτ. (4.40) Then the Gronwall's inequality implies that for 0 ≤ t ≤ T * \|X(t; ξ 0 )\| ≤ R exp ∇ u L ∞ L ∞ T * T * ≤ C 0 (R, T * ), (4.41) where C 0 (R, T * ) depends on the generic constant C, T * , (ϕ 0 , φ 0 , u 0 ) and R. Then for any 0 < R < +∞, it follows from (4.25) that (ϕ t , φ t ) ∈ L ∞ ([0, T * ]; H 2 ), w t ∈ L ∞ ([0, T * ]; H 1 ) ∩ L 2 ([0, T * ]; H 2 ).(ϕ, φ) ∈ C([0, T * ]; H s ′ ∩ weak-H 3 ), w ∈ C([0, T * ]; H s ′ ∩ weak-H 3 ). (4.43) Using the same arguments as in the proof of Lemma 4.2-4.3, one has Step 2.3: Uniqueness. Let (ϕ i , W i ) = (ϕ i , φ i , w i ), i = 1, 2, be two solutions obtained in Step 2.1, and lim t→0 sup ϕ(t) 3 ≤ ϕ 0 3 , lim t→0 sup φ(t) 3 ≤ φ 0 3 ,(4.ϕ = ϕ 1 − ϕ 2 , φ = φ 1 − φ 2 , w = w 1 − w 2 , W = W 1 − W 2 . Then ϕ 1 = ϕ 2 follows from ϕ t + v · ∇ϕ = 0, and W 1 = W 2 follows from ϕ 1 , u). W t + 3 j=1 A * j (V, u)∂ j W + ϕ 2 1 L(w) = G(W ,1 + ϕ 0 3 + + φ 0 3 + u 0 Ξ ≤ c 0 . Let (ϕ 0 , W 0 = (φ 0 , w 0 )) with the regularities (ϕ 0 , φ 0 ) ∈ C([0, T * ]; H 3 ), w 0 ∈ C([0, T * ]; H s ′ ) ∩ L ∞ ([0, T * ]; H 3 ), ϕ 0 ∇ 4 w 0 ∈ L 2 ([0, T * ]; L 2 ) for any s ′ ∈ [2, 3) be the solution to the following problem:                            X t + u · ∇X = 0 in (0, +∞) × R 3 , Y t + u · ∇Y = 0 in (0, +∞) × R 3 , Φ t − X 2 △Φ = 0 in (0, +∞) × R 3 , (X, Y, Φ)\| t=0 = (ϕ 0 , φ 0 , 0) in R 3 , (X, Y, Φ) → (0, 0, 0) as \|x\| → +∞, t > 0. (4.47) Choose a time T * * ∈ (0, T * ] small enough such that for 0 ≤ t ≤ T * * , \|\|ϕ 0 (t)\|\| 2 3 + \|\|φ 0 (t)\|\| 2 3 + \|\|w 0 (t)\|\| 2 3 + t 0 \|ϕ 0 ∇ 4 w 0 \|ds ≤ c 2 1 . (4.48) Now the existence, uniqueness and time continuity can be proved as follows. Step 1: Existence. Let (h, ψ, v) = (ϕ 0 , φ 0 , w 0 ) and (ϕ 1 , W 1 ) be a classical solution to (4.25). Then we construct approximate solutions (ϕ k+1 , W k+1 ) = (ϕ k+1 , φ k+1 , w k+1 ) inductively as follows: Assume that (ϕ k , W k ) has been defined for k ≥ 1, and let (ϕ k+1 , W k+1 ) be the unique solution to (4.25) with (h, V ) replaced by (ϕ k , W k ) as:                                ϕ k+1 t + (w k + u) · ∇ϕ k+1 + δ − 1 2 ϕ k div(w k + u) = 0, W k+1 t + 3 j=1 A * j (W k , u)∂ j W k+1 + (ϕ k+1 ) 2 L(w k+1 ) = H(ϕ k+1 ) · Q(w k + u) + G(W k+1 , ϕ k+1 , u), (ϕ k+1 , W k+1 )\| t=0 = (ϕ 0 , W 0 ), x ∈ R 3 , (ϕ k+1 , W k+1 ) → (0, 0) as \|x\| → +∞, t > 0. (4.49) Here the sequence (ϕ k , W k ) satisfies the uniform a priori estimate (4.24) for 0 ≤ t ≤ T * * , and is also compactly supported as long as it exists. Now we prove the convergence of the whole sequence (ϕ k , W k ) to a limit (ϕ, W ) in some strong sense. Set ϕ k+1 = ϕ k+1 − ϕ k , W k+1 = (φ k+1 , w k+1 ) ⊤ , φ k+1 = φ k+1 − φ k , w k+1 = w k+1 − w k , then it follows from (4.49) that                                        ϕ k+1 t + (w k + u) · ∇ϕ k+1 + w k · ∇ϕ k = − δ − 1 2 (ϕ k div(w k−1 + u) + ϕ k divw k ), W k+1 t + 3 j=1 A * j (W k , u)∂ j W k+1 + (ϕ k+1 ) 2 L(w k+1 ) = − 3 j=1 A j (W k )∂ j W k − ϕ k+1 (ϕ k+1 + ϕ k )L(w k ) + H(ϕ k+1 ) − H(ϕ k ) · Q(w k + u) + H(ϕ k ) · Q(w k ) −B(∇ u, W k+1 ) − D(ϕ k+1 (ϕ k+1 + ϕ k ), ∇ 2 u). (4.50) First, multiplying (4.50) 1 by 2ϕ k+1 and integrating over R 3 yield that d dt \|ϕ k+1 \| 2 2 ≤C \|∇w k \| ∞ + \|∇ u\| ∞ \|ϕ k+1 \| 2 2 + C\|ϕ k+1 \| 2 \|w k \| 2 \|∇ϕ k \| ∞ + C\|ϕ k+1 \| 2 \|∇w k−1 \| ∞ + \|∇ u\| ∞ \|ϕ k \| 2 + \|ϕ k divw k \| 2 ≤Cν −1 \|ϕ k+1 (t)\| 2 2 + ν \|w k \| 2 2 + \|ϕ k \| 2 2 + \|ϕ k divw k \| 2 2 ,(4.51) where 0 < ν ≤ 1 10 is a constant to be determined. Second, multiplying (4.50) 2 by 2W k+1 and integrating over R 3 , one has d dt \|W k+1 \| 2 + 2α\|ϕ k+1 ∇w k+1 \| 2 2 + 2(α + β)\|ϕ k+1 divw k+1 \| 2 2 = (W k+1 ) ⊤ divA * (W k , u)W k+1 − 2 3 j=1 (W k+1 ) ⊤ A j (W k )∂ j W k − 2 δ − 1 δ ∇(ϕ k+1 ) 2 · Q(w k+1 ) · w k+1 − 2 ϕ k+1 (ϕ k+1 + ϕ k )Lw k · w k+1 + 2 ∇(ϕ k+1 (ϕ k+1 + ϕ k )) · Q(w k + u) · w k+1 + 2 ∇(ϕ k ) 2 · Q(w k ) · w k+1 − 2 B(∇ u, W k+1 ) · W k+1 − 2 ϕ k+1 (ϕ k+1 + ϕ k )L( u) · w k+1 := 8 i=1 J i . (4.52) The terms J 1 -J 8 above can be estimated as follows J 1 = (W k+1 ) ⊤ divA * (W k , u)W k+1 ≤ C\|∇(w k + u)\| ∞ \|W k+1 \| 2 2 ≤ C\|W k+1 \| 2 2 , J 2 = − 2 3 j=1 A j (W k )∂ j W k · W k+1 ≤C\|∇W k \| ∞ \|W k \| 2 \|W k+1 \| 2 ≤ Cν −1 \|W k+1 \| 2 2 + ν\|W k \| 2 2 , (4.53) J 3 = − 2 δ − 1 δ ∇(ϕ k+1 ) 2 · Q(w k+1 ) · w k+1 ≤C\|∇ϕ k+1 \| ∞ \|ϕ k+1 ∇w k+1 \| 2 \|w k+1 \| 2 ≤ C\|W k+1 \| 2 2 + α 20 \|ϕ k+1 ∇w k+1 \| 2 2 , J 4 = − 2 ϕ k+1 (ϕ k+1 + ϕ k )Lw k · w k+1 ≤C\|ϕ k+1 \| 2 \|ϕ k+1 w k+1 \| 6 \|Lw k \| 3 + C\|ϕ k+1 \| 2 \|ϕ k Lw k \| ∞ \|w k+1 \| 2 ≤C\|ϕ k+1 \| 2 2 + α 20 \|ϕ k+1 ∇w k+1 \| 2 2 + C(1 + \|ϕ k ∇ 4 w k \| 2 2 )\|W k+1 \| 2 2 , J 5 =2 ∇(ϕ k+1 (ϕ k+1 + ϕ k )) · Q(w k + u) · w k+1 ≤C(\|∇ 2 w k \| 6 + \|∇ 2 u\| 6 )\|ϕ k+1 w k+1 \| 3 \|ϕ k+1 \| 2 + C(\|ϕ k ∇ 2 w k \| ∞ + \|ϕ k ∇ 2 u\| ∞ )\|w k+1 \| 2 \|ϕ k+1 \| 2 + C(\|∇w k \| ∞ + \|∇ u\| ∞ )\|ϕ k+1 \| 2 \|ϕ k+1 ∇w k+1 \| 2 + J * 5 , J 6 =4 ϕ k ∇ϕ k · Q(w k ) · w k+1 ≤C\|∇ϕ k \| ∞ \|ϕ k ∇w k \| 2 \|w k+1 \| 2 ≤ ν\|ϕ k ∇w k \| 2 2 + Cν −1 \|w k+1 \| 2 2 , J 7 = − 2 B(∇ u, W k+1 ) · W k+1 ≤ C\|∇ u\| ∞ \|W k+1 \| 2 2 , J 8 = − 2 ϕ k+1 (ϕ k+1 + ϕ k )L u · w k+1 ≤ C\|ϕ k+1 \| 2 \|ϕ k+1 + ϕ k \| ∞ \|L u\| ∞ \|w k+1 \| 2 ,(4.54) where one has used the inequality (10.17) for the term \|ϕ k Lw k \| ∞ , and J * 5 = − 2 i,j ϕ k+1 (ϕ k − ϕ k+1 + ϕ k+1 )Q ij (w k + u)∂ i (w k+1 ) (j) ≤C(\|∇w k \| ∞ + \|∇ u\| ∞ ) \|∇w k+1 \| ∞ \|ϕ k+1 \| 2 2 + \|ϕ k+1 ∇w k+1 \| 2 \|ϕ k+1 \| 2 . (4.55) Then it follows from (4.52)-(4.55) and Young's inequality that d dt \|W k+1 \| 2 + α\|ϕ k+1 ∇w k+1 \| 2 2 ≤C(ν −1 + \|ϕ k ∇ 4 w k \| 2 2 )\|W k+1 \| 2 2 + C\|ϕ k+1 \| 2 2 + ν(\|ϕ k ∇w k \| 2 2 + \|ϕ k \| 2 2 + \|W k \| 2 2 ). (4.56) Finally, define Γ k+1 (t) = sup s∈[0,t] \|W k+1 (s)\| 2 2 + sup s∈[0,t] \|ϕ k+1 (s)\| 2 2 . It follows from (4.51) and (4.56) that d dt \|W k+1 \| 2 + \|ϕ k+1 (t)\| 2 2 + α\|ϕ k+1 ∇w k+1 \| 2 2 ≤E k ν (\|W k+1 \| 2 2 + \|ϕ k+1 \| 2 2 ) + ν(\|ϕ k ∇w k \| 2 2 + \|ϕ k \| 2 2 + \|W k \| 2 2 ), for some E k ν such that t 0 E k ν (s)ds ≤ C + C ν t for 0 ≤ t ≤ T * * . Then Gronwall's inequality implies that Γ k+1 + t 0 α\|ϕ k+1 ∇w k+1 \| 2 2 ds ≤ ν t 0 \|ϕ k ∇w k \| 2 2 ds + tν sup s∈[0,t] \|W k \| 2 2 + \|ϕ k \| 2 2 exp (C + C ν t). Choose ν 0 > 0 and T * ∈ (0, min(1, T * * )) small enough such that ν 0 exp C = 1 8 min 1, α , exp(C ν 0 T * ) ≤ 2, which implies that ∞ k=1 Γ k+1 (T * ) + T * 0 α\|ϕ k+1 ∇w k+1 \| 2 2 dt ≤ C < +∞. Thus, by the above estimate for Γ k+1 (T * ) and (4.24), the whole sequence (ϕ k , W k ) converges to a limit (ϕ, W ) = (ϕ, φ, w) in the following strong sense: (ϕ k , W k ) → (ϕ, W ) in L ∞ ([0, T * ]; H 2 (R 3 )). (4.57) Due to (4.24) and the lower-continuity of norm for weak convergence, (ϕ, W ) still satisfies (4.24). Now (4.57) implies that (ϕ, W ) satisfies problem (4.1) in the sense of distribution. So the existence of a classical solution is proved. Step 2: Uniqueness. Similarly to the proof of Lemma 4.4, for any t ∈ [0, T * ], the solution (ϕ, φ, w) is still compactly supported with respect to the space variable x, and the size of their supports is uniformly bounded by the constant C 0 (R, T * ). Let (ϕ 1 , W 1 ) and (ϕ 2 , W 2 ) be two solutions to (4.1) satisfying the uniform a priori estimate (4.24). Set ϕ = ϕ 1 − ϕ 2 , W = (φ, w) = (φ 1 − φ 2 , w 1 − w 2 ). Then according to (4.50), (ϕ, W ) solves the following system                      ϕ t + (w 1 + u) · ∇ϕ + w · ∇ϕ 2 + δ − 1 2 (ϕdiv(w 1 + u) + ϕ 2 divw) = 0, W t + 3 j=1 A * j (W 1 , u)∂ j W + ϕ 2 1 L(w) = − 3 j=1 A j (W )∂ j W 2 −ϕ(ϕ 1 + ϕ 2 )L(w 2 ) + H(ϕ 1 ) − H(ϕ 2 ) · Q(W 2 ) +H(ϕ 1 ) · Q(W ) − B(∇ u, w) − D(ϕ(ϕ 1 + ϕ 2 ), ∇ 2 u). (4.58) Using the same arguments as in the derivation of (4.51)-(4.56), and letting Λ(t) = \|W (t)\| 2 2 + \|ϕ(t)\| 2 2 , one can get that d dt Λ(t) + C\|ϕ 1 ∇w(t)\| 2 2 ≤ I(t)Λ(t),(4.59) for some I(t) such that t 0 I(s)ds ≤ C for 0 ≤ t ≤ T * . So the Gronwall's inequality yields ϕ = φ = w = 0. Then the uniqueness is obtained. Step 3. The time-continuity can be obtained via the same arguments as in Lemma 4.4. 5. Global-in-time well-posedness with compactly supported initial density under (P 0 ) or (P 1 ) In this section, we will consider the global-in-time well-posedness of classical solutions to the Cauchy problem (4.1) with compactly supported (ϕ 0 , φ 0 ). To this end, we first rewrite (4.1) as                        ϕ t + w · ∇ϕ + δ − 1 2 ϕdivw = − u · ∇ϕ − δ − 1 2 ϕdiv u, W t + 3 j=1 A j (W )∂ j W + ϕ 2 L(w) = H(ϕ) · Q(w + u) + G * (W, ϕ, u), (ϕ, φ, w)(t = 0, x) = (ϕ 0 , W 0 ) = (ϕ 0 , φ 0 , 0), x ∈ R 3 , (ϕ, φ, w) → (0, 0, 0) as \|x\| → ∞ for t ≥ 0, (5.1) where G * (W, ϕ, u) = − B(∇ u, W ) − 3 j=1 u (j) ∂ j W − D(ϕ 2 , ∇ 2 u). (5.2) Then the main result in this section can be stated as: for any constant s ′ ∈ [2, 3). Moreover, when (P 2 ) holds, the smallness assumption on ϕ 0 could be removed. We will prove Theorem 5.1 in the following five Subsections 5.1-5.5 via establishing a uniform-in-time weighted estimates under the condition (P 0 ) or (P 1 ) (see Theorem 1.4 and Remark 1.2). The proof for the cases (P 2)-(P 3) will be given in Section 6. Introduce a time weighed norm Z(t) for classical solutions as Y k (t) =\|∇ k W (t)\| 2 , Y 2 (t) = 3 k=0 (1 + t) 2γ k Y 2 k (t), U k (t) =\|∇ k ϕ(t)\| 2 , U 2 (t) = 3 k=0 (1 + t) 2δ k U 2 k (t), Z 2 (t) =Y 2 (t) + U 2 (t), γ k = k − n, δ k = k − m,(5.4) with (n, m) to be determined in Subsections 5.3-5.4. Denote also Z(0) = Z 0 , Y (0) = Y 0 , U (0) = U 0 . Hereinafter, C ≥ 1 will denote a generic constant depending only on fixed constants (α, β, γ, A, δ, κ), but independent of (ϕ 0 , W 0 ), which may be different from line to line, and C 0 > 0 denotes a generic constant depending on (C, ϕ 0 , W 0 ). Specially, C(l) (or C 0 (l)) denotes a generic positive constant depending on (C, l) (or (C, ϕ 0 , W 0 , l)). It then follows from Lemma 10.2 that: Lemma 5.1. \|W (t)\| ∞ ≤C(1 + t) 2n−3 2 Y (t), and \|∇W (t)\| ∞ ≤ C(1 + t) 2n−5 2 Y (t), \|ϕ(t)\| ∞ ≤C(1 + t) 2m−3 2 U (t), and \|∇ϕ(t)\| ∞ ≤ C(1 + t) 2m−5 2 U (t). 5.1. Energy estimates on W . First, applying ∇ k to (5.1) 2 , multiplying by ∇ k W and integrating over R 3 , one can get 1 2 d dt \|∇ k W \| 2 2 + αϕ 2 \|∇ k+1 w\| 2 + (α + β)ϕ 2 \|div∇ k w\| 2 =R k (W ) + S k (W, u) + L k (W, ϕ, u) + Q k (W, ϕ, u), (5.5) where R k (W ) = − ∇ k W · ∇ k 3 j=1 A j (W )∂ j W − 3 j=1 A j (W )∂ j ∇ k W + 1 2 3 j=1 ∇ k W · ∂ j A j (W )∇ k W, S k (W, u) = − ∇ k W · ∇ k B(∇ u, W ) + 1 2 3 j=1 ∂ j u (j) ∇ k W · ∇ k W − ∇ k W · ∇ k 3 j=1 u (j) ∂ j W − 3 j=1 u (j) ∂ j ∇ k W , L k (W, ϕ, u) = − ∇ϕ 2 · S(∇ k w) − ∇ k (ϕ 2 Lw) − ϕ 2 L∇ k w · ∇ k w + ∇ k (ϕ 2 L u) · ∇ k w ≡: L 1 k + L 2 k + L 3 k ,(5.6) and Q k (W, ϕ, u) = ∇ϕ 2 · Q(∇ k w) + ∇ k (∇ϕ 2 · Q(w)) − ∇ϕ 2 · Q(∇ k w) · ∇ k w + ∇ k ∇ϕ 2 · Q( u) · ∇ k w ≡: Q 1 k + Q 2 k + Q 3 k . (5.7) The right hand side of (5.5) will be estimated in the next lemmas. Lemma 5.2 (Estimates on R k and S k ). R k (W )(t, ·) ≤C\|∇W \| ∞ Y 2 k , (5.8) k + r 1 + t Y 2 k + S k (W, u)(t, ·) ≤C 0 Y k Z(1 + t) −γ k −2 ,(5.9) for k = 0, 1, 2, 3, where the constant r is given by: r = − 1 2 if γ ≥ 5 3 , or 3 2 γ − 3 if 1 < γ < 5 3 . (5.10) Proof. Step 1: Estimates on R k . Noticing that R k is a sum of terms as ∇ k W · ∇ l W · ∇ k+1−l W for 1 ≤ l ≤ k, then (5.8) is obvious when k = 0, or 1. For k = 0, 1, one can apply Lemma 10.5 to ∇W to get that \|∇ j W \| p j ≤ C\|∇W \| 1−2/p j ∞ \|∇ k W \| 2/p j 2 , for p j = 2 k − 1 j − 1 . (5.11) If l = k and l = 1, since 1/p l + 1/p k−l+1 = 1 2 , then Hölder's inequality implies \|∇ k W ∇ l W ∇ k+1−l W \| ≤ \|∇ k W \| 2 \|∇ l W \| p l \|∇ k+1−l W \| p k−l+1 ≤ C\|∇W \| ∞ \|∇ k W \| 2 2 . The other cases could be handled similarly. Thus (5.8) is proved. Step 2: Estimates on S k . The intergrand of S k (W, u) in (5.6) 2 can be rewritten as s k (W, u) = − ∇ k W · B(∇ u, ∇ k W ) + 1 2 3 j=1 ∂ j u (j) ∇ k W · ∇ k W − ∇ k W · ∇ k (B(∇ u, W )) − B(∇ u, ∇ k W ) − ∇ k W · ∇ k 3 j=1 u (j) ∂ j W − 3 j=1 u (j) ∂ j ∇ k W ≡: s 1 k + s 2 k ,(5.12) where s 1 k is a sum of terms with a derivative of order one of u, and s 2 k is a sum of terms with a derivative of order at least two for u. Step 2.1: Estimates on S 1 k = s 1 k . Let ∇ k = ∂ β 1 β 2 ...β i ...β k with β i = 1, 2, 3. Decompose S 1 k as: S 1 k = − ∇ k W · B(∇ u, ∇ k W ) + 1 2 3 j=1 ∂ j u (j) ∇ k W · ∇ k W − ∂ β 1 ...β k W · k i=1 3 j=1 ∂ β i u (j) ∂ j ∂ β 1 ...β i−1 β i+1 ...β k W = I 1 + I 2 + I 3 ,(5.13) where, from Proposition 2.1, I 1 -I 3 are given by I 1 = − 3(γ − 1) 2(1 + t) ∇ k φ · ∇ k φ − 1 1 + t ∇ k w · ∇ k w + G 1 , I 2 = 3 2 1 1 + t Y 2 k + G 2 , I 3 = − k 1 + t Y 2 k + G 3 , \|G j \| ≤ C 0 (1 + t) 2 Y 2 k , j = 1, 2, 3. (5.14) Therefore, S 1 k (W, u)(t, ·) ≤ C 0 (1 + t) 2 Y 2 k − A k 1 + t ∇ k w · ∇ k w − B k 1 + t ∇ k φ · ∇ k φ ≤C 0 Y k Z(1 + t) −γ k −2 − k + r 1 + t Y 2 k ,(5.15) where A k = k − 1 2 , B k = 3 2 γ − 3 + k, r = min A k , B k − k. (5.16) Then (5.9) for S 1 k follows from (5.15)-(5.16). Step 2.2: Estimates on S 2 k = s 2 k . s 2 k is a sum of the terms as E 1 (W ) = ∇ k W · ∇ l u · ∇ k+1−l W for 2 ≤ k ≤ 3 and 2 ≤ l ≤k; E 2 (W ) = ∇ k W · ∇ l+1 u · ∇ k−l W for 1 ≤ k ≤ 3 and 1 ≤ l ≤k. Hence, S 2 1 (W ) ≤C\|W \| ∞ \|∇ 2 u\| 2 \|∇W \| 2 ≤ C 0 Y 1 Z(1 + t) −2−γ 1 , S 2 2 (W ) ≤C \|∇ 2 u\| ∞ \|∇W \| 2 + \|∇ 3 u\| 2 \|W \| ∞ \|∇ 2 W \| 2 ≤ C 0 Y 2 Z(1 + t) −2−γ 2 , S 2 3 (W ) ≤C \|∇ 2 u\| ∞ \|∇ 2 W \| 2 + \|∇ 3 u\| 2 \|∇W \| ∞ + \|∇ 4 u\| 2 \|W \| ∞ \|∇ 3 W \| 2 ≤C 0 Y 3 Z(1 + t) −2−γ 3 ,(5.17) which imply that S 2 k (W ) ≤ C 0 Y k Z(1 + t) −2−γ k , for k = 1, 2, 3. This, together with (5.15), yields (5.9)-(5.10). Lemma 5.3 (Estimates on L k ). For any suitably small constant η > 0, there are two constants C(η) and C 0 (η) such that L k (W ) ≤η\|ϕ∇ k+1 w\| 2 2 δ 3k + C(η)(1 + t) 2m−5−γ k Z 3 Y k + C 0 (1 + t) 2m−n−4.5−γ k Z 2 Y k + C 0 (η)(1 + t) 2m−10 Z 2 . (5.18) Proof. Step 1: Estimates on L 1 k . It is easy to check that, L 1 k ≤C\|ϕ\| ∞ \|∇ϕ\| ∞ \|∇ k+1 w\| 2 \|∇ k w\| 2 ≤ C(1 + t) 2m−5−γ k Z 3 Y k , for k ≤ 2, L 1 3 ≤C\|ϕ∇ 4 w\| 2 \|∇ϕ\| ∞ \|∇ 3 w\| 2 ≤ η\|ϕ∇ 4 w\| 2 2 + C(η)(1 + t) 2m−5−γ 3 Z 3 Y 3 . (5.19) Step 2: Estimates on L 3 k . If k = 0 or 1, one has L 3 0 ≤C\|ϕ\| ∞ \|∇ 2 u\| ∞ \|ϕ\| 2 \|w\| 2 ≤ C 0 (1 + t) 2m−n−4.5−γ 0 Z 2 Y 0 , L 3 1 ≤C \|∇ϕ\| 2 \|∇ 2 u\| ∞ + \|ϕ\| ∞ \|∇ 3 u\| 2 \|ϕ\| ∞ \|∇w\| 2 ≤C 0 (1 + t) 2m−n−4.5−γ 1 Z 2 Y 1 . (5.20) For the case k = 2, decompose L 3 2 L 3 2 (0, 2) + L 3 2 (1, 1) + L 3 2 (2, 0). One can get L 3 2 (0, 2) ϕ 2 ∇ 2 L u · ∇ 2 w ≤ C\|ϕ\| 2 ∞ \|∇ 2 L u\| 2 \|∇ 2 w\| 2 ≤C 0 (1 + t) 2m−n−4.5−γ 2 Z 2 Y 2 , L 3 2 (1, 1) ∇ϕ 2 · ∇L u · ∇ 2 w ≤ C\|ϕ\| ∞ \|∇ϕ\| ∞ \|∇L u\| 2 \|∇ 2 w\| 2 ≤C 0 (1 + t) 2m−n−4.5−γ 2 Z 2 Y 2 , L 3 2 (2, 0) ∇ 2 ϕ 2 · L u · ∇ 2 w ≤ C(\|∇ϕ\| 2 ∞ \|L u\| 2 + \|ϕ\| ∞ \|∇ 2 ϕ\| 2 \|L u\| ∞ )\|∇ 2 w\| 2 ≤C 0 (1 + t) 2m−n−4.5−γ 2 Z 2 Y 2 . (5.21) For the case k = 3, decompose L 3 3 L 3 3 (0, 3) + L 3 3 (1, 2) + L 3 3 (2, 1) + L 3 3 (3, 0). Then, by integration by parts, one can obtain L 3 3 (0, 3) ϕ 2 · ∇ 3 L u · ∇ 3 w ≤ C\|ϕ\| ∞ \|∇ 2 L u\| 2 \|ϕ∇ 4 w\| 2 + \|∇ϕ\| ∞ \|∇ 3 w\| 2 ≤η\|ϕ∇ 4 W \| 2 2 + C 0 (1 + t) 2m−n−4.5−γ 3 Z 2 Y 3 + C 0 (η)(1 + t) 2m−10 Z 2 , L 3 3 (1, 2) ∇ϕ 2 · ∇ 2 L u · ∇ 3 w ≤ C\|ϕ\| ∞ \|∇ϕ\| ∞ \|∇ 2 L u\| 2 \|∇ 3 w\| 2 ≤C 0 (1 + t) 2m−n−4.5−γ 3 Z 2 Y 3 , L 3 3 (2, 1) ∇ 2 ϕ 2 · ∇L u · ∇ 3 w ≤ C \|∇ϕ\| 2 ∞ \|∇ 3 u\| 2 + \|ϕ\| ∞ \|∇ 2 ϕ\| 6 \|∇ 3 u\| 3 \|∇ 3 w\| 2 ≤C 0 (1 + t) 2m−n−4.5−γ 3 Z 2 Y 3 , L 3 3 (3, 0) ∇ 3 ϕ 2 · L u · ∇ 3 w ≤ C \|ϕ\| ∞ \|∇ 3 ϕ\| 2 + \|∇ϕ\| ∞ \|∇ 2 ϕ\| 2 \|L u\| ∞ \|∇ 3 w\| 2 ≤C 0 (1 + t) 2m−n−4.5−γ 3 Z 2 Y 3 ,(5.22) with η > 0 being any sufficiently small constant. It follows from (5.20)-(5.22) that L 3 k ≤η\|ϕ∇ k+1 w\| 2 2 δ 3k + C 0 (1 + t) 2m−n−4.5−γ k Z 2 Y k + C 0 (η)(1 + t) 2m−10 Z 2 . (5.23) Step 3: Estimates on L 2 k . If k = 1, one gets L 2 1 = ∇ϕ 2 · Lw · ∇w ≤ C\|ϕ\| ∞ \|∇ϕ\| ∞ \|∇ 2 w\| 2 \|∇w\| 2 ≤ C(1 + t) 2m−5−γ 1 Z 3 Y 1 . (5.24) Next for k = 2, decompose L 2 2 L 2 2 (1, 1) + L 2 2 (2, 0). In a similar way for L 1 2 , one has L 2 2 (1, 1) ∇ϕ 2 · ∇Lw · ∇ 2 w ≤ C(1 + t) 2m−5−γ 2 Z 3 Y 2 , L 2 2 (2, 0) ∇ 2 ϕ 2 · Lw · ∇ 2 w ≤ C(\|∇ϕ\| 2 ∞ \|∇ 2 w\| 2 + \|ϕ\| 6 \|∇ 2 w\| 6 \|∇ 2 ϕ\| 6 )\|∇ 2 w\| 2 ≤C(1 + t) 2m−5−γ 2 Z 3 Y 2 . (5.25) At last for k = 3, decompose L 2 3 L 2 3 (1, 2) + L 2 3 (2, 1) + L 2 3 (3, 0). In a similar way for L 1 3 , one can get L 2 3 (1, 2) ∇ϕ 2 · ∇ 2 Lw · ∇ 3 w ≤ η\|ϕ∇ 4 w\| 2 2 + C(η)(1 + t) 2m−5−γ 3 Z 3 Y 3 , L 2 3 (2, 1) ∇ 2 ϕ 2 · ∇Lw · ∇ 3 w ≤ C(\|∇ϕ\| 2 ∞ \|∇ 3 w\| 2 + \|∇ 2 ϕ\| 3 \|ϕ∇Lw\| 6 )\|∇ 3 w\| 2 ≤η\|ϕ∇ 4 w\| 2 2 + C(η)(1 + t) 2m−5−γ 3 Z 3 Y 3 , L 2 3 (3, 0) ∇ 3 ϕ 2 · Lw · ∇ 3 w ≤C\|∇ϕ\| ∞ \|∇ 2 ϕ\| 3 \|∇ 3 w\| 2 \|∇ 2 w\| 6 + C\|ϕ∇ 3 w\| 6 \|∇ 2 w\| 3 \|∇ 3 ϕ\| 2 ≤η\|ϕ∇ 4 w\| 2 2 + C(η)(1 + t) 2m−5−γ 3 Z 3 Y 3 . (5.26) Then combining the estimates (5.24)-(5.26) yields L 2 k ≤η\|ϕ∇ k+1 w\| 2 2 δ 3k + C(η)(1 + t) 2m−5−γ k Z 3 Y k . (5.27) Thus (5.18) follows from above three steps. Lemma 5.4 (Estimates on Q k ). For any suitably small constant η > 0, there are two constants C(η) and C 0 (η) such that Q k (W, u) ≤η\|ϕ∇ k+1 w\| 2 2 δ 3k + C(η)(1 + t) 2m−5−γ k Z 3 Y k + C 0 (1 + t) 2m−n−3.5−γ k Z 2 Y k + C 0 (η)(1 + t) 2m−10 Z 2 + (4α + 6β)δ (δ − 1)(1 + t) \|∇ 3 ϕ\| 2 \|ϕ∇ 3 divw\| 2 . (5.28) Proof. Step 1: Estimates on Q 1 k . In a similar way for L 1 k , it is easy to get Q 1 k ≤η\|ϕ∇ k+1 w\| 2 2 δ 3,k + C(1 + t) 2m−5−γ k Z 3 Y k . Step 2: Estimates on Q k 3 . If k = 0, one has Q 3 0 ≤C\|ϕ\| ∞ \|∇ u\| 6 \|∇ϕ\| 3 \|w\| 2 ≤ C 0 (1 + t) 2m−n−4.5−γ 0 Z 2 Y 0 . (5.29) For k = 1, in a similar way for L 3 1 , one obtains 0). Then as for L 3 2 (1, 1) and L 3 2 (2, 0), one has Q 3 1 ≤C \|ϕ\| ∞ \|∇ 2 ϕ\| 3 \|∇ u\| 6 + \|∇ϕ\| 2 6 \|∇ u\| 6 + \|ϕ\| ∞ \|∇ϕ\| 2 \|∇ 2 u\| ∞ \|∇w\| 2 ≤C 0 (1 + t) 2m−n−4.5−γ 1 Z 2 Y 1 . (5.30) Next for k = 2, decompose Q 3 2 Q 3 2 (0, 2) + Q 3 2 (1, 1) + Q 3 2 (2,Q 3 2 (0, 2) C ∇ϕ 2 · ∇ 3 u · ∇ 2 w ≤ C 0 (1 + t) 2m−n−4.5−γ 2 Z 2 Y 2 , Q 3 2 (1, 1) C ∇ 2 ϕ 2 · ∇ 2 u · ∇ 2 w ≤ C 0 (1 + t) 2m−n−4.5−γ 2 Z 2 Y 2 , Q 3 2 (2, 0) C ∇ 3 ϕ 2 · ∇ u · ∇ 2 w ≤ C \|∇ 3 ϕ\| 2 \|ϕ\| ∞ + \|∇ 2 ϕ\| 6 \|∇ϕ\| 3 \|∇ u\| ∞ \|∇ 2 w\| 2 ≤C 0 (1 + t) 2m−n−3.5−γ 2 Z 2 Y 2 . (5.31) For k = 3, let Q 3 3 Q 3 3 (0, 3) + Q 3 3 (1, 2) + Q 3 3 (2, 1) + Q 3 4 (3, 0). Then as for L 3 3 (1, 2), L 3 3 (2, 1) and L 3 3 (3, 0), one can get 2 i=0 Q 3 3 (i, 3 − i) C ∇ϕ 2 · ∇ 4 u · ∇ 3 w + C ∇ 2 ϕ · ∇ 3 u · ∇ 3 w + C ∇ 3 ϕ 2 · ∇ 2 u · ∇ 3 w ≤ C 0 (1 + t) 2m−n−4.5−γ 3 Z 2 Y 3 . (5.32) Finally, it remains to handle the term Q 3 3 (3, 0) defined below, for which some additional information is needed. Using integration by parts and Proposition 2.1, one can get Q 3 3 (3, 0) δ δ − 1 3 i,j=1 α∇ 3 ∂ j ϕ 2 (∂ i u (j) + ∂ j u (i) ) + β∇ 3 ∂ i ϕ 2 div u · ∇ 3 w (i) =Q 3 3 (A) − δ δ − 1 3 i,j=1 α∇ 3 ϕ 2 (∂ i u (j) + ∂ j u (i) ) · ∇ 3 ∂ j w (i) − δ δ − 1 β∇ 3 ϕ 2 div u · ∇ 3 divw =Q 3 3 (A) + Q 3 3 (B) + Q 3 3 (D) − δ (δ − 1)(1 + t) (4α + 6β)ϕ∇ 3 ϕ · ∇ 3 divw ≤Q 3 3 (A) + Q 3 3 (B) + Q 3 3 (D) + (4α + 6β)δ (δ − 1)(1 + t) \|∇ 3 ϕ\| 2 \|ϕ∇ 3 divw\| 2 , where Q 3 3 (A) =C ∇ 3 ϕ 2 · ∇ 2 u · ∇ 3 w = C ϕ∇ 3 ϕ + ∇ϕ · ∇ 2 ϕ · ∇ 2 u · ∇ 3 w ≤C \|ϕ\| ∞ \|∇ 3 ϕ\| 2 + \|∇ϕ\| ∞ \|∇ 2 ϕ\| 2 \|∇ 2 u\| ∞ \|∇ 3 w\| 2 ≤C 0 (1 + t) 2m−n−4.5−γ 3 Z 2 Y 3 ,(5.33) and Q 3 3 (B) =C ∇ϕ · ∇ 2 ϕ · ∇ u · ∇ 4 w =C ∇ϕ · ∇ 3 ϕ + ∇ 2 ϕ · ∇ 2 ϕ · ∇ u + ∇ϕ · ∇ 2 ϕ · ∇ 2 u · ∇ 3 w ≤C \|∇ϕ\| ∞ \|∇ 3 ϕ\| 2 \|∇ u\| ∞ + \|∇ 2 ϕ\| 6 \|∇ 2 ϕ\| 6 \|∇ u\| 6 + \|∇ϕ\| 3 \|∇ 2 u\| ∞ \|∇ 3 w\| 2 ≤C 0 (1 + t) 2m−n−3.5−γ 3 Z 2 Y 3 , Q 3 3 (D) = − δ (δ − 1)(1 + t) 2 3 i,j=1 2αϕ∇ 3 ϕ(K ij + K ji ) · ∇ 3 ∂ j w (i) − δ (δ − 1)(1 + t) 2 3 i=1 2βϕ∇ 3 ϕK ii · ∇ 3 divw ≤η\|ϕ∇ 4 w\| 2 2 + C 0 (η)(1 + t) 2m−10 Z 2 . (5.34) Then combining the estimates (5.29)-(5.34) yields Q 3 k ≤η\|ϕ∇ k+1 w\| 2 2 δ 3k + C 0 (1 + t) 2m−n−3.5−γ k Z 2 Y k + C 0 (η)(1 + t) 2m−10 Z 2 + (4α + 6β)δ (δ − 1)(1 + t) \|∇ 3 ϕ\| 2 \|ϕ∇ 3 divw\| 2 . (5.35) Step 3: Estimates on Q 2 k . For k = 1, direct estimates give Q 2 1 =C ϕ∇ 2 ϕ + ∇ϕ · ∇ϕ · ∇w · ∇w ≤C(\|ϕ\| 3 \|∇ 2 ϕ\| 6 \|∇w\| ∞ + \|∇ϕ\| 2 ∞ \|∇w\| 2 \|∇w\| 2 ≤ C(1 + t) 2m−5−γ 1 Z 3 Y 1 . (5.36) For k = 2, in a similar way for L 2 2 (2, 0), one gets easily Q 2 2 =C ∇ 3 ϕ 2 · ∇w + ∇ 2 ϕ 2 · ∇ 2 w · ∇ 2 w ≤C \|ϕ\| ∞ \|∇ 3 ϕ\| 2 \|∇w\| ∞ + \|∇w\| ∞ \|∇ϕ\| 6 \|∇ 2 ϕ\| 3 \|∇ 2 w\| 2 + C(1 + t) 2m−5−γ 2 Z 3 Y 2 ≤ C(1 + t) 2m−5−γ 2 Z 3 Y 2 . (5.37) For k = 3, as for L 2 3 (3, 0) and L 2 3 (2, 1), it follows from integration by parts that Q 2 3 =C ∇ 4 ϕ 2 · ∇w + ∇ 3 ϕ 2 · ∇ 2 w + ∇ 2 ϕ 2 · ∇ 3 w · ∇ 3 w ≤η\|ϕ∇ 4 w\| 2 2 + C(η)(1 + t) 2m−5−γ 3 Z 3 Y 3 + Q 2 3 (A), (5.38) with Q 2 3 (A) C ∇ 3 ϕ 2 · ∇w · ∇ 4 w = C ϕ∇ 3 ϕ + ∇ϕ · ∇ 2 ϕ · ∇w · ∇ 4 w ≤C\|∇ 3 ϕ\| 2 \|∇w\| ∞ \|ϕ∇ 4 w\| 2 + Q 2 3 (B) ≤η\|ϕ∇ 4 w\| 2 2 + C(η)(1 + t) 2m−5−γ 3 Z 3 Y 3 + Q 2 3 (B), (5.39) where the term Q 2 3 (B) can be estimated by using integration by parts again, Q 2 3 (B) C ∇ϕ · ∇ 2 ϕ · ∇w · ∇ 4 w =C ∇ 2 ϕ · ∇ 2 ϕ · ∇w + ∇ϕ · ∇ 3 ϕ · ∇w + ∇ϕ · ∇ 2 ϕ · ∇ 2 w · ∇ 3 w ≤C \|∇ 2 ϕ\| 2 6 \|∇w\| 6 + \|∇ 3 ϕ\| 2 \|∇ϕ\| ∞ \|∇w\| ∞ + \|∇ϕ\| 6 \|∇ 2 ϕ\| 6 \|∇ 2 w\| 6 \|∇ 3 w\| 2 ≤C(1 + t) 2m−5−γ 3 Z 3 Y 3 .Q 2 k ≤η\|ϕ∇ k+1 w\| 2 2 δ 3k + C(η)(1 + t) 2m−5−γ k Z 3 Y k . (5.41) Then (5.28) follows directly from above estimates. It follows from (5.5) and Lemmas 5.2-5.4 that Lemma 5.5. 1 2 d dt Y 2 k + k + r 1 + t Y 2 k + α \|ϕ∇ k+1 w\| 2 + (α + β) \|ϕ∇ k divw\| 2 ≤C(η)(1 + t) 2m−5−γ k Z 3 Y k + C 0 (1 + t) 2m−n−3.5−γ k + (1 + t) n−2.5−γ k Z 2 Y k + C 0 (η) (1 + t) 2m−10 Z 2 + (1 + t) −γ k −2 ZY k + η\|ϕ∇ k+1 w\| 2 2 δ 3k + (4α + 6β)δ (δ − 1)(1 + t) \|∇ 3 ϕ\| 2 \|ϕ∇ k divw\| 2 δ 3k . (5.42) 5.2. Estimates on ϕ. ϕ is estimated in the following lemma. Lemma 5.6. 1 2 d dt \|∇ k ϕ\| 2 2 + 3 2 δ − 3 + k 1 + t \|∇ k ϕ\| 2 2 ≤C(1 + t) n−2.5−δ k Z 2 U k + C 0 (1 + t) −2−δ k ZU k + δ − 1 2 \|ϕ∇ 3 divw\| 2 \|∇ 3 ϕ\| 2 δ 3k . (5.43) Proof. Step 1. Applying ∇ k to (5.1) 1 , multiplying by ∇ k ϕ and integrating over R 3 , one gets 1 2 d dt \|∇ k ϕ\| 2 2 = w · ∇ k+1 ϕ + δ − 1 2 ∇ k ϕdivw · ∇ k ϕ + S * k (ϕ, u) − Λ k 1 − Λ k 2 , (5.44) where S * k (ϕ, u) is given in Step 2 below, and Λ k 1 = ∇ k (w · ∇ϕ) − w · ∇ k+1 ϕ · ∇ k ϕ, Λ k 2 = δ − 1 2 ∇ k (ϕdivw) − ∇ k ϕdivw · ∇ k ϕ. Integration by parts yields immediately that w · ∇ k+1 ϕ + δ − 1 2 ∇ k ϕdivw · ∇ k ϕ ≤ C\|∇w\| ∞ \|∇ k ϕ(t)\| 2 2 . (5.45) Step 2: Estimates on S * k = s * k with the integrand defined as s * k (ϕ, u) = − δ − 1 2 ∇ k ϕ · ∇ k ϕdiv u + 1 2 div u∇ k ϕ · ∇ k ϕ − δ − 1 2 ∇ k ϕ · ∇ k (ϕdiv u) − div u∇ k ϕ − ∇ k ϕ · ∇ k u · ∇ϕ − u · ∇ k+1 ϕ ≡: s * 1 k + s * 2 k , (5.46) where s * 1 k is a sum of terms with a derivative of order one of u, and s * 2 k is a sum of terms with a derivative of order at least two of u. Step 2.1: Estimates on S * 1 k = s * 1 k . Let ∇ k = ∂ β 1 ...β k with 0 ≤ β i = 1, 2, 3. S * 1 k = − δ − 1 2 ∇ k ϕ · ∇ k ϕdiv u + 1 2 div u∇ k ϕ · ∇ k ϕ − ∂ β 1 ...β k ϕ · k i=1 k j=1 ∂ β i u (j) ∂ j ∂ β 1 ...β i−1 β i+1 ...β k ϕ = J 1 + J 2 + J 3 , (5.47) where, by Proposition 2.1, J 1 -J 3 are estimated by J 1 = − 3(δ − 1) 2(1 + t) U 2 k + N 1 , J 2 = 3 2 1 1 + t U 2 k + N 2 , J 3 = − k 1 + t U 2 k + N 3 , \|N j \| ≤ C 0 (1 + t) 2 U 2 k , j = 1, 2, 3. (5.48) Therefore, it holds that S * 1 k (W, u)(t, ·)dx ≤ C 0 (1 + t) 2 U 2 k − 3 2 δ − 3 + k 1 + t U 2 k . (5.49) Step 2.2: Estimates on S * 2 k = s * 2 k . s * 2 k is a sum of the terms defined as E 1 (ϕ) = ∇ k ϕ · ∇ l u · ∇ k+1−l ϕ for 2 ≤ k ≤ 3 and 2 ≤ l ≤ k; E 2 (ϕ) = ∇ k ϕ · ∇ l+1 u · ∇ k−l ϕ for 1 ≤ k ≤ 3 and 1 ≤ l ≤ k. Then it holds that S * 2 1 (ϕ) ≤C\|ϕ\| 6 \|∇ 2 u\| 2 \|∇ϕ\| 3 ≤ C 0 (1 + t) −2−δ 1 ZU 1 , S * 2 2 (ϕ) ≤C(\|∇ 2 u\| ∞ \|∇ϕ\| 2 + \|∇ 3 u\| 2 \|ϕ\| ∞ )\|∇ 2 ϕ\| 2 ≤ C 0 (1 + t) −2−δ 2 ZU 2 , S * 2 3 (ϕ) ≤C \|∇ 2 u\| ∞ \|∇ 2 ϕ\| 2 + \|∇ 3 u\| 6 \|∇ϕ\| 3 + \|∇ 4 u\| 2 \|ϕ\| ∞ \|∇ 3 ϕ\| 2 ≤C 0 (1 + t) −2−δ 3 ZU 3 ,(5.50) which implies immediately that S * 2 k (ϕ) ≤ C 0 (1 + t) −2−δ k ZU k , for k = 1, 2, 3. Step 3: Estimates on Λ k 1 + Λ k 2 . It follows from Lemma 5.2 and Hölder's inequality that Finally, set Λ 1 1 +Λ 1 2 ≤ C\|∇w\| ∞ \|∇ϕ(t)\| 2 2 ≤ C(1 + t) n−2.5−δ 1 Z 2 U 1 , Λ 2 1 +Λ 2 2 ≤ C(\|∇ϕ\| ∞ \|∇ 2 w(t)\| 2 + \|∇w\| ∞ \|∇ 2 ϕ(t)\| 2 )\|∇ 2 ϕ(t)\| 2 ≤C(1 + t) n−2.5−δ 2 Z 2 U 2 , Λ 3 1 ≤C \|∇ϕ\| ∞ \|∇ 3 w(t)\| 2 + \|∇ 2 ϕ\| 6 \|∇ 2 w(t)\| 3 + \|∇w\| ∞ \|∇ 3 ϕ(t)\| 2 \|∇ 3 ϕ(t)\| 2 ≤C(1 + t) n−2.5−δ 3 Z 2 U 3 , Λ 3 2 ≤C \|∇ϕ\| ∞ \|∇ 3 w(t)\| 2 + \|∇ 2 ϕ\| 6 \|∇ 2 w(t)\| 3 + δ − 1 2 \|ϕ∇ 3 divw\| 2 \|∇ 3 ϕ\| 2 ≤C(1 + t) n−2.5−δ 3 Z 2 U 3 + δ − 1 2 \|ϕ∇ 3 divw\| 2 \|∇ 3 ϕ\| 2 .H(A * , A, B) = α(A * ) 2 + (α + β)A 2 + 3 2 δ − 3 + m 1 + t B 2 − δ − 1 2 (1 + t) n−m + 2δ δ − 1 (2α + 3β)(1 + t) m−n−1 AB, A * =(1 + t) γ 3 \|ϕ∇ 4 w\| 2 , A = (1 + t) γ 3 \|ϕ∇ 3 divw\| 2 , B = (1 + t) δ 3 \|∇ 3 ϕ\| 2 ,(5.52) then the following lemma holds: Lemma 5.7. There exist some positive constants b m , such that 1 2 d dt Z 2 + b m 1 + t (Z 2 − B 2 ) + H(A * , A, B) + α 2 k=0 (1 + t) 2γ k \|ϕ∇ k+1 w(t)\| 2 2 + (α + β) 2 k=0 (1 + t) 2γ k \|ϕ∇ k divw(t)\| 2 2 ≤η(1 + t) 2γ 3 \|ϕ∇ 4 w\| 2 2 + C(η)(1 + t) 2m−5 Z 4 + C 0 (1 + t) 2m−n−3.5 Z 3 + C 0 (1 + t) n−2.5 Z 3 + C 0 (η) (1 + t) 2m−2n−4 + (1 + t) −2 Z 2 ,b m =    min n − 0.5, 3 2 δ − 3 + m if γ ≥ 5 3 , min 3γ 2 − 3 + n, 3 2 δ − 3 + m if 1 < γ < 5 3 , (5.54) where η > 0 is any suitably small constant. Proof. (5.53) can be obtained by multiplying (5.42) and (5.43) by (1 + t) 2γ k and (1 + t) 2δ k respectively and summing the resulting inequalities together. 5.3. Proof of Theorem 5.1 under the condition (P 0 ). Set M 1 = 2α + 3β 2α + β , M 2 = −3δ + 1 + 1 2 M 3 , M 3 = (δ − 1) 2 4(2α + β) + 4δ 2 (2α + β) (δ − 1) 2 M 2 1 + 2M 1 δ, M 4 = 1 2 min 3γ − 3 2 , −M 2 − 1 2 , 1 + M 2 , Φ(A, B) =(2α + β)A 2 + 3 2 δ − 3 + m 1 + t B 2 − δ − 1 2 (1 + t) n−m + 2δ δ − 1 (2α + 3β)(1 + t) m−n−1 AB. (5.55) Lemma 5.8. Let condition (P 0 ) and M 1 > 0 hold. Then for m = n + 0.5 = 3, there exist some positive constants ν * , b * and ǫ * , such that 1 2 d dt Z 2 + (1 − ν * )b * 1 + t Z 2 ≤ C(1 + t) 1+ǫ * Z 4 + C 0 (1 + t) −1−ǫ * Z 2 , (5.56) where ǫ * = 1 2 min 3γ − 3 2 , −M 2 − 1 2 , 1 > 0, ν * = min 3γ − 3 4(3γ − 1) , −M 4 − 1 6δ − M 3 , 1 10 > 0, b * =    min 2, 3 2 δ − 1 4 M 3 > 1 if γ ≥ 5 3 , min 3γ 2 − 0.5, 3 2 δ − 1 4 M 3 > 1 if 1 < γ < 5 3 . (5.57) Moreover, there exists a constant Λ(C 0 ) such that Z(t) is globally well-defined in [0, +∞) if Z 0 ≤ Λ(C 0 ). Proof. Step 1. First, note that Φ(A, B) =aA 2 + bB 2 − cAB = a A − c 2a B 2 + dB 2 , (5.58) with a =(2α + β), b = 3 2 δ − 3 + m 1 + t , c = δ − 1 2 (1 + t) n−m + 2δ δ − 1 (2α + 3β)(1 + t) m−n−1 , d =(1 + t) −1 3 2 δ − 3 + m − 1 4 (δ − 1) 2 4(2α + β) (1 + t) 2n−2m + 4δ 2 (2α + β) (δ − 1) 2 M 2 1 (1 + t) 2m−2n−2 + 2M 1 δ(1 + t) −1 . Then according to Lemma 10.10 and (5.53), for any ν ∈ (0, 1), one has which implies that 1 2 d dt Z 2 + b m 1 + t (Z 2 − B 2 ) + να(A * ) 2 + (1 − ν)dB 2 + ν b − c 2 4(α + β) B 2 + α 2 k=0 (1 + t) 2γ k \|ϕ∇ k+1 w(t)\| 2 2 + (α + β) 2 k=0 (1 + t) 2γ k \|ϕ∇ k divw(t)\| 2 2 ≤η(A * ) 2 + C(η)((1 + t) 2m−5 + (1 + t) 2m−5−2γ 3 )Z 4 + C 0 (1 + t) 2m−n−3.5 + (1 + t) n−2.5 Z 3 + C 0 (η) (1 + t) 2m−2n−4 + (1 + t) −2 Z 2 .d = (1 + t) −1 d * (m) = (1 + t) −1 3 2 δ − 3 + m − 1 4 M 3 . (5.61) Choose ǫ * = 1 2 min{ 3γ−3 2 , −M 2 −1 2 , 1 10 } > 0. Then it follows from (5.59) that 1 2 d dt Z 2 + b m 1 + t (Z 2 − B 2 ) + να(A * ) 2 + (1 − ν) d * (m) 1 + t B 2 + ν 1 + t d * − 1 4 M 3 α α + β B 2 ≤η(A * ) 2 + C(η)((1 + t) 2m−5+ǫ * + (1 + t) 2m−5−2γ 3 )Z 4 + C 0 (η)(1 + t) −1−ǫ * Z 2 . (5.62) Step 2. On one hand, for fixed δ > 1, a necessary condition to gurantee that the set Π = {(α, β)\|α > 0, 2α + 3β > 0, and M 2 < −1} is not empty is that M 1 < 3 2 − 1 δ . On the other hand, for fixed α, β, γ, δ, there always exists a sufficiently large number m such that d * (m) > 0. Indeed, one needs that d * (m) = 3 2 δ − 3 + m − 1 4 M 3 = − 1 2 M 2 + m − 5 2 > 0. Here for the choice m = n + 0.5 = 3, one has therefore d * = 3 2 δ − 1 4 M 3 . , it hold that 1 + ǫ * − 2(1 − ν * )d * = M 4 + 2ν * d * < − 1, (1 − ν)d * + ν d * − 1 4 M 3 α α + β ≥(1 − ν * )d * . (5.63) Based on this observation, we choose η = να 100 , which, together with (5.53), m = n+0.5 = 3 and (5.58), implies that 1 2 d dt Z 2 + (1 − ν * )b * 1 + t Z 2 + να 2 (1 + t) 2γ 3 \|ϕ∇ 4 w(t)\| 2 2 + α 2 k=0 (1 + t) 2γ k \|ϕ∇ k+1 w(t)\| 2 2 + (α + β) 2 k=0 (1 + t) 2γ k \|ϕ∇ k divw(t)\| 2 2 ≤C(1 + t) 1+ǫ * Z 4 + C 0 (1 + t) −1−ǫ * Z 2 . (5.64) Therefore, d dt Z + (1 − ν * )b * 1 + t Z ≤ C(1 + t) 1+ǫ * Z 3 + C 0 (1 + t) −1−ǫ * Z. (5.65) According to Proposition 2.2, and ǫ * > 0, 1 + ǫ * − 2(1 − ν * )b * < −1, then Z(t) satisfies Z(t) ≤ (1 + t) −(1−ν * )b * exp C 0 −ǫ * (1 + t) −ǫ * − 1 Z −2 0 − 2C t 0 (1 + s) M exp 2C 0 −ǫ * (1 + t) −ǫ * − 1 ds 1 2 , (5.66) where M = 1 + ǫ * − 2(1 − ν * )b * < −1. Moreover, Z(t) is globally well-defined for t ≥ 0 if and only if 0 < Z 0 < 1 2C t 0 (1 + s) M exp 2C 0 −ǫ * (1 + t) −ǫ * − 1 ds 1 2 . (5.67) Moreover, Z(t) ≤ C 0 (1 + t) −(1−ν * )b * for all t ≥ 0, (5.68) which implies that Y k (t) ≤ C 0 (1 + t) −γ k −(1−ν * )b * , and U k (t) ≤ C 0 (1 + t) −δ k −(1−ν * )b * for ∀ t ≥ 0. (5.69) Finally, it follows from (5.64) and (5.68) that for k = 0, 1, 2, 3: (1 + t) 2γ k \|ϕ∇ k+1 w\| 2 ds ≤ C 0 . (5.70) 5.4. Proof of Theorem 5.1 under the condition (P 1 ). For this case, we first choose 2n − 2m = −(1 + ǫ) < −1 for sufficiently small constant 0 < ǫ < 1 2 min 1, 3γ − 3 . Then it follows from Lemma 5.7 that Lemma 5.9. There exist some positive constants b m , C and C 0 such that The proof is routine and thus omitted. The rest of the proof for Theorem 5.1 under the condition (P 1 ) is similar to the one in Subsection 5.3 for the condition (P 0 ). 1 2 d dt Z 2 + b m 1 + t Z 2 + α 2 3 k=0 (1 + t) 2γ k \|ϕ∇ k+1 w(t)\| 2 2 ≤C(1 + t) 2n−4+ǫ Z 4 + C 0 (1 + t) −1−ǫ Z 2 ,(5. 6. Global-in-time well-posedness with compactly supported density under (P 2 ) or (P 3 ) Denote ι = (δ − 1)/(γ − 1). It is always assumed that ι ≥ 2 or = 1. Based on the analysis of the previous section, it remains to estimate the following two terms: L k (W, u) = − ∇φ 2ι · S(∇ k w) − ∇ k (φ 2ι Lw) − φ 2ι L∇ k w · ∇ k w + ∇ k (φ 2ι L u) · ∇ k w ≡: L 1 k + L 2 k + L 3 k , Q k (W, u) = ∇φ 2ι · Q(∇ k w) + ∇ k (∇φ 2ι · Q(w)) − ∇φ 2ι · Q(∇ k w) · ∇ k w + ∇ k ∇φ 2ι · Q( u) · ∇ k w ≡: Q 1 k + Q 2 k + Q 3 k . 6.1. Estimates on L k and Q k . Lemma 6.1. For any suitably small constant η > 0, there are two constants C(η) and C 0 (η) such that L k (W ) ≤η\|φ ι ∇ k+1 w\| 2 2 δ 3,k + C(η)(1 + t) 2ιn−3ι−2−γ k Y 2ι+1 Y k + C 0 (1 + t) 2ιn−3ι−n−1.5−γ k Y 2ι Y k + C 0 (η)(1 + t) 2ιn−3ι−7 Y 2ι , Q k (W,ū) ≤η\|φ ι ∇ k+1 w\| 2 2 δ 3,k + C(η)(1 + t) 2ιn−3ι−2−γ k Y 2ι+1 Y k + C 0 (1 + t) 2ιn−3ι−n−0.5−γ k Y 2ι Y k + C 0 (η)(1 + t) 2ιn−3ι−5 Y 2ι . (6.1) Proof. Now we give only the corresponding estimates on Q k . The estimates on L k can be obtained similarly. Step 1: Estimates on Q k 1 . Direct estimates yield that Q 1 k ≤C\|φ\| 2ι−1 ∞ \|∇φ\| ∞ \|∇ k+1 w\| 2 \|∇ k w\| 2 ≤ C(1 + t) 2ιn−3ι−2−γ k Y 2ι+1 Y k , for k ≤ 2, Q 1 3 ≤C\|φ ι ∇ 4 w\| 2 \|φ\| ι−1 ∞ \|∇φ\| ∞ Y 3 ≤ η\|φ ι ∇ 4 w\| 2 2 + C(η)(1 + t) 2ιn−3ι−2−γ 3 Y 2ι+1 Y 3 . (6.2) Step 2: Estimates on Q k 3 . If k = 0 or 1, one has Q 3 0 ≤C 0 \|φ\| 2ι−2 ∞ \|∇φ\| ∞ \|∇ u\| ∞ \|φ\| 2 \|w\| 2 ≤ C(1 + t) 2ιn−3ι−n−0.5−γ 0 Y 2ι Y 0 , Q 3 1 ≤C\|φ\| 2ι−1 ∞ \|∇φ\| 2 \|∇ 2 u\| ∞ \|∇w\| 2 ≤ C 0 (1 + t) 2ιn−3ι−n−0.5−γ 1 Y 2ι Y 1 . (6.3) For k = 2, decompose Q 3 2 Q 3 2 (L) + Q 3 2 (2, 0), which can be estimated as Q 3 2 (L) C (∇φ 2ι · ∇ 3 u + ∇ 2 φ 2ι · ∇ 2 u) · ∇ 2 w ≤C \|φ\| ∞ \|∇φ\| ∞ \|∇ 3 u\| 2 + \|∇φ\| 2 ∞ \|∇ 2 u\| 2 + \|φ\| ∞ \|∇ 2 φ\| 2 \|∇ 2 u\| ∞ \|φ\| 2ι−2 ∞ \|∇ 2 w\| 2 ≤C 0 (1 + t) 2ιn−3ι−n−1.5−γ 2 Y 2ι Y 2 , (6.4) and Q 3 2 (2, 0) C ∇ 3 φ 2ι · ∇ u · ∇ 2 w ≤ C \|∇φ\| ∞ \|∇φ\| 2 6 \|∇ u\| 6 + \|φ\| ∞ \|∇ 2 φ\| 6 \|∇φ\| 6 \|∇ u\| 6 + \|φ\| 2 ∞ \|∇ 3 φ\| 2 \|∇ u\| ∞ \|φ\| 2ι−3 ∞ \|∇ 2 w\| 2 ≤C 0 (1 + t) 2ιn−3ι−n−0.5−γ 2 Y 2ι Y 2 . (6.5) For k = 3, set Q 3 3 Q 3 3 (L) + Q 3 3 (3, 0), where Q 3 3 (L) =C ∇φ 2ι · ∇ 4 u + ∇ 2 φ 2ι · ∇ 3 u + ∇ 3 φ 2ι · ∇ 2 u · ∇ 3 w ≤C \|φ\| 2 ∞ \|∇φ\| ∞ \|∇ 4 u\| 2 + \|φ\| ∞ ∇φ\| 2 ∞ \|∇ 3 u\| 2 + \|φ\| 2 ∞ \|∇ 2 φ\| 6 \|∇ 3 u\| 3 + \|∇φ\| 2 ∞ \|∇φ\| 2 + \|φ\| ∞ \|∇ 2 φ\| 2 \|∇φ\| ∞ + \|φ\| 2 ∞ \|∇ 3 φ\| 2 \|∇ 2 u\| ∞ \|φ\| 2ι−3 ∞ \|∇ 3 w\| 2 ≤C 0 (1 + t) 2ιn−3ι−n−1.5−γ 3 Y 2ι Y 3 , Q 3 3 (3, 0) =C ∇ 4 φ 2ι · ∇ u · ∇ 3 w = C ∇ 3 φ 2ι · ∇ 2 u · ∇ 3 w + ∇ u · ∇ 4 w) ≤C 0 (1 + t) 2ιn−3ι−n−1.5−γ 3 Y 2ι Y 3 + Q 3 3 (A), (6.6) with Q 3 3 (A) C ∇ 3 φ 2ι · ∇ u · ∇ 4 w ≤C \|∇ 3 φ\| 2 \|φ\| ι−1 ∞ + \|∇φ\| 3 \|∇ 2 φ\| 6 \|φ\| ι−2 ∞ \|∇ u\| ∞ \|φ ι ∇ 4 w\| 2 + Q 3 3 (B) ≤η\|φ ι ∇ 4 w\| 2 2 + C 0 (η)(1 + t) 2ιn−3ι−5 Y 2ι + Q 3 3 (B),(6.7) where the term Q 3 3 (B) can be estimated by using integration by parts again, Q 3 3 (B) C φ 2ι−3 ∇φ · ∇φ · ∇φ · ∇ u · ∇ 4 w ≤C \|∇φ\| 3 ∞ \|φ\| 2ι−4 ∞ \|∇φ\| 2 + \|∇ 2 φ\| 6 \|∇φ\| 2 6 \|φ\| 2ι−3 ∞ \|∇ u\| ∞ \|∇ 3 w\| 2 + C(1 + t) 2ιn−3ι−n−1.5−γ 3 Y 2ι Y 3 ≤ C 0 (1 + t) 2ιn−3ι−n−0.5−γ 3 Y 2ι Y 3 . (6.8) Then (6.3)-(6.8) give that Q 3 k ≤η\|φ ι ∇ k+1 w\| 2 2 + C 0 (1 + t) 2ιn−3ι−n−0.5−γ k Y 2ι Y k + C 0 (η)(1 + t) 2ιn−3ι−5 Y 2ι . (6.9) Step 3: Estimates on Q 2 k . For k = 1, one has Q 2 1 =C φ 2ι−1 ∇ 2 φ + φ 2ι−2 ∇φ · ∇φ · ∇w · ∇w ≤C\|φ\| 2ι−2 ∞ \|φ\| ∞ \|∇ 2 φ\| 6 \|∇w\| 3 + \|∇φ\| 2 ∞ \|∇w\| 2 \|∇w\| 2 ≤C(1 + t) 2ιn−3ι−2−γ 1 Y 2ι+1 Y 1 .Q 2 2 =C ∇ 3 φ 2ι · ∇w + ∇ 2 φ 2ι · ∇ 2 w · ∇ 2 w ≤C \|φ\| ∞ \|∇φ\| ∞ \|∇w\| ∞ \|∇ 2 φ\| 2 + \|φ\| 2 ∞ \|∇w\| ∞ \|∇ 3 φ\| 2 + \|∇φ\| 2 ∞ \|∇φ\| 3 \|∇w\| 6 + \|φ\| ∞ \|∇φ\| 2 ∞ \|∇ 2 w\| 2 + \|∇ 2 w\| 6 \|∇ 2 φ\| 3 \|φ\| 2 ∞ \|φ\| 2ι−3 ∞ \|∇ 2 w\| 2 ≤ C(1 + t) 2ιn−3ι−2−γ 2 Y 2ι+1 Y 2 . (6.11) For k = 3, using integration by parts, one can get Q 2 3 =C ∇ 4 φ 2ι · ∇w + ∇ 3 φ 2ι · ∇ 2 w + ∇ 2 φ 2ι · ∇ 3 w · ∇ 3 w ≤C\|φ\| 2ι−3 ∞ \|∇φ\| 6 \|∇φ\| ∞ \|∇φ\| 6 + \|φ\| ∞ \|∇ 2 φ\| 6 \|∇ 2 w\| 6 \|∇ 3 w\| 2 + C\|φ ι ∇ 3 w\| 6 \|∇ 2 w\| 3 \|φ\| ι−1 ∞ \|∇ 3 φ\| 2 + C\|φ\| ι−1 ∞ (\|φ\| ι−1 ∞ \|∇φ\| 2 ∞ \|∇ 3 w\| 2 + \|∇ 2 φ\| 3 \|φ ι ∇ 3 w\| 6 )\|∇ 3 w\| 2 ≤η\|φ ι ∇ 4 w\| 2 2 + C(η)(1 + t) 2ιn−3ι−2−γ 3 Y 2ι+1 Y 3 + Q 2 3 (A), (6.12) with Q 2 3 (A) =C ∇ 3 φ 2ι · ∇w · ∇ 4 w ≤C \|∇ 3 φ\| 2 \|φ\| ∞ + \|∇φ\| 3 \|∇ 2 φ\| 6 \|φ\| ι−2 ∞ \|∇w\| ∞ \|φ ι ∇ 4 w\| 2 + Q 2 3 (B) ≤C(η)(1 + t) 2ιn−3ι−2−γ 3 Y 2ι+1 Y 3 + η\|φ ι ∇ 4 w\| 2 2 + Q 2 3 (B),(6.13) where the term Q 2 3 (B) can be estimated by using integration by parts again, Q 2 3 (B) =C φ 2ι−3 ∇φ · ∇φ · ∇φ · ∇w · ∇ 4 w ≤C \|∇φ\| 2 ∞ \|∇φ\| 2 + \|∇ 2 φ\| 6 \|∇φ\| 3 \|φ\| ∞ \|φ\| 2ι−4 ∞ \|∇φ\| ∞ \|∇w\| ∞ \|∇ 3 w\| 2 + C\|φ\| 2ι−3 ∞ \|∇φ\| 3 ∞ \|∇ 2 w\| 6 \|∇ 3 w\| 2 ≤C(1 + t) 2ιn−3ι−2−γ 3 Y 2ι+1 Y 3 . (6.14) Then the estimates (6.10)-(6.13) lead to Q 2 k ≤η\|φ ι ∇ k+1 w\| 2 2 + C(η)(1 + t) 2ιn−3ι−2−γ k Y 2ι+1 Y k . (6.15) 6.2. Derivation of a global-in-time well-posedness for an ordinary differential inequality. Then, based on the above energy estimates, one can derive the following lemma. Lemma 6.2. There exist some positive constants a = r + n, C 0 and C such that dY (t) dt + a 1 + t Y (t) ≤ C 0 (1 + t) −1−η Y (t) + C(1 + t) 2ιn−3ι−1+2θη−η Y 2ι+1 ,(6.16) holds for any sufficiently small constant 0 < η < 3ι(γ−1) 2ι−1 . Moreover, there exists a constant Λ(C 0 ) such that Y (t) is globally well-defined in [0, +∞) if Y 0 ≤ Λ(C 0 ). Proof. First, it follows from Lemmas 5.2, 6.1 and (5.5) that 1 2 d dt Y 2 k + k + r 1 + t Y 2 k + 1 2 α \|φ ι ∇ k+1 w\| 2 ≤CY Y k (1 + t) −γ k −2 + C(1 + t) n−2.5 Y Y 2 k + C(1 + t) 2ιn−3ι−2−γ k Y 2ι+1 Y k + C 0 (1 + t) 2ιn−3ι−n−0.5−γ k Y 2ι Y k + C 0 (1 + t) 2ιn−3ι−5 Y 2ι . (6.17) Second, multiplying (6.17) by (1 + t) 2γ k and simplifying by Y give dY (t) dt + a 1 + t Y (t) ≤C 0 (1 + t) −2 Y + C(1 + t) n−2.5 Y 2 + C(1 + t) 2ιn−3ι−2 Y 2ι+1 + C 0 (1 + t) 2ιn−3ι−n−0.5 Y 2ι + C 0 (η)(1 + t) 2ιn−3ι−2n+1 Y 2ι−1 ≤C 0 (1 + t) −1−η Y (t) + C(1 + t) 2ιn−3ι−1+2θη−η Y 2ι+1 (6.18) for any sufficiently small constant 0 < η < 1. A sufficient condition for the global existence of solutions to (6.18) is K a,ι,η = 2ιn − 3ι − 2 + 1 + 2ιη − η − 2ιa < −1. Indeed, it follows from the definition of a and ι ≥ 2 or ι = 1 that K a,ι,η =    −2ι − 1 − η + 2ιη < −1 if γ ≥ 5 3 ; 3ι(1 − γ) + η(2ι − 1) < −1 if 1 < γ < 5 3 ,(6.19) for the sufficiently small constant 0 < η < 3ι(γ−1) 2ι−1 . Then, (6.18)-(6.19), Proposition 2.2 and Lemma 5.2 give Y (t) ≤ (1 + t) −a exp C 0 η (1 − (1 + t) −η ) Z −2ι 0 − 2ιC t 0 (1 + s) Ka,ι,η exp 2ιC 0 η 1 − (1 + s) −η ds 1 2ι . (6.20) Thus, Y (t) is defined for t ≥ 0 if and only if 0 < Y 0 < 1 2ιC t 0 (1 + s) Ka,ι,η exp 2ιC 0 η 1 − (1 + s) −η ds 1 2ι . (6.21) Moreover, it holds that Y (t) ≤ C 0 (1 + t) −r−n , and Y k (t) ≤ K(1 + t) −k−r for all t ≥ 0. (6.22) Now we are ready to prove Theorem 5.1. Step 1: Estimates on ϕ∇ 4 w. Note that (6.17) and (6.22) for k = 3 imply that 1 2 d dt Y 2 3 + 3 + r 1 + t Y 2 3 + 1 2 α \|φ ι ∇ 4 w\| 2 ≤ C(1 + t) −8.5−3r .(6.23) Multiplying (6.23) by (1 + t) 2e on both sides gives 1 2 i. e., e ≤ 3 + r, one can obtain d dt ((1 + t) 2e Y 2 3 ) + 3 + r − e 1 + t ((1 + t) 2e Y 2 3 ) + 1 2 α(1 + t) 2e \|φ ι ∇ 4 w\| 2 ≤C(1 + t) −8.5−3r+2e .(1 + t) 2e Y 2 3 + 1 2 α t 0 (1 + t) 2e \|φ ι ∇ 4 w\| 2 ds ≤ C 0 . (6.25) Step 2: Estimates on ϕ = φ ι . Here we always assume that ι ≥ 2 in the rest of this proof. Due to (6.22), one can get \|∇ k ϕ\| 2 ≤ C(1 + t) −1.5ι−rι+1.5−k . (6.26) Applying ∇ 3 to (5.1) 1 , multiplying by ∇ 3 ϕ and integrating over R 3 , then similar to Lemma 5.6, one can get 1 2 d dt \|∇ 3 ϕ\| 2 2 + 3δ 2(1 + t) \|∇ 3 ϕ\| 2 2 ≤C 0 (1 + t) −r−2.5 \|∇ 3 ϕ\| 2 2 + C(1 + t) −0.75ι−0.5rι−r−3.25 \|∇ 3 ϕ(t)\| 3/2 2 + C(1 + t) −1.5ι−rι−3.5 \|∇ 3 ϕ\| 2 + C\|ϕ∇ 4 w\| 2 \|∇ 3 ϕ(t)\| 2 . (6.27) Then, multiplying (6.27) by (1 + t) 2p and simplifying by G 2 = (1 + t) 2p \|∇ 3 ϕ\| 2 2 lead to 1 2 dG 2 dt + 3 2 δ − p 1 + t G 2 ≤C 0 (1 + t) −r−2.5 + (1 + t) 2p−2e G 2 + C(1 + t) −0.75ι−0.5rι−r−3.25+0.5p G 3/2 + C(1 + t) −1.5ι−rι−3.5+p G + C(1 + t) 2e \|ϕ∇ 4 w\| 2 2 . (6.28) Choose p satisfying the following conditions: 3 2 δ − p > 0, −0.75ι − 0.5rι − r − 3.25 + 0.5p < − 1, 2p − 2e < −1, −1.5ι − rι − 3.5 + p < − 1. Then (6.28) yields (1 + t) 2p \|∇ 3 ϕ\| 2 2 ≤ C 0 . Global-in-time well-posedness without compactly supported assumption In this section, we will extend the global-in-time well-posedness in Theorem 5.1 to the more general case whose initial mass density ρ 0 is still small, but is not necessary to be compactly supported, which can be stated as: for any constant s ′ ∈ [2, 3). Moreover, when (P 2 ) holds, the smallness assumption on ϕ 0 could be removed. In the following subsections, we will prove Theorem 7.1 under the condition (P 0 ). The proof for other cases is similar, and so is omitted. 7.1. Existence. According to Theorem 5.1, for the initial data (ϕ R 0 , φ R 0 , w 0 ) = (ϕ 0 F (\|x\|/R), φ 0 F (\|x\|/R), 0), there exists the unique global regular solution (ϕ R , φ R , w R ) satisfying: \|∇ k φ R (t)\| 2 + \|∇ k w R (t)\| 2 ≤C 0 (1 + t) −(1−ν * )b * +2.5−k ; (1 + t) k−2.5 ϕ R ∇ k+1 w R L 2 L 2 t ≤ C 0 , \|∇ k ϕ R (t)\| 2 ≤C 0 (1 + t) −(1−ν * )b * +3−k ,(7.2) for any positive time t > 0, with C 0 independent of R. Due to (7.2) and the following relations:                      ϕ R t = −(w R + u) · ∇ϕ R − δ − 1 2 ϕdiv(w R + u), φ R t = −(w R + u) · ∇φ R − γ − 1 2 φ R div(w R + u), w R t = −w R · ∇w R − γ − 1 2 φ R ∇φ R − (ϕ R ) 2 Lw R +∇(ϕ R ) 2 · Q(w R + u) − u · ∇w R − w R · ∇ u − (ϕ R ) 2 L u,(7.3) it holds that for any finite constant R 0 > 0 and finite time T > 0, ϕ R t H 2 (B R 0 ) + φ R t H 2 (B R 0 ) + w R t H 1 (B R 0 ) + t 0 ∇ 2 w R t 2 L 2 (B R 0 ) ds ≤ C 0 (R 0 , T ), (7.4) for 0 ≤ t ≤ T , where the constant C 0 (R 0 , T ) > 0 depending on C 0 , R 0 and T . Since (7.2) and (7.4) are independent of R, there exists a subsequence of solutions (still denoted by) (ϕ R , φ R , w R ) converging to a limit (ϕ, φ, w) in the sense: (ϕ R , φ R , w R ) → (ϕ, φ, w) strongly in C([0, T ]; H 2 (B R 0 )). (7.5) For k = 0, 1, 2, 3, denote a k = −(1 − ν * )b * + 2.5 − k, b k = −(1 − ν * )b * + 3 − k, c k = k − 2.5. Again due to (7.2), there exists a subsequence of solutions (still denoted by) (ϕ R , φ R , w R ) converging to the same limit (ϕ, φ, w) as above in the following weak* sense (for k = 0, 1, 2, 3): (1 + t) b k ϕ R ⇀ (1 + t) b k ϕ weakly* in L ∞ ([0, T ]; H 3 (R 3 )), (1 + t) a k φ R , w R ⇀ (1 + t) a k φ, w) weakly* in L ∞ ([0, T ]; H 3 (R 3 )). (7.6) Combining the strong convergence in (7.5) and the weak convergence in (7.6) shows that (ϕ, φ, w) also satisfies the corresponding estimates (7.2) and (for k = 0, 1, 2, 3): (1 + t) c k ϕ R ∇ k+1 w R ⇀ (1 + t) c k ϕ∇ k+1 w weakly in L 2 ([0, T ] × R 3 ). (7.7) It is then obvious that (ϕ, W ) is a weak solution to problem (4.1) in the sense of distribution. 7.2. Uniqueness. Let (ϕ 1 , W 1 ) and (ϕ 2 , W 2 ) be two solutions to (4.1) satisfying the uniform a priori estimates (7.2). Set F N = F (\|x\|/N ), and ϕ = ϕ 1 − ϕ 2 , W = (φ, w) = (φ 1 − φ 2 , w 1 − w 2 ), ϕ N = ϕF N , W N = W F N = (φ N , w N ), then (ϕ N , W N ) solves the following problem                                                  ϕ N t + (w 1 + u) · ∇ϕ N + w N · ∇ϕ 2 + δ − 1 2 (ϕ N div(w 2 + u) + ϕ 1 divw N ) = ϕ(w 1 + u) · ∇F N + δ − 1 2 ϕ 1 w · ∇F N , W N t + 3 j=1 A * j (W 1 , u)∂ j W N + ϕ 2 1 L(w N ) = − 3 j=1 A j (W N )∂ j W 2 − ϕ N (ϕ 1 + ϕ 2 )L(w 2 ) + F N H(ϕ 1 ) − H(ϕ 2 ) · Q(W 2 ) +H(ϕ 1 ) · Q(W N ) − B(∇ u, w N ) − D(ϕ N (ϕ 1 + ϕ 2 ), ∇ 2 u) + 3 j=1 A * j (W 1 , u)W ∂ j F N + ϕ 2 1 (L(w N ) − F N L(w)) − H(ϕ 1 ) · Q(W ) · ∇F N , (ϕ N , W N )\| t=0 = (0, 0), x ∈ R 3 , (ϕ N , W N ) → (0, 0) as \|x\| → +∞, t > 0. (7.8) It is not hard to obtain that d dt \|ϕ N \| 2 2 ≤ C(\|∇w 1 \| ∞ + \|∇w 2 \| ∞ + \|∇ u\| ∞ )\|ϕ N \| 2 2 + C\|∇ϕ 2 \| ∞ \|w N \| 2 \|ϕ N \| 2 + C\|ϕ 1 divw N \| 2 \|ϕ N \| 2 + I 1 N , d dt \|W N \| 2 2 + 1 2 α ϕ 2 1 \|∇w N \| 2 ≤ C 1 + W 1 2 3 + W 2 2 3 + ∇ u 2 3 \|W N \| 2 2 + C \|ϕ 2 ∇ 2 w 2 \| ∞ \|w N \| 2 + \|∇ 2 w 2 \| 3 \|ϕ 1 ∇w N \| 2 + \|∇ϕ 1 \| ∞ \|w N \| 2 \|ϕ N \| 2 + I 2 N ,(7.9) where the error terms I 1 N -I 2 N are given and estimated by I 1 N = N ≤\|x\|≤2N 1 N (\|w 1 \| + \| u\|)\|ϕ\| 2 + \|ϕ\|\|w\|\|ϕ 1 \| dx ≤C (\|w 1 \| ∞ + \|∇ u\| ∞ ) ϕ L 2 (R 3 /B N ) + \|ϕ 1 \| ∞ w L 2 (R 3 /B N ) ϕ L 2 (R 3 /B N ) , I 2 N = N ≤\|x\|≤2N 1 N (\|W 1 \| + \| u\|)\|W \| 2 + \|ϕ 1 \| 2 \|w\| 2 + \|ϕ 1 \| 2 \|w\|\|∇w\| dx + N ≤\|x\|≤2N 1 N \|ϕ 1 + ϕ 2 \|\|ϕ\|\|∇w 2 \| + \|∇ϕ 1 \|\|ϕ 1 ∇w\| \|w\|dx ≤C(\|W 1 \| ∞ + \|∇ u\| ∞ ) W 2 L 2 (R 3 /B N ) + C\|ϕ 1 \| 2 ∞ w 2 L 2 (R 3 /B N ) + C(\|ϕ 1 \| ∞ + \|∇ϕ 1 \| ∞ ) w L 2 (R 3 /B N ) \|ϕ 1 ∇w\| 2 + C\|ϕ 1 + ϕ 2 \| ∞ \|w 2 \| ∞ w L 2 (R 3 /B N ) ϕ L 2 (R 3 /B N ) , for 0 ≤ t ≤ T . Using the same arguments as in the derivation of (4.51)-(4.56), and letting Second, in terms of (ϕ, φ, u), one has     Λ N (t) = \|W N (t)\| 2 2 + α 2C \|ϕ N (t)\| 2 2 , one can have        d dt Λ N (t) + α 2 \|ϕ 1 ∇w N (t)\| 2 2 ≤ J(t)Λ N (t) + I 1 N (t) + I 2 N (t),   φ t + u · ∇φ + γ − 1 2 φdivu = 0, u t + u · ∇u + γ − 1 2 φ∇φ + ϕ 2 Lu = ∇ϕ 2 · Q(u). (8.2) Case 1: 1 < min{γ, δ} ≤ 3. We assume that 1 < γ ≤ 3, and the other cases can be dealt with similarly. Since ρ = φ 2 γ−1 and 2 γ−1 ≥ 1, so ρ ∈ C 1 ((0, T ) × R 3 ). Thus, (ρ, u) solves the Cauchy problem (1.1)-(1.7) in the classical sense. Case 2: min{γ, δ} > 3. For definiteness, we assume that γ > 3, and the other cases could be dealt with similarly. Since φ = ρ γ−1 2 Multiplying (8.2) 1 by ∂ρ ∂φ (t, x) = 2 γ−1 φ 3−γ γ−1 (t, x) ∈ C((0, T ) × R 3 )and 2 γ−1 > 0, so ρ ∈ C((0, T ) × R 3 ). It follows from (8.2) 2 and γ−1 2 > 1 that ρ t + div(ρu) = 0, when ρ(t, x) > 0. (8.3) Now the whole space could be divide into two domains: the vacuum domain V 0 and its complement V p (t). Then it holds that for any smooth f , t 0 Vp(s) ρf t + ρu · ∇f dxds = Vp(0) ρ 0 f (0, x)dx, (8.4) which means that t 0 R 3 ρf t + ρu · ∇f dxdt = R 3 ρ 0 f (0, x)dx.ρ(t, x) ≥ 0, ∀(t, x) ∈ [0, T ] × R 3 . That is to say, (ρ, u) satisfies the Cauchy problem (1.1)-(1.7) in the sense of distributions and has the regularities shown in Definition 1.2, which means that the Cauchy problem(1.1)-(1.7) has a unique regular solution (ρ, u). Proof of Theorem 1.6 In this section, we prove Theorem 1.6 by modifying the proof of Theorem 1.4. For the Cauchy problem (1.1)-(1.2) with (1.6)-(1.7), if divT = αρ δ △u, it can be rewritten into                    ϕ t + w · ∇ϕ + δ − 1 2 ϕdivw = − u · ∇ϕ − δ − 1 2 ϕdiv u, W t + 3 j=1 A j (W )∂ j W − ϕ 2 △w = G * (W, ϕ, u), (ϕ, φ, w)(t = 0, x) = (ϕ 0 , φ 0 , 0), x ∈ R 3 , (ϕ, φ, w) → (0, 0, 0) as \|x\| → ∞ for t ≥ 0, (9.1) where G * (W, ϕ, u) = − B(∇ u, W ) − 3 j=1 u (j) ∂ j W − 0 ϕ 2 △ u . (9.2) Let γ k = k − 5 2 and δ k = k − m, where m will be determined in the end of this section, and Z(t) be defined in (5.4). Then one has Lemma 9.1. There exist some positive constants b m , such that dZ dt (t) + b m 1 + t Z(t) ≤    C(1 + t) 2m−5 Z 3 (t) + C 0 (1 + t) 2m−9 Z if m ∈ [3.5, 4), C(1 + t) 2m−5 Z 3 (t) + C 0 (1 + t) 5−2m Z if m ∈ (3, 3.5),(9. 3) where b m is given by (5.54). The proof is similar to that of Lemma 5.7 with Q k = 0. So the details are omitted here. 9.1. We treat the case m ∈ [3.5, 4) first. Lemma 9.2. Let δ > 1, γ > 4/3 and m ∈ [3.5, 4). For the following problem      dZ(t) dt + b m 1 + t Z(t) = C 1 (1 + t) 2m−5 Z 3 (t) + C 2 (1 + t) 2m−9 Z, Z(x, 0) = Z 0 , (9.4) there exists a constant Λ such that Z(t) is globally well-defined in [0, +∞) if Z 0 ≤ Λ. Proof. Set M = 2m − 5 − 2b m . Due to Proposition 2.2 and its proof, a sufficient condition for the global existence of solution is M < −1, and 2m − 8 < 0, which requires that M < −1 if γ ≥ 5 3 , or 4 3 ≤ 2 3 m − 1 < γ < 5 3 . (9.5) Therefore, by choosing Z 0 small enough, one can obtain the global existence of the solution to (9.4). Moreover, Z(t) ≤ C 0 (1 + t) −bm for all t ≥ 0. (9.6) Next we prove Theorem 1.6 for the cases γ > 4/3. First, according to Lemma 9.1, when m ∈ [3.5, 4), the following inequality holds dZ dt (t) + b m 1 + t Z(t) ≤C(1 + t) 2m−5 Z 3 (t) + C 0 (1 + t) 2m−9 Z. (9.7) Then it follows from Lemma 9.2 that there exists a constant Λ such that Z(t) is globally well-defined in [0, +∞) if Z 0 ≤ Λ, and (9.6) holds. According to Lemma 5.5 and its proof with Q k = 0, it is easy to see that α 3 k=1 t 0 (1 + t) 2γ k \|ϕ∇ k+1 w\| 2 ds ≤ C 0 , for t ≥ 0. (9.8) The rest of the proof is similar to that for Theorem 1.4, and so is omitted. 9.2. We now treat the case m ∈ (3, 3.5). Lemma 9.3. Let δ > 1, γ > 1 and m ∈ (3, 3.5). For the following problem      dZ dt (t) + b m 1 + t Z(t) = C(1 + t) 2m−5 Z 3 (t) + C 0 (1 + t) 5−2m Z, Z(x, 0) = Z 0 , (9.9) there exists a constant Λ such that Z(t) is globally well-defined in [0, +∞) if Z 0 ≤ Λ. Proof. Set M = 2m − 5 − 2b m . Due to Proposition 2.2 and its proof, a sufficient condition for a global solution with small data is M < −1, and 6 − 2m < 0, which means that m must belong to (3, 3.5), and requires that M < −1 if γ ≥ 5 3 , or 1 < 2m 3 − 1 < γ < 5 3 . (9.10) Therefore, by choosing Z 0 small enough, one can obtain the global existence of the solution to (9.9). Moreover, (9.6) still holds. Next we prove Theorem 1.6 for the cases γ > 1. First, according to Lemma 9.1, when m ∈ (3, 3.5), it holds that dZ dt (t) + b m 1 + t Z(t) ≤C(1 + t) 2m−5 Z 3 (t) + C 0 (1 + t) 5−2m Z. (9.11) Then from Lemma 9.3, there exists a constant Λ such that Z(t) is globally well-defined in [0, +∞) if Z 0 ≤ Λ, and (9.6) holds. According to Lemma 5.5 and its proof with Q k = 0, it is easy to see that (9.8) holds. The rest of the proof is similar to that for Theorem 1.4, and so is omitted. Appendix In the first part of this appendix, we list some lemmas which were used frequently in the previous sections. The rest of the appendix will be devoted to show the proofs of Propositions 2.1-2.2 and some special Sobolev inequalities. 10.1. Some lemmas. The first one is the well-known Gagliardo-Nirenberg inequality. Lemma 10.1. [13] For p ∈ [2, 6], q ∈ (1, ∞), and r ∈ (3, ∞), there exists some generic constant C > 0 that may depend on q and r such that for f ∈ H 1 (R 3 ), and g ∈ L q (R 3 ) ∩ D 1,r (R 3 ), it holds that \|f \| p p ≤ C\|f \| Some common versions of this inequality can be written as \|u\| 6 ≤ C\|u\| D 1 , \|u\| ∞ ≤ C ∇u 1 , \|u\| ∞ ≤ C u W 1,r , for r > 3. where C is a positive constant that may depend on p. The next one can be found in Majda [23]. ∀s ≥ 1, if f, g ∈ W s,a ∩ W s,b (R 3 ), then it holds that \|∇ s (f g) − f ∇ s g\| r ≤ C s \|∇f \| a \|∇ s−1 g\| b + \|∇ s f \| b \|g\| a ,(10. 4) \|∇ s (f g) − f ∇ s g\| r ≤ C s \|∇f \| a \|∇ s−1 g\| b + \|∇ s f \| a \|g\| b , (10.5) where C s > 0 is a constant depending only on s, and ∇ s f (s > 1) is the set of all ∇ ζ x f with \|ζ\| = s. Here ζ = (ζ 1 , ζ 2 , ζ 3 ) ∈ R 3 is a multi-index. Next, the interpolation estimate, product estimate, composite function estimate and so on are given in the following four lemmas. (1) For functions f, g ∈ H s ∩ L ∞ and \|ν\| ≤ s, there exists a constant C s depending only on s such that ∇ ν (f g) s ≤ C s (\|f \| ∞ \|∇ s g\| 2 + \|g\| ∞ \|∇ s f \| 2 ).(10.7) (2) Assume that g(u) is a smooth vector-valued function on Ω, u(x) is a continuous function with u ∈ H s ∩ L ∞ , u(x) ∈ Ω 1 and Ω 1 ⊆ Ω. Then for s ≥ 1, there exists a constant C s depending only on s such that \|∇ s g(u)\| 2 ≤ C s ∂g ∂u s−1,Ω 1 \|u\| s−1 ∞ \|∇ s u\| 2 . (10.8) As a consequence of the Aubin-Lions Lemma, one has (c.f. [29]), Lemma 10.8. [29] Let X 0 , X and X 1 be three Banach spaces satisfying X 0 ⊂ X ⊂ X 1 . Suppose that X 0 is compactly embedded in X and that X is continuously embedded in X 1 . (1) Let G be bounded in L p (0, T ; X 0 ) with 1 ≤ p < +∞, and ∂G ∂t be bounded in L 1 (0, T ; X 1 ). Then G is relatively compact in L p (0, T ; X). (2) Let F be bounded in L ∞ (0, T ; X 0 ) and ∂F ∂t be bounded in L q (0, T ; X 1 ) with q > 1. Then F is relatively compact in C(0, T ; X). The following lemma is useful to improve weak convergence to strong convergence. Lemma 10.9. [23] If the function sequence {w n } ∞ n=1 converges weakly to w in a Hilbert space X, then it converges strongly to w in X if and only if w X ≥ lim sup n→∞ w n X . The last lemma will be used in the proof shown in Section 5. where the term J * can be controlled by J * ≤ η\|ϕ∇ 4 w\| 2 2 + C(η)(1 + t) 2m−5−2γ 3 Z 4 , for any constant η > 0 small enough, and the constant C(η) > 0. Proof. According to the definition of div, one directly has \|ϕ∇ 3 divw\| 2 2 = 3 i=1 \|ϕ∇ 3 ∂ i w (i) \| 2 2 + 2 3 i,j=1, i<j ϕ 2 ∇ 3 ∂ i w (i) · ∇ 3 ∂ j w (j) .(10.10) Via the integration by parts, one can obtain that J ij = ϕ 2 ∇ 3 ∂ i w (i) · ∇ 3 ∂ j w (j) = ϕ 2 ∇ 3 ∂ i w (j) · ∇ 3 ∂ j w (i) + ∂ i ϕ 2 ∇ 3 ∂ j w (i) ∇ 3 w (j) − ∂ j ϕ 2 ∇ 3 ∂ i w (i) ∇ 3 w (j) . (10.11) It is easy to see that J * ij = ∂ i ϕ 2 ∇ 3 ∂ j w (i) ∇ 3 w (j) − ∂ j ϕ 2 ∇ 3 ∂ i w (i) ∇ 3 w (j) ≤C\|∇ϕ\| ∞ \|ϕ∇ 4 w\| 2 \|∇ 3 w\| 2 ≤η\|ϕ∇ 4 w\| 2 2 + C(η)(1 + t) 2m−5−2γ 3 Z 4 , Proof. (I) Let G(t, x) = ∇ u(t, x) and G 0 (x 0 ) = ∇u 0 (x 0 ). Then G(t, X(t; x 0 )) = I d + tG 0 (x 0 ) −1 G 0 (x 0 ). Note that I d + tG 0 (x 0 ) −1 = det(I d + tG 0 ) −1 (adj(I d + tG 0 )) ⊤ , where adj(I d + tG 0 ) stands for the adjugate of (I d + tG 0 ). Then it holds that \|∇ x u\| ∞ ≤ (1 + t\|∇u 0 \| ∞ ) d−1 (1 + tκ) d \|∇ x 0 u 0 \| ∞ . (10.13) In the right-hand side term, one has the norm of the following terms to estimate: • a)J 1 = (I d + tG 0 ) −1 Λ k (I d + tG 0 ) −k−1 ; • b)J 2 = ∇ j x G(I d + tG 0 ) j−s ∐ k i =1 t∇ k i −1 G 0 (I d + tG 0 ) −k , k i =1 (k i − 1) = k − j, where s is the number of k j = 1. For each of these terms, it needs to apply first the induction hypothesis and consider the L ∞ -norm in space for the terms with t. It needs to use IH(j) for j ≤ k − 1 to show that b) is a product of (∇ j G 0 ) β j with j jβ j = k and of tI d , I d + tG 0 , (I d + tG 0 ) −1 , such that the L ∞ -norm of the terms with t is bounded by a constant times (1 + t) −(k+2) . Then one can find an upper bound in L 2 -norm for ∐ 1≤j≤k ∇ j G 0 β j with j jβ j = k, by using Gagliardo-Nirenberg inequality. Then one gets for j jβ j = k: \|J 1 \| 2 ≤ C(1 + t) −k−2 \|Λ k \| 2 ≤ C(1 + tκ) −k−2 \| ∐ ∇ j G 0 β j \| 2 ≤C(1 + t) −k−2 c k+1 0 , \|J 2 \| 2 ≤ C(1 + t) −k+j \|∇ j x G ∐ k i =1 ∇ k i −1 G 0 \| 2 ≤C(1 + t) −k−2 c k+1 0 . To conclude, one obtains the upper bound in IH(k) which depends on κ, \|G 0 \| ∞ , and \|∇ k G 0 \| 2 for 1 ≤ k ≤ m, that is to say on κ and u 0 Ξ . Finally, it needs to make a change of variables to obtain: \|∇ j x G(t, ·)\| 2 ≤ (1 + t) d 2 \|∇ j x G(t, X(t; ·))\| 2 . Therefore (ii) is true for l ∈ N, l ≥ 2 since ∇ x G = ∇ 2 u. Then by interpolation one obtains the result for all l ∈ R, l ≥ 2. (III) Since m − 1 > d 2 , one gets that ∇ 2 u 0 ∈ L ∞ . Thus ∇ 2 u ∈ L ∞ , and ∇ x G(t, X(t; x 0 )) = −(I d + tG 0 ) −1 ∇G 0 (x 0 )(I d + tG 0 ) −2 . (10.14) Using the estimates obtained in (I), we obtain: \|∇ x G(t, X(t; x 0 ))\| ∞ = O((1 + t) −3 ). (IV) If u 0 (0) = 0, then it is obvious that u(t, 0) = 0 for any t ≥ 0, and u(t, x) = u(t, 0) + ∇ u(t, ax) · x = ∇ u(t, ax) · x, for any (t, x) ∈ [0, T ] × R 3 , where a ∈ [0, 1] is some constant. Thus (2.2) is proved. Proof of Proposition 2.2. Proof. The solution Z(t) of the Cauchy problem (2.4) can be solved as: Z(t) = (1 + t) −b exp C 2 D 2 +1 (1 + t) D 2 +1 − 1 Z −(a−1) 0 − (a − 1)C 1 t 0 (1 + s) M exp (a−1)C 2 D 2 +1 (1 + t) D 2 +1 − 1 ds (1 + t) D 2 +1 − 1 ds in the presence of vacuum region, there is another possibility of finite singularity defined as follows. Definition 1.1. The non-empty open set V ⊂ R 3 is called a hyperbolic singularity set of Theorem 1 . 6 . 16Let (1.20) hold with β = 0. If the viscous term divT is given by αρ δ △u, then under the initial conditions shown in Theorem 1.4, there exists a unique regular solution (ρ, u) in [0, T ] × R 3 to the Cauchy problem (1.1)-(1.2) with (1.6)-(1.7). Moreover, if 1 < min(γ, δ) ≤ 3, the solution (ρ, u) solves the Cauchy problem (1.1)-(1.2) with (1.6)-(1.7) in [0, T ] × R 3 classically. Lemma 3. 1 . 1Let 1 < min{γ, δ} ≤ 2, and (ρ, u) be the regular solution obtained in Theorem 1.1 with \|P(0)\| > 0, then Lemma 4 . 1 . 41Let η > 0 and (A1)-(A3) in Theorem 4.1 hold. Then there exists a unique classical solution . Assume that the initial data (ϕ 0 , φ 0 , u 0 ) satisfy conditions (A1)-(A3) shown in Theorem 4.1. Then there exist a time T * = T * (α, β, A, γ, δ, ϕ 0 , W 0 ) > 0 independent of R and a unique classical solution 39) and (4.42) imply that for any s ′ ∈ [0, 3), 44) which, together with Lemma 10.9 and (4.43), implies that (ϕ, φ) is right continuous at t = 0 in H 3 space. The time reversibility of equations in (4.25) for (φ, ϕ) yields that (ϕ, φ) ∈ C([0, T * ]; H 3 ). (4.45) . Proof of Theorem 4.1. The proof is based on the classical iteration scheme and the existence results for the linearized problem obtained in §4.2. As in §4.2, we define constants c 0 and c 1 , and assume that Theorem 5. 1 . 1Let (1.20) and any one of conditions (P 0 )-(P 3 ) hold. If the initial data(ϕ 0 , φ 0 , u 0 ) satisfies (A 1 )-(A 3 ) in Theorem 4.1 and φ 0 3 + ϕ 0 3 ≤ D 0 (α, β, δ, A, γ, κ, u 0 Ξ ),where D 0 > 0 is some constant depending on (α, β, δ, A, γ, κ, u 0 Ξ ), then for any T > 0,there exists a unique global classical solution (ϕ, φ, u) in [0, T ] × R 3 to (4.1) satisfying (ϕ, φ) ∈ C([0, T ]; H 3 ), w ∈ C([0, T ]; H s ′ ) ∩ L ∞ ([0, T ]; H 3 ), ϕ∇ 4 w ∈ L 2 ([0, T ]; L 2 ), (5.3) constant b m given by 2.2, in order to obtain the uniform estimates on Z, one needs 2m − 2n − 2 = 2n − 2m = −1, (5.60) 3 > 1 and M 4 = ǫ * + M 2 < −1, so for ν * = min 3γ−3 4(3γ−1) , −M 4 −1 6δ−M 3 , 1 20 and ν = min 1 200 , 4d * v * (α+β) M 3 α 71)with the constant b m given by (5.54). r − e ≥ 0, −8.5 − 3r + 2e < −1, Theorem 7 . 1 . 71Let (1.20) and any one of the conditions (P 0 )-(P 3 ) hold. If initial data(ϕ 0 , φ 0 , u 0 ) satisfies (A 1 )-(A 2 ),then for any positive time T > 0, there exists a unique global classical solution (ϕ, φ, u) in [0, T ] × R 3 to the Cauchy problem (4.1) satisfying (ϕ, φ, w) ∈ C([0, T ]; H s ′ loc ) ∩ L ∞ ([0, T ]; H 3 ), ϕ∇ 4 w ∈ L 2 ([0, T ]; L 2 ), (7.1) 2 + I 2 N )dt → 0, as N → +∞,and Gronwall's inequality that ϕ = φ = w = 0. Then the uniqueness is obtained.7.3. Time continuity. This follows from the uniform estimates (7.2) and equations in (4.1). 8. Proof of Theorem 1.4 Based on Theorem 7.1, now we are ready to give the global well-posedness of the regular solution to the original problem (1.1)-(1.7), i.e., the proof of Theorem 1.4. Moreover, we will show that this regular solution satisfies (1.1) classically in positive time (0, T * ] when 1 < min(γ, δ) ≤ 3.Proof. First, Theorem 7.1 shows that there exists a unique global classical solution (ϕ, φ, w) ) = (ϕ, φ) ∈ C 1 ((0, T ) × R 3 ), and (u, ∇u) ∈ C((0, T ) × R 3 ).(8.1) on both sides yields the continuity equation, (1.1) 1 . While multiplying (8.2) 2 by φ 2 γ−1 = ρ(t, x) ∈ C 1 ((0, T ) × R 3 ) on both sides gives the momentum equations, (1.1) 2 . ρ(t, x) gives the momentum equations, (1.1) 2 . Thus, (ρ, u) satisfies the Cauchy problem (1.1)-(1.7) in the sense of distributions. Finally, it is easy to show that 10.2. [13] Let p > 3/2 and f ∈ H p (R 3 ). Then \|f \| ∞ ≤ C\|f \| Lemma 10. 3 . [ 23 ] 323Let r, a and b be constants such that 1 ≤ a, b, r ≤ ∞. Lemma 10.4.[23] Let u ∈ H s , then for any s ′ ∈ [0, s], there exists a constant C s depending only on s such thatu s ′ ≤ C s u Lemma 10.5. [13] Let r ≥ 0, i ∈ [0, r], and f ∈ L ∞ ∩ H r .Then ∇ i f ∈ L 2r/i , and there some generic constants C i,r > 0 such that\|∇ i f \| 2r/i ≤ C i,r \|f \| Lemma 10.6. [23] Let functions u, v ∈ H s and s > 3 2 , then u · v ∈ H s , and there exists a constant C s depending only on s such that uv s ≤ C s u s v s . Lemma 10.7. [23] Lemma 10 . 10 . 1010Let (ϕ, φ, w) be the regular the solution to the Cauchy problem (4.1), and Z be the time weighted energy defined in (5.4). One has\|ϕ∇ 3 divw\| 2 2 ≤ \|ϕ∇ 4 w\| 2 2 + J * ,(10.9) constant η > 0 small enough, and the constant C(η) > 0, which, along with (10.10)-(10.11), quickly implies (10.9). 10.2. Proof of Proposition 2.1. Here in this subsection, set f Ξ = \|∇f \| ∞ + ∇ 2 f m−2 for any m ≥ 2. M = D 1 − (a − 1)b < −1.Thus, according to (2.3), Z(t) is globally well-defined for t ≥ 0 if and only if0 < Z 0 < 1 (a − 1)C 1 t 0 (1 + s) M exp (a−1)C 2 D 2 +1 . (10.16) Therefore, by choosing Z 0 small enough, one can obtain the global existence of our Cauchy problem (2.4). RewriteG(t, X(t; x 0 )) = 1SinceThen for t large enough, one has \|t −1 G −1 0 (x 0 )\| < 1 for all x 0 , andThen any eigenvalue λ G of G has the following formwhere λ is an eigenvalue of G 0 .(II) Let H(t, x 0 ) = G(t, X(t; x 0 )). By induction, it is easy to show that, for k ≥ 1:where Λ k is a sum of products of t(I d + tG 0 ) −1 and ∇ j G 0 , j ∈ {0, 1, ..., k}, appearing β j times with j jβ j = k.On the one hand, by induction, it is also easy to show that, for k ≥ 1:On the other hand, one can also show that for all j ≥ 1 by induction: IH(j). ∇ j x G is a sum of terms which are products in a certain order of : (I d + tG 0 ) −1 , tI d , I d + tG 0 or ∇ l G 0 appearing β l times, with l lβ l = j. Moreover, the L ∞ -norm of the terms with t is bounded by a constant times (1 + t) −(j+2) , and \|∇ j x G(X(t; ·), t)\| 2 ≤ C j (1 + t) −(j+2) , with C j = C(κ, j, u 0 Ξ ). Suppose IH(k-1), then ∇ j x G(t, X(t; x 0 )) Some diffusive capillary models of Korteweg type. 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Well-Posedness and Singularity Formation of Compressible Isentropic Navier-Stokes Equations. S Zhu, Shanghai Jiao Tong UniversityPH.D ThesisS. Zhu, Well-Posedness and Singularity Formation of Compressible Isentropic Navier-Stokes Equa- tions, PH.D Thesis, Shanghai Jiao Tong University, 2015. Zhu) Mathematical Institute. Z P Xin ; Shatin, N T , Hong Kong ; Shatin, N T , Hong Kong, The Institute of Mathematical Sciences. Oxford OX2 6GG, UK; Clayton, 3800, AustraliaThe Chinese University of Hong Kong ; University of Oxford ; School of Mathematical Sciences, Monash University ; The Chinese University of Hong KongThe Institute of Mathematical Sciences. E-mail address: [email protected](Z.P. Xin) The Institute of Mathematical Sciences, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong. E-mail address: [email protected] (S. G. Zhu) Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK; School of Mathematical Sciences, Monash University, Clayton, 3800, Australia; The Institute of Mathematical Sciences, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong. E-mail address: [email protected]	[]
[ "Multi-dimensional Network Security Game: How do attacker and defender battle on parallel targets?", "Multi-dimensional Network Security Game: How do attacker and defender battle on parallel targets?" ]	[ "Yuedong Xu [email protected] ", "John C S Lui \nDepartment of Computer Science & Engineering\nThe Chinese University of Hong Kong\nChina\n", "\nDepartment of Electronic Engineering\nFudan University\nChina\n" ]	[ "Department of Computer Science & Engineering\nThe Chinese University of Hong Kong\nChina", "Department of Electronic Engineering\nFudan University\nChina" ]	[]	In this paper, we consider a new network security game wherein an attacker and a defender are battling over "multiple" targets. This type of game is appropriate to model many current network security conflicts such as Internet phishing, mobile malware or network intrusions. In such attacks, the attacker and the defender need to decide how to allocate resources on each target so as to maximize his utility within his resource limit. We model such a multidimensional network security game as a constrained nonzero sum game. Two security breaching models, the productform and the proportion-form, are considered. For each breaching model, we prove the existence of a unique Nash equilibrium (NE) based on Rosen's theorem and propose efficient algorithms to find the NE when the games are strictly concave. Furthermore, we show the existence of multiple NEs in the product-form breaching model when the strict concavity does not hold. Our study sheds light on the strategic behaviors of the attacker and the defender, in particular, on how they allocate resources to the targets which have different weights, and how their utilities as well as strategies are influenced by the resource constraints.	null	[ "https://arxiv.org/pdf/2202.09755v1.pdf" ]	739,557	2202.09755	7a830c4bce3034e91e828c6ba42f9f7b129d0cb4	Multi-dimensional Network Security Game: How do attacker and defender battle on parallel targets? 20 Feb 2022 Yuedong Xu [email protected] John C S Lui Department of Computer Science & Engineering The Chinese University of Hong Kong China Department of Electronic Engineering Fudan University China Multi-dimensional Network Security Game: How do attacker and defender battle on parallel targets? 20 Feb 2022 In this paper, we consider a new network security game wherein an attacker and a defender are battling over "multiple" targets. This type of game is appropriate to model many current network security conflicts such as Internet phishing, mobile malware or network intrusions. In such attacks, the attacker and the defender need to decide how to allocate resources on each target so as to maximize his utility within his resource limit. We model such a multidimensional network security game as a constrained nonzero sum game. Two security breaching models, the productform and the proportion-form, are considered. For each breaching model, we prove the existence of a unique Nash equilibrium (NE) based on Rosen's theorem and propose efficient algorithms to find the NE when the games are strictly concave. Furthermore, we show the existence of multiple NEs in the product-form breaching model when the strict concavity does not hold. Our study sheds light on the strategic behaviors of the attacker and the defender, in particular, on how they allocate resources to the targets which have different weights, and how their utilities as well as strategies are influenced by the resource constraints. I. Introduction The economics of network security has become a thriving concern in fixed line and mobile Internet. Due to the popularity of e-commerce and other online services, malicious attacks have evolved into profit driven online crimes in the forms of Internet phishing, network intrusion, mobile malware etc. Although security defence is essential, the networking community is still witnessing an increased number of global attacks. Part of reasons are the economic benefits on performing attacks by hackers as well as the inadequate protection against the persistent attacks. Therefore, economic studies beyond the technological solutions are vitally important to reveal the behaviors of the defenders and the malicious attackers, and game theory serves as a well suited mathematical tool to bring about this fundamental understanding. A prominent application of game theory in security is intrusion detection where an attacker exploits system vulnerabilities and a defender monitors the events occurring in a network strategically [4] [6]. Recent advances of network security games have two features. One is called uncertainty that incorporates incomplete information of players [6] and stochastic properties of players or environments [7]. The other is called interdependency in which the actions of players may affect other players. This type of interactions are sometimes regarded as network effects with positive or negative externality [9], [10], [15]. In this work, we explore a new type of network security game which is characterized by multi-dimensional attacks. We are motivated by three facts. Firstly, the effectiveness of attack or defence depends on the amount of resources that are used. The resource is an abstract representation of manpower, machines, technologies, etc. For instance, many resources are needed to create malicious websites in phishing attacks, or to camouflage malicious apps in mobiles, or to recruit zombie machines in DDoS attacks, or to probe server vulnerabilities in intrusion attacks. However, one needs to note that resource is not free for the attacker and the defender. Secondly, the attacker and the defender usually possess limited resources. For instance, the number of active bots that a botmaster can manipulate is usually constrained to a few thousands [22]. Thirdly, the attacker can assaults multiple targets for better economic returns. These targets may represent different banks in the Internet phishing attack [11], or different Android apps in mobile malware, or different servers in network intrusion attacks. These targets vary in values or importances. Attacking (resp. protecting) more targets requires a larger amount of resources, which may exceed the resource budget of the attacker (resp. defender). As a consequence, the conflicts on multiple targets are conjoined whenever the attacker or the defender has limited amount of resources. This transforms the decision making in network security issues into myopic constrained optimization problems. We propose a non-zero sum game-theoretic framework to characterize the constrained resource allocation between an attacker and a defender. The utility of the attacker is modeled as the profit, which is equivalent to the loss of victims minus the costs of attack resources. The utility of the defender is modeled as the loss of victims plus the costs of defence resources. Both players aim to optimize their individual utilities. We express the loss of victims on a target as a product of its weight and the security breaching probability. Two breaching models are considered; one is the product-form of attack and defence efficiencies, the other is the proportion-form of attack and defence efficiencies. In our work, we focus on the following questions: 1) How does a player select targets to attack/defend and how does he allocate resources to heterogenous targets at the NE? 2 ) How do the resource limits of the players influence the NE and their performance at the NE? This work provides important insights into the multidimensional network security issues. In the product-form breaching model, both players allocate positive resources to the subsets of more valuable targets at the NE. For any two targets protected by the defender, he always allocates more resources to one with a higher value. While the attacker may allocate more resources to the more important targets, or evade the well-protected valuable targets, depending on the defender's relative ineffectiveness of defence (RID). We also show the existence of multiple NEs that yield different utilities to the players in some special scenarios. The attacker and the defender may place positive amount of resources to more targets when they acquire larger resource budgets. When both players do not possess sufficient amount of resources, anyone of them may improve his utility if his resource limit increases. However, when the defender always has sufficient amount of resources, the increase of attacker's resource limit can lead to an arm race in which both players may obtain smaller utilities at the NE. In the proportion-form breaching model, the attacker and the defender allocate resources on all the targets. Each player allocates more resources to more valuable targets. The resource insufficient player can improve his utility by acquiring more resources. Our major contributions are summarized as below: -We propose a novel network security game framework that captures the competition on multiple targets simultaneously. -We present efficient algorithms to find the unique NE when the objective functions are strictly concave. -We also show the existence of multiple NEs when the objective functions are not strictly concave. -We provide important insights on how the attacker(s) and the defender allocate resources to heterogeneous targets under resource constraints, and how the NE(s) is (are) influenced by the resource constraints. The remainder of this paper is as follows. Section II describes the game model. Section III carries out the analysis of the NE as well as the impact of resource limits on the NE. Section IV presents a linear intrusion detection game analysis. We analyse the NE of the proportion-form breaching model in Section V. Section VI surveys the related works and Section VII concludes. II. Game Model and Basic Properties In this section, we present a game-theoretic model for network security issues. The game contains two players, one being an attacker and the other being a defender, they simultaneously compete on multiple targets. We first provide some salient features of current network security attacks, and then present our model formulation. A. Motivation We are motivated by new features of network attacks and defences that are not well captured by existing works (e.g. [6] and reference therein). Most of state-of-the-art researches focus on the one dimensional strategies (i.e., monitoring probability of intrusion, channel access probability or insurance adoption of a node). Such game models are insufficient to characterize the modern days security attacks such as phishing and mobile malware, etc. Here, we present some salient features of network attacks that lead to our game formulation. First, the attackers and the defenders are resource constrained. Resources are defined in a variety of forms. For instance, in the fast-flux phishing attack, the hijacked IP address is one type of resources of the attackers. In a mobile malware attack, the attacker's resources can be the technology and the manpowers used to spoof the security check mechanism of the third-party apps markets. In DDos attacks, a botmaster is usually able to control only a few thousands active bots [22]. Similarly, the defender needs to allocate resources such as technologies and manpowers to detect and remove these attacks. In general, both the attacker and the defender only possess limited resources. Second, the efficiencies of attacks and defences depend on how many resources are allocated. While existing works (e.g. references in [6]) assume that the payoffs of the attacker and the defender are determined by whether the target is attacked or defended. We take phishing attack as an example. By creating more malicious websites, the phishing attacker is able to seduce more users and to perform more persistent attacks. If the defender allocates more resources to perform proactive detection, more malicious sites will be ferreted out in zero-day, and the attack time window will be reduced. Similarly, if more efforts are spent to create malicious Android apps, the attacker can carry out more effective camouflage, thus gaining more profits through stealing private information or sending premium SMS imperceptibly. As a countermeasure, the defender will install these apps on his cloud and examine their suspicious events for a certain amount of time. Last but not least, the attacker and the defender battle not on a single, but rather, multiple targets. Attackers are profit-driven. They are inclined to attack many targets in parallel. The targets are specified as different E-banks in phishing, different apps in mobile malware attacks and different servers in network intrusions. Note that the targets vary in their valuations, so the attacker and the defender may allocate different amount of resources to them. To attack (resp. protect) multiple targets, more resources are required. How to perform parallel attacks becomes a challenging problem when players have certain resource limits. All these motivate our study on the strategic allocation of limited resources by the players on multiple targets simultaneously. B. Models Let us start with the basic security game which consists of two players, an attacker A and a defender D. The attacker launches attacks on N targets (or "battlefields" interchangeably) which we denote as B = {B 1 , · · ·, B N }. The target B i is associated with a weight w i (i = 1, · · ·, N ). When w i >w j , B i is more valuable than B j . Without loss of generality, we rank all targets from 1 to N in the descending order of their weights (i.e. w i >w j if i<j). Attacking a target may consume some resources such as manpower to design malware, social engineering techniques to camouflage them, or dedicate many compromised machines for attacks. Defending a target needs manpower, investment in technology, and computing facilities etc. Here, we monetarize different types of resources. Let c be the price of per-unit of A's resources, and letĉ be that of D's resources. We next define two important terms that form the utilities of the attacker and the defender. • Attack efficiency. Let x i be the amount of resources spent by A on B i , and let f (x i ) be the corresponding attack efficiency on target B i . Here, f (·) reflects the ability of the attacker to intrude a system, or to camouflage the malware, etc. We assume that f (x i ) is a differentiable, strictly increasing and concave function with respect to (w.r.t.) x i . The concavity means that the increment of attack efficiency decreases when A further increases x i . Without loss of generality, we let f (0) = 0 and 0 ≤ f (x i ) ≤ 1. • Defence efficiency. Denote y i as the resources that D uses to detect and remove the attacks on target B i . Let g(y i ) be the defence efficiency when D allocates y i to B i . We assume that g(y i ) is a differentiable, strictly increasing and concave function of y i with g(0) = 0 and 0 ≤ g(y i ) ≤ 1. For the sake of convenience, we define a complementary functiong(y i ), the defence inefficiency, which hasg(y i ) = 1 − g(y i ). Then, g(·) is a decreasing and convex function. It is very difficult to capture the loss of victims (also the revenue of the attacker) due to the obscure interaction between the attack efficiency of A and the defence efficiency of D. Here, we formulate two simplified breaching models, one is named a "product-form" model and the other is named a "proportion-form" model. Denote by p i the breaching probability of target B i . Then, there exist • Product-form model: p i = f (x i )g(y i ); • Proportion-form model: p i = f (xi) f (xi)+g(yi) . In the product-form model, the change of attack (resp. defence) efficiency causes a linear change of breaching probability. For mobile phishing attacks, the defence efficiency can be regarded as the probability of detecting malware, and the attack efficiency represents the ratio of victims defrauded by the attacker. Then, the breaching probability can be taken as a product of attack efficiency and defence inefficiency. A classic example of the product-form model is the matrix-form intrusion detection game where f (x i ) and g(y i ) are linear functions [6]. The attack efficiency denotes the probability of performing an attack and the defence efficiency denotes the probability of performing a detection action. In reality, the resources of the attacker and the defender have a coupled effect on the security of a target. The increase of attack efficiency might not yield a linearly augmented breaching probability. However, it is very difficult to quantify their coupling. Here, we present a proportion-form breaching model that generalizes the cyber-security competition in [21] and the DDoS attacks on a single target in [12]. The breaching probability increases with the attack efficiency, while at a shrinking speed. In practice, both A and D have limited resource budgets which we denote by X A and Y D respectively, with 0 < X A , Y D < ∞. Our focus is to unravel the allocation strategies of the players on multiple targets with the consideration of resource limits. To achieve this goal, we make the following assumption on the attack and defence efficiencies. Assumption: lim xi→∞ f (x i ) = 1 and lim yi→∞ g(y i ) = 1 in the product-form model if not mentioned explicitly. Late on, we consider the linear f (x i ) and g(y i ) that generalize intrusion detection game to multiple targets. As a consequence of attacking B i , A receives an expected revenue of w i p i . Let U A be the aggregate profit of A on all the N targets. We have U A = N i=1 w i p i −c N i=1 x i . The attacker A is usually profit driven and is assumed to be risk-neutral. His purpose is to maximize U A under the resource cap X A . Then, the constrained resource allocation problem is expressed as max {xi} N i=1 U A subject to N i=1 x i ≤ X A .(1) The defender D's objective is to minimize the revenue of the attacker A with the consideration of his resource budget. Let U D be the disutility of D given by U D = − N i=1 w i p i −ĉ N i=1 y i . Whenĉ (resp. c) is 0, D (resp. A) has a use-it-or-lose-it cost structure such that he will utilize all his resources. The resource allocation problem of D can be formulated as: max {yi} N i=1 U D subject to N i=1 y i ≤ Y D .(2) Noticing that A and D have conflicting objectives, we model the resource allocation problem as a two-player noncooperative game and we denote it as G. Let H be a convex hull expressed as {(x i , y i )\|x i ≥ 0, y i ≥ 0, N i=1 x i ≤ X A , N i=1 y i ≤ Y D }. In what follows, we define a set of concepts for the game. Definition 1: Nash Equilibrium: Let x = (x 1 , · · · , x N ) and y = (y 1 , · · · , y N ) be the feasible resource allocations by A and D in the convex hull H respectively. An allocation profile S = {x * , y * } is a Nash equilibrium (NE) if U A (x * , y * ) ≥ U A (x, y * ) and U D (x * , y * ) ≥ U D (x * , y) for any x = x * and y = y * . Definition 2: [8] (Concave game) A game is called concave if each player i chooses a real quantity in a convex set to maximize his utility u i (x i , x −i ) where u i (x i , x −i ) is concave in x i . Theorem 1: [8] (Existence and Uniqueness) A concave game has a NE. Let M be a n×n matrix function in which M ij =ϕ i ∂ 2 ui ∂xi∂xj , for some constant choices of ϕ i >0. If M +M T is strictly negative definite, then the NE is unique. Theorem 2: The multi-dimensional security game G has a unique NE for the product-form breaching model if the attack and defence efficiencies are strictly concave, and for the proportion-form breaching model. All the proofs in this work can be found in the appendix. III. Nash Equilibrium and Influence of Resource Limits for Product-form Model In this section, we propose an algorithm to find the NE and present its properties. Furthermore, we analyze how the resource limits X D and Y D influence the allocation strategies of the attacker and the defender. A. Solving NE for the Generalized Game In the previous section, we have shown the existence of a unique NE in the multi-dimensional security game G. However, we have not stated how to derive the NE, which is nontrivial in fact. Define (x * , y * ) as the NE of G. We show that (x * , y * ) has the following property. Theorem 3: There exist non-negative variables λ and ρ such that −w i f (x * i )g ′ (y * i ) −ĉ = ρ if y * i > 0 ≤ ρ if y * i = 0 ,(3)w i f ′ (x * i )g(y * i ) − c = λ if x * i > 0 ≤ λ if x * i = 0 ,(4) where λ ≥ 0 if N i=1 x * i = X A λ = 0 if N i=1 x * i < X A and(5)ρ ≥ 0 if N i=1 y * i = Y D ρ = 0 if N i=1 y * i < Y D .(6) Herein, λ and ρ are viewed as shadow prices of violating the resource limits. From Theorem 3, one can see that x * i and y * i may take on 0, which occurs when A or D decides not to attack or defend target B i . Our main question here is that given X A and Y D , how λ and ρ are solved at the NE? Before answering this question, we state the sets of targets with positive resources of A and D at the NE. Lemma 1: Let K A be the number of targets with positive resources of A, and K D be that with positive resources of D at the NE. We have i) the set of targets being attacked is {B 1 , · · · , B KA } and the set of targets being defended is {B 1 , · · · , B KD }; ii) K A ≥ K D . Remark: The utility of Lemma 1 is that it greatly reduces the space of searching K D and K A , which is essential for us to compute the values of λ, ρ, x * i and y * i at the NE. In fact, we only need to test at most (N +1)(N +2)/2 possible sets of targets. Define two inverse functions h D (·) := {g ′ } −1 (·) and h A (·) := {f ′ } −1 (·). At the NE, the resources used by A and D on a target are given by x * i =      h A ( c+λ wig(y * i (λ,ρ)) ) ∀ i ≤ K D h A ( c+λ wig(0) ) ∀ K D <i≤K A 0 ∀ i > K A ,(7)y * i = h D ( −(ρ+ĉ) wif (x * i (λ,ρ)) ) ∀ i ≤ K D 0 ∀ i > K D .(8) In what follows, we define a set of notations w.r.t. the total resources (denoted as Tot Res) used by both players at the NE in Table I. The pair (X suf A , Y suf D ) denote the sufficient amount of resources needed by A and D when λ and ρ are both 0. If both X A >X suf A and Y D >Y suf D hold, A and D have some unused resources at the NE. Then, the strategies of A and D on one target are independent of the other targets. We can partition the plane of (X A , Y D ) into four domains: D 1 ) X A ≥X suf A and Y D ≥Y suf D ; D 2 ) X A <X suf A and Y D ≥Ŷ suf D ; D 3 ) X A ≥X suf A and Y D <Y suf D ; D 4 ) none of the above. If (X A , Y D ) ∈ D 1 , the consumed resources of A and D at the NE are X suf A and Y suf D respectively. If (X A , Y D ) ∈ D 2 , the resources of A are insufficient. Then, A uses X A resources and D usesŶ suf D at the NE. If (X A , Y D ) ∈ D 3 , the resources of D are insufficient. Then, A usesX suf A resources and D uses Y D at the NE. If (X A , Y D ) ∈ D 4 , A uses X A and D uses Y D resources at the NE. The partition of (X A , Y D ) enables us to understand when the attacker (resp. the defender) possesses sufficient amount of resources for the attack (resp. defence). X * A := N i=1 x * i (Tot Res used by A at the NE) Y * D := N i=1 y * i (Tot Res used by D at the NE) X suf A Tot Res used by A at the NE with λ=ρ=0 Y suf D Tot Res used by D at the NE with λ=ρ=0 X suf A Tot Res needed by A at the NE to let λ=0, given Y D < Y suf D (i.e. ρ>0) Y suf A Tot Res needed by D at the NE to let ρ=0, given X A < X suf A (i.e. λ>0) TABLE I NOTATIONS OF TOTAL AMOUNT OF RESOURCES The remaining challenge on deriving the NE is how λ and ρ are found for the given K A and K D . Intuitively, we can solve λ and ρ based on Eqs. (5)(6)(7)(8). However, there does not exist an explcit expression in general. We propose a bisection algorithm in Fig. 1 to search λ and ρ. The basic idea is to express ρ as two functions of λ, ρ 1 (λ) obtained from Eqs. (5)(7)(8) and ρ 2 (λ) obtained from Eqs. (6)(7)(8), and then compute their intersection. To guarantee that the bisection algorithm can find feasible λ and ρ if they exist, we show the monotonicity of ρ 1 (λ) and ρ 2 (λ) in the following lemma. Lemma 2: Suppose that feasible λ and ρ (i.e. λ, ρ≥0) exist for the fixed K A and K D at the NE. The following properties hold i) if λ is 0, there has a unique ρ ≥ 0; ii) if ρ is 0, there has a unique λ; iii) ρ 1 (λ) is a strictly increasing function and ρ 2 (λ) is a strictly decreasing function. The monotonicity property enables us to use bisection algorithm to check the existence of the pair (λ, ρ) and solve them if they exist. When X A and Y D are sufficient, the NE can be directly computed via eqs. (7) and (8). When the resources of either A or D are insufficient, the NE is found by the lines 5∼17 in Fig.1. When both players have insufficient resources, the NE is obtained by the lines 18∼26. The complexity order of finding the sets with positive resource allocation is merely O(N 2 ). Input: N , X A , Y D , w i , c,ĉ, f (·) and g(·); Output: K A , K D , λ, ρ, x * i and y * i 1: Initialize K A = K D = N 2: Let λ=ρ=0, compute y * i , x * i using eqs. (7),(8) for all i; 3: Compute X suf A := N i=1 x * i and Y suf D = N i=1 y * i ; 4. If both X A ≥ X suf A and Y D ≥ Y suf D , exit; 5: For K A ≥ 1 6: K D = K A 7: For K D ≥ 1 8: If X A ≤ X suf A 9: Find λ by letting ρ = 0 and X * A = X A via (7)(8); 10: Elseif Y D ≤ Y suf D 11: Find ρ by letting λ = 0 and Y * D = Y D via (7)(8); 12: End; 13: If x * i ≥ 0, y * i ≥ 0, exit; 14: K D = K D − 1 15: End 16: K A = K A − 1 17: End 18: For K A ≥ 1 19: K D = K A = N 20: For K D ≥ 1 21: Compute the fixed point (ρ, λ) which solves (7) and (8) by setting Y * D =Y D and X * A =X A ; Given new pair (λ, ρ), compute y * i and x * i via (7) and (8); 22: If x * i ≥ 0, y * i ≥ 0, exit; 23: K D = K D − 1 24: End 25: K A = K A − 1 26: End Fig. 1. Algorithm to find K A , K D , λ, ρ, x * i and y * i at the NE B. Properties of NE Given the resource limits X A , Y D and other system parameters, we now know the way that the unique NE is computed. Our subsequent question is how a player disposes resources on heterogeneous targets at the NE. Lemma 3: The NE (x * , y * ) satisfies the following properties: • y * i ≥ y * j for 1 ≤ i < j ≤ K D ; • x * i ≥ x * j for K D < i < j ≤ K A ; • i) x * i > x * j ifg ′ (y) g(y) is strictly increasing w.r.t. y, ii) x * i = x * j ifg ′ (y) g(y) is a constant, and iii) x * i < x * j ifg ′ (y) g(y) is strictly decreasing w.r.t. y for all 1≤i<j≤K D . The first property manifests that D is inclined to allocate more resources to the targets with higher weights at the NE. The second property means that if two targets are not protected by D at the NE, A allocates more resources to the one of higher value. However, it is uncertain whether A allocates more (or less) resources to a high (or lower) value target among the top K D targets with positive resources of D. We next use three examples to highlight that all the possibilities can happen. These examples differ in the choice of (complementary) defence efficiency functions. We define a new term, "relative ineffectiveness of defence (RID)", as the expressiong ′ (y) g(y) . Note that the first-order derivativeg ′ (y) reflects how fast (i.e. the slope)g(y) decreases with the increase of y. RID reflects the relative slope that the increase of y reducesg(y). Ifg ′ (y) g(y) is decreasing in y, further increasing y makesg(y) decreases faster and faster. On the contrary, ifg ′ (y) g(y) is increasing in y, further increasing y only results in a smaller and smaller relative reduction ofg(y) (considering the sign ofg ′ (y)). We suppose that A and D allocate positive resources to B 1 and B 2 . Example 1 (InvG): f (x)=1−(1+x) −a and g(y)= 1 1+θy . The following defence inefficiency equality holds,g ′ (y) g(y) = −θ 1+θy . Then, we obtain wi wj = ( 1+xi 1+xj ) 2(1+a) 1−(1+xi) −a 1−(1+xj ) −a . Due to w i > w j , it is easy to show x i > x j by contradiction. Example 2 (ExpG): f (x)=1−(1+x) −a and g(y)= exp(−θy). The expressiong ′ (y) g(y) is equal to −θ. According to the KKT conditions in Theorem 3, there has ( 1+xi 1+xj ) 1+a 1−(1+xi) −a 1−(1+xj ) −a = 1. The above equation holds only upon x i = x j . Example 3 (QuadG): f (x)=1 − (1 + x) −a andg(y) = (1 − θy) 2 . There existsg ′ (y) g(y) = − 2θ 1−θy . Theorem 3 yields wj wi = ( 1+xi 1+xj ) 1+a ( 1−(1+xi) −a 1−(1+xj ) −a ) 2 . Then, there has x * i < x * j . Remark 2: For InvG-likeg(y), RID is strictly increasing. The attacker's best strategy is to allocate more resources to more important targets. In a word, the attacker and the defender have a "head-on confrontation". For ExpG-likẽ g(y), RID is a constant. The attacker sees a number of equally profitable targets. For QuadG-likeg(y), RID is a decreasing function. The attacker tries to avoid the targets that are effectively protected by the defender. Intuitively, when a player does not possess sufficient resources, he will gain a higher utility if his resource limit increases. This is true in a variety of cases. Suppose that not all the targets are attacked by A. When X A increases, A can at least gain more profits by allocating the extra resources to the targets that are not under attack. We next present a counter-intuitive example. Suppose that A and D allocate positive amount of resources to all the targets at the NE. The resources of A are insufficient while those of Y D are sufficient, that is, λ > 0 and ρ = 0. When X A increases, it is easy to show by contradiction that λ decreases and x i increases. Due to the equality −w i f (x i )g ′ (y i ) =ĉ in the KKT conditions, y i also becomes larger. The utility of the attacker on target B i at the NE is given by w i f (x i )g(y i ) − cx i = −ĉg (yi) g ′ (yi) − cx i . If RID of the defender,g ′ (y) g(y) , is a constant or a decreasing function of y i , the expression −ĉg (yi) g ′ (yi) is a constant or decreases as y i increases. Hence, the utility of the attacker on target B i decreases when X A increases. Remark 3: When the defender's resources are insufficient, the attacker gains more profits by acquiring more resources and allocating them to more important targets. When the defender's resources are sufficient, the attacker may explore new targets to attack, other than using all the resources to battle with the resource sufficient defender at the NE. C. Visualizing Whether a Target Is Attacked or Protected From Theorem 3, one can see that x * i and y * i may take on 0, which occurs when A or D decides not to attack or defend target B i . We next show the regions of λ and ρ upon which x * i or y * i hits 0. There are four possibilities, i) (3) and (4) yield the region R 1 (λ, ρ), wherein both A and D give up target B i . x * i = 0 and y * i = 0; ii) x * i > 0 and y * i = 0; iii) x * i = 0 and y * i > 0; and iv) x * i > 0 and y * i > 0. We denote R ++ := {λ ≥ 0; ρ ≥ 0}. Case (i): Eqs.R 1 (λ, ρ) = {λ ≥ max(w i f ′ (0)g(0) − c, 0); ρ ≥ max(−w i f (0)g ′ (0) −ĉ, 0)}. Case (ii): Given x * i = h A ( c+λ wig(0) ) , we obtain the region R 2 (λ, ρ) wherein A attacks but D gives up target B i : R 2 (λ, ρ) = {0 ≤ λ < w i f ′ (0)g(0) − c; ρ > max(−w i f h A ( c+λ wig(0) ) g ′ (0) −ĉ, 0)}; Case (iii): Substituting y * i by h D (− ρ+ĉ wif (0) ) in Eq. (8), we obtain the region R 3 (λ, ρ) where A gives up while D defends target B i : R 3 (λ, ρ) = {0 ≤ ρ < −w i f (0)g ′ (0) − c; λ > max(w i f ′ (0)g h D (− ρ+ĉ wif (0) )), 0)}; Due to f (0) := 0, ρ does not possess a valid value, so the region R 3 is empty. Case (iv): We have the region R 4 = R ++ \{R 1 ∪R 2 ∪R 3 }. For any i = 1, · · ·, N , x * i and y * i contain two unknown variables λ and ρ. Hence, in case iv), we can rewrite x * i and y * i by x * i (λ, ρ) and y * i (λ, ρ). Remark 4: The physical meanings of R 1 to R 4 are as follows: i) if (λ, ρ) ∈ R 1 , both A and D do not allocate resources to this target; ii) if (λ, ρ) ∈ R 2 , A attacks this target while D decides not to defend it; iii) if (λ, ρ) ∈ R 3 , A does not attack this target while D defends it; iv) if (λ, ρ) ∈ R 4 , A attacks this target and D defends it. The purpose of defining R 1 to R 4 is that we can gain some insights into the impacts of system parameters (e.g., X A , Y D , w i , c andĉ) on the NE without directly solving the NE. Here, for any pair (λ, ρ), the increase of λ means the decrease of X A , and the increase of ρ means the decrease of Y D . This property is derived in the proof of Lemma 2. Let us illustrate R 1 ∼ R 4 by using a simple example. 0 1 0 1 λ ρ 1−c−ĉ 1−c−ĉ 1−c R 2 R 4 R 1 Fig. 2. (λ, ρ) regions of Ex.5 Example 4: f (x)=1− exp(−x);g(y)= exp(−y); w i =1. It is easy to obtain h A (x)= − log(x) and h D (y)=− log(−y). Substituting these expressions to R 1 -R 4 in the above equations, we derive the regions of (λ, ρ) by Fig. 2 shows these regions with parameters c andĉ. R 1 is not empty. This implies that both A and D do not allocate resource to this target when λ is larger than 1−c. A large λ corre -sponds to the situation that X A is relatively small compared with the optimally needed resources for the attacks. The increase of λ drives a point in R 4 to migrate to R 2 or even R 1 . This means that the defender and the attacker may give up this target in sequence when X A becomes more and more scarce. R 1 (λ, ρ) = {λ ≥ max(1 − c, 0), ρ ≥ 0}; R 3 = ∅ and R 2 (λ, ρ) = {0 ≤ λ < 1 − c, 0 ≤ ρ < 1 − c −ĉ− λ}; R 4 = R ++ \ {R 1 ∪ R 2 }. IV. A Linear Intrusion Detection Game for Product-form Model In this section, we investigate the existence and uniqueness of NE of an intrusion detection game where the attack and defence efficiencies are linear functions. A. A Matrix-form Game We study a matrix-form multi-dimensional intrusion detection game. The payoff matrix on target B i is shown in Fig.3 where A (resp. N A) denotes "attack" (resp. "not attack") strategy, and D (resp. N D) denotes "defend" (resp. "not defend") strategy. Here, w i denotes the loss of victims for the pair-wise strategies (A, N D) and γw i denotes that for (A, D) with γ ∈ (0, 1). Let c andĉ be the costs of the "attack" and the "defend" strategies. Note thatĉ refers to not only the cost of resources, but also the cost of performance such as QoS or false alarm of benign events. We consider the mixed strategies of A and D in which A attacks target B i with probability x i and D detects this target with probability y i . Each player only has one action on all the targets, which yields the resource constraints: N i=1 x i ≤ X A ≤ 1, N i=1 y i ≤ Y D ≤ 1 and 0 ≤ x i , y i ≤ 1. D N D A (γw i − c, −γ 1 w i −ĉ) (w i − c, −w i ) N A (0, −ĉ) (0, 0) Fig. 3. Payoff Matrix To make the game non-trivial, we let γw i ≤ c and w i > c ∀i, i.e. the loss of victims is greater than the cost of the attacker on an unprotected target, and is less than this cost on a protected target. Given the attack probabilities {x i } N i=1 and the detection probabilities {y i } N i=1 , the utilities of A and D can be derived easily, U A = w i x i − (1 − γ)w i x i y i − cx i , U D = −w i x i + (1 − γ)w i x i y i −ĉy i . The above utility functions fall in the category of our product-form game with f (x) := x andg(y) := 1 − (1 − γ)y. The resource constraints hold naturally because the sum of attack probabilities is no larger than 1, and the sum of detection probabilities is also no larger than 1. For the sake of simplicity, we denote a new variable asγ := 1 − γ. B. Computing NE We take the derivatives of U A (resp. U D ) over x i (resp. y i ) and obtain dU A /dx i = w i − w iγ y i − c, dU D /dy i = w iγ x i −ĉ. The existence of a NE is guaranteed by the concavity of the game. Before diving into the solution of the NE, we present a property of the sets of targets that are attacked or defended at the NE. Lemma 4: The sets of targets with positive resources at the NE are given by i) {B 1 , · · · , B KA } for the attacker and {B 1 , · · · , B KD } for the defender; ii) either K A = K D or K A = K D + 1. Lemma 4 is the sufficient condition of the existence of NE. Similar to Lemma 1, A and D allocate resources to the subsets of more important targets. The difference lies in that A may allocate resources to more targets than D when f (·) and g(·) are nonlinear functions, but to at most one more target than D when f (·) and g(·) are our linear functions. We proceed to find the NE by considering different regions of X A and Y D in the following theorem. Theorem 4: The multi-dimensional intrusion detection game admits a NE as below • P A (k)<X A <P A (k+1) and Y D >P D (k+1) for 0≤k≤N −1. The NE is uniquely determined by x * i =     ĉ wiγ , ∀ i ≤ k X A − k j=1ĉ wj , i=k+1 0, ∀ i>k+1(9)y * i = (1 − w k+1 wi ) 1 γ , ∀ i≤k 0, ∀ i > k .(10) Here, the sum over an empty set is 0 conventionally. • P D (k)<Y D <P D (k+1) and X A >P A (k) for 1≤k≤N . The NE is uniquely determined by x * i = ( k j=1 wi wj ) −1 X A , ∀ i ≤ k 0, ∀ i > k (11) y * i = ( k j=1 wi wj ) −1 Y D − 1 γ k + 1 γ , ∀i≤k 0, ∀i>k . (12) • X A >P A (N ) and Y D >P D (N +1). The NE is uniquely determined by x * i =ĉ w iγ , y * i = 1 γ − c w iγ , ∀ 1≤i≤N.(13)• X A =P A (k) and Y D ≥P D (k) for 1≤k≤N . Denote byỸ D an arbitrary real number in the range [P D (k), min{Y D , P D (k+1)}]. A NE is given by x * i = ĉ wiγ , ∀ i ≤ k 0, ∀ k+1≤i≤N(14)y * i = 1 γ +( k j=1 wi wj ) −1 (Ỹ D −k 1 γ ), ∀ i ≤ k 0, ∀ i > k .(15) • Y D =P D (k) and P A (k−1)≤X A ≤P A (k) for 2≤k≤N . Denote byX A an arbitrary real number in the range [P A (k−1), X A ]. A NE is given by x * i = ( k j=1 wi wj ) −1X A , ∀ i ≤ k 0, ∀ i>k+1(16)y * i = (1− w k+1 wi ) 1 γ , ∀ i ≤ k 0, ∀ i > k .(17) Here, P A (k) and P D (k) are defined as P A (0):=0, P A (k):= k i=1ĉ wiγ , ∀ 1≤k≤N ; P D (1)=0, P D (k):= k−1 i=1 1 γ (1− w k wi ), and P D (N +1):= 1 γ N − N i=1 c wiγ . We illustrate the relationship between NE and resource limits in Fig.4. When f (·) and g(·) are linear, the best response of a player becomes a step-like function. The feasible domain of (X A , Y D ) is partitioned into three parts: i) D 1 -sufficient X A and sufficient Y D ; ii) D 2insufficient X A and sufficient Y D ; iii) D 4 -insufficient X A and insufficient Y D . The total consumed resources at the NEs for D 1 and D 2 are located in the step-like boundary curve. When X A or Y D take some special values, the boundary curve illustrates the existence of multiple NEs. In the horizontal boundary, different NEs bring the same utility to the attacker, but different utilities to the defender. In the vertical boundary, the utilities of the defender are the same, while those of the attacker are different. Let us take a look at an example with X A =ĉ w1γ and Y D >(1− w2 w1 ) 1 γ . Two NEs on target B 1 can be (x * 1 , y * 1 ) (1) =(ĉ w1γ , 0) and (x * 1 , y * 1 ) (2) =(ĉ w1γ , (1− w2 w1 ) 1 γ ). Both A and D do not allocate resources to other targets. The utility of D is given by U D =− 1 γĉ at the both NEs. The utilities of A are given by U (1) A =x * 1 (w 1 −c) and U(2) A =x * 1 (w 2 −c) at the two NEs. At the first NE, B 1 is the most profitable to A. At the second NE, B 1 and B 2 are equally profitable. In both NEs, A cannot gain more profits by switching to another allocation strategy unilaterally. Besides, the total consumed resources for D 4 can be mapped to an arbitrary point in this domain, in which both players have insufficient resources. Remark 5: We summarize the salient properties of the NEs for linear attacking efficiency and linear uptime as below. 1) The targets with x * i > 0 are equally profitable to A such that A has no incentive to change his strategy. 2) D prefers to allocate more resources to the more valuable targets. As a countermeasure, A allocates more resources to the targets that are not effectively protected by D. 3) The NE is not unique with some special choices for X A and Y D . If multiple NEs exist for a given pair (X A , Y D ), they yield the same utility for one player, but different utilities for the other player. V. Nash Equilibrium for Proportion-form Model In this section, we analyze the NE strategy of the players on different targets for the proportion-form breaching model. Nash Equilibrium and its Properties: We define (x * , y * ) as the NE of the game for the proportion-form model. Here, we relax the constraints to be f (·), g(·)≥0 (unlike 0≤f (·), g(·)≤1 in the product-form model). The breaching probability in the proportion-form model cannot exceed 1. Based on the KKT conditions, (x * , y * ) is given by the following theorem. Theorem 5: There exist non-negative variables λ and ρ such that w i f (x * i )g ′ (y * i ) (f (x * i ) + g(y * i )) 2 −ĉ = ρ if y * i > 0 ≤ ρ if y * i = 0 ,(18)w i f ′ (x * i )g(y * i ) (f (x * i ) + g(y * i )) 2 − c = λ if x * i > 0 ≤ λ if x * i = 0 ,(19) with the slackness conditions in Eq. (5) and (6). As the first step to find the NE, we need to investigate how many targets will be attacked by A and defended by D. The following lemma shows that both A and D allocate resources to all the targets in B. Lemma 5: At the NE, there have x * i > 0 and y * i > 0 for all i = 1, · · · , K if f (·) and g(·) are concave and strictly increasing with f (0) = 0 and g(0) = 0. Lemma 5 simplifies the complexity to obtain the NE strategy because we do not need to test whether a target will be attacked or defended. Then, the equalities in Eqs. (18) and (19) hold. Similarly, we partition (X A , Y D ) into four domains to fine the NE: D 1 ) X A ≥X suf A and Y D ≥Y suf D ; D 2 ) X A <X suf A and Y D ≥Ŷ suf D ; D 3 ) X A ≥X suf A and Y D <Y suf D ; D 4 ) none of the above. The method to find the NE contains the similar steps as those of the algorithm in Fig.1. We need to check whether (X A , Y D ) is located in a domain from D 1 to D 4 one by one. We next study how A and D allocate resources to different targets, given the resource limits X A and Y D . The NE strategy satisfies the following properties. Lemma 6: A and D always allocate more resources to the more important targets, i.e. x * i >x * j and y * i >y * j if w i >w j . Remark 6: In comparison to the product-form breaching model, the players in the proportion-form breaching model always allocate more resources to the more valuable targets. For the generalized proportion-form breaching model, it is usually difficult to analyze how the NE and the utilities at the NE are influenced by the resource limits. Therefore, we consider two specific examples with explicit functions f (·) and g(·). Example 5: Let f (x) = x a and g(y) = y a in the breaching probability model with 0<a≤1. Then, for the four cases w.r.t. the sufficiency of X A and Y D , there have: D 1 ): X A ≥ X suf A = N i=1 wia( ĉ c ) a (1+( ĉ c ) a ) 2 ·c and Y D ≥ Y suf A = N i=1 wia( ĉ c ) a (1+( ĉ c ) a ) 2 ·ĉ . The increase of X A or Y D does not influence the NE and the utilities of A and D. D 2 ): X A < X suf A and Y D ≥Ŷ suf D = N i=1 wia( c+λ c ) a (1+( c+λ c ) a ) 2 ·ĉ . where λ is determined by (1 + ( c+λ c ) a ) 2 ·ĉ a (c + λ) 1−a X A = N i=1 w i a. Due to 0<a≤1, λ is a strictly decreasing function of X A . As X A grows, x * i and y * i increase accordingly. Then, the utilities of D and A are given by U D = − N i=1 ( w iĉ a (c + λ) a +ĉ a −ĉy * i ); U A = N i=1 ( w iĉ a (c+λ) a +ĉ a − w i ac(c+λ) (a−1)ĉa ((c+λ) a +ĉ a ) 2 ). It is obvious to see that U D is a decreasing function of the attacker's resource X A . We take the first-order derivative of U A over λ. However, U A does not necessarily increase when X A grows. Let us take a look at a special situation with a=1. We then take the first order derivative of U A over λ and obtain dU A dλ = N i=1 w iĉ (c−λ−ĉ) (c+λ +ĉ) 3 .(20) When c<λ+ĉ, U A is a decreasing function of λ, and hence an increasing function of X A . Otherwise, U A decreases as X A increases. This implies that A always benefits from obtaining more resources if his cost is smaller than that of D. When A's cost is larger than D's, more resources may lead to a reduced utility of A. U D = − N i=1 w i (X A ) a (X A ) a + (Y D ) a −ĉY D ; U A = N i=1 w i (X A ) a (X A ) a + (Y D ) a − cX A(21) When X A increases, U D decreases accordingly. However, increasing X A does not necessarily bring a higher utility to A. Similarly, increasing Y D yields a worse utility to A, but not necessarily resulting a higher utility to D. VI. Related Work Today's network attacks have evolved into online crimes such as phishing and mobile malware attacks. The attackers are profit-driven by stealing private information or even the money of victims. Authors in [5] measured the uptime of malicious websites in phishing attacks to quantify the loss of victims. Sheng et al. provided the interviews of experts in [18] to combat the phishing. A number of studies proposed improved algorithms to filter the spams containing links to malicious websites in [19], [20]. In mobile platforms, users usually publish root exploits that can be leveraged by malicious attackers. Authors in [1] proposed a new cloud-based mobile botnets to exploit push notifcation services as a means of command dissemination. They developed a stress test system to evaluate the effectiveness of the defence mechanisms for Android platform in [2]. Felt et al. surveyed the behavior of current mobile malware and evaluated the effectiveness of existing defence mechanism in [3]. Game theoretic studies of network security provide the fundamental understandings of the decision making of attackers and defenders. Authors in [7] used stochastic game to study the intrusion detection of networks. More related works on the network security game with incomplete information and stochastic environment can be found in [6], [13]. Another string of works studied the security investment of nodes whose security level depended on the his security adoption and that of other nodes connected to him. Some models did not consider the network topology [9] and some others studied either fixed graph topologies [14] or the Poisson random graph [10], [15]. Among the studies of network security game, [16], [4], [17] are closely related to our work. In [16], authors used the standard Colonel Blotto game to study the resource allocation for phishing attacks. An attacker wins a malicious website if he allocates more resources than the defender, and loses otherwise. This may oversimplify the competition between an attacker and a defender. Our work differs in that the attackers perform attacks on multiple non-identical banks or e-commerce companies, and the competition is modeled as a non-zero sum game that yields a pure strategy. In [4], the authors formulated a linearized model for deciding the attack and monitoring probabilities on multiple servers in network intrusion attacks. Altman et al. in [17] studied a different type of multi-battlefield competition in wireless jamming attack that provides important insights of power allocation on OFDM channels. VII. Conclusion In this work, we formulate a generalized game framework to capture the conflict on multiple targets between a defender and an attacker that are resource constrained. A product-form and a proportion-form security breaching models are considered. We prove the existence of a unique NE, and propose efficient algorithms to search this NE when the game is strictly concave. Our analysis provides important insights in the practice of network attack and defence. For the product-form breaching model, i) the defender always allocates more resources to the more important target, while the attacker may not follow this rule; ii) when the defender has sufficient amount of resources, more resources of the attacker might not bring a better utility to him; iii) when the game is not strictly concave, there may exist multiple NEs that yield different utilities of the players. For the proportion-form breaching model, iv) both the attacker and the defender allocate more resources to more important targets; v) a resource insufficient player causes a reduction of his opponent's utility, while not necessarily gaining a better utility by himself when his resource limit increases. SUPPLEMENT: PROOFS OF LEMMAS AND THEOREMS YUEDONG XU, JOHN C.S. LUI . Proof of Theorem 2 Proof: We prove the existence and uniqueness of the NE for the product-form and the proportion-form breaching models separately. Product-form: The second-order derivatives of U A over x can be expressed as ∂ 2 U A ∂x 2 i = w i f ′′ (x i )g(y i ) < 0, and ∂ 2 U A ∂x i ∂x j = 0, ∀i, j. The second-order derivatives of U D over y are given by ∂ 2 U D ∂y 2 i = −w i f (x i )g ′′ (y i ) < 0, and ∂ 2 U D ∂y i ∂y j = 0, ∀i, j. Since f (x i ) is strictly concave w.r.t. x i , U A is a concave function of the strategy profile {x i , i = 1, · · · , N }. Based on Rosen's theorem [8], there always exists a NE in the game G. The matrix M on a target is given by M = w i ϕ 1 f ′′ (x)g(y) ϕ 1 f ′ (x)g ′ (y) −ϕ 2 f ′ (x)g ′ (y) −ϕ 2 f (x)g ′′ (y) .(22) Then, there has M +M T = w i 2ϕ 1 f ′′ (x)g(y) (ϕ 1 −ϕ 2 )f ′ (x)g ′ (y) (ϕ 1 −ϕ 2 )f ′ (x)g ′ (y) −2ϕ 2 f (x)g ′′ (y) .(23) Suppose ϕ 1 = ϕ 2 > 0. Because f ′′ (x) < 0 and g ′′ (y) > 0, then matrix −(M +M T ) is positive definite. Hence, M +M T is negative definite, resulting in the unique NE in the game G. Proportion-form: The second-order derivatives of U A over x can be expressed as ∂ 2 U A ∂x 2 i = w i g(y i ) f ′′ (x i )(f (x i )+g(y i ))−2(f ′ (x i )) 2 (f (x i ) + g(y i )) 3 < 0 and ∂ 2 UA ∂xi∂xj = 0, ∀i, j, due to f ′′ (x i ) < 0. The secondorder derivatives of U D over y can be expressed as ∂ 2 U D ∂y 2 i = w i f (x i ) g ′′ (x i )(f (x i )+g(y i ))−2(g ′ (x i )) 2 (f (x i ) + g(y i )) 3 < 0 and ∂ 2 UD ∂yi∂yj = 0, ∀i, j, due to g ′′ (y i ) < 0. Hence, G is a concave game that admits a NE. The matrix M on a target is given by M = ϕ 1 ∂ 2 UA ∂x 2 i ϕ 1 ∂ 2 UA ∂xi∂yi ϕ 2 ∂ 2 UD ∂xi∂yi ϕ 2 ∂ 2 UD ∂y 2 i .(24) Because of ∂ 2 UA ∂xi∂yi = − ∂ 2 UD ∂xi∂yi , if we let ϕ 1 = ϕ 2 > 0, the expression M +M T is obtained by M +M T = 2ϕ 1 ∂ 2 UA ∂x 2 i 0 0 2ϕ 2 ∂ 2 UD ∂y 2 i .(25) It is obvious to see that M +M T is negative definite. Hence, in the proportion-form breaching model, there exists a unique NE. Proof of Theorem 3 Proof: Recall that U A is concave in x and U D is concave in y. Then, the best responses of A and D are the solutions to two convex optimization problems. Let λ and ρ be Lagrange multipliers of A and D respectively. Let L D (y, ρ) be the Lagrangian function of the defender D. We have L D (y, ρ) = − N i=1 w i f (x i )g(y i ) −ĉ N i=1 y i +ρ(Y D − N i=1 y i ). (26) Our first step is to find the optimal y j as a function of ρ. Taking the derivative over y i , we obtain dL D (y, ρ) dy i = −w i f (x i )g ′ (y i )−(ĉ+ρ), ∀i=1, · · ·, N. (27) The optimal resource allocated to target i, or y i , satisfies the following condition ρ = −w i f (x i )g ′ (y i ) −ĉ(28) when y i is greater than 0. If y i = 0 and dL d (y,ρ) dyi > 0, we have: ρ > −w i f (x i )g ′ (y i ) −ĉ.(29) Whenĉ is 0, the left hand of Equation (3) is positive for any x i > 0. Hence, ρ is always positive if there is at least one target with x i > 0 at the NE. This means that D will consume all the resources Y D . Whenĉ > 0, the Karush-Kuhn-Tucker (KKT) conditions give rise to ρ ≥ 0 if N i=1 y * i = Y D , ρ = 0 if N i=1 y * i < Y D .(30) Following the same approach, we define the Lagrangian function of A as L A (x, λ) = N i=1 w i f (x i )g(y i ) − c N i=1 x i +λ(X A − N i=1 x i ).(31) The first-order derivatives are given by dL A (x, ν) dx i = w i f ′ (x i )g(y i ) − (c + λ), ∀i = 1, · · · , N. (32) If x * i is non-zero, the above derivative equals to 0. Otherwise, L A (x, λ) is a strictly decreasing function of x i such that x * i := 0. The Lagrange multiplier λ also satisfies the slackness condition. Proof of Lemma 1 Proof: Consider two targets i, j with w i > w j . We assume that x * i = 0 and x * j > 0 at the NE. The utility received by A is better if it shifts some x * j to the i th target. This contradicts the assumption that the game is at the NE. Hence, A only attacks K A targets with the descending order of their weights. We next assume y * i = 0 and y * j > 0. There exists an inequality −g ′ (y * j ) < −g ′ (0). According to Theorem 3, we have −w j f (x * j )g ′ (y * j ) = −w i f (x * i )g ′ (y * i ). Then, we can conclude x * j > x * i such that f ′ (x * j ) < f ′ (x * i ). The KKT condition in Equation (4) shows w j f ′ (x * j )g(y * j ) > w i f ′ (x * i )g(0). Because f ′ (x * j ) < f ′ (x * i ) , w i > w j and g(y * j ) <g(0), the inequality does not hold. Hence, D attacks K D with the descending order of the weights. For the claim K A ≥ K D , this can be inferred from our preceding analysis since D will not allocate resources to an target without being attacked by A when f (0) = 0. Proof of Lemma 2 Proof: To search (λ, ρ), we need to consider three different cases step by step: 1) λ>0 and ρ=0, 2) λ=0 and λ>0 and 3) λ>0 and ρ>0. Here, the change of λ and ρ does not alter K A and K D at the NE. Recall that h A (·) is a decreasing function and h D (·) is an increasing function. For simplicity, we let ↑ denote "increase" and let ↓ denote "decrease". The symbol ⇒ denotes "give rise to". Step 1: λ>0 and ρ=0. When λ ↑, x * i ↓ for K D <i ≤ K A . For 1≤i≤K D , there are two possibilities, x * i ↑ or x * i ↓. In what follows, we will show that x * i is strictly decreasing. We assume that x * i ↑ as λ ↑. According to Eqs. (7) and (8), we have the following relationships for all 1≤i≤K D : λ ↑⇒ x * i ↑⇒ f (x * i ) ↑⇒ −ĉ w i f (x * i ) ↑⇒ h D ( −ĉ w i f (x * i ) ) ↑ ⇒ y * i ↑⇒g(y * i ) ↓⇒ c+λ w ig (y * i ) ↑⇒ h A ( c+λ w ig (y * i ) ) ↓⇒ x * i ↓ which causes a self contradiction. Therefore, as λ increases, x * i cannot increases. It is easy to validate that x * i cannot remain the same. Thus, x * i is a strictly decreasing function of λ. According to the slackness condition, there has KA i=1 x * i = X A . If there exists a feasible λ to satisfy this equality, λ should be unique. A bisection algorithm can find the solution. Step 2: λ=0 and ρ>0. We assume that y * i ↑ when ρ ↑. Then, the following relationship holds: ρ ↑⇒ y * i ↑⇒g(y * i ) ↓⇒ c w ig (y * i ) ↑⇒ h A ( c w ig (y * i ) ) ↓ ⇒ x * i ↓⇒ f (x * i ) ↓⇒ −(ĉ+ρ) w i f (x * i ) ↓⇒ h D ( −(ĉ+ρ) w i f (x * i ) ) ↓⇒ y * i ↓ which contradicts to the assumption. Similarly, we can show that y * i cannot remain unchanged. Therefore, when ρ increases, y * i is strictly decreasing for all 1≤i≤K D . The slackness condition gives rise to KD i=1 y * i = Y D . Then, we can use the bisection algorithm to find ρ if it exists. Step 3: λ>0 and ρ>0. We consider two cases: K A = K D and K A > K D . Recall that the implicit function ρ 1 (λ) is obtained from Eqs. (5)(7)(8) and the implicit function ρ 2 (λ) is obtained from Eqs. (6)(7)(8). Step 3.1 K A =K D . When λ increases, there are two cases due to the constraint KD i=1 x * i =X A . One is that x * i does not change for all 1≤i≤K D . The other is that there exist two targets B i and B j (1≤i, j≤K D ) in which x * i increases and x * j decreases. If x * i does not change for all 1≤i≤K D , the following relationships hold λ ↑⇒g(y * i ) ↑⇒ y * i ↓, ∀1≤i≤K D . Because of ρ>0, there must have KD i=1 y * i =Y D , which contradicts to the conclusion y * i decreases for all 1≤i≤K D . Therefore, the case that x * i (1≤i≤K D ) does not change is not true. We next turn to the second case that x * i increases and x * j decreases when λ increases. The following relationships hold λ ↑⇒ x * i ↑⇒ f (x * i ) ↑⇒ − 1 w i f (x * i ) ↑ . If ρ increases or remains the same, we continue the induction by −1 w i f (x * i ) ↑⇒ −(c+ρ) w i f (x * i ) ↑⇒ h D ( −(c+ρ) w i f (x * i ) ) ↑⇒ y * i ↑ ⇒g(y * i ) ↓⇒ c+λ w ig (y * i ) ↑⇒ h A ( c+λ w ig (y * i ) ) ↓⇒ x * i ↓ . The condition x * i ↑ contradicts to the conclusion x * i ↓. Therefore, ρ must decreases when λ increases. In a word, ρ 1 (λ) is a strictly decreasing function. According to the slackness condition in Eq. (6), there has KD i=1 y * i =Y D . When λ increases, there are also two cases w.r.t. y * i . One is that y * i does not change for 1≤i≤K D . The other is that there exist two targets B i and B j (1≤i, j≤K D ) in which y * i increases and y * j decreases. If y * i does not change for 1≤i≤K D , the following relationships hold λ ↑⇒ c+λ w ig (y * i ) ↑⇒ h A ( c+λ w ig (y * i ) ) ↓⇒ x * i ↓⇒ −1 w i f (x * i ) ↓ . Because y * i does not change, ρ must increase. For the second case, when y * i increases, we obtain the following relationships λ * i ↑⇒g(y * i ) ↓⇒ c+λ w ig (y * i ) ↑⇒ h D ( c+λ w ig (y * i ) ) ↓ ⇒ x * i ↓⇒ f (x * i ) ↑⇒ −1 w i f (x * i ) ↓ . If ρ decreases or remains the same, there must have −1 w i f (x * i ) ↓⇒ −(ĉ+ρ) w i f (x * i ) ↓⇒ h D ( −(ĉ+ρ) w i f (x * i ) ) ↓⇒ y * i ↓, which contradicts to the condition y * i ↑. Hence, ρ must increase in this case. As a consequence, the implicit function ρ 2 (λ) is a strictly increasing function. Step 3.2 K A >K D . The slackness condition in Eq. (5) is expressed as X A = KD i=1 h A ( c+λ w ig (y * i (λ, ρ)) ) + KA i=KD+1 h A ( c+λ w ig (0) ). (33) When λ increases, the expression h A ( c+λ wig(0) ) is strictly decreasing for K D+1 ≤i≤K A . This implies that x * i decreases for K D+1 ≤i≤K A . Due to the constraint KA i=1 x * i =X A , x * i increases in at least one target B i for 1≤i≤K D . In other word, the case that x * i does not change with the increase of λ does not happen. Then, following the analysis in the Step 3.1, we can see that ρ 1 (λ) is a strictly decreasing function and ρ 2 (λ) is a strictly increasing function. This concludes the proof. Proof of Lemma 3 Proof: We prove this lemma by contradiction. When the both players allocate resource to targets B i and B j at the NE, there exists w i w j f (x * i ) f (x * j )g ′ (y * i ) g ′ (y * j ) = w i w j f ′ (x * i ) f ′ (x * j )g (y * i ) g(y * j ) = 1.(34) If y * i < y * j , the following inequality holds g ′ (y * i ) <g ′ (y * j ) < 0 because g(·) is strictly convex. The above inequality yields g ′ (y * i ) g ′ (y * j ) > 1. Combined with Eq.(34), we obtain f (x * i ) f (x * j ) < 1. Since f (·) is strictly increasing and strictly concave, there have x * i < x * j and 0 < f ′ (x * j ) < f ′ (x * i ). Then, we can conclude w i w j f ′ (x * i ) f ′ (x * j )g (y * i ) g(y * j ) > 1,(35) which contradicts to Eq.(34). Therefore, if D allocates resource to targets B i and B j , (i < j), at the NE, there must have y * i > y * j . Eq.(34) can be rewritten as f (x * i ) f ′ (x * i )g ′ (y * i ) g(y * i ) = f (x * j ) f ′ (x * j )g ′ (y * i ) g(y * j ) . (36) Wheng ′ (y) g(y) is a constant, there exists f (x * i ) f ′ (x * i ) = f (x * j ) f ′ (x * j ) . If x * i > x * j , there have f (x * i ) > f (x * j ) and 0 ≤ f ′ (x * i ) < f (x * j ) . This gives rise to the inequality f (x * i ) f ′ (x * i ) > f (x * j ) f ′ (x * j ) , which contradicts to the above equality. It is also easy to show that the relationship x * i < x * j also contradicts to the above equality. Hence, we obtain x * i = x * j . We next suppose thatg ′ (y) g(y) is an increasing function of y. Given y * i > y * j for 1 ≤ i<j≤K D , we obtaing ′ (x * j ) g(x * j ) <g ′ (x * i ) g(x * i ) < 0. Then, eq.(36) yields f (x * i ) f ′ (x * i ) > f (x * j ) f ′ (x * j ) , or equivalently x * i > x * j . Similarly, wheng ′ (y) g(y) is a strictly decreasing function of y, there must have x * i < x * j . Proof of Lemma 4 Proof: This lemma is proved by contradiction. We consider even more general functions: f (x) = b 1 x and g(y) = b 2 − b 3 y. In the intrusion detection game, we let b 1 = b 2 = 1 and b 3 > 0. i). We assume x * i =0 and x * j >0 at the NE for two targets B i and B j with w i >w j . The best response of D must satisfy y * i =0. Then, the following inequality holds dU A dx i \| xi=x * i =w i b 1 b 2 −c>w j b 1 b 2 −w j b 2 b 3 y * j −c= dU A dx j \| xj =x * j . A obtains a higher profit if he transfers the resource on B j to B i . Thus, it is not a NE. We further assume y * i =0 and y j >0 at the NE for two targets B i and B j with w i >w j . The marginal profits on B i and B j satisfy w i b 1 b 3 x * i < w j b 1 b 3 x * j . The above inequality gives rise to x * i <x * j because of w i >w j . When y * i =0 and y * j >0, the marginal profits of A on B i and B j satisfy dUA dxi \| xi=x * i > dUA dxi \| xj=x * j . Then, A has a larger utility if he moves the resource on B j to B i . This contradicts to the claim x * i <x * j . Thus, it is not a NE. To sum up, A allocates resources to the top K A targets and D allocates resources to the top K D targets. It is also very intuitive to validate K D <K A . ii). We assume K A >K D +1 at the NE. Let B i and B j be two targets for K D <i, j<N . The marginal profits of A on B i and B j satisfy dU A dx i = w i b 1 b 2 −c, dU A dx j = w j b 1 b 2 − c. Because of w i =w j , A can obtain a larger utility by aggregating the resources to the more profitable target. Thus, it is not a NE. To sum up, K A and K D must satisfy K D ≤ K A ≤ K D +1. This concludes the proof. Proof of Theorem 4 Proof: The proof utilizes the conclusions of lemma 4. According to the properties of the NE, there have w i b 1 b 2 −w i b 1 b 3 y * i −c = λ ≥ 0, ∀ 1≤i≤K A ; (37) w KD b 1 b 2 − w KD b 1 b 3 y * KD ≥ w KD +1 b 1 b 2 if K D <N ;(38) w i b 1 b 3 x * i −ĉ = ρ ≥ 0, ∀ 1≤i≤K D ; (39) w i b 1 b 3 x * KD +1 −ĉ ≤ ρ, if K D <N.(40)x * i =ĉ +ρ wib1b3 , ∀ i ≤ K D ≤ĉ +ρ wK D +1b1 b3 , i=K D +1 (K D <N ) ,(41)y * i = b2 b3 − c+λ wib1b3 , ∀ i ≤ K D ≤ (1 − wK D +1 wi ) b2 b3 , ∀ i≤K D (K D <N ) .(42) Before commencing the analysis, we recall the following notations: P A (k) and P D (k) are defined as P A (0):=0, P A (k):= k i=1ĉ wib1b3 , ∀ 1≤k≤N ; P D (1)=0, P D (k):= k−1 i=1 b2 b3 (1− w k wi ), and P D (N +1):= b2 b3 N − N i=1 c wib1b3 . i). We first prove the following claim via three steps: • P A (k)<X A <P A (k+1) and Y D >P D (k+1) for 0≤k≤N −1. The NE is uniquely determined by x * i =     ĉ wib1b3 , ∀ i ≤ k X A − k j=1ĉ wj b1b2 , i=k+1 0, ∀ i>k+1(43)y * i = (1 − w k+1 wi ) b2 b3 , ∀ i≤k 0, ∀ i > k .(44) Step 1.1 K D cannot be less than k We assume K D <k. If λ>0, A allocates all of his resources on the top K A targets, that is, X A = KD i=1ĉ +ρ w i b 1 b 3 + x * KD +1 ≤ KD +1 i=1ĉ +ρ w i b 1 b 3 . (45) Because of k i=1ĉ wib1b3 <X A < k+1 i=1ĉ wib1b3 , there has k i=1ĉ w i b 1 b 3 < KD+1 i=1ĉ +ρ w i b 1 b 3 .(46) Due to the condition K D <k, the above inequality gives rise to ρ>0, which means that the marginal utility of D is positive. Thus, D allocates all the resources to the top K D targets. According to the expression of NE, the total resources allocated by D on K D targets satisfy Y D ≤ KD i=1 (1 − w KD+1 w i ) b 2 b 3 .(47) This contradicts to the condition Y D > k j=1 (1− w k+1 wj ) b2 b3 when K D <k. If λ=0, the marginal utility on any target B i that has no resource of D is given by w i b 1 b 2 − c > 0 (i>K D ). A obtains a larger utility by shifting resources to any unprotected target, which is a feasible NE. Therefore, K D cannot be less than k. Step 1.2 K D cannot be larger than k We assume K D > k. The total amount of resources used by A at the NE is given by KD+1 i=1 x * i = KD i=1ĉ +ρ w i b 1 b 3 +x * KD +1 ≥ KD i=1ĉ w i b 1 b 3 . Due to the conditions K D >k and X A < k+1 i=1ĉ wib1b3 , we obtain KD+1 i=1 x * i >X A , which is not true. Hence, K D cannot be larger than k. Step 1.3 K D is equal to k In the above analysis, we observe that λ must satisfy λ ≥ w k+1 b 1 b 2 − c,(48) given the condition k<N . Otherwise, A can perform better by moving the resources to the (k+1) th target. Since λ>0, A fully utilizes his resources. We then consider the value of ρ. When ρ>0, D allocates all the resources to the top k targets. This yields Y D = b 2 b 3 k − k i=1 c+λ w i b 1 b 3 .(49) Submitting (48) to (49), we obtain the condition Y D ≤ k i=1 (1− w k+1 wi ) b2 b3 . This contradicts to the initial con- dition Y D > k i=1 (1− w k+1 wi ) b2 b3 . Hence, ρ cannot be greater than 0. When ρ = 0, the NE strategies of A and D can be easily solved by (43) and (44). ii) We next prove the second claim. • P D (k)<Y D <P D (k+1) and X A >P A (k) for 1≤k≤N . The NE is uniquely determined by x * i = ( k j=1 wi wj ) −1 X A , ∀ i ≤ k 0, ∀ i > k(50)y * i = ( k j=1 wi wj ) −1 Y D − b2 b3 k + b2 b3 , ∀i≤k 0, ∀i>k .(51) Step 2.1 K D cannot be less than k We assume K D <k. If λ>0, we obtain the condition ρ>0 following the expression in (45). This means that D allocates Y D resources to the top K D targets. Then, there has the following inequality at the NE Y D ≤ KD i=1 (1 − w KD+1 w i ) b 2 b 3 .(52) Note that the feasible region of Y D is Y D > k−1 j=1 (1− w k wj ) b2 b3 . Because of K D <k, there has Y D > k−1 j=1 (1− w k w j ) b 2 b 3 ≥ KD j=1 (1− w k w j ) b 2 b 3 ≥ KD j=1 (1− w KD +1 w j ) b 2 b 3 . (53) The inequality (52) contradicts to (53), which means that λ cannot be greater than 0. If λ=0, all the resources of A will be moved to target B KD+1 . Then, this is not a NE. Therefore, K D cannot be less than k. Step 2.2 K D cannot be larger than k We assume K D >k with conditioned on k<N . The total amount of resources used by D at the NE satisfy KD i=1 y * i = KD i=1 ( b 2 b 3 − c+λ w i b 1 b 3 )(54) There must have λ≥w KD+1 b 1 b 2 −c if A allocates positive resources to target B KD . Considering the additional condition K D >k, Eq. (54) yields KD i=1 y * i ≥ KD i=1 (1− w KD +1 w i ) b 2 b 3 > k i=1 (1− w k+1 w i ). (55) The resource limit of D should satisfy Y D ≥ KD i=1 y * i . However, the inequality (55) contradicts to the condition Y D < k j=1 (1− w k+1 wj ) b2 b3 . Therefore, K D cannot be larger than k. Step 2.3 K D is equal to k We consider two scenarios separately, k<N and k = N . If k<N , there must have λ ≥ w k+1 b 1 b 2 − c according to Eq. (48). If the equality λ = w k+1 b 1 b 2 − c holds, the total amount of resources used by D at the NE is given by k i=1 (1− w k+1 wi ) b2 b3 . This contradicts to the range of Y D . Hence, there only has λ>w k+1 b 1 b 2 −c, which means that both A and D allocate positive resources to k targets. Since λ>w k+1 b 1 b 2 −c, there exists X A = k i=1 x * i = k i=1ĉ +ρ wib1b3 . Because X A > k i=1ĉ wib1b3 , ρ must be positive. Hence, by letting X A = k i=1 x * i and Y D = k i=1 y * i , we can directly solve the NE as x * i = ( k j=1 w i w j ) −1 X A , ∀ i ≤ k(56)y * i =( k j=1 w i w j ) −1 Y D − b 2 b 3 k + b 2 b 3 , ∀i≤k(57) If k=N , there has λ≥0. Here, when λ=0, the total amount of resources utilized by D at the NE is given by N i=1 y * i = b2 b3 N − N i=1 c wib1b3 . Because Y D < b2 b3 N − N i=1 c wib1b3 , there has N i=1 y * i >Y D , which is not true. Hence, λ is always greater than 0. It is easy to conclude ρ>0 since X A > ci wib1b3 are fully utilized at the NE. Now we are clear that both X A and Y D are disposed on all N targets. The NE can be computed in the same way as that in Eqs. (56) and (57). iii) We then prove the third claim. • X A >P A (N ) and Y D >P D (N +1), the NE is given by x * i =ĉ w i b 1 b 3 and b 2 b 3 − c w i b 1 b 3 .(58) To prove this claim, we only need to show that λ and ρ are both 0 at the NE. We still prove it by contradiction. If λ>0, all the resources of A are allocated to these N targets. Because X A is larger than N i=1ĉ wib1b3 , ρ must be positive in the marginal utility functions. As a countermeasure, D allocates all the resources to defend these targets. However, after D allocates all of his resources, the marginal utilities of A become negative due to Y D > b2 b3 N − N i=1 c wib1b3 . The best strategy of A is to give up all the targets. Hence, either λ and ρ cannot be 0 at the NE. The only possible NE must satisfy λ=ρ=0, which leads to the expression of the NE in Eq. (58). iv) We continue to prove the fourth claim. • X A =P A (k) and Y D ≥P D (k) for 1≤k≤N . DenoteỸ D as any real value in the range [P D (k), min{Y D , P D (k+1)}]. There exist multiple NEs given by x * i = ĉ wib1b3 , ∀ i ≤ k 0, ∀ k+1≤i≤N (59) y * i = b2 b3 +( k j=1 wi wj ) −1 (Ỹ D −k b2 b3 ), ∀ i ≤ k 0, ∀ i > k(60) When x * i is taken asĉ wib1b3 for i≤k, the marginal utilities of D are always 0 on the targets from B 1 to B k . This means that D cannot obtain a better utility by unilaterally changing his strategy. In this scenario, A does not change his allocation strategy as long as his marginal utilities on the targets from B 1 to B k are the same and are nonnegative. LetỸ D be the total amount of resources utilized by D at the NE. There must have k i=1 y * i = k i=1 ( b 2 b 3 − c+λ w i b 1 b 3 ) =Ỹ D .(61) Therefore, the strategy of D is obtained by (62) and y * i = 0 for i>k. Note thatỸ D cannot be larger than P D (k+1). Otherwise, the marginal utilities of A on B 1 to B k become negative such that A gives up these targets. y * i = b 2 b 3 +( k j=1 w i w j ) −1 (Ỹ D −k b 2 b 3 ), ∀ 1≤i≤k v.) We finally prove the fifth claim. • Y D =P D (k) and P A (k−1)≤X A ≤P A (k) for 2≤k≤N . We denoteX A in the range [P A (k−1), X A ]. There exist multiple NEs given by x * i = ( k j=1 wi wj ) −1X A , ∀ i ≤ k 0, ∀ i > k+1(63)y * i = (1− w k+1 wi ) b2 b3 , ∀ i ≤ k 0, ∀ i > k(64) When y * i is taken as b2 b3 (1− w k+1 wi ), the marginal utilities of A on targets from B 1 to B k are all 0. Then, A cannot improve his utility by individually changing his strategy. At the NE, the marginal utilities of D on targets from B 1 to B k should be non-negative and identical. LetX A be the amount of resources used by A at the NE. There exist w i x * i =w j x * j >0 for all i, j ≤ k and k i=1 x * i =X A . Hence, the NE strategy of A is given by x * i = ( k j=1 wi wj ) −1X A for i ≤ k and x * i = 0 for i > k. This concludes the proof. Proof of Theorem 5 The proof follows that of Theorem 3. Let λ and ρ be the Lagrange multipliers of A and D respectively. Let L D (y, ρ) be the Lagrange function of the defender D that has L D (y, ρ) = − N i=1 w i f (x i ) f (x i ) + g(y i ) −ĉ N i=1 y i +ρ(Y D − N i=1 y i ). (65) We take the derivative of L D (y, ρ) over y i and obtain dL D (y, ρ) dy i = w i f (x i )g ′ (y i ) (f (x i ) + g(y i )) 2 − (ĉ + ρ), ∀i. (66) Here, L D (y, ρ) is optimized in two ways. If the above derivative is 0, there exists a non-zero resource allocation strategy, i.e. y * i > 0. If the above derivative is less than 0, then y * i is 0. Similarly, we can find the conditions for the attacker to maximize his utility. For the sake of redundancy, we omit the detailed proof. Proof of Lemma 5 Proof: According to Theorem 2, these exists a unique NE with the proportion-form breaching model. We next show by contradiction that x i cannot be 0 on any target B i at the NE. Suppose x * i = 0 on target B i . Then, there has f (x * i ) = 0 such that y * i is 0. When target B i is not protected by D, the best response of A is to allocate an arbitrarily small amount of resources to this target. Hence, (0, 0) is not an equilibrium strategy for A and D. Therefore, A and D allocate positive resources to all the targets at the NE. Proof of Lemma 6 Fig. 4 . 4Sufficiency of X A and Y D with linear f (·) and g(·) D 3 ): X A >X suf A and Y D ≥ Y suf D .This case is symmetric to that of D 2 ), which is not analyzed here. D 4 ): both X A and Y D are insufficient.In this domain, all the resources of A and D are utilized. Then, there have x Y D . The utilities of A and D are given by Here, Eq.(37) means that the marginal utilities of A are non-negative and are the same on the top K A targets. Eq. (38) means that the marginal utility of A on any top K D target is larger than that on target B KD+1 . This guarantees the condition K D ≤K A ≤K D +1. Eq. (39) ensures that D allocates positive resources to the top K D targets. Eq. (40) means that D does not allocate resources to B KD +1 . The above conditions give rise to the solution to the NE, Proof: Consider two targets B i and B j with w i > w j . The following equations hold at the NE. w i f ′ (x i )g(y i ) (f (x i ) + g(y i )) 2 = w j f ′ (x j )g(y j ) (f (x j ) + g(y j )) 2 = c + λ;(67)The above equations yield the following relationshipWe prove this lemma by contradiction. Let us assume that there has y i < y j . Because g(·) is a concave and strictly increasing function, we have g(y i ) < g(y j ) and g ′ (y i ) > g ′ (y j ). The right hand of Eq.(69) is greater than 1. Then, there must have x i < x j in the left hand of Eq.(69). We define two functions, f 1 (x, y) and f 2 (x, y), whereWe take the derivatives of f 1 (x, y) and f 2 (x, y) over x and y respectively.(73)The signs of ∂f1 ∂y and ∂f2 ∂x depend on whether f (x) is greater than g(y) or not. 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[ "Randomness Requirement on CHSH Bell Test in the Multiple Run Scenario Randomness Requirement on CHSH Bell Test in the Multiple Run Scenario 2", "Randomness Requirement on CHSH Bell Test in the Multiple Run Scenario Randomness Requirement on CHSH Bell Test in the Multiple Run Scenario 2", "Randomness Requirement on CHSH Bell Test in the Multiple Run Scenario Randomness Requirement on CHSH Bell Test in the Multiple Run Scenario 2", "Randomness Requirement on CHSH Bell Test in the Multiple Run Scenario Randomness Requirement on CHSH Bell Test in the Multiple Run Scenario 2" ]	[ "Xiao Yuan [email protected] \nCenter for Quantum Information\nInstitute for Interdisciplinary Information Sciences\nTsinghua University\n100084BeijingChina\n", "Zhu Cao \nCenter for Quantum Information\nInstitute for Interdisciplinary Information Sciences\nTsinghua University\n100084BeijingChina\n", "Xiongfeng Ma \nCenter for Quantum Information\nInstitute for Interdisciplinary Information Sciences\nTsinghua University\n100084BeijingChina\n", "Xiao Yuan [email protected] \nCenter for Quantum Information\nInstitute for Interdisciplinary Information Sciences\nTsinghua University\n100084BeijingChina\n", "Zhu Cao \nCenter for Quantum Information\nInstitute for Interdisciplinary Information Sciences\nTsinghua University\n100084BeijingChina\n", "Xiongfeng Ma \nCenter for Quantum Information\nInstitute for Interdisciplinary Information Sciences\nTsinghua University\n100084BeijingChina\n" ]	[ "Center for Quantum Information\nInstitute for Interdisciplinary Information Sciences\nTsinghua University\n100084BeijingChina", "Center for Quantum Information\nInstitute for Interdisciplinary Information Sciences\nTsinghua University\n100084BeijingChina", "Center for Quantum Information\nInstitute for Interdisciplinary Information Sciences\nTsinghua University\n100084BeijingChina", "Center for Quantum Information\nInstitute for Interdisciplinary Information Sciences\nTsinghua University\n100084BeijingChina", "Center for Quantum Information\nInstitute for Interdisciplinary Information Sciences\nTsinghua University\n100084BeijingChina", "Center for Quantum Information\nInstitute for Interdisciplinary Information Sciences\nTsinghua University\n100084BeijingChina" ]	[]	The Clauser-Horne-Shimony-Holt inequality test is widely used as a mean of invalidating the local deterministic theories and a tool of device independent quantum cryptographic tasks. There exists a randomness (freewill) loophole in the test, which is widely believed impossible to be closed perfectly. That is, certain random inputs are required for the test. Following a randomness quantification method used in literature, we investigate the randomness required in the test under various assumptions. By comparing the results, one can conclude that the key to make the test result reliable is to rule out correlations between multiple runs. PACS numbers: arXiv:1409.7875v1 [quant-ph]	10.1103/physreva.91.032111	[ "https://arxiv.org/pdf/1409.7875v1.pdf" ]	115,952,688	1409.7875	29f033f2cc523e3f6539ad5d825bbd12e13ba3db	Randomness Requirement on CHSH Bell Test in the Multiple Run Scenario Randomness Requirement on CHSH Bell Test in the Multiple Run Scenario 2 28 Sep 2014 Xiao Yuan *[email protected] Center for Quantum Information Institute for Interdisciplinary Information Sciences Tsinghua University 100084BeijingChina Zhu Cao Center for Quantum Information Institute for Interdisciplinary Information Sciences Tsinghua University 100084BeijingChina Xiongfeng Ma Center for Quantum Information Institute for Interdisciplinary Information Sciences Tsinghua University 100084BeijingChina Randomness Requirement on CHSH Bell Test in the Multiple Run Scenario Randomness Requirement on CHSH Bell Test in the Multiple Run Scenario 2 28 Sep 2014 The Clauser-Horne-Shimony-Holt inequality test is widely used as a mean of invalidating the local deterministic theories and a tool of device independent quantum cryptographic tasks. There exists a randomness (freewill) loophole in the test, which is widely believed impossible to be closed perfectly. That is, certain random inputs are required for the test. Following a randomness quantification method used in literature, we investigate the randomness required in the test under various assumptions. By comparing the results, one can conclude that the key to make the test result reliable is to rule out correlations between multiple runs. PACS numbers: arXiv:1409.7875v1 [quant-ph] Introduction Historically, Bell tests [1] are proposed for distinguishing quantum theory from local hidden variable models (LHVMs) [2]. In a general picture, a Bell test involves multiple parties who randomly choose inputs and generate outputs with pre-shared physical resources. Based on the probability distributions of inputs and outputs, an inequality, called Bell's inequality, is defined. Any Bell test is meaningful only if all LHVMs satisfy the Bell's inequality; while in quantum mechanics, such inequality can be violated via certain quantum settings. Experimental observation of the violation of Bell's inequality would show that LHVMs are not sufficient to describe the world, and other theories, such as the quantum mechanics, are demanded. In this work, we focus on the bipartite scenario and investigate one of the most wellknown Bell tests, the Clauser-Horne-Shimony-Holt (CHSH) inequality [3]. As shown in Fig. 1(a), two space-like separated parties, Alice and Bob, randomly choose input settings x and y from an input set I = {0, 1} and generate outputs bits a and b based on their inputs and pre-shared quantum (ρ) and classical (λ) resources, respectively. The probability distribution p(a, b\|x, y), obtaining outputs a and b conditioned on inputs x and y, are determined by specific strategies of Alice and Bob. By assuming that the input settings x and y are chosen fully randomly and equally likely, we define the CHSH inequalities by convex combining the probability distribution p(a, b\|x, y) according to S = a,b,x,y (−1) a⊕b+x·y p(a, b\|x, y) ≤ S C = 2, (1.1) where the plus operation ⊕ is modulo 2, · is numerical multiplication, and S C is the (classical) bound of Bell value S for all LHVMs. Similarly, there is an achievable bound S Q = 2 √ 2 for the quantum theory [4]. In this case, a violation of the classical bound S C indicates the need for alternative theories other than LHVMs, such as quantum theory. For general no signalling (NS) theories [5], we denote the corresponded upper bound as S N S = 4. It is straightforward to see that S N S ≥ S Q ≥ S C . In practice, the technique of violating a Bell's inequality can be applied to other quantum information tasks, such as, device independent quantum key distribution [6,7] and randomness expansion [8,9]. Security proofs of these tasks are generally independent of the realization devices or correctness of quantum theory, but relies on violating a Bell's inequality. For instance, we can consider the devices of Alice and Bob as black boxes. In this case, we can assume, in the worse scenario, that an adversary Eve, instead of Alice and Bob, performs measurements as shown in Fig. 1(b). Because the two parties are space-like separated, the probability distribution generated in this way is always within the scope of LHVMs, that is, p(a, b\|x, y) = p(a\|x, λ 1 )p(b\|y, λ 1 ). Therefore, Eve cannot fake a violation of any Bell tests, which intuitively explains the security of the device independent tasks. Since the first experiment in the early 1980s [10], lots of lab demonstrations of the CHSH inequality has been presented. These experiment results show explicit violations of the LHVMs bound S C , and meanwhile, suffer from a few technical and inherent Bob, x and y, are decided by perfect random number generators (RNGs), which produce uniformly distributed random numbers; (b) The measurement devices are controlled by an adversary Eve through local hidden variables λ 1 ; (c) The input random numbers are additionally controlled by some local hidden variable λ 2 , which is accessible to Eve. x a Alice y b Bob RNG RNG p a b x y ( , \| , ) x a y b p a x λ q x y λ ( \| ) ( \| ) ( , \| ) , 1 1 2 p b y λ , λ 2 a) c) x a y b λ 1 RNG RNG p a x λ ( \| ) ( \| ) , 1 1 p b y λ , b) Eve λ ρ , Eve λ 1 Eve Eve RNG RNG loopholes, which might invalidate the conclusions. Two well-known technical obstacles are due to the locality loophole and the detection efficiency loophole, which can be closed with more delicately designed experiments and developed instruments [11,12,13]. In contrast to the technical loopholes, there also exists an inherent loophole that cannot be closed perfectly in Bell tests -the input settings may not be chosen randomly. In the worst case, the inputs can be all predetermined, which makes it possible to violate the Bell inequalities even with LHVMs. In this case, a witness of violating a Bell's inequality does not imply the demands for non-LHVM theories and the Bell test becomes meaningless. On the other hand, without the quantum theory or violation of Bell's inequalities, one cannot get provable randomness. Therefore, the assumption of true input randomness are indispensable in Bell tests because we cannot prove or disprove its existence. Practically, suppose the input settings are partially controlled by an adversary Eve, who wants to convince Alice and Bob a violation of Bell's inequality with classical settings, as shown in Fig. 1(c). In this case, we model the imperfect randomness by assuming that the input settings x and y are chosen according to some probability distribution q(x, y\|λ 2 ), where λ 2 are local variables available to Eve. Now, the probability distribution p(a, b\|x, y) are defined by p(a, b\|x, y) = p A (a\|x, λ 1 )p B (b\|y, λ 1 )q(x, y\|λ 2 ). (1.2) We can therefore rewrite the CHSH inequality, given in Eq. (1.1), as S = 4 λ a,b,x,y (−1) a⊕b+x·y p A (a\|x, λ)p B (b\|y, λ)q(x, y\|λ)q(λ),(1.3) where the factor 4 is because of the average probability of choosing the input settings x and y, q(x, y) = λ q(x, y\|λ)q(λ), is required to be 1/4, and the hidden variables λ 1 and λ 2 are combined as λ. Notice that, in the extreme (deterministic) case where q(x, y\|λ) = 0 or 1 for all x, y, the local hidden variables λ deterministically control the input settings. Then Eve is able to violate Bell tests to an arbitrary value with LHVMs. On the other hand, if Eve has no control of the input settings where q(x, y\|λ) = 1/4 for all x, y, she cannot fake a violation at all. Therefore, a meaningful question to ask is how one can assure that a violation of the CHSH inequality is not caused by Eve's attack on imperfect input randomness. That is, we want to know what the requirement of the input randomness is to guarantee that an observed violation truly stems from quantum effects. In the following, we first introduce the quantification of input randomness and review previous works on this question in Section 2. Then we study a simplified case to gain the intuition behind Eve's optimal strategy in Section 3. Finally, we investigate the randomness requirement of the CHSH test and conclude our result in Section 4. Randomness Requirement Let us start with quantifying the input randomness. Here, we make use of the randomness parameter P adopted in Ref. [14] to fulfill such an attempt, other tools such as the Santha-Vazirani source [15] may work similarly. The parameter P is defined to be the maximum probability of choosing the inputs conditioned on the hidden variable λ, P = max x,y,λ q(x, y\|λ). (2.1) With this definition, the larger P is, the less input randomness, the more control Eve has, and the easier for her to fake a quantum violation with LHVMs. In the CHSH test, P takes values in the regime of [1/4, 1]. When P = 1, it represents the case that Eve has the most control of Alice and Bob's inputs, that is, the local hidden variable λ can determine at least one set of values of x and y. When P = 1/4, it corresponds to the case of complete randomness, where the adversary have no prior information on the inputs. Note that the definition of P essentially follows the min-entropy, which is widely used to quantify randomness of a random variable X in information theory, H min = − log [max x prob(X = x)]. Intuitively, given complete randomness where P = 1/4, the value S with LHVMs are bounded by S C as shown in Eq. (1.1); while given the most dependent (on λ) randomness where P = 1, the value S with LHVMs could reach the mathematical maximum, S N S in the CHSH test. Then it is interesting to check the maximal S value for P ∈ (1/4, 1) with LHVMs. In this work, we are interested in when the adversary can fake a quantum violation given certain randomness P . We thus exam the lower bound P Q of P such that the Bell test result can reach the quantum bound S Q with an optimal LHVM. This lower bound P Q puts a minimal randomness requirement in a Bell test experiment. Only if the freedom of choosing inputs satisfies P < P Q , can one claim that the Bell test is free of the randomness loophole. Recently, lots of efforts have been spent to investigate such requirement of randomness needed to guarantee the correctness of Bell tests [16,17,18,14,19,20]. These works analyze under different conditions. One condition is about whether the input settings are correlated or not in different runs. We call it single run, referring to the case that the input settings are independent for different runs, and multiple run referring to otherwise. The other condition is about whether the random inputs of Alice and Bob are correlated. Conditioned on these different assumptions of the input randomness, the lower bound P Q that allows LHVMs to saturate the quantum bound S Q in the CHSH Bell test is summarized in Table 1. In the single run scenario, the optimal strategy for Eve reaches S = 24P − 4 and S = 8P in the case that Alice's and Bob's input settings are correlated and uncorrelated, respectively [16,14]. To achieve the maximum quantum violation S Q = 2 √ 2, the critical randomness requirement is shown in Table 1. It is worth mentioning that if one has randomness P ≥ P N S = 1/3 and P ≥ P N S = 1/2 for the case of correlated and uncorrelated, respectively, Eve is able to recover arbitrary NS correlations. In a more realistic scenario, the multiple run case, the input settings in different runs are generally correlated. Denote N to be the number of test runs, x i (y i ) and a i (b i ) to be the input and output of Alice (Bob) for the ith run, where i = 1, 2, . . . , N , respectively. In the multiple run scenario, the input settings of Alice and Bob can be further correlated by q(x 1 , x 2 , . . . , x N , y 1 , y 2 , . . . , y N \|λ), (2.2) Therefore, similar to the definition of Eq. (1.3), we define the CHSH test in the multiple run case, S = 4 N λ a,b,x,y (−1) a⊕b+x·y p A (a\|x, λ)p B (b\|y, λ)q(x, y\|λ)q(λ), (2.3) where a = (a 1 , a 2 , . . . , a N ), b = (b 1 , b 2 , . . . , b N ), x = (x 1 , x 2 , . . . , x N ), y = (y 1 , y 2 , . . . , y N ), and · is inner product of two vectors, x and y. Now, we can define the input randomness parameter, as an extension of Eq. (2.1), P = max x,y,λ q(x, y\|λ) 1/N . (2.4) It is obvious that the adversary is easier to fake a violation of a Bell test with LHVMs with increasing number of runs N . This is because the adversary can take advantage of additional dependence of the inputs in different runs. It has been shown that with randomness P ≥ P Q = 0.258, Eve is able to fake the maximum quantum violation S Q [19]. This result [19] puts a very strict requirement on the RNGs to guarantee a faithful CHSH test. A meaningful remaining question is thus to consider the case of multiple run but uncorrelated scenario. As all Bell experiments must run many times to sample the probability distribution, it is reasonable and also practical to consider a joint attack by Eve. On the other hand, the uncorrelated assumption is also reasonable in many realistic cases, where the experiment instruments of Alice and Bob are manufactured independently. In fact, if the inputs are determined by cosmic photons that are causally disconnected from each other, there should be no correlations between the input randomness of Alice and Bob [21]. Considering uncorrelated inputs of Alice and Bob, q(x, y\|λ) = q A (x\|λ)q B (y\|λ),(2.5) we want to investigate the optimal attack with restricted randomness input P in the following CHSH inequality, S = 4 N λ a,b,x,y (−1) a⊕b+x·y p A (a\|x, λ)p B (b\|y, λ)q A (x\|λ)q B (y\|λ)q(λ).(2.6) That is, we want to maximize Eq. (2.6) with the constraint of Eq. (2.4). In particular, we are interested to see when this maximal value can reach S Q = 2 √ 2. Single run case We first review the optimal strategy in the single run scenario [14] to get an intuition behind the optimal attack of the adversary. Hereafter, we mainly focus on the scenario that Alice and Bob's inputs are uncorrelated as defined in Eq. (2.5). Thus, what we want is to maximize the CHSH value S, S = λ q(λ)S λ ,(3.1) where S λ = 4 a,b,x,y (−1) a⊕b+x·y p A (a\|x, λ)p B (b\|y, λ)q A (x\|λ)q B (y\|λ),(3.2) with restricted randomness P , given in Eq. (2.1). Since any probabilistic LHVM, that is, p A (a\|x, λ)p B (b\|y, λ), could be realized by a convex combination of deterministic ones [22], it is therefore sufficient to only consider deterministic LHVMs. Due to the symmetric definition of the CHSH inequality, we only need to consider a specific strategy of p A (0\|x, λ) = p B (0\|y, λ) = 1, and p A (1\|x, λ) = p B (1\|y, λ) = 0 for some given λ, and all the other ones works similarly. By substituting the special strategy into Eq. (3.2), we get S λ = 4 [q A (0)q B (0) + q A (0)q B (1) + q A (1)q B (0) − q A (1)q B (1)] .(3.3) Suppose P A = max x,λ {q A (x\|λ)}, P B = max y,λ {q B (x\|λ)}, and hence P = P A P B , S λ can be maximized to S λ ≤ 4 [1 − 2(1 − P A )(1 − P B )] = 8(P A + P B − P ) − 4. (3.4) Given P , S λ is supper bounded by S λ ≤ 8P,(3.5) where the equality holds when P B = 1/2 and P A = 2P . Thus, the optimal strategy with LHVMs is S = 8P . Note that, when the input settings are fully random, P = 1/4, the optimal strategy of LHVMs is S = 2, which recovers the original LHVMs bound S C . It is easy to see that, to saturate the quantum bound S Q = 2 √ 2, the randomness should be at least P Q = S Q /8 = √ 2/4 ≈ 0.354, as shown in Table 1. In the single run case, we only need to consider deterministic strategies of p(a, b\|x, y) due to the symmetric definition of the CHSH inequality. We also take advantage of this property in the derivation of the multiple run case. In addition, we can see that the optimal strategy of LHVMs is to choose x or y fully randomly and the other one as biased as possible. This biased optimal strategy is counter-intuitive since the adversary do not need to control the inputs of both parties, but only those of one party. We show that this counter-intuitive feature does not hold in the optimal strategy in the multiple run case. Multiple run case Now we consider the multiple run scenario with uncorrelated input randomness. That is, optimizing Eve's LHVM strategy Eq. (2.6) with constraints defined in Eq. (2.4). Similar to the single run case, from the symmetric argument, we also only need to consider one specific value of λ in the strategy: p A (0\|x, λ) = p B (0\|y, λ) = 1, and p A (1\|x, λ) = p B (1\|y, λ) = 0. Given the probabilities of Alice's and Bob's inputs, q A (x\|λ), q B (y\|λ), the CHSH value for this specific λ is given by Eq. (2.6), S λ = 4   1 − 2 N x,y∈{0,1} N x · yq A (x\|λ)q B (y\|λ)   . (4.1) Our attempt is therefore to maximize Eq. (4.1) with constraints q A (x\|λ)q B (y\|λ) ≤ P N ,(4.2) for all q A (x\|λ) and q B (y\|λ). Since in the single run scenario, the optimal strategy requires only one party with biased conditional probability, we first analyze the case with only Alice's inputs biased and Bob's inputs uniformly distributed. Then we investigate the case where the inputs of both parties are biased. We can see that the one party biased strategy is not optimal in the multiple run case, even when N = 2. One party Biased In the case when Eve only (partially) controls one of the inputs, say Alice's, the probability of Alice's input string q A (x\|λ) is biased and Bob's input string is uniformly distributed, that is, q B (y\|λ) = 1 2 N . (4.3) The randomness is characterized by Eq. (2.4), after substituting Eq. (4.3), P = P A 2 , (4.4) where P A is defined by P A = max λ,x q A (x\|λ) 1/N . Then, the CHSH value, Eq. (4.1), becomes S λ = 4   1 − 1 N 2 N −1 x,y∈{0,1} N x · yq A (x\|λ)   . (4.5) Denote the number of bit 1 in an N string a as L 1 (a). Given the number of bit 1 in x, k A = L 1 (x), we can sum over y, y∈{0,1} N x · y = k A j=1 2 N −k A j k A j = 2 N −1 k A ,(4.6) and group the summation of x according to k A , S λ = 4   1 − 1 N N k A =0 L 1 (x)=k A q A (x\|λ)k A   , (4.7) One only need to consider the LHVMs whose probabilities of q A (x\|λ) with the same k A are the same. Otherwise, we can always take an average of q A (x\|λ) with the same k A without increasing the randomness parameter P . Thus we can rewrite S λ as S λ = 4 1 − 1 N N k A =0 q k A (x\|λ) N k A k A , (4.8) with normalization requirement N k A =0 q k A (x\|λ) N k A = 1, (4.9) and constraints defined in Eq. (4.2). The optimization of Eq. (4.8) can be solved sufficiently via linear programming. Intuitively, to maximize S λ with given P defined in Eq. (4.4), we can simply assign q k A (x\|λ) that has large k A be 0 and that has smaBll k A be (2P ) N . Suppose there exists an integer l such that P can be written as P = 1 2 l k A =0 N k A −1/N ,(4.S = 4   1 − 1 N N k A =0 1 2 l k A =0 N k A −1/N N k A k A   . (4.11) For a general case where an integer l cannot be found satisfying Eq. (4.10), we can first find an integer l such that, 1 2 l+1 k A =0 N k A −1/N < P ≤ 1 2 l k A =0 N k A −1/N . (4.12) Then we can assign q k A (x\|λ) to be q k A (x\|λ) =        (2P ) N k A ≤ l 1− l k A =0 (2P ) N ( N k A ) −1/N ( N l+1 ) k A = l + 1 0 k A > l + 1 (4.13) For finite N , one can numerically solve the problem according to Eq. (4.13). As shown in Fig. 2, the optimal strategy for N = 1, 10, 100 are calculated. With increasing N , the optimal value S increases and hence a valid Bell test requires a smaller P (more randomness). Figure 2: Optimal value of the CHSH test for given randomness P with various rounds N based on only one party biased randomness. The solid line is the optimal strategy for N → ∞, which upper bounds all finite N rounds. Note that the curve is not smooth for finite runs N because the optimal strategy q k A defined in Eq. (4.13) jumps in l. With N grows larger, the curve tends to be smoother. In the case of N → ∞, we can derive an analytic bound for all finite N strategies. By following the technique used in Ref. [19], we first can estimate P defined in Eq. (4.12) with the limit of N → ∞ by, where is an arbitrary positive number. Then we can substitute Eq. (4.15) into Eq. (4.14), and get a relation between optimized CHSH value S and the corresponding randomness parameter P , lim N →∞ P = 1 2ll (1 −l) 1−l ,P = 1 2 4 − S 4 (4−S)/4 S 4 (S/4) . (4.16) By substituting the quantum bound S Q = 2 √ 2 into Eq. (4.16), we can get the critical randomness requirement to be P Q = 0.273. Note that, although Eve only control Alice's input settings, she can still fake a quantum violation with sufficiently low randomness, which is lower than the single run case even when Alice's and Bob's inputs are correlated. Thus we show that the randomness is more demanded for the conditions of multiple/single run compared to the correlation between Alice and Bob. Both parties biased Now we consider a general attack, where Eve controls both inputs of Alice and Bob. In this case, we need to optimize Eq. (4.1) with constraints defined in Eq. (4.2). Similarly, we also group the summation of x and y according to the corresponded number of bit 1, k A = L 1 (x) and k B = L 1 (y), S λ = 4   1 − 2 N N k A ,k B =0 L 1 (x)=k A L 1 (y)=k B q A (x\|λ)q B (y\|λ)x · y   . (4.17) Now, if we assume that q A (x\|λ) (q B (y\|λ)) has the same value for equal k A (k B ), we can sum over x and y for given k A and k B , k A ,k B x · y = N k A min{k A ,k B } j=max{1,k A +k B −N } j k A j N − k A k B − j = N k A k A N − 1 k B − 1 = k A k B N N k A N k B . (4.18) We can then get the S value to be S λ = 4 1 − 2 N 2 N k A ,k B =0 q k A (x\|λ) N k A q k B (y\|λ) N k B k A k B ,(4.19) with the constraints of q A (x\|λ) and q B (y\|λ), N k A =1 q k A (x\|λ) N k A = 1, N k B =1 q k B (y\|λ) N k B = 1. (4.20) It is worth mentioning that the assumption that q A (x\|λ) (q B (y\|λ)) takes the same value for equal k A (k B ) is not obviously equivalent to the original optimization problem defined in Eq. (4.17). We thus take this step as an additional assumption, and conjecture it to be true for certain cases of the input randomness. The problem defined in Eq. (4.19) with constraints of Eq. (4.20) cannot be solved by linear programming directly, as to the nonlinear terms q k A (x\|λ)q k B (y\|λ). However, we can still optimize it with similar methods used in the previous section. Define the maximum randomness on each side P A = [max λ,x q k A (x\|λ)] 1/N , P B = [max λ,y q k B (y\|λ)] 1/N . (4.21) To maximize S λ , we can do it first for the Alice side, q k A , and then Bob side q k B . By doing so, it is not hard to see that S λ is maximized by assigning q k A that has small number of k A to be P A and that has large number of k A to be 0, and similarly for q k B . Thus we need to first find l A and l B for Alice and Bob, such that, l A +1 k A =0 N k A −1/N < P A ≤ l A k A =0 N k A −1/N l B +1 k B =0 N k B −1/N < P B ≤ l B k B =0 N k B −1/N . (4.22) Then we can assign q k A (x\|λ) and q k B (y\|λ) to be q k A (x\|λ) =          (P A ) N k A ≤ l A 1− l A k A =0 P N A ( N k A ) −1/N ( N l A +1 ) k A = l A + 1 0 k A > l A + 1 q k B (y\|λ) =          (P B ) N k B ≤ l B 1− l B k B =0 P N B ( N k B ) −1/N ( N l B +1 ) k B = l B + 1 0 k B > l B + 1 (4.23) to optimize S λ defined in Eq. (4.19). For finite N , we can also numerically solve the optimization problem defined in Eq. (4.19). As shown in Fig. 3. The value S increases with the number of runs N , thus the strategy with infinite rounds puts a bound on the strategy with finite rounds. In the case of N → ∞, we can also find analytical relation between optimized S and the corresponded P . Similarly, we can first estimate P A and P B defined in Eq. (4.22) with the limit of N → ∞ by lim N →∞ P A =ll A A (1 −l A ) 1−l A , lim N →∞ P B =ll B B (1 −l B ) 1−l B ,(4.24) wherel A = l A /N andl B = l B /N , and S according to S = 4 − 8l AlB . (4.25) As we still have to optimize over all possible P A and P B that satisfies P A P B = P , we cannot get a direct analytic formula like in Eq. (4.16), while we can still numerically solve and plot it in Fig. 3. To reach a maximum quantum violation S Q = 2 √ 2 with a LHVM, the randomness is required to be P ≥ P Q ≈ 0.264. Discussion We take an additional assumption in the derivation of the both parties biased case, thus the obtained bound P Q ≈ 0.264 is still an upper bound of a general optimal attack. As we already know, the randomness requirement for the worst case, that is, multiple run with Alice and Bob correlated, is strictly bounded by P Q ≈ 0.258 [19]. Thus, we know that the tight P Q for the case of multiple run but Alice and Bob uncorrelated should lie in the regime of [0.258, 0.264]. To gain intuition why we take the additional assumption, first notice that what we want is to minimize the average contribution of x · y in Eq. (4.17). In our case, where P is near 1/4, q A (x\|λ) and q B (y\|λ) can be regarded as an approximately flat distribution. On average, the x (y) contains less number of 1s will contribute more to S, which means we should assign the corresponded probability q A (x\|λ) (q A (y\|λ)) bigger in order to maximize S. As q A (x\|λ) (q A (y\|λ)) is upper bounded by P A (P B ), an intuitive optimal strategy is then to let q A (x\|λ) (q A (y\|λ)) be P A (P B ) for x (y) contains less number of 1s, and be 0 for the ones contains more number of 1s. As q A (x\|λ) (q A (y\|λ)) should also satisfy the normalization condition (Eq. (4.20)), we can simply follow the strategy defined in Eq. (4.23) to realize the intuition, which on the other hand satisfies the assumption we take. Follow the above intuition, we conjecture the assumption to be true for certain cases of N . That is, for finite N , we conjecture it to be true when equalities are taken in Eq. (4.22) for both P A and P B . On the other hand, we want to emphasize that for a finite N , the assumption will not generally hold in the optimal strategy if the equalities in Eq. (4.22) are not fulfilled. For example, if the probability of l A + 1 and l B + 1 in Eq. (4.23) is not 0 but very small, we should not take all q A (x\|λ) and q B (y\|λ) equally as q k A and q k B , especially for the case of L 1 (x) = l A + 1 and L 1 (y) = l B + 1 , respectively. In fact, there do exists a cleverer assignment of q A (x\|λ) and q B (y\|λ) such that only x and y that gives small x · y get probability instead of all of x and y that L 1 (x) = l A + 1 and L 1 (B) = l B + 1. However, with increasing runs N , this kind of clever attack stops working as for the equalities can be more approximately satisfied with larger N . Therefore, we also conjecture the assumption to be true for all possible P with N goes to infinity. As we can see, our obtained P Q ≈ 0.264 is already very close to the worst case value that is 0.258, we can therefore conclude that the multiple run correlation is already a strong resource for the adversary, no matter whether Alice and Bob are correlated or not. In addition, as we know that the bound P Q for the most loose case, that is, single run and Alice Bob uncorrelated, is given to be 0.354 [14], we also suggest that the key loophole of the input randomness is the correlation between multiple runs instead of correlation of Alice and Bob. Conclusion In this work, we consider the randomness requirement of CHSH test in the multiple run scenario. By considering an adversary Eve who independently controls the input randomness of Alice and Bob, we investigate the minimum randomness requirement to guarantee that a violation of the CHSH inequality is not due to Eve's attack (LHVM). Considering that Eve controls only Alice's input but leaves Bob's input uniformly distributed, we found the randomness Eve need to control to fake a quantum violation is P Q = 0.273. And the randomness required when controlling both Alice and Bob is P Q ≤ 0.264. By comparing the results to the ones listed in Table. 1, we conclude that the key randomness loophole is due to the correlation between multiple runs. As the randomness requirement which considers multiple run attack is not easy to realize in real experiments, we thus suggest the experiments to rule correlations of the input settings from different runs. To guarantee the securities of the device independent tasks, we also suggest that one should check whether there is correlation between random inputs from different runs. For further research, we are interested to know whether there exists Bell inequalities that suffers less from the randomness loophole. By assuming different kinds of assumptions, the randomness requirement behaves different. Recently, by considering a nonzero lower bound for the input random probability p(x, y\|λ), Pütz et al. show a Bell inequality which suffers from very little randomness loophole [23]. That is, any adversary cannot fake a quantum violation as long as the lower bound of p(x, y\|λ) is nonzero regardless of its upper bound P defined in Eq. (2.1). Therefore, it is interesting to investigate the multiple run randomness requirement of the CHSH inequality with additional assumption. Figure 1 : 1Bell tests in the bipartite scenario. (a) The inputs of Alice and Figure 3 : 3Possible optimal value of the CHSH test for given randomness P with various rounds N based on uncorrelated input. The solid line corresponds the strategy for N → ∞, which upper bounds all finite N cases. The curves are not smooth for finite N as for similar reasons like in the one party biased case, and it tends to be smooth with N → ∞. 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[ "New implementation of hybridization expansion quantum impurity solver based on Newton-Leja interpolation polynomial", "New implementation of hybridization expansion quantum impurity solver based on Newton-Leja interpolation polynomial" ]	[ "Li Huang \nBeijing National Laboratory for Condensed Matter Physics\nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina\n\nScience and Technology on Surface Physics and Chemistry Laboratory\nP.O. Box 718-35621907Mianyang, SichuanChina\n", "Xi Dai \nBeijing National Laboratory for Condensed Matter Physics\nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina\n" ]	[ "Beijing National Laboratory for Condensed Matter Physics\nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina", "Science and Technology on Surface Physics and Chemistry Laboratory\nP.O. Box 718-35621907Mianyang, SichuanChina", "Beijing National Laboratory for Condensed Matter Physics\nInstitute of Physics\nChinese Academy of Sciences\n100190BeijingChina" ]	[]	We introduce a new implementation of hybridization expansion continuous time quantum impurity solver which is relevant to dynamical mean-field theory. It employs Newton interpolation at a sequence of real Leja points to compute the time evolution of the local Hamiltonian efficiently. Since the new algorithm avoids not only computationally expansive matrix-matrix multiplications in conventional implementations but also huge memory consumptions required by Lanczos/Arnoldi iterations in recently developed Krylov subspace approach, it becomes advantageous over the previous algorithms for quantum impurity models with five or more bands. In order to illustrate the great superiority and usefulness of our algorithm, we present realistic dynamical mean-field results for the electronic structures of representative correlated metal SrVO3.	null	[ "https://arxiv.org/pdf/1205.1708v1.pdf" ]	117,371,839	1205.1708	eec0384275505763c12edde565be2b8f268e5729	New implementation of hybridization expansion quantum impurity solver based on Newton-Leja interpolation polynomial May 2012 (Dated: May 1, 2014) Li Huang Beijing National Laboratory for Condensed Matter Physics Institute of Physics Chinese Academy of Sciences 100190BeijingChina Science and Technology on Surface Physics and Chemistry Laboratory P.O. Box 718-35621907Mianyang, SichuanChina Xi Dai Beijing National Laboratory for Condensed Matter Physics Institute of Physics Chinese Academy of Sciences 100190BeijingChina New implementation of hybridization expansion quantum impurity solver based on Newton-Leja interpolation polynomial May 2012 (Dated: May 1, 2014)numbers: 0270Ss7110Fd7130+h7110Hf We introduce a new implementation of hybridization expansion continuous time quantum impurity solver which is relevant to dynamical mean-field theory. It employs Newton interpolation at a sequence of real Leja points to compute the time evolution of the local Hamiltonian efficiently. Since the new algorithm avoids not only computationally expansive matrix-matrix multiplications in conventional implementations but also huge memory consumptions required by Lanczos/Arnoldi iterations in recently developed Krylov subspace approach, it becomes advantageous over the previous algorithms for quantum impurity models with five or more bands. In order to illustrate the great superiority and usefulness of our algorithm, we present realistic dynamical mean-field results for the electronic structures of representative correlated metal SrVO3. I. INTRODUCTION The rapid development of efficient numerical and analytical methods for solving quantum impurity models has been driven in recent years by the great success of dynamical mean-field theory (DMFT) 1,2 and its non-local extensions. 3 In the framework of DMFT, the momentum dependence of self-energy is neglected, then the solution of general lattice model may be obtained from the solution of an appropriately defined quantum impurity model plus a self-consistency condition. Both the non-local extensions of DMFT 3 and realistic DMFT (i.e, local density approximation combined with dynamical mean-field theory, LDA+DMFT) calculations, 2 involve multi-site or multi-orbital quantum impurity models, whose solutions are computationally expensive and in practice the bottleneck of the whole calculations. Therefore it is of crucial importance to develop fast, reliable, and accurate quantum impurity solvers. The multi-site or multi-orbital nature of the most relevant quantum impurity models favors quantum Monte Carlo methods. In the past two decades, perhaps the most commonly used impurity solver is the wellknown Hirsch-Fye quantum Monte Carlo (HFQMC) algorithm. 1-4 This solver is numerically exact, but computationally expensive. Furthermore it suffers inevitable systematic error which is introduced by time discretization procedure, and thus is not suitable for solving the quantum impurity models under low temperature. Very recently, important progresses have been achieved with the development of various continuous time quantum Monte Carlo (CTQMC) impurity solvers, which are based on stochastic sampling of diagrammatic expansion of the partition function. 5,6 According to the differences in perturbation expansion terms, these CTQMC quantum impurity solvers can be classified into two types: weak coupling (also named as interaction expansion) and strong coupling (also named as hybridization expansion) implementations. The weak coupling CTQMC impurity solver was first proposed by Rubtsov et al.,7,8 who expanded the partition function in the interaction terms. This is the method of choice for cluster calculations of relatively simple models at small interactions, because the computational effort scales as the cube of the system size. As an useful complement, Werner and Millis proposed 9,10 another powerful and flexible CTQMC impurity solver, which is based on a diagrammatic expansion in the impurity-bath hybridization and the local interactions are treated exactly. Since this algorithm perturbs around an exactly solved atomic limit, it is particularly efficient at moderate and strong interactions. Furthermore, due to its ability to provide the information about atomic states, 11 hybridization expansion quantum impurity solver is the desirable tool for LDA+DMFT calculations for strongly correlated materials. However, since the Hilbert space of the local problems grows exponentially with the number of sites or orbitals, the computational effort scales exponentially, rather than cubically with system size. Hence the applications of hybridization expansion quantum impurity solver in LDA+DMFT calculations are severely constrained, especially for those multi-orbital impurity models with rotationally invariant interaction terms. With this obstacle in mind, in this paper we present a new efficient implementation for hybridization expansion quantum impurity solver, which enables reliable and fast simulations for multi-orbital models with up to seven orbitals on modern computer clusters or GPU-enable workstations. The rest of this paper is organized as follows: In Sec.II a brief introduction to the conventional implementation 9,11 and newly developed Krylov subspace approach 12 for the hybridization expansion quantum impurity solver in general matrix formalism is provided. In Sec.III the new algorithm based on Newton interpolation at real Leja points (for simplicity, in the following of this paper we just name it as Newton-Leja interpolation or Newton-Leja algorithm) is presented in details. Then in Sec.IV we discuss the truncation approximation, accuracy, and performance issues of the new algorithm and compare it with competitive Krylov subspace approach. In Sec.V the LDA+DMFT calculated results for typical strongly correlated metal SrVO 3 by using the Newton-Leja algorithm as an impurity solver are illustrated. Section VI serves as a conclusion and outlook. In Appendix, concise introductions for the socalled Newton interpolation and Leja points are available as well. II. HYBRIDIZATION EXPANSION QUANTUM IMPURITY SOLVER The general quantum impurity model to be solved in the framework of single site DMFT can be given by the following Hamiltonian: 1,2 H qim = − a,σ (µ − ∆ a )n a,σ + H hyb + H bath + H int . (1) Here a labels orbital, σ labels spin, µ is the chemical potential, ∆ a is the energy level shift for orbital a from a crystal field splitting, and n a,σ is the occupation operator. H hyb , H bath , and H int denote impurity-bath hybridization, bath environment, and interaction terms, respectively. The basic idea of CTQMC impurity solvers is very simple. 5,6 One begins from a general Hamiltonian H = H a + H b , which is split into two parts labeled by a and b, writes the partition function Z = Tre −βH in the interaction representation with respect to H a , and expands in powers of H b , thus Z = k (−1) k k! ( k i=1 β 0 dτ i )Tr[e −βHa T k i=1 H b (τ i )]. (2) Here T is the time-ordering operator. The trace Tr [...] evaluates to a number and diagrammatic Monte Carlo method enables a sampling over all orders k, all topologies of paths and diagrams, and all times τ 1 , · · · , τ k in the same calculation. Because this method is formulated in continuous time from the beginning, time discretization errors which are severe in Hirsch-Fye algorithm 4 do not have to be controlled any more. This continuous time method does not rely on an auxiliary field decomposition and a particular partitioning of the Hamiltonian into "interacting" and "non-interacting" parts. In principles, the only requirement is that one may decompose the Hamiltonian in such a way that the time evolution associated with H a and the contractions of operators H b may easily be evaluated. Thus there are several variations of CTQMC impurity solvers. 5, 6 We note that all of the continuous time diagrammatic expansion algorithms are based on the same general idea, there are only significant differences in the specifics of how the expansions are arranged, the measurements are done, and the errors are controlled. In this paper, we focus on the hybridization expansion algorithm merely. In the hybridization expansion algorithm, 9-11 based on Eq.(1) and Eq.(2), H b is taken to be the impurity-bath hybridization term H hyb and H a = H bath + H loc , where H loc = H int − a,σ (µ − ∆ a )n a,σ .(3) Since H hyb = pj (V j p c † p d j + V j * p d † j c p ) =H hyb +H † hyb ,(4) contains two terms which create and annihilate electrons on the impurity, respectively, only even powers of the expansion and contributions with equal numbers ofH hyb andH † hyb can yield a nonzero trace. Inserting theH hyb andH † hyb operators explicitly into Eq. (2) and then separating the bath operators (c † p and c p ) and impurity operators (d † j and d j ), we finally obtain Z Z bath = ∞ k=0 β 0 dτ 1 ... β τ k−1 dτ k β 0 dτ ′ 1 ... β τ k ′ −1 dτ ′ k b1...b k b ′ 1 ...b ′ k Tr d e −βH loc T k i=1 d bi (τ i )d † b ′ i (τ ′ i ) det∆.(5) Here we define the bath partition function Z bath = Tre −βH bath = σ p (1 + e −βǫp ),(6) and ∆ is a k × k matrix with elements ∆ lm = ∆ j l jm (τ l − τ m ), where ∆ lm (τ ) = α V l * α V m α 1 + e −βǫα × e ǫα(τ −β) , τ > 0 −e ǫατ , τ < 0.(7) Next the diagrammatic Monte Carlo technique can be used to sample Eq.(5). The two basic actions required by ergodicity are the insertion and removal of a pair of creation and annihilation operators. Additional updates keeping the order k constant are typical not required for ergodicity but may speed up equilibrium and improve the sampling efficiency. In the Monte Carlo simulation, the most time consuming part is to calculate the following trace factor: 5,6 w loc = b1...b k b ′ 1 ...b ′ k Tr d e −βH loc T k i=1 d bi (τ i )d † b ′ i (τ ′ i ) .(8) If H loc is diagonal in the occupation number basis defined by the d † a and d a operators, a separation of "flavors" (spin, site, orbital, etc.) as in the segment formalism 10 is possible, and the computational efficiency is fairly satisfactory. Conversely, if H loc is not diagonal in the occupation number basis, the calculation of w loc becomes more involved and challenging. The conventional strategy proposed by Werner and Millis et al. 9,11 is to evaluate the trace factor (see Eq. (8)) in the eigenstate basis of H loc , because in this basis the time evolution operators e −τ H loc become diagonal and can be computed easily. On the other hand, in this representation the d α and d † α operators become complicated matrices. Hence this algorithm involves many multiplications of matrices whose size is equal to the dimension of the Hilbert space of H loc . In order to facilitate the task of multiplying these operator matrices it is crucial to arrange the eigenstates according to some carefully chosen good quantum numbers. 11 Then the evaluation of the trace is reduced to block matrix multiplications. With this trick, the present state of the art is that five spin degenerate bands can be treated exactly. However, since the matrix blocks are dense and the largest blocks grow exponentially with system size, the simulation of bigger models becomes extreme expensive and is only doable if the size of the blocks is severely truncated. Various truncation and approximation schemes provide limited access to larger problems, but as the number of orbitals is increased the difficulties rapidly become insurmountable. 5,6 Recently, Läuchli and Werner 12 present an implementation of the hybridization expansion impurity solver which employs sparse matrix exact diagonalization technique to compute the time evolution of the local Hamiltonian H loc and then evaluate the weight of diagrammatic Monte Carlo configurations. They propose to adopt the occupation number basis, in which the creation and annihilation operators can easily be applied to any given states and in which the sparse nature of H loc matrix can be exploited during the imaginary time evolution by relying on mature Krylov subspace iteration method. [13][14][15] Their implementation is based on very efficient sparse matrix algorithm for the evaluation of matrix exponentials applied to a general vector, i.e., exp(−τ H loc )\|ν . At first the algorithm try to construct a Krylov subspace K p (\|ν ) = span{\|ν , H loc \|ν , H 2 loc \|ν , ..., H p loc \|ν }, (9) by Lanczos or Arnoldi iterations, and then approximate the full matrix exponential of the Hamiltonian projected onto the Krylov subspace K p (\|ν ). Here p means the dimension of the built Krylov subspace. It has been shown rigorously that these Krylov subspace iteration algorithms converge rapidly as a function of p. 14 Since this implementation involves only matrix-vector multiplications of the type d † \|ν , d\|ν , and H loc \|ν with sparse operators d † , d, and symmetric matrix H loc , and is thus doable in principle even for systems for which the multiplication of dense matrix blocks becomes prohibitively expensive or for which the matrix blocks will not even fit into the memory anymore. Their algorithm avoids computationally expensive matrix-matrix multiplications and becomes advantageous over the conventional implementation for models with five or more bands. Laüchli et al. 12 have illustrated the power and usefulness of the Krylov subspace approach with dynamical mean-field results for a given five-band model which captures some aspects of the physics of the iron-based superconductors. III. NEWTON-LEJA INTERPOLATION METHOD The spirit of our new implementation for hybridization expansion quantum impurity solver is quite similar with previous Krylov subspace approach. 12 We just adopt the occupation number basis and exploit the sparse nature of d † α and d α operators by applying them to any given states as well. But the kernel of our implementation is to evaluate the time evolution of sparse symmetric matrix H loc , i.e. exp(−τ H loc ), by Newton interpolation at a sequence of real Leja points, instead of the Krylov subspace approach. Our algorithm inherits all the advantages of Krylov subspace approach, and is significantly superior to the latter on efficiency and memory consumption aspects. Consequently our implementation will be very promising to replace the Krylov subspace approach. How to efficiently evaluate the matrix exponentials of local Hamiltonian H loc applied to any given vectors, i.e., exp(−τ H lov )\|ν , is the essential ingredient in hybridization expansion quantum impurity solver. We note that fast evaluation of the matrix exponential functions, just like exp(τ A)v and ϕ(τ A)v, is the key building block of the so-called "exponential integrators" in engineering mathematics and has received a strong impulse in recent years. [16][17][18] Here A ∈ R n×n , v ∈ R n , τ is arbitrary time step, and ϕ(z) = exp(z) − 1 z .(10) To this respect, most authors regard Krylov-like as the methods of choice. 19 Nevertheless, an alternative class of polynomial interpolation methods has been developed since the beginning, 20 which is based on direct interpolation or approximation of the matrix exponential functions on the eigenvalue spectrum (or the field of values) of the relevant matrix A. Despite of a preprocessing stage needed to get a rough estimation of some marginal eigenvalues, the latter is competitive with Krylov-like methods in several instances, namely on large scale, sparse, and in general asymmetric matrices, usually arising from the spatial discretization of parabolic partial differential equations (PDEs). 21 Among others, the Newton interpolation based on real Leja points method (for a brief review of these two concepts, please refer to Appendix) has shown very attractive computational features. [21][22][23][24][25] Given a general matrix A and a general vector v, Newton interpolation approximates the exponential propagator as ϕ(τ A)v ∼ p m (τ A)v,(11) with m polynomial expansion order and p m (z) Newton interpolating polynomial of ϕ(z) at a sequence of real Leja points {ξ k } in a compact subset of the complex plane containing the eigenvalue spectrum of matrix A: [21][22][23][24] p m (τ A)v = m j=0 d j Ω j v,(12) and Ω j = j−1 k=0 (τ A − ξ k I).(13) Here I is the unit matrix, and d j is the corresponding divided differences 25 for ϕ(z). Observe that once ϕ(τ A) is computed, then exp(τ A)v = τ Aϕ(τ A)v + v.(14) Thus in practice it is numerically convenient to interpolate the function ϕ(τ A) at first. By replacing general matrix A with −H loc and general vector v with \|ν , and using Eq.(11)- (14), our algorithm rests on Newton interpolation of exp(−τ H lov )\|ν at a sequence of Leja points on the real focal interval, say [α, β], of a suitable ellipse containing the eigenvalue spectrum of local Hamiltonian −H loc . The use of real Leja points is suggested by the fact that on such a well defined ellipse they can give a stable interpolant, superlinearly convergent to entire functions due to the analogous scalar property, 24,26 i.e., lim m→∞ sup \|\|ϕ(A)v − p m (A)v\|\| 1/m 2 = 0.(15) In real arithmetic, a key step is given by estimating at low cost the reference focal interval [α, β] for the eigenvalue spectrum of −H loc matrix. In present works we adopt the simplest estimation given directly by the famous Gershgorin's theorem. Now let us describe the full workflow for the trace evaluation in some more details. In the initial stage of the hybridization expansion quantum impurity solver, the following steps are necessary: (1) Calculate the low-lying eigenstates of H loc by Lanczos iteration algorithm or exact diagonalization technique. 1 Decide which eigenstates should be kept in the trace calculation and the other high-lying eigenstates are discarded. (2) Determine the real focal interval [α, β] of a suitable ellipse containing the eigenvalue spectrum of matrix −H loc by using the Gershgorin's theorem. (3) Compute a sequence of real Leja points {ξ k , k = 0, ..., m − 1} on the interval [α, β] by using the fast Leja points (FLPs) algorithm 26 and initialize the Newton-Leja algorithm. Depending on the simulation results gathered in present works the average degree for Newton interpolation m ∼ 15, thus 64 real Leja points are enough to guarantee excellent convergence. Then, in the actual calculation of a trace factor, we proceed as follows: (4) Select an eigenstate as retained before, and propagate it to the first time evolution operator. Since the initial state is an eigenstate of H loc , it is simply multiplied by an exponential factor for the first time interval. (5) Apply the creation or annihilation operator on the propagated state by using the efficient sparse matrix-vector multiplication technique. (6) Propagate the current state to next time evolution operator using the Newton-Leja algorithm as described above. The Newton-Leja algorithm turns out to be quite simple and efficient, and its time complexity is very similar with the Krylov subspace approach. According to Eq.(12) and Eq.(13), matrix-matrix multiplications are practically avoided, and the most important arithmetic is sparse matrix-vector multiplication. Furthermore, being based on vector recurrences in real arithmetic, its storage occupancy and computational cost are very small, and it results more efficient than Krylov subspace approach on large scale problems. 21 In addition, this algorithm is very well structured for a parallel implementation, as it has been demonstrated in the references. 23 It is worth to mention that except for the traditional parallelism strategy for random walking and Markov chain in the Monte Carlo algorithm, in a fine-grained parallelism algorithm the (4) ∼ (8) steps can be easily parallelized over the retained eigenstates with multi-thread technique in modern multi-core share memory computers. Further acceleration by using CUDA-GPU technology is another very promising research area. In the next section we should address the performance and reliability issues of this new algorithm. IV. BENCHMARK In this section, we try to benchmark our new algorithm and compare the calculated results with other existing implementations for hybridization expansion quantum impurity solvers. Three aspects, including truncation error, reliability, and efficiency are mainly discussed. Just like Krylov subspace approach, the truncation approximation can be adopted by our algorithm as well. And the high accuracy and superior performance of our algorithm are proved with extensive test cases. A spin degenerated three-band Hubbard model, with the following local Hamiltonian with rotationally invariant interactions 5 H loc = − a,σ (µ − ∆ a )n a,σ + a U n a,↑ n a,↓ + a>b,σ [U ′ n a,σ n b,−σ + (U ′ − J)n a,σ n b,σ ] − a<b J ′ (d † a,↓ d † b,↑ d b,↓ d a,↑ + h.c.) − a<b J ′ (d † b,↑ d † b,↓ d a,↑ d a,↓ + h.c.),(16) is used as a toy model to examine our implementation. Here U ′ = U − 2J and Hund's coupling parameter J ′ = J = U/4. All the orbitals have equal bandwidth 4.0 eV, and a semicircular density of states is chosen. The chemical potential µ is fixed to keep the system at half filling and the crystal field splitting ∆ α is set to be zero. Unless it is specifically stated, this model is used throughout this section. We solve this toy model in the framework of single site DMFT using variant implementations of hybridization expansion quantum impurity solvers, including conventional matrix implementation, 5,6 Krylov subspace approach, 12 and our Newton-Leja algorithm. A. Truncation approximation In the calculation of trace factor, some high-lying eigenstates with negligible contributions can be abandoned in advance to improve the computational effi-ciency. At low temperature region this truncation approximation is practical, however, at high temperature region it must be used with great care. We solve the predefined three-band model (see Eq.(16)) at extreme high temperature (β = 10, T ∼ 1100 K) and moderate interaction strength (U = 2.0 eV) to explore the influence of truncation approximation to the single particle Green's function. At first we can obtain the exact solutions by using the conventional matrix algorithm without any truncations. Then we run the simulation again by using the Newton-Leja algorithm with O(1) level (in which only the 4-fold degenerated ground states are retained) and O(2) level (in which not only the 4-fold degenerated ground states but also the 28-fold degenerated first excited states are kept) truncations respectively. The calculated results are shown in Fig.1. It is apparent that under O(1) level truncation there are significantly systematic deviations between the approximate and exact results, while under O(2) level truncation the deviations can be ignored safely. It has been suggested by Laüchli et al. 12 that the truncation approximation to the ground state vectors is legitimate just for temperatures which are ≤ 1% of the bandwidth. Therefore in this case the temperature is too high to apply the O(1) level truncation, but the O(2) level truncation is still acceptable. Finally, It should be pointed out that for this model the computational speed with O(2) level truncation is at least twice faster than that of full calculation without any truncations. B. Accuracy of Newton-Leja algorithm Next we try to demonstrate the accuracy of the new approach in a wider parameter range. The top panel of Fig.2 shows the measured single particle Green's function G(τ ) for β = 30 and different values of the Coulomb interaction strength U . Since the model temperature (T ∼ 390 K) meets the truncation criterion, 12 , both O(1) and O(2) level truncations are valid. Of course, the O(2) level truncation can give finer results at the cost of performance, so in these simulations we apply the O(1) level truncation merely. In this figure the open symbols were computed with the conventional matrix method without any truncations and used to test the precision of the Newton-Leja algorithm. Not surprisingly, essentially perfect agreement between the two methods is found for all relevant interaction strengths. The bottom panel of Fig.2 illustrates the calculated results for U = 4.0 eV at different values of inverse temperature β. The O(1) level truncation is adopted for the Newton-Leja algorithm. The exact solutions obtained by traditional matrix method are shown as a comparison. It is noticed that when the inverse temperature β is 10, the deviations between traditional matrix implementation and Newton-Leja algorithm are distinct. The temperature is lower, the smaller the deviation. And for β ≥ 20, the deviations can be considered negligible. In other words, their results become indistinguishable at temperatures which are ≤ 1% of the bandwidth. Nevertheless, the Newton-Leja algorithm with carefully chosen approximations is demonstrated to be controllable, reliable and consistent with original implementations, and can be widely used in standard DMFT calculations. C. Efficiency of Newton-Leja algorithm Efficiency is always a major concern for newly developed quantum impurity solver, so it is urgent for us to explore the performance of Newton-Leja algorithm. Based on considerable testing, we found that when the system size is not large enough, the Newton-Leja algorithm is less efficiency than general matrix formulation, 5,6 even the Krylov subspace approach. 12 Whereas, when the system size is large enough, e.g., five-band or bigger model, the situation completely opposite. The Krylov subspace approach outperforms the conventional matrix method and Newton-Leja algorithm is better than the Krylov subspace approach. In this subsection, we try to compare the efficiencies between Newton-Leja algorithm and Krylov subspace approach, and determine the system size for which the former is superior to the latter. Since the key building blocks for both Newton-Leja algorithm and Krylov subspace approach are the calculations of time evolution operators, we just compare their efficiencies by the evaluation of exp(−∆τ H loc )\|ν . Here \|ν is a randomly generated vector and time step ∆τ is set to be 0.5 and 5.0, corresponding to typically short and long time intervals, respectively. The multi-orbital local Hamiltonian H loc is constructed by using Eq. (16), and the interaction strength U is fixed to be 4.0 eV and J ′ = J = U/4. Both the maximum allowable dimension for Krylov subspace p and number for real Leja points m are fixed to be 64, which are sufficient to obtain convergent and accurate results. In order to eliminate the influence of fluctuating measurement data, we repeated every benchmark for 20 times and then evaluated the average time consumption per evaluation of matrix exponentials. The benchmark results for multi-orbital systems with n =1, 2, · · · , 7 (n labels the number of bands) are displayed in Fig.3. As can be seen from the figure, when the system size is small or moderate (n ≤ 4) the Newton-Leja algorithm exhibits worse performance than Krylov subspace ap- proach. However, when increasing the system size continually (5 ≤ n ≤ 7, indicated in this figure by pink zone) the Newton-Leja algorithm is the winner. For instance, at n = 5 and ∆τ = 5.0 the Newton-Leja algorithm is almost ten times faster than the Krylov subspace approach. More impressively, at n = 7 and ∆τ = 5.0, the Krylov subspace approach is slower than Newton-Leja algorithm about even four orders of magnitude. We note that for short time interval the performances for both implementations are close, but as for n ≥ 5 the Newton-Leja algorithm still exhibits better performance. According to these benchmarks, it is tentatively suggested that the Newton-Leja algorithm is much more suitable for the systems with five or more orbitals than Krylov subspace approach. Next we concentrate our attentions to the five-band model, which plays an important role in the underlying physics of transition metal compounds, and make further benchmarks for our new implementation. The average time consumption per evaluation of matrix exponentials exp(−∆τ H loc )\|ν ) is used again as a measurement of efficiency. The model parameters are consistent with previous settings except the time interval ∆τ = 0.1 ∼ 3.5. The maximum number of real Leja points m is fixed to be 64, while the maximum allowable dimension of Krylov subspace p is varied from 16 to 64. The benchmark results are shown in Fig.4. As is illustrated in this figure, at short (∆τ < 0.8) and long (∆τ > 1.8) time intervals, the Newton-Leja algorithm shows better efficiency. It is apparent that when ∆τ > 2.0, actually no decline in efficiency for Newton-Leja algorithm is seen, i.e., the efficiency has nothing to do with the length of time interval. That is because the average degree of Newton interpolation (i.e. number of Leja points used in Newton interpolation) almost remain unchanged. According to our experiences, m = 15 ∼ 20 is suitable for most of the five, six, and seven-band models in general parameter ranges. As is mentioned above, provided limiting m and a well defined ellipse containing the eigenvalue spectrum of matrix −H loc , a stable interpolant, superlinearly convergent to matrix exponentials exp(−∆τ H loc )\|ν ) is guaranteed by the Leja points method (see Eq. (15)). 24,26 So we can decrease the maximum number of real Leja points m further to reduce the memory consumption and obtain higher efficiency. As for the Krylov subspace approach, at short time interval region the simulation with smaller Krylov subspace exhibits better performance, while at long time interval region the contrary is indeed true. It is very easy to be understood: at long time interval region, the Krylov subspace algorithm requires larger dimension of subspace to obtain fast convergence speed, and it is hardly to achieve convergence with small dimension of Krylov subspace only when more iterations are done. However, it is impossible to increase the dimension of Krylov subspace infinitely, the oversize of Krylov subspace will deteriorate the performance of Krylov approach quickly. Finally, two concrete cases are provided to demonstrate the superior performance of Newton-Leja algorithm. Let's consider typical three-band and five-band Hubbard models respectively. The moderate Coulomb interaction strength U = 4.0 eV, and inverse temperature β = 20 or 40 (T ∼ 580 K or 290 K) are chosen. These two models are solved separately by hybridization expansion quantum impurity solvers based on Newton-Leja algorithm and Krylov subspace approach with the same computational parameters, and the computational times are gathered and compared with each other. The benchmark results are summarized in table I. It is confirmed again that for the three-band model Krylov subspace approach is more efficient than Newton-Leja algorithm, yet (Color online) The single particle spectral functions for vanadium 3d states in SrVO3 obtained by LDA+DMFT calculations. As for the Newton-Leja algorithm, the rotationally invariant interaction terms are taken into considerations, while for segment algorithm 10 only the Ising-like densitydensity interaction terms are treated. The spectral function is obtained from imaginary time Green's function G(τ ) by using maximum entropy method, 35 and the calculated results are cross-checked by using recently developed stochastic analytical continuation method. 36 Previous LDA+DMFT results, 29 in which traditional HFQMC is used as an impurity solver, are shown in this figure as a comparison. for the five-band model Newton-Leja algorithm exhibits much better performance, which is consistent with previous benchmark results. Thus by any measure, our new implementation is better than Krylov subspace approach at low temperature and large system size. V. APPLICATION In this section, in order to illustrate the usefulness of Newton-Leja algorithm, we used it as a quantum impurity solver in the non-self-consistent LDA+DMFT calculations for representative transition metal oxide SrVO 3 . SrVO 3 is a well-known t 1 2g e 0 g metal. It is a good test case for LDA+DMFT calculations because it is cubic and nonmagnetic and also the t 2g bands are isolated from both e g and oxygen 2p bands in the LDA band structure. Numerous theoretical calculations (including LDA+DMFT) [27][28][29][30] and experiments [31][32][33][34] have been done on this compound. This is thus an ideal system to benchmark our implementation. The LDA+DMFT framework employed in present works has been described in the literatures. 29,37,38 The ground state calculations have been carried out by using the projector augmented wave (PAW) method with the ABINIT package. The cutoff energy for plane wave expansion is 20 Ha, and the k-mesh for Brillouin zone integration is 12 × 12 × 12. The low-energy effective LDA Hamiltonian is obtained by applying a projection onto maximally localized Wannier function (MLWF) orbitals including all the vanadium 3d and oxygen 2p orbitals, which is described in details in reference [29]. That would correspond to a 14 × 14 p − d Hamiltonian which is a minimal model 39 required for a correct description of the electronic structure of SrVO 3 . The LDA+DMFT calculations presented below have been done for the experimental lattice constants (a 0 = 7.2605 a.u). All the calculations were preformed in paramagnetic state at the temperature of 1160 K (β = 10). The Coulomb interaction is taken into considerations merely among vanadium 3d orbitals. In the present work, we choose U = 4.0 eV and J = 0.65 eV, which are accordance to previous LDA+DMFT calculations. 29 We adopt the around mean-field (AMF) scheme proposed in reference [29] to deal with the double counting energy. The effective quantum impurity problem for the DMFT part was solved by two implementations of hybridization expansion CTQMC quantum impurity solver supplemented with recently developed orthogonal polynomial representation algorithm. 40 The former implementation is based on the segment representation 10 and only the Ising-like density-density interaction terms are treated. The latter implementation is based on the Newton-Leja algorithm, and the local Hamiltonian is with rotationally invariant interactions (see Eq. (16)). The maximum entropy method 35 was used to perform analytical continuation to obtain the impurity spectral function from imaginary time Green's function G(τ ) of vanadium 3d states. In the present simulations, each LDA+DMFT iteration took 40000000 Monte Carlo steps per process. Since the segment representation hybridization expansion impurity solver 10 is extreme efficient, it took less 5 hour to finish 30 LDA+DMFT iterations by using a 8-cores Xeon CPU. Though O(2) level approximation is adopted during the simulation, the Newton-Leja algorithm is much more time consumption and took about 12 ∼ 13 hours to finish single LDA+DMFT iteration by using 64 cores in a Xeon cluster. The calculated single particle spectral functions for vanadium 3d states in SrVO 3 are shown in Fig.5. The calculated results within density-density interaction are consistent with previous LDA+DMFT simulations. [28][29][30] But the results obtained by classical segment representation and Newton-Leja algorithm display remarkable differences. For examples, the upper and lower Hubbard bands in t 2g sub-bands obtained by Newton-Leja algorithm with rotationally invariant interactions are more apparent. And the quasiparticle resonance peak around the Fermi level exhibits clearly shoulder structure, while that shoulder peak is absent in the segment picture (density-density interaction case). Since the vanadium e g sub-bands are isolated from t 2g states and located above the Fermi level, so they show roughly similar peak structures for both two implementations of hybridization expansion quantum impurity solvers. Nevertheless, based on our calculated results, it is suggested that the spinflip and pair-hopping terms in rotationally invariant interactions may play a key role in understanding the subtle electronic structure of SrVO 3 around the Fermi level, which has been ignored by previous theoretical calculations. VI. CONCLUSIONS We have presented an alternate implementation of the hybridization expansion quantum impurity solver which makes use of the Newton-Leja interpolation to evaluate the weight of Monte Carlo configurations. The new implementation inherits all the advantages of previously developed Krylov subspace approach with less memory consumption and better convergence control. It shows tremendous growth in computational performance over Krylov subspace approach and conventional matrix implementation at low temperature and large system size, and provides a controlled and efficient way which enables the LDA+DMFT study of transition metal compounds, or even actinide compounds with realistic interactions. To demonstrate the power of the new implementation, we used it as an impurity solver in the LDA+DMFT calculations for typical strongly correlated metal SrVO 3 . The obtained impurity spectral function of vanadium t 2g states shows apparent distinctions with previous calculated results. It is argued that the full Hund's exchange may has a big impact on the fine energy spectrum near the Fermi level. The generalization of Newton-Leja algorithm to high performance CUDA-GPU architecture is underway. ( 7 ) 7Go back to step 5 if more creation and annihilation operators are present. (8) Add the contribution of the propagated state to the trace. (9) Go back to step 4 until all retained eigenstates have been considered in the trace. FIG. 1 . 1(Color online) Single particle Green's function G(τ ) of a three-band Hubbard model with β = 10 and U = 2.0 eV computed by the conventional matrix implementation (open squares, exact results) and the Newton-Leja algorithm (filled squares for O(1) level truncation and open circles for O(2) level truncation of the trace). FIG. 2 . 2(Color online) Comparison between the single particle Green's functions of a three-band model computed with the traditional matrix method (open symbols, without any truncations) and those computed with the Newton-Leja algorithm (full symbols, with O(1) level truncation). Upper panel: the calculated single particle Green's functions at different interaction strengths U and β is fixed to 30. Lower panel: the calculated single particle Green's functions at U = 4.0 eV and different inverse temperatures β. FIG. 3 . 3(Color online) Efficiencies (time consumption per evaluation of matrix exponentials exp(−∆τ H loc )\|ν is used as metric) of the Krylov subspace approach and Newton-Leja algorithm as a function of system size. The local Hamiltonian H loc with rotationally invariant interaction at half-filling is defined in Eq.(16), where U = 4.0 eV and J/U = 0.25. ∆τ = 0.5 and 5.0 for typically short and long time intervals, respectively. The initial vector \|ν is generated randomly. Both the maximum number of real Leja points for Newton-Leja algorithm m and maximum allowable dimension of Krylov subspace for Krylov approach p are 64. The results obtained by these two algorithms are consistent with each other within the machine precision. In the pink zone, the efficiency of Newton-Leja algorithm is clearly superior to Krylov subspace approach. FIG . 4. (Color online) Efficiencies (time consumption per evaluation of matrix exponentials exp(−∆τ H loc )\|ν is used as metric) of the Krylov subspace approach and Newton-Leja algorithm as a function of time interval for the five-band model with rotationally invariant interaction (U = 4.0 eV, J/U = 0.25).The chemical potential µ is fixed to fulfill the half-filling condition. Randomly generated vector \|ν is used as an initial state. The maximum number of real Leja points m is fixed to be 64, while the maximum allowable size of Krylov subspace p is varied from 16 to 64. The convergence criterion for all algorithms are the same. When ∆τ > 3.5, it is very difficult to obtain converged solutions for Krylov subspace approach by current settings. FIG. 5. (Color online) The single particle spectral functions for vanadium 3d states in SrVO3 obtained by LDA+DMFT calculations. As for the Newton-Leja algorithm, the rotationally invariant interaction terms are taken into considerations, while for segment algorithm 10 only the Ising-like densitydensity interaction terms are treated. The spectral function is obtained from imaginary time Green's function G(τ ) by using maximum entropy method, 35 and the calculated results are cross-checked by using recently developed stochastic analytical continuation method. 36 Previous LDA+DMFT results, 29 in which traditional HFQMC is used as an impurity solver, are shown in this figure as a comparison. online) The first nine asymmetric real Leja points in [-2:2] interval generated by "Fast Leja Points" algorithm.26 The labels denote the sequences of Leja points. TABLE I . IThe average computational times per DMFT iteration for typical three-band and five-band Hubbard models solved by hybridization expansion quantum impurity solvers based on Newton-Leja algorithm and Krylov subspace approach respectively. In current simulations, each DMFT iteration took 40000000 Monte Carlo steps per processor, and the results were averaged over 16 processors.method three-band model five-band model U = 4.0 eV and β = 20 Newton-Leja, m = 64 3.23h 29.07h Krylov, p = 64 0.46h 36.80h U = 4.0 eV and β = 40 Newton-Leja, m = 64 6.12h 67.32h Krylov, p = 64 0.78h 399.36h A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg, Rev. Mod. Phys. 68, 13 (1996). 2 G. Kotliar, S. Y. Savrasov, K. Haule, V. S. Oudovenko, O. Parcollet, and C. A. Marianetti, Rev. Mod. Phys. 78, 865 (2006). ACKNOWLEDGMENTSWe acknowledge financial support from the National Science Foundation of China and that from the 973 program of China under Contract No.2007CB925000 and No.2011CBA00108. All the LDA+DMFT calculations have been performed on the SHENTENG7000 at Super-computing Center of Chinese Academy of Sciences (SC-CAS).Appendix A: Leja pointsFor the reader's convenience, here we briefly recall the definition of real Leja points. Sequences of Leja points {ξ j } ∞ j=0 for the compact set K are defined recursively as follows: if ξ 0 is an arbitrary fixed point in K, then ξ j are generated recursively in such a way thatAn efficient algorithm for computing a sequence of real Leja points, the so-called "Fast Leja Points (FLPs)", has been proposed in reference[26]. InFig.6the first nine Leja points in [-2,2] interval are shown as a illustration. The Leja sequences are attractive for interpolation at high-degree, in view of the stability of the corresponding algorithm in the Newton interpolation. 22Appendix B: Newton interpolationGiven a set of m + 1 data pointswhere no two x j are the same, the so-called Newton interpolation polynomial is defined as followswith d j the divided difference 25 and the Newton basis polynomial Ω j (x) defined asIt is apparent that Eq.(12)and Eq.(13) are the matrix forms of Eq.(B2) and Eq.(B3), respectively. 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[ "Finding the Right Recipe for Low Resource Domain Adaptation in Neural Machine Translation", "Finding the Right Recipe for Low Resource Domain Adaptation in Neural Machine Translation" ]	[ "Virginia Adams [email protected] \nNVIDIA\n\n", "Sandeep Subramanian [email protected] \nNVIDIA\n\n", "Mike Chrzanowski [email protected] \nNVIDIA\n\n", "Oleksii Hrinchuk [email protected] \nNVIDIA\n\n", "Oleksii Kuchaiev [email protected] \nNVIDIA\n\n", "CASanta Clara \nNVIDIA\n\n" ]	[ "NVIDIA\n", "NVIDIA\n", "NVIDIA\n", "NVIDIA\n", "NVIDIA\n", "NVIDIA\n" ]	[]	General translation models often still struggle to generate accurate translations in specialized domains. To guide machine translation practitioners and characterize the effectiveness of domain adaptation methods under different data availability scenarios, we conduct an in-depth empirical exploration of monolingual and parallel data approaches to domain adaptation of pre-trained, third-party, NMT models in settings where architecture change is impractical. We compare data centric adaptation methods in isolation and combination. We study method effectiveness in very low resource (8k parallel examples) and moderately low resource (46k parallel examples) conditions and propose an ensemble approach to alleviate reductions in original domain translation quality. Our work includes three domains: consumer electronic, clinical, and biomedical and spans four language pairs -Zh-En, Ja-En, Es-En, and Ru-En. We also make concrete recommendations for achieving high in-domain performance and release our consumer electronic and medical domain datasets for all languages and make our code publicly available.	10.48550/arxiv.2206.01137	[ "https://arxiv.org/pdf/2206.01137v1.pdf" ]	249,282,646	2206.01137	b3507c29bc40558f58bcd17af5e527fcbbebe98d	Finding the Right Recipe for Low Resource Domain Adaptation in Neural Machine Translation Virginia Adams [email protected] NVIDIA Sandeep Subramanian [email protected] NVIDIA Mike Chrzanowski [email protected] NVIDIA Oleksii Hrinchuk [email protected] NVIDIA Oleksii Kuchaiev [email protected] NVIDIA CASanta Clara NVIDIA Finding the Right Recipe for Low Resource Domain Adaptation in Neural Machine Translation General translation models often still struggle to generate accurate translations in specialized domains. To guide machine translation practitioners and characterize the effectiveness of domain adaptation methods under different data availability scenarios, we conduct an in-depth empirical exploration of monolingual and parallel data approaches to domain adaptation of pre-trained, third-party, NMT models in settings where architecture change is impractical. We compare data centric adaptation methods in isolation and combination. We study method effectiveness in very low resource (8k parallel examples) and moderately low resource (46k parallel examples) conditions and propose an ensemble approach to alleviate reductions in original domain translation quality. Our work includes three domains: consumer electronic, clinical, and biomedical and spans four language pairs -Zh-En, Ja-En, Es-En, and Ru-En. We also make concrete recommendations for achieving high in-domain performance and release our consumer electronic and medical domain datasets for all languages and make our code publicly available. Introduction The prevalence of pre-trained models has fueled exciting academic and industry progress in natural language processing. It has allowed practitioners to re-use computationally expensive training steps and bypass the most inaccessible portion of model training (Wolf et al., 2019). In neural machine translation (NMT), these general pre-trained models still struggle with translating domain specific material and require further tuning to achieve desired performance. In this work, we focus on methods for adapting off-the-shelf, third party, pre-trained translation models in which no additional architectural changes or edits to the model's pre-training scheme are possible. Intuitively, domain adaptation using clean, in-domain parallel data should provide the best results. However, such data is often hard and expensive to obtain. Monolingual in-domain data is more abundant and, at the cost of translation quality, can be used to generate synthetic parallel data. We aim to elucidate which domain adaptation approaches of off-the-shelf translation models best suit various low data resource scenarios to yield the highest in-domain translation quality. We explore the benefits and trade-offs of domain adaptation methods in combination and isolation. While setting up our experiments, we found English indomain monolingual data to be much more readily available than in-domain data for other languages. Collecting high quality monolingual electronic, medical, and biomedical domain data for adapting out of English translation models (En→) proved to be difficult to the extent that we limit our study to models translating into English (→En). For all experiments, the source language is one of Russian, Chinese, Spanish, or Japanese and the target language is always English. Similarly, as the source language is always non-English, we limit the scope of our work to scenarios with differing access to indomain parallel and target side monolingual data. We examine domain adaptation approaches under three in-domain data availability scenarios: parallel data only, target side monolingual data only, and both parallel and target side monolingual data. We compare parallel in-domain fine-tuning, mixeddomain fine-tuning (Zhang et al., 2019), traditional back-translation (Sennrich et al., 2016a;Edunov et al., 2018), tagged back-translation (Caswell et al., 2019), and in-domain language model shallow fusion across scenarios where applicable. See Table 1 for a breakdown of data availability conditions and the fixed architecture adaptation methods that can be applied to each. Further, we use of domain classifiers to mine additional in-domain parallel data -adding dimension to the quantity verses quality trade off encountered arXiv:2206.01137v1 [cs.CL] Jun 2022 This Study In-Domain Data Scenario Adaptation Approaches Parallel Source Mono Target Mono FT SF BT ST TBT TST Table 1: Data Resource Scenarios and Corresponding Possible Adaptation Methods. Adaptation approaches include 1) FT -Finetuning, 2) SF -Shallow Fusion decoding with in-domain language models, 3) BT -Backtranslation, 4) ST -Self-training, 5) TBT -Tagged Backtranslation, 6) TST -Tagged Self-training in back-translation discussions. Finally, we suggest an ensemble approach to mitigate degradation in original domain performance. Contributions Our main contributions include: 3 Related Work Low-Resource Machine Translation Solutions to low resource machine translation range from transfer learning approaches and data mining strategies like the ones we focus on in this paper, to meta learning approaches Gu et al. (2018), pretraining methods Song et al. (2019), multilingual Zoph et al. (2016) strategies, and methods making strategic use of monolingual source and target data (Zhang and Zong, 2016). He et al. (2016) proposed a dual learning reinforcement learning solution to low-resource machine translation that takes advantage of large amounts of both source and target monolingual data. Ahmadnia and Dorr (2019) combines dual learning with self-training and co-learning using the synthetic translation examples produced during the dual learning round trip translations to further train their translation models. Domain Adaptation Strategies Domain adaptation is a widely studied research area with strategies that vary by data access, training objective, and architectural changes. As we focus purely on data centric strategies for adapting off-the-shelf NMT models, so other adaptation approaches like adapter methods Houlsby et al. (2019), differential adaptation, and deep fusionDou et al. (2019a) that require access and/or edits to the NMT model's architecture fall out of the scope of this work. In section 4 we give a detailed introduction to each domain adaption method we explore. Empirical Studies of Fixed-Architecture Domain Adaptation There are a couple of existing empirical comparisons of domain adaptation methods using LSTM neural machine translation models. Chu et al. (2017) explores mixed domain fine-tuning and compares different in-domain up-sampling strategies to mitigate overfitting on generally low resource parallel domain data. Our work is most similar to that of . In their empirical study, compares fine-tuning NMT models on parallel mixed domain data with fine-tuning models on data that was synthetically generated via backtranslation. Though they propose a single domain adaptation method for RNN based models in which they combine back-translation, mixed-domain finetuning, and shallow fusion strategies, they do not explore iterative combinations of these approaches and therefore do not give strong evidence for one method over another. They also don't consider tagged back-translation, multi-domain ensembling, or additional data mining strategies as we do in this work. (Saunders, 2021) and (Chu and Wang, 2018) perform literary surveys on domain adaptation approaches for neural machine translation. Other works have explored domain adaptation under one of the three situations we compare in our investigation. Sun et al. (2019) studies training and adapting unsupervised translation models with exclusively monolingual data. They use cross-lingual language model pre-training (Conneau and Lample, 2019) to initialize their unsupervised neural machine translation (UNMT) models, then train and fine-tune their models according to different scenarios modulating the presence or absence of in-domain and out-of-domain source and target monolingual data. Methods We focus on the efficacy of domain adaptation approaches for pre-trained models with access to different combinations of parallel and monolingual target language data. We assume access to outof-domain NMT models in both language directions, but narrow our study to improving in-domain performance in the Other Language → English direction, using English → Other Language models solely for back-translation. We empirically compare domain adaptation methods separately and together. We only consider adaptation of a fixedarchitecture base models. Fine-Tuning We characterize the compromise between minimizing general domain degradation and improving in-domain performance in our parallel data approaches. We also experiment with fine-tuning baseline models on solely parallel in-domain data and on a mix of original and in-domain data (Zhang et al., 2019). Back-Translation In back-translation (Sennrich et al., 2016a;Edunov et al., 2018;Lample et al., 2018), target side monolingual data is used to generate synthetic parallel data. A reverse direction translation model translates the target language into the source language, often using sampling instead of greedy decoding to increase translation diversity. The forward di-rection translation model is fine-tuned on this generated parallel data. The reverse direction translation model can be used as is, or fine-tuned with available domain data before back-translation (Kumari et al., 2021;Artetxe et al., 2018). In tagged back-translation (Caswell et al., 2019) a special token (e.g. <BT>) is prepended before the synthetically generated source sentence. This tag helps the model differentiates noisy synthetic translations from ground truth examples. Shallow Fusion Decoding Shallow fusion (Gulcehre et al., 2015;Lample et al., 2018;Dou et al., 2019b) combines the next token probability predicted by a pre-trained language model possessing parameters φ t with the next token probability predicted by the NMT model's decoder θ t at every time step t. The generated translation benefits from the fluidity and target language knowledge of the language model while relying on the NMT decoder for semantic content. The two probabilities are added with a language model coefficient λ LM scaling the language model's contribution. P (y t \|y <t , x) = P N M T (y t \|y <t , x; θ t ) +λ LM * P LM (y t \|y <t ; φ t )(1) The language model is fine-tuned on target side monolingual data before shallow fusion decoding. Ensemble We propose using an ensemble of fine-tuned models with the base translation model to gain the benefits of adaptation across domains while maintaining high original domain performance. k indicates the total number of models in the ensemble, we average their probability distributions over the next token at every decoding time step t. P (y t \|y <t , x; θ 1 . . . θ k ) = 1 k k i=1 P (y t \|y <t , x; θ i ) Here P (y t \|y <t , x; θ i ) is the probability of target token y at time step t for a single NMT model i given the input tokens x and previously generated tokens y <t . Datasets We define low resources as falling between 5k and 9k in-domain parallel training examples and moderately low resource has an order of magnitude more data, in this case around 47k examples. We create low resource consumer electronic and medical domain datasets for each language pair. We also gathered in-domain monolingual data for the medical and consumer electronic domains. Final data totals for each language, split, and domain are listed in Table 2. We make the datasets and dataset creation code publicly available. 1 Parallel Consumer Electronic Dataset We used human generated translations from consumer electronic websites to construct the consumer electronic dataset. We crawled multilingual versions of XXXX 2 website, matching translated versions of each page via their URLs. To convert document level translations into aligned sentences, we separated sentences using NLTK's sentence splitter 3 for English, Spanish, and Russian. We used the Spacy 4 library's Chinese splitter to separate Mandarin sentences and the Konoha 5 library to split Japanese sentences. We then used the Vecalign library 6 ( Thompson and Koehn, 2019) in conjunction with the Language-Agnostic SEntence Representations (LASER) multilingual embedding library (Artetxe and Schwenk, 2019) to align translated document pairs on a sentence level. We selected sentence pairs within a set cosine distance range of 0.07 to 0.6 for the training split, where we define cosine distance as (1 -cosine similarity). For the validation and test splits, we used a narrower cosine distance range of 0.1 to 0.5 and removed overlapping validation and test examples from the train split. We manually cleaned the 1 Anonymized 2 Website anonymized for review 3 https://www.nltk.org/api/nltk.tokenize.html 4 https://spacy.io/models/zh 5 https://github.com/himkt/konoha 6 https://github.com/thompsonb/vecalign validation and test splits-separating examples containing multiple sentences and removing sentence fragments lacking a clear meaning. Parallel Medical Dataset Parallel translations of medical domain data were gathered from translated pdfs publicly provided by the NIH U.S. National Library of Medicine 7 . An identical process to the one used for the consumer electronic dataset was employed to create the parallel medical train, validation, and test splits. Parallel Biomedical Dataset We use the publicly available WMT'20 biomedical shared task train split for our Ru ↔ En biomedical domain data. To explore the benefits of noisy parallel data, we also mine additional parallel in-domain data from the out-of-domain En ↔ Ru WMT'21 News dataset. Here, noise comes from potential domain misclassification instead of from erroneous translation as with back-translation. To collect this data, we trained English and Russian biomedical domain classifiers. Each classifier utilized a pre-trained BERT Base style encoder (Devlin et al., 2018) with added classification layers. Our Russian domain classifier used RuBERT Base (Kuratov and Arkhipov, 2019). An equal amount of 45K negative and positive classification examples were collected from the parallel En ↔ Ru WMT'21 news task training data and the WMT'20 Biomedical Shared Task train set respectively. We classified the English half of the entire 26M parallel En ↔ Ru WMT'21 news task training data, saving all sentences with predicted biomedical domain probabilities over 50%. We then used our Russian classifier to predict biomedical domain probabilities for the Russian half of the parallel data. We averaged the classifier scores from the English and Russian domain classifiers and used this averaged score as our final selection criteria. See Table 5 for data totals corresponding to different probability score cutoffs. Monolingual Data We trained consumer electronic and medical domain binary classifiers to select in-domain monolingual data from the cc100 dataset the positive class and an equal amount of randomly sampled cc100 data was collected for the negative. After a total of 500k English sentences were classified as in-domain, the top 200k, 50k and n (where n is commensurate with parallel data totals for a given domain) examples with the highest in-domain probabilities were used in experiments. 6 Experimental Setup Base Models We start by training strong baseline models for all four language pairs: Spanish, Chinese, Russian and Japaneses to English. We train our models on WMT'21 news data. Table 3 shows initial SacreBLEU (Post, 2018) results of our models on WMT'20 test sets as well as in-domain test sets. Our models are based on the transformer large architecture (Vaswani et al., 2017). As suggested in Shoeybi et al. (2019), we move the layer normalization step for every transformer block to before each multi-head attention and feed forward sub-layer. The NMT models have 240M parameters. They took between 22 and 24 hours to train on 64 Tesla-V100 32GB GPUs with a per GPU batch size of 16k tokens. We use an initial learning rate between 1e-4 and 5e-4 with between 8k and 30k warm-up steps and an Adam (Kingma and Ba, 2015) optimizer. We use byte-pair encoding (BPE) (Sennrich et al., 2016b) to create our NMT vocabularies. The BPE model was trained using the original domain data. The Zh → En, Ja → En, and Ru → En translation models have separate encoder and decoder vocabularies, while our Es → En model shares a single vocabulary between the encoder and decoder. Each vocabulary has 32k tokens. Our reverse direction base models (En → Other Language) used for back-translation experiments were trained in the same manner and with the same transformer architecture as our baseline forward direction models. Language Models Our language models use a similar 16-layer transformer decoder architecture to Radford et al. (2019) with the same pre-layer normalization edit recommended by Shoeybi et al. (2019) as in our base NMT models. Though all the language models are English, they are each distinctly trained for every language pair to ensure the decoder and language models have the same tokenizer vocabulary. They are all trained on News Crawl 9 English data, then fine-tuned on the English half of the in-domain parallel datasets separately. Adaptation We fixed the fine-tuning learning rates to be between 1e-5 and 5e-6. Models were fine-tuned on 1 Tesla-V100 16GB GPU until in-domain validation BLEU scores plateaued. BLEU plateau occurred after only 1 epoch for Es-En fine-tuning experiments with a batch size of 1024 tokens. Zh-En, Ja-En, Ru-En models' validation BLEU stopped improving after 15-20 epochs, while the Ru-En models for the biomedical domain finished training after 1 epoch. We back-translate our monolingual data described in section 5.4 with our reverse direction models generating synthetic parallel data from the top 200k, top 50k, and top n (where n equals the number parallel examples for that language pair and domain) monolingual examples. The top n and top 50k parallel examples are a higher quality subset of the 200k examples, allowing us to characterize the impact of quantity verses quality of back-translated data in a low resource environment. We fine-tune our base models exclusively on back-translated data for our target side monolingual experiments and on a mix of human-translated and back-translated data for our combined parallel and target monolingual experiments. In-Domain Parallel Results The in-domain parallel results averaged across all language pairs and across the consumer electronic and medical domains are displayed in Table 4. Mixed-domain fine-tuning has slightly lower indomain performance on average (a -0.6 difference) compared with fine-tuning on in-domain parallel data only. Mixed domain fine-tuning does help maintain original domain performance. We see an average original domain BLEU score of 27.4 for models tuned on mixed domain data and an average score of 23.8 for those fine-tuned on indomain data only. Shallow fusion decoding with an in-domain language model boosts performance for all languages and domains. Mitigating Original Domain Degradation via Ensembling We ensemble our fine-tuned in-domain parallel and baseline models together. When ensembled, baseline performance remains within 0.7 BLEU of its original score across all languages. This is a huge improvement over the 10+ BLEU score drop seen when fine-tuning on the consumer electronic domain. No ensemble out performs their single fine-tuned model counterparts when evaluated on in-domain data. Nevertheless, the ensemble still achieves a several BLEU point improvement in each domain over the baseline and the average BLEU score across all domains is much higher when additionally comparing against any single model's out-of-domain performance. These results indicate that, when translating mixed domain or unknown domain data, ensembling in-domain models should lead to higher quality translations-even when domains are drastically different (e.g. the consumer electronic and medical domains). Figure 1 presents the original vs. new domain trade-off for the consumer electronic and medical domains averaged over all language pairs. Figure 1b highlights the advantage of ensembling. The x-axis values in 1b are the combined average consumer electronic and medical domain BLEU scores irrespective of the domain for which each model was fine-tuned. Benefits of Mined In-Domain Parallel Data Fine-tuning the baseline Ru → En model with combined mined and original parallel data increased performance over fine-tuning with original data alone by 0.2 and 0.7 BLEU. A higher domain probability cutoff threshold, favoring reduced indomain noise over larger data quantity, resulted in a 0.5 BLEU score difference between the two models trained with mined data. It should be noted that the additional parallel data was mined from the parallel Ru → En training set used to train the baseline model. Though the model saw all mined examples during initial baseline training, viewing these in-domain examples again during the finetuning stage still increased in-domain performance over fine-tuning on purely unseen data. See Table 5 for a result breakdown. Target Side Monolingual Results In the bottom half of Table 4 we see that fine-tuning a base model on high quality back-translated data comes within approximately 3 BLEU points of finetuning on human translated in-domain parallel data on average. When analyzing individual models, the best Ja → En monolingual model matched the performance of the in-domain parallel model for the medical domain and surpassed it by 0.7 BLEU points in the consumer electronic domain. Back-Translated Quantity vs. Quality Trade-Off We compare fine-tuning on back-translated data mined from cc100 with fine-tuning on the backtranslated English half of each in-domain parallel dataset. Across the language pairs, there seems to be no major difference in performance between models fine-tuned with 200k, 50k, or top n totals of back-translated cc100 data. They each reach an average BLEU score between 35 and 36 as seen in Table 4. When base models are fine-tuned on the back-translated target half of the original indomain parallel datasets, the model's performance increased by an average of 3.3 BLEU compared to the cc100 back-translation experiments. Even with over 20x less data, fine-tuning on clean (in terms of domain accuracy) back-translated examples out scores utilizing noisier data. This point is also illustrated in Figure 2. Shallow Fusion Across the board shallow fusion leads to within 1.0 BLEU score increase compared to the baseline scores in each domain. For Ru → En, Es → En, and Ja → En shallow fusion with in-domain language models also increases original domain performance within 1.0 BLEU point of their original WMT'20 scores. This shows that even language models fine-tuned on out of domain data still have an advantageous impact when used for shallow fusion decoding. In-Domain Parallel + Target Side Monolingual Results We experimented with a number of approaches to combining back-translated data with in-domain parallel data. A summary of results for these experiments can be seen in Table 6. We first used our baseline reverse direction model to back-translate the top 50k cc100 sentences from each domain. Baseline models fine-tuned on a mix of this data and in-domain parallel data improved an average of 6.2 BLEU points from the baseline. We then tried fine-tuning our reverse direction model on our parallel domain data before back-translation. Combining this back-translated data with human translated parallel-data resulted in another +0.7 BLEU increase on average. Next we experimented with tagged back-translation. We prepended a special back-translation token (< BT >) to the beginning of every synthetic back-translated input from our previous iteration. Tagging back-translated exam- In-Domain BLEU (c) Es → En Figure 2: In-Domain BLEU scores after fine-tuning the baseline model on back-translated data. The green points correspond to scores from models fine-tuned on the back-translated target-half of the in-domain parallel datasets. The pink points are from models fine-tuned on back-translated cc100 data. Models with scores shown in green saw smaller volumes of high quality synthetic data compared to those in pink. Base models were fine-tuned on a mix of in-domain parallel data and back-translated Top50k cc100 data. SF stands for shallow fusion and BT stand for backtranslation. "w/ Tuned" indicates where target side monolingual data was back-translated with a reverse direction model that has been fine-tuned on parallel in-domain data. Methods using human translated parallel data alone out preformed those combining backtranslated and human translated parallel data. ples actually slightly decreased the average BLEU score by -0.1 compared to not adding tags. Finally, we used in-domain shallow fusion decoding at inference time with our model fine-tuned via tagged back-translation for a +0.6 average performance boost. Despite our efforts, we found none to be as effective as fine-tuning on purely in-domain data or a mix of in-domain and out-of-domain parallel data. 10 Recommendations 5. It is better to mine a moderate amount of parallel data over a larger amount of in-domain monolingual data. Conclusion We conduct an empirical study comparing parallel and monolingual data approaches to domain adaptation in NMT. We made recommendations on how to achieve the best in-domain translation performance with access to low resource parallel and/or monolingual domain data. Additionally, we explored model ensembling to reduce regression of original domain performance and the benefits of mined in-domain parallel data. We hope this work can guide others in their creation of high quality domain specific machine translation systems. Figure 1 : 14: A summary of results for parallel only and target monolingual only experiments. Each value is the BLEU score for the corresponding adaptation method averaged over every language pair and over the consumer electronic and medical domains. Ensemble denotes the scores for an ensemble of the baseline model and models fine-tuned on in-domain data. SF stands for shallow fusion. BT stands for back translation. TopN models were fine-tuned on an amount of back-translated cc100 data equal to the number of examples in their corresponding in-domain datasets. Original vs. new domain performance trade-off across parallel adaptation methods. (a) shows the average original domain performance as a function of the average in-domain BLEU score for each new domain across all languages, capturing this trade-off when translating one new domain at a time. (b) displays the average in-and-out of domain BLEU scores for each adaptation method over all language pairs, encapsulating trade off trends when translating text from multiple new domains simultaneously. • A systematic empirical comparison of domain adaptation approaches of third-party, fixed architecture transformer-based NMT models• A simple ensemble method to preserve origi- nal domain performance while gaining trans- lation ability across new domains • An effective low resource parallel data aug- mentation approach to improve in-domain per- formance • The release of consumer electronic and clini- cal domain datasets across Russian → English, Chinese → English, Spanish → English, and Japanese → English translation pairs and our code. Domain Language Pair Train Val TestElectronic Zh → En 7,041 475 479 Ja → En 6,777 452 460 Es → En 6,973 421 430 Ru → En 7,276 478 522 Medical Zh → En 8,760 448 446 Ja → En 5,399 460 461 Es → En 8,494 434 437 Ru → En 5,401 507 493 Biomedical Ru → En 46,782 279 - Table 2 : 2Total parallel examples for each split of each language pair. 8 . When training the classifiers, target side in-domain data was used forLanguage pair WMT CE Medical Biomed Zh → En 24.5 34.5 29.9 - Ja → En 19.8 36.1 26.8 - Es → En 39.9 46.1 50.1 - Ru → En 36.2 25.6 27.7 38.5 Table 3 : 3SacreBLEU scores of baseline models on WMT'20 for all language pairs except Es → En, and in-domain test sets for all languages. The Es → En scores are on WMT'12. ScenarioAdaptation Method New-Domain Avg. Original Domain Avg.Parallel Only Baseline 34.6 30.1 Ensemble 39.3 29.4 Mixed-Domain 42.1 27.4 In-Domain 42.7 23.8 In-Domain + SF 43.0 25.8 Target Monolingual Only BT Top200k cc100 35.4 28.4 BT Top50k cc100 35.9 27.3 BT TopN cc100 35.8 28.9 BT Target Half 39.0 27.3 BT Target Half + SF 40.0 27.2 Table Table 5 : 5The performance increase from adding mined parallel data to the biomedical Ru → En fine-tuning set. "cutoff" is the domain classifier probability threshold and "total" is the train set size with mined examples added. Table 6 : 6In-domain parallel + target side monolingual results for Ru→En BLEU scores averaged across the consumer electronic, medical, and biomedical domains. Table 7 : 7Detailed Ja → En in-domain parallel results. SF stands for shallow fusion.Languages Domain Model Description In-Domain Original Domain Zh → En Consumer Electronic Baseline 34.5 24.5 Ensemble Across Domains 39.8 22.1 Mixed-Domain Finetune 41.0 20.3 In-Domain Finetune 42.1 14.2 In-Domain Finetune + SF 42.2 14.1 Medical Baseline 29.9 24.5 Ensemble Across Domains 41.0 22.1 Mixed-Domain Finetune 44.8 20.7 In-Domain Finetune 44.7 14.4 In-Domain Finetune + SF 45.0 19.5 Table 8 : 8Detailed Zh → En in-domain parallel results. SF stands for shallow fusion.Languages Domain Model Description In-Domain Original Domain Es → En Consumer Electronic Baseline 46.1 39.9 Ensemble Across Domains 51.8 39.5 Mixed-Domain Finetune 54.6 37.6 In-Domain Finetune 56.4 33.7 In-Domain Finetune + SF 56.6 33.7 Medical Baseline 50.1 39.9 Ensemble Across Domains 54.1 39.5 Mixed-Domain Finetune 55.2 37.7 In-Domain Finetune 55.3 36.5 In-Domain Finetune + SF 55.2 36.1 Table 9 : 9Detailed Es → En in-domain parallel results. SF stands for shallow fusion.Languages Domain Model Description In-Domain Original Domain Ru → En Consumer Electronic Baseline 25.6 36.2 Ensemble Across Domains 29.5 35.9 Mixed-Domain Finetune 35.5 31.9 Mixed-Domain Finetune + SF 35.8 32.2 In-Domain Finetune 35.9 23.6 In-Domain Finetune + SF 36.1 23.2 Medical Baseline 27.7 36.2 Ensemble Across Domains 31.9 35.9 Mixed-Domain Finetune 39.2 32.3 Mixed-Domain Finetune + SF 39.4 32.5 In-Domain Finetune 38.7 31.6 In-Domain Finetune + SF 39.2 31.8 Biomedical Baseline 38.5 36.2 Ensemble Across Domains 39.0 35.9 Mixed-Domain Finetune 41.3 37.0 Mixed-Domain Finetune + SF 41.6 37.1 In-Domain Finetune 42.0 32.8 In-Domain Finetune + SF 41.7 32.4 Table 10 : 10Detailed Ru → En in-domain parallel results. SF stands for shallow fusion.Table 11: Detailed Ru → En in-domain parallel + target monolingual results. BT stands for backtranslation and SF stands for shallow fusion.Languages Domain Model Description In-Domain Original Domain Ru → En Consumer Electronic Baseline 25.6 36.2 In-Domain + Baseline BT 32.4 33.3 In-Domain + Finetuned BT 34.4 25.8 In-Domain + Tagged Finetuned BT 34.2 21.8 In-Domain + Tagged Finetuned BT + SF 34.8 22.1 Medical Baseline 27.7 36.2 In-Domain + Baseline BT 36.8 26.2 In-Domain + Finetuned BT 37.3 27.1 In-Domain + Tagged Finetuned BT 37.9 20.2 In-Domain + Tagged Finetuned BT + SF 38.2 20.0 Biomedical Baseline 38.5 36.2 In-Domain + Baseline BT 41.1 33.8 In-Domain + Finetuned BT 40.9 34.6 In-Domain + Tagged Finetuned BT 40.2 34.6 In-Domain + Tagged Finetuned BT + SF 41.0 34.8 Table 12 : 12Detailed Ja → En in-domain target monolingual results. BT stands for backtranslation and SF stands for shallow fusion.Table 13: Detailed Zh → En in-domain target monolingual results. BT stands for backtranslation and SF stands for shallow fusion.Languages Domain Model Description In-Domain Original Domain Zh → En Consumer Electronic Baseline 34.5 24.5 Baseline + SF 34.5 23.8 BT Top 200k 35.5 25.2 BT Top 50k 35.5 25.2 BT Top 50k + SF 35.5 24.2 BT Top CE Total 35.8 25.1 BT CE Target 38.2 26.2 BT CE Target + SF 38.4 24.7 Medical Baseline 29.9 24.5 Baseline + SF 29.7 20.2 BT Top 200k 33.6 24.8 BT Top 50k 35.6 17.2 BT Top 50k + SF 36.2 15.5 BT Top Medical Total 34.6 20.1 BT Medical Target 39.2 20.1 BT Medical Target + SF 42.0 19.5 Table 14 : 14Detailed Es → En in-domain target monolingual results. BT stands for backtranslation and SF stands for shallow fusion.Table 15: Detailed Ru → En in-domain target monolingual results. 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[ "On coefficients of the interior and exterior polynomials", "On coefficients of the interior and exterior polynomials" ]	[ "Xiaxia Guan \nSchool of Mathematical Sciences\nXiamen University\n361005XiamenFujianChina\n", "Xian'an Jin \nSchool of Mathematical Sciences\nXiamen University\n361005XiamenFujianChina\n" ]	[ "School of Mathematical Sciences\nXiamen University\n361005XiamenFujianChina", "School of Mathematical Sciences\nXiamen University\n361005XiamenFujianChina" ]	[]	The interior polynomial and the exterior polynomial are generalizations of valuations on (1/ξ, 1) and (1, 1/η) of the Tutte polynomial T G (x, y) of graphs to hypergraphs, respectively. The pair of hypergraphs induced by a connected bipartite graph are abstract duals and are proved to have the same interior polynomial, but may have different exterior polynomials. The top of the HOMFLY polynomial of a special alternating link coincides with the interior polynomial of the pair of hypergraphs induced by the Seifert graph of the link. Let G = (V ∪ E, ε) be a connected bipartite graph. In this paper, we mainly study the coefficients of the interior and exterior polynomials. We prove that the interior polynomial of a connected bipartite graph is interpolating. We strengthen the known result on the degree of the interior polynomial for connected bipartite graphs with 2-vertex cuts in V or E. We prove that interior polynomials for a family of balanced bipartite graphs are monic and the interior polynomial of any connected bipartite graph can be written as a linear combination of interior polynomials of connected balanced bipartite graphs. The exterior polynomial of a hypergraph is also proved to be interpolating. It is known that the coefficient of the linear term of the interior polynomial is the nullity of the bipartite graph, we obtain a 'dual' result on the coefficient of the linear term of the exterior polynomial: if G − e is connected for each e ∈ E, then the coefficient of the linear term of the exterior polynomial is \|V \| − 1. Interior and exterior polynomials for some families of bipartite graphs are computed.	null	[ "https://arxiv.org/pdf/2201.12531v1.pdf" ]	246,430,622	2201.12531	6e14cc7494e13b765a0a183d4b8c76405067ffd4	On coefficients of the interior and exterior polynomials Xiaxia Guan School of Mathematical Sciences Xiamen University 361005XiamenFujianChina Xian'an Jin School of Mathematical Sciences Xiamen University 361005XiamenFujianChina On coefficients of the interior and exterior polynomials Preprint submitted to Elsevier arXiv:2201.12531v1 [math.CO] 29 Jan 20221 Corresponding authorHypergraphBipartite graphInterior polynomialExterior poly- nomialCoefficient MSC(2020) 05C3105C65 The interior polynomial and the exterior polynomial are generalizations of valuations on (1/ξ, 1) and (1, 1/η) of the Tutte polynomial T G (x, y) of graphs to hypergraphs, respectively. The pair of hypergraphs induced by a connected bipartite graph are abstract duals and are proved to have the same interior polynomial, but may have different exterior polynomials. The top of the HOMFLY polynomial of a special alternating link coincides with the interior polynomial of the pair of hypergraphs induced by the Seifert graph of the link. Let G = (V ∪ E, ε) be a connected bipartite graph. In this paper, we mainly study the coefficients of the interior and exterior polynomials. We prove that the interior polynomial of a connected bipartite graph is interpolating. We strengthen the known result on the degree of the interior polynomial for connected bipartite graphs with 2-vertex cuts in V or E. We prove that interior polynomials for a family of balanced bipartite graphs are monic and the interior polynomial of any connected bipartite graph can be written as a linear combination of interior polynomials of connected balanced bipartite graphs. The exterior polynomial of a hypergraph is also proved to be interpolating. It is known that the coefficient of the linear term of the interior polynomial is the nullity of the bipartite graph, we obtain a 'dual' result on the coefficient of the linear term of the exterior polynomial: if G − e is connected for each e ∈ E, then the coefficient of the linear term of the exterior polynomial is \|V \| − 1. Interior and exterior polynomials for some families of bipartite graphs are computed. Introduction The Tutte polynomial T G (x, y) [11] is an important and well-studied branch of graph and matroid theory, having wide applications from knot theory to statistical mechanics. Motivated by the study of the HOMFLY polynomial [1,9], which is a generalization of the celebrated Jones polynomial [2] in knot theory, of special alternating links, Kálmán introduced the interior polynomial I H (x) and the exterior polynomial X H (y) [3] (see Definitions 10 and 11) via interior and exterior activities of hypertrees (firstly introduced as the left or right degree vector in [8]). The two polynomials generalized valuations of T G (1/x, 1) and T G (1, 1/y) of the Tutte polynomials to hypergraphs, respectively. Kálmán and Postnikov proved that the interior polynomial and the exterior polynomial are two invariants of hypergraphs by a straightforward argument and indirect approach by counting Ehrhart-type lattice points in [3] and [5]. Moreover, in [5] they also showed that the interior polynomial is a Tutte-type invariant of bipartite graphs via the Ehrhart polynomial of the root polytope Q and the h-vector of any triangulation of Q, in other words, the pair of hypergraphs induced by a bipartite graph have the same interior polynomials. In [4,5], the authors established a relation between the top of the HOMFLY polynomial of any special alternating link and the interior polynomial of the Seifert graph (which is a bipartite graph) of the link. Recently, Kato [7] introduced the signed interior polynomial of signed bipartite graphs and extended the relation to any oriented links, and in [6] Kálmán and Tóthmérész defined two one-variable generating functions I and X using two 'embedding activities' and proved that the generating function of internal embedding activities coincides with the interior polynomial. In this paper, we mainly study the coefficients of the interior and exterior polynomials of hypergraphs. We prove that the interior polynomial and the exterior polynomial of connected bipartite graphs are both interpolating, that is, if I G (x) = m i=0 a i x i and X G (y) = n j=0 b j y j of a connected bipartite graph G, then a i = 0 and b j = 0 for any i = 0, 1, · · · , n and j = 0, 1, · · · , m. Kálmán [3] showed that the degree of interior polynomial of G is at most min{\|V \|, \|E\|} − 1 for a bipartite graphs G = (V ∪ E, ε), and the degree of exterior polynomial of a hypergraph H = (V, E) is at most \|E\| − 1. In this paper we study the degree of the interior polynomial of connected bipartite graphs with 2-vertex cuts in E and prove that for a connected bipartite graph G = (V ∪ E, ε) wtih a 2-vertex cut {e 1 , e 2 } in E, then the degree of the interior polynomial of G is at most min{\|E\| − 1, \|V \| − t + 1}, where t is the number of connected components of G − {e 1 , e 2 }. We show that interior polynomials for a family of balanced bipartite graphs are monic. It is well-known that the Tutte polynomial of graphs has a deletion-contraction formula. Using deletion-contraction relations of the interior polynomials repeatedly, we show that the interior polynomial of any connected bipartite graph can be represented as a linear combination of the interior polynomials of connected balanced bipartite graphs. Kálmán [3] proved that both the interior polynomial and the exterior polynomial have constant term 1 for any connected bipartite graph. Moveover, they proved that the coefficient of the linear term in the interior polynomial for any connected bipartite graph is the nullity of the bipartite graph. In this paper, we prove that if G − e is connected for each e ∈ E, then the coefficient of the linear term of the exterior polynomial is \|V \| − 1. As far as we know that the interior polynomial and exterior polynomial of only few bipartite graphs are known. In this paper, we also compute the interior polynomial and exterior polynomial for some families of bipartite graphs such as trees, even cycles, complete bipartite graphs and so on. The paper is organized as follows. In Section 2 we give preliminaries on the two polynomials and make some necessary preparations. In Section 3, we study interpolatory property, the degree, the monic property and the representation of the interior polynomial. Interpolatory property and the coefficient of the linear term of the exterior polynomial are studied in Section 4. In Section 5, the interior polynomial and exterior polynomial for some families of bipartite graphs are computed. Preliminaries In this section, we will give some definitions and summarise some known results. Definition 1. A hypergraph is a pair H = (V, E), where V is a finite set and E is a finite multiset of non-empty subsets of V . Elements of V are called vertices and elements of E are called hyperedges, respectively, of the hypergraph. A hypergraph is a generalization of a graph with loops and multiple edges allowed. For a hypergraph H, its associated bipartite graph BipH is defined as follows. Definition 2. The sets V and E are the colour classes of the bipartite graph BipH, and an element v of V is connected to an element e of E in BipH if and only if v ∈ e. Clearly, BipH is simple, that is, it has no multiple edges in BipH (and clearly, it has no loops). If H is a graph, then BipH is the subdivision of H, but two multiple edges obtained by subdividing a loop should be replaced by a single edge. Conversely, a pair of hypergraphs H = (V, E) and H = (E, V ) can be recovered from a bipartite graph (without multiple edges) if we specify V and E as vertex set of the hypergraph, respectively. These two hypergraphs H = (V, E) and H = (E, V ) are called abstract duals. We say a hypergraph H = (V, E) is connected if BipH is connected. Throughout the paper we consider connected hypergraphs. Definition 3. Let H = (V, E) be a connected hypergraph. A hypertree in H is a function f : E → N = {0, 1, 2, · · · } so that a spanning tree τ of its associated bipartite graph BipH can be found with the degree of e in τ , d τ (e) = f (e) + 1 for each e ∈ E. We call that τ realises or induces f . We denote the set of all hypertrees in H with B H . Hypertrees generalize spanning trees of graphs in the sense that an edge e is in the tree if and only if f (e) = 1 and not in the tree if and only if f (e) = 0. As a generalization of the rank of a graph, the following parameter µ(E ) is introduced for each E ⊂ E. Definition 4. Let G = (V ∪ E, ε) be a connected bipartite graph. For a subset E ⊂ E, let G\| E denote the bipartite graph formed by E , all edges of G incident with elements of E and their endpoints in V . We denote µ(E ) = 0 for E = ∅, and µ(E ) = \| E \| − c(E ) for E = ∅, where E = V ∩ (G\| E ) and c(E ) is the number of connected components of G\| E . The following facts hold for hypertrees of a hypergraph. (1) 0 ≤ f (e) ≤ d BipH (e) − 1 for all e ∈ E; (2) e∈E f (e) = \|V \| − 1; ( Definition 9. Let H be a connected hypergraph. Let f be a hypertree and e, e be two hyperedges of H. We say f is the hypertree so that a transfer of valence is possible from e to e if the function f obtained from f by decreasing f (e) by 1 and increasing f (e ) by 1 is also a hypertree. We also say that f and f are related by a transfer of valence from e to e . Now we introduce (internal and external) activity and inactivity of hypertrees of a hypergraph with a fixed totally ordering on E. 3) e∈E f (e) ≤ µ(E ) for all E ⊂ E. Definition 10. Let H = (V, E) be a connected hypergraph and f be a hypertree. A hyperedge e ∈ E is internally active with respect to the hypertree f if one cannot decrease f (e) by 1 and increase f (e ) of a hyperedge e smaller than e by 1 so that another hypertree results. We say that a hyperedge e ∈ E is internally inactive with respect to the hypertree f if it is not internally active. Let ι(f ) and ι(f ) denote the number of internally active hyperedges and internally inactive hyperedges, respectively, with respect to f . These two values are called the internal activity and internal inactivity, respectively, of f . Similar to internal activity and inactivity, there are external activity and inactivity of a hypertree in a hypergraph with a fixed totally ordering on E. Definition 11. Let H = (V, E) be a connected hypergraph and f be a hypertree. A hyperedge e ∈ E is externally active with respect to the hypertree f if one cannot increase f (e) by 1 and decrease f (e ) for some hyperedge e < e by 1 so that another hypertree results. We say that a hyperedge e ∈ E is externally inactive with respect to f if it is not externally active. Let (f ) and (f ) denote the number of externally active hyperedges and externally inactive hyperedges, respectively, with respect to f . These two values are called the external activity and external inactivity, respectively, of f . We have that ι(f )+ι(f ) = (f )+ (f ) = \|E\| by definitions above. Now, we can define the interior polynomial and the exterior polynomial of hypergraphs as follows. Kálmán and Postnikov proved that both the interior polynomial and the exterior polynomial are well-defined in [3] and [5], i.e, they do not depend on the order on E. If G = (V, E) is a graph with the Tutte polynomial T G (x, y), then its (viewed as a hypergraph) interior polynomial is x \|V \|−1 T (1/x, 1), and its exterior polynomial is y \|E\|−\|V \|+1 T (1, 1/y). Moreover, the interior polynomial is an invariant of bipartite graphs [5]. However, the two abstract dual hypergraphs induced by a bipartite graph may have different exterior polynomials. When we say the exterior polynomial of a bipartite graph, we shall specify which colour class is the set of hyperedges. In addition, it is not difficult to see that multiple edges in a bipartite graph do not affect the set of hypertrees and so do the two polynomials. Recall that for a hypergraph H its associated bipartite graph BipH is always simple. Definition 15. The degree of the polynomial f is the maximum of its supp(f ). The polynomial f is called interpolating if its supp(f ) is an integer interval [n 1 , n 2 ] of all integers from n 1 to n 2 , inclusive. The interior polynomial In this section, we study interpolatory property of the interior polynomial, the degree of the interior polynomial for bipartite graphs having 2-vertex cuts in E or V , monic property of interior polynomials for balanced bipartite graphs and the representation of the interior polynomial of a bipartite graph as a linear combination of those of some balanced bipartite graphs. Since I H (x) = I H (x) for abstract dual hypergraphs H and H, without loss of generality we assume that vertices in E of a connected bipartite graph G = (V ∪ E, ε) will be regarded as hyperedges of the hypergraph H. Interpolatory property In this subsection, we will show that the interior polynomial of any connected bipartite graph is interpolating. We need the following lemmas. Lemma 16 ( [3]). Let G = (V ∪ E, ε) be a connected bipartite graph. Let f be a hypertree of G. Then for any non-empty subset E ⊂ E, if E is not tight at f , then f is a hypertree so that a transfer of valence is possible from some element of E \ E to some element of E . Lemma 17 ( [3]). Let G = (V ∪ E, ε) be a connected bipartite graph, and f be a hypertree of G. Let e i ∈ E (i = 1, 2, 3). If e 1 can transfer valence to e 2 and e 2 can transfer valence to e 3 with respect to f , then e 1 can transfer valence to e 3 with respect to f . Lemma 18. Let G = (V ∪ E, ε) be a connected bipartite graph, and let f be a hypertree of G. Suppose that e, e ∈ E and e = e . Then e can transfer valence to e with respect to f if and only if f (e) = 0, and every subset E ⊂ E, which contains e and does not contain e, is not tight at f . Proof. The necessity is obvious. For sufficiency, let us take E 1 = {e }. Since E 1 is not tight at f some element of E \ E 1 can transfer valence to e for f by Lemma 16. Let U 1 be the set consisting of all elements of E \ E 1 that can transfer valence to e for f . If e ∈ U 1 , then the conclusion is true. If e / ∈ U 1 , then we take E 2 = U 1 ∪ E 1 . Note that E 2 is not tight at f . Then some element of E \ E 2 can transfer valence to some element of E 2 for f by Lemma 16. Let U 2 be the set consisting of all elements of E \ E 2 that can transfer valence to some element of E 2 for f . It is obvious that E 1 is a proper subset of E 2 . Moreover, all elements of E 3 = U 2 ∪ E 2 (except for e ) can transfer valence to e for f by Lemma 17, and E 2 is a proper subset of E 3 . Continue the above process, we will eventually obtain that e can transfer valence to e for f . Lemma 19. Let G = (V ∪E, ε) be a connected bipartite graph, and let f be a hypertree of G. Given an order on E, then the hyperedge e ∈ E is internally inactive with respect to f if and only if f (e) = 0, and there exists a hyperedge e < e so that every subset E ⊂ E, which contains e and does not contain e, is not tight at f . Proof. The hyperedge e is internally inactive with respect to f , if and only if for some hyperedge e < e, e can be transferred valence from e with respect to f by the definition. e can transfer valence to e with respect to f if and only if f (e) = 0, and every subset E ⊂ E, which contains e and does not contain e, is not tight at f by Lemma 18. Thus, the conclusion is true. Remark 20. Lemmas 18 and 19 can also be obtained by using Theorem 5 (3) directly. Let G = (V ∪ E, ε) be a connected bipartite graph, and e 1 , e 2 ∈ E and e 1 = e 2 . Let f 1 and f 2 be hypertrees of G with f 1 (e 1 ) < f 2 (e 1 ) and f 1 (e) = f 2 (e) for all e ∈ E and e = e 1 , e 2 . The following lemma is known. Lemma 21 ( [3]). If f 1 is a hypertree such that valence can be transferred from e 2 to e, then f 2 is a hypertree such that valence can be transferred from e 1 to e. Lemma 22. Given an order on E, and suppose that the hyperedge e > e 1 . If e is internally active with respect to f 1 , then it is also internally active with respect to f 2 . Proof. If the hyperedge e is internally active with respect to f 1 , then (i) f 1 (e) = 0 or (ii) there exists a subset U ⊂ E, which contains e and does not contain e, is tight at f 1 for any hyperedge e < e by Lemma 19. If (i) holds, e can not be e 2 , then f 2 (e) = f 1 (e) = 0, clearly, e is internally active with respect to f 2 . If (ii) is true, then in particular, there exists a subset U 1 ⊂ E, which contains e 1 and does not contain e, is tight at f 1 since e 1 < e. Take U = e <e U . Then the subset U ⊂ E, which contains any e < e and does not contain e, is tight at f 1 by Theorem 7. Since e 1 ∈ U we have g∈U f 1 (g) ≤ g∈U f 2 (g) . Thus U is also tight at f 2 . This implies that e cannot transfer valence to any e < e in f 2 , that is, e is internally active with respect to f 2 . Lemma 23. Let G = (V ∪E, ε) be a connected bipartite graph. If there exists a hypertree with internal inactivity k in G, then there exists a hypertree with internal inactivity k − 1 in G for any integer k ≥ 1. Proof. Given an order on E, assume that f 1 is a hypertree with internal inactivity k in G, and e m is the smallest internally inactive hyperedge with respect to f 1 . Let f be a hypertree with internal inactivity k and the hyperedge e m is the smallest internally inactive hyperedge with respect to f so that the entry f (e m ) of e m is smallest. Let e n be the smallest hyperedge that can be transferred valence from e m with respect to f . Then there is the hypertree g with g(e m ) = f (e m ) − 1, g(e n ) = f (e n ) + 1 and g(e) = f (e) for all e = e m , e n . Next we prove that the hyperedge e is internally active with respect to g if and only if e is internally active with respect to f for every hyperedge e = e m and will show that g is a hypertree with internal inactivity k − 1. For any hyperedge e with e < e n , since e is internally active with respect to f , we have (i) f (e) = 0 or (ii) there exists E 1 ⊂ E, which contains e and does not contain e, is tight at f for each hyperedge e < e by Lemma 19. If (i) holds, then g(e) = f (e) = 0, clearly, e is internally active with respect to g. Assume that (ii) is true. Note that e < e < e n < e m . We have that e m can not transfer valence to e with respect to f . Then there exists E 2 ⊂ E, which contains e and does not contain e m , is tight at f by Lemma 18. Take E = E 1 ∩ E 2 . Then E , which contains e and does not contain e m and e, is tight at f by Theorem 7. Note that e∈E f (e) ≤ e∈E g(e) since e m / ∈ E . We have that E is tight at g, that is, there exists E ⊂ E, which contains e and does not contain e, is tight at g for each hyperedge e < e. Thus, by Lemma 19, e is internally active with respect to g. For the hyperedge e n , we claim e n is internally active with respect to g. Assume that the opposite is true. Then there is a hyperedge e < e n that can be transferred valence from e n with respect to g. For the hyperedge e with e > e n that is internally active with respect to f (in fact, e is internally active with respect to f if e n < e < e m ), e is also internally active with respect to g by Lemma 22. For the hyperedge e > e m that is internally inactive with respect to f , we claim that e is internally inactive with respect to g. Otherwise, e is internally active with respect to f by Lemma 22, a contradiction. It is obvious that e m is internally active with respect to g by the choice of the hypertree f . In fact, if e m is internally inactive with respect to g, then g will be a hypertree with internal inactivity k and the hyperedge e m is the smallest internally inactive hyperedge with respect to g. Since f (e m ) > g(e m ), it contradicts the choice of the hypertree f . Thus, g is a hypertree with internal inactivity k − 1. Remark 24. In the proof of Lemma 23, the claim that e n is internally active with respect to g can be obtained by using Lemma 21. Theorem 25. The interior polynomial of any connected bipartite graph is interpolating. Proof. It follows directly from Lemma 23. Degree In this subsection, we will strengthen the result on the degree of the interior polynomial. It has been considered by Kálmán in [3], and the author obtained the following result. Theorem 26 ( [3]). Let G = (V ∪ E, ε) be a connected bipartite graph. Then the degree of the interior polynomial of G is at most min{\|E\| − 1, \|V \| − 1}. The following properties hold for the interior polynomial and the exterior polynomial of bipartite graphs. Theorem 27 ([3]). Let G 1 = (V 1 ∪ E 1 , ε 1 ) and G 2 = (V 2 ∪ E 2 , ε 2 ) be two connected disjoint bipartite graphs. (1) Let G be the connected bipartite graph obtained by identifying a vertex v 1 ∈ V 1 and a vertex v 2 ∈ V 2 or identifying a vertex e 1 ∈ E 1 and a vertex e 2 ∈ E 2 . Then I G (x) = I G 1 (x)I G 2 (x) and X G (y) = X G 1 (y)X G 2 (y). (2) Let G be the connected bipartite graph obtained by identifying one edge (v 1 , e 1 ) of G 1 and one edge (v 2 , e 2 ) of G 2 , where v 1 ∈ V 1 , v 2 ∈ V 2 , e 1 ∈ E 1 , e 2 ∈ E 2 , v 1 and v 2 are identified, and e 1 and e 2 are identified. Then I G (x) = I G 1 (x)I G 2 (x) and X G (y) = X G 1 (y)X G 2 (y). Now we consider the case that a connected bipartite graph G = (V ∪E, ε) has a 2-vertex cut in E. Proof. We consider vertices on E as hyperedges of hypergraph H. Given an order on E, let f be a hypertree and τ be a spanning tree inducing f in G. Since G−{e 1 , e 2 } has t connected components G 1 , G 2 , · · · , G t , there is at least one edge of τ joining each G i (i = 1, 2, · · · , t) to one of the hyperedges e 1 and e 2 , and there is at least a connected component G i (i = 1, 2, · · · , t) incident with both e 1 and e 2 in the spanning tree τ . Hence d τ (e 1 ) + d τ (e 2 ) ≥ t + 1. Since d τ (e) = f (e) + 1 for all e ∈ E, f (e 1 ) + f (e 2 ) ≥ t − 1. There are two cases. Case 1. f (e 1 ) + f (e 2 ) ≥ t. In this case, recall that Similarly there are at most \|V \| − t internally inactive hyperedges in E\{e 1 , e 2 } with respect to f . Without loss of generality, we assume that e 1 < e 2 . Then we claim that e 1 is internally active with respect to f . Otherwise, there is a hypertree g with g(e 1 ) = f (e 1 ) − 1 and g(e) = f (e) + 1 for some hyperedge e < e 1 and g(e ) = f (e ) for all e = e, e 1 . We have that g(e 1 ) + g(e 2 ) = t − 2, a contradiction. Hence there are still at most \|V \| − t + 1 internally inactive hyperedges with respect to f . A bipartite graph G = (V ∪ E, ε) is said to be balanced if \|V \| = \|E\|. Corollary 29. Let G = (V ∪ E, ε) be a connected balanced bipartite graph with \|V \| = \|E\| = n. If G has a 2-vertex cut {u 1 , u 2 } in E or V so that G − {u 1 , u 2 } has at least three connected components, then the coefficient of the term x n−1 in I G (x) is 0. Proof. It follows from Theorem 28 and the fact that taking abstract dual does not affect the interior polynomial. Similar to Theorem 28, we can prove the following more general result, and the proof is left for the readers. Theorem 30. Let G = (V ∪ E, ε) be a connected bipartite graph. If it has m pairs of vertex-disjoint 2-vertex cuts {{v 2i−1 , v 2i }\|i = 1, 2, · · · , m} in V and n pairs of vertex-disjoint 2-vertex cuts {{e 2i−1 , e 2i }\|i = 1, 2, · · · , n} in E, then the degree of the interior polynomial of G is at most min{\|E\| − t + 2m − 1, \|V \| − k + 2n − 1}, where t = m i=1 t i , k = n i=1 k i , t i is the number of connected components of G − {v 2i−1 , v 2i } for i = 1, 2, · · · , m, and k i is the number of connected components of G − {e 2i−1 , e 2i } for i = 1, 2, · · · , n. Monic property In this subsection we consider the monic property of interior polynomials for balanced bipartite graphs. Lemma 31. Let G = (V ∪ E, ε) be a connected balanced bipartite graph with \|V \| = \|E\| = n, and E = {e 1 , e 2 , · · · , e n }. Then the coefficient of the term x n−1 of the interior polynomial I G (x) of G is non-zero, if and only if each f i (i = 1, 2, · · · , n) is a hypertree of G, where f i (e i ) = 0 and f i (e j ) = 1 for all j = i. Moreover, the coefficient of the term x n−1 of the interior polynomial I G (x) of G is in fact at most 1. Proof. Without loss of generality, we assume that e 1 < e 2 < · · · < e n . The sufficiency follows from the fact that f 1 is a hypertree with internal inactivity n − 1. For necessity, if f is a hypertree with internal inactivity n − 1, then f has exactly one zero entry and n − 1 one entries since n i=1 f (e i ) = n − 1 and to ensue that internal inactivity of f is n − 1, each f i must be a hypertree of G. Note that f 2 , · · · , f n and other hypertrees of G with at least two zero entries have internal inactivity at most n − 2, hence the coefficient of the term x n−1 of the interior polynomial I G (x) for G is at most 1. It is well-known that a graph is 2-connected if and only if it has an ear decomposition. Theorem 32. Let G = (V ∪ E, ε) be a 2-connected balanced bipartite graph with \|V \| = \|E\| = n. If it has an ear decomposition such that each ear is from some element of V to some element of E, then the coefficient of the term x n−1 of the interior polynomial I G (x) for G is 1. Proof. Suppose that E = {e 1 , e 2 , · · · , e n }. By Lemma 31, we only need to show that each f i (i = 1, 2, · · · , n) with f i (e i ) = 0 and f i (e j ) = 1 for all j = i is a hypertree of G. We shall prove the theorem by induction on the number of ears in the ear decomposition of G. If the number of ears is zero, then G = C 2n . Suppose C 2n = v 1 e 1 v 2 e 2 · · · v n e n . Let T i be the spanning tree of C 2n obtained from C 2n by removing an edge incident with e i for every i = 1, 2, · · · , n. Then the hypertree f i induced by T i satisfies f i (e i ) = 0 and f i (e) = 1 for all e = e i . Assume that the theorem holds for any 2-connected bipartite graph G = (V ∪ E , ε ) whose ear decomposition has k ears. Suppose that G is obtained from G by adding the k + 1-th ear P with length 2l + 1 from some element of V to some element e j of E . We order hyperedges of E firstly then l hyperedges on P . By induction hypothesis, we suppose that τ i (i = 1, 2, · · · , \|E\| ) is the spanning tree of G inducing f i , where f i is the hypertree with f i (e i ) = 0 and f i (e) = 1 for e ∈ E \ e i . Now we show that f i (i = 1, 2, · · · , \|E\| + l) with f i (e i ) = 0 and f i (e) = 1 for e = e i is a hypertree of G. Let be the edge adjacent to e j in P . For i = 1, 2, · · · , \|E\| , τ i = τ i +P − is a spanning tree of G and it exactly induces the hypertree f i . For i = \|E\| + 1, \|E\| + 2, · · · , \|E\| + l, τ i = τ j + P − i is a spanning tree of G and it exactly induces the hypertree f i , where i is an edge adjacent to e i on P . An example illustrating the proof is given in Figure 1. Linear representation In this subsection, using deletion-contraction relations repeatedly, we show that the interior polynomial of any connected bipartite graph can be represented as a linear combination of the interior polynomials of connected balanced bipartite graphs. Theorem 33 ( [3]). If u ∈ V ∪ E is a vertex of valence 1 in the bipartite graph G, then I G (x) = I G\u (x) and X G (y) = X G\u (y). Let G = (V ∪ E, ε) be a connected bipartite graph and e ∈ E. We denote by G \ e the bipartite graph obtained from G by removing e and all edges incident with e. Moveover, we denote by G/e the bipartite graph obtained from G \ e by identifying all vertices adjacent to e and replacing all multi-edges by single ones. Theorem 34 ( [3]). If u ∈ V ∪ E is a vertex of valence 2 in the bipartite graph G and G \ u is connected, then I G (x) = I G\u (x) + xI G/u (x). If e ∈ E and d G (e) = 2, then X G (y) = yX G\e (y) + X G/e (y). For a connected bipartite graph G = (V ∪ E, ε), we use G t to denote the bipartite graph obtained from G by adding t vertices V = {v 1 , v 2 , · · · , v t } to V and joining them to the same pair of vertices e 1 , e 2 ∈ E for some positive integer t and we use G to denote the bipartite graph obtained from G by identifying e 1 and e 2 , and replacing all multi-edges by single ones. See Figure 2. We have the following Lemma 35. Lemma 35. I G (x) = I Gt (x) − txI G (x). Proof. Note that G = G t \ V and I Gt/v 1 (x) = I Gt\v 1 /v 2 (x) = I Gt\v 1 \v 2 /v 3 (x) = · · · = I G (x) by Theorem 33. By Theorem 34, we have that I Gt (x) = I Gt\v 1 (x) + xI Gt/v 1 (x) = I Gt\v 1 \v 2 (x) + xI Gt\v 1 /v 2 (x) + xI G (x) = I Gt\v 1 \v 2 (x) + 2xI G (x) = · · · = I G (x) + txI G (x). Then I G (x) = I Gt (x) − txI G (x). Theorem 36 ([3]). For any connected bipartite graph G = (V ∪ E, ε), the coefficient of the linear term in the interior polynomial of G is n(G) = \|ε\| − (\|V \| + \|E\|) + 1. For a connected bipartite graph G = (V ∪ E, ε), if G i (i = 1, 2) is the bipartite graph obtained from G by adding m i new vertices to V and joining them to the same pair of vertices e i , e i ∈ E for some positive integer m i . Let G i be the bipartite graph obtained from G by identifying e i and e i and replacing all multi-edges by single ones for i = 1, 2. We know that I G (x) = I G 1 (x)−m 1 xI G 1 (x) = I G 2 (x)−m 2 xI G 2 (x) by Lemma 35. Since the coefficient of the linear term in I G i (x) is n(G i ) for i = 1, 2 by Theorem 36, we know that if m 1 n(G 1 ) = m 2 n(G 2 ), then the coefficients of the quadratic terms in the interior polynomials I G 1 (x) and I G 2 (x) are equal. In particular, we have: Corollary 37. If m 1 = m 2 and \|N G (e 1 ) ∩ N G (e 1 )\| = \|N G (e 2 ) ∩ N G (e 2 )\|, then the coefficients of the quadratic terms in the interior polynomials I G 1 (x) and I G 2 (x) are equal. Proof. This is because \|N G (e 1 ) ∩ N G (e 1 )\| = \|N G (e 2 ) ∩ N G (e 2 )\| will imply n(G 1 ) = n(G 2 ). For a connected bipartite graph G = (V ∪E, ε), without loss of generality, we assume that \|V \| ≤ \|E\| = n. Let t = \|E\| − \|V \|. Denote by G n the bipartite graph obtained from G by adding t vertices to V and joining them to the same pair of vertices e 1 , e 2 ∈ E. Denote by G 1 the bipartite graph obtained from G by identifying e 1 and e 2 and replacing all multi-edges by single ones. Next, we recursively define two families of bipartite graphs as follows. Let G n−i = (E n−i ∪ V n−i , ε n−i ) be the bipartite graph obtained from G i = (E i ∪ V i , ε i ) by adding t − i vertices to V i and joining them to the same pair of vertices e i 1 , e i 2 ∈ E i for any i = 1, 2, · · · , t. Let G i+1 be the bipartite graph obtained from G i by identifying e i 1 and e i 2 and replacing all multi-edges by single ones for any i = 1, 2, · · · , t − 1. Corollary 38. With notations above, we have I G (x) = t i=0 t! (t−i)! (−1) i x i I G n−i (x). Proof. If t = 0, then the corollary is obvious. If t = 0, then by Lemma 35, we have that I G i (x) = I G n−i (x)−(t−i)xI G i+1 (x) for any i = 1, 2, · · · , t. Combin- ing with I G (x) = I Gn (x)−txI G 1 (x), we have I G (x) = t i=0 t! (t−i)! (−1) i x i I G n−i (x). Note that \|V n−i \| = \|E n−i \| = n−i for all i = 1, 2, · · · , t by the construction of G n−i , we have: Theorem 39. The interior polynomial of any connected bipartite graph can be written as a linear combination of the interior polynomials of connected balanced bipartite graphs. The exterior polynomial In this section, we study interpolatory property and the coefficient of the linear term of the exterior polynomial. Without loss of generality, we assume that vertices of E will be regarded as hyperedges for a connected bipartite graph G = (V ∪ E, ε) in this section. Interpolatory property In this subsection, we will show that the exterior polynomial of any connected bipartite graph is interpolating. We also need some lemmas similar to the interior polynomial. Lemma 41. Given an order on E, if a hyperedge e > e 1 is externally active with respect to f 1 , then it is also externally active with respect to f 2 . Proof. If the hyperedge e is externally active with respect to f 1 , then for any hyperedge e < e, (i) f 1 (e ) = 0 or (ii) there exists a subset U ⊂ E, which contains e and does not contain e , is tight at f 1 by Lemma 40. We first claim that e 2 cannot transfer valence to e for f 2 . Otherwise, by Lemma 21, e 1 can transfer valence to e for f 1 , which implies that the hyperedge e is externally inactive with respect to f 1 as e > e 1 , a contradiction. If f 1 (e ) = 0 (e can not be e 1 ) and e = e 2 , then e cannot transfer valence to e for f 2 since f 2 (e ) = f 1 (e ) = 0. If (ii) is true, then in particular, there exists a subset U 1 ⊂ E, which contains e and does not contain e 1 , is tight at f 1 , since e 1 < e. Take U = U ∩ U 1 (it is possible that U = U 1 ). We have that the subset U ⊂ E, which contains e, and does not contain e and e 1 , is tight at f 1 by Theorem 7. Note that e∈U f 1 (e) ≤ e∈U f 2 (e). We have that U is tight at f 2 . It implies that e cannot transfer valence to e for f 2 . Thus, e is externally active with respect to f 2 . Lemma 42. Let G = (V ∪E, ε) be a connected bipartite graph. If there exists a hypertree with external inactivity k for G, then there exists a hypertree with external inactivity k − 1 in G for any integer k ≥ 1. Proof. Given an order on E, assume f 1 is a hypertree with external inactivity k in G and e m is the smallest hyperedge that is externally inactive with respect to f 1 . Let f be a hypertree with external inactivity k and the hyperedge e m is the smallest externally inactive hyperedge with respect to f so that the entry f (e m ) of e m is biggest. Let e n be the smallest hyperedge that can transfer valence to e m . Then there is the hypertree g with g(e m ) = f (e m ) + 1, g(e n ) = f (e n ) − 1 and g(e) = f (e) for all e = e m , e n . Next we prove that the hyperedge e is externally active with respect to g if and only if e is externally active with respect to f for every hyperedge e = e m . For any hyperedge e with e < e n , since e is externally active with respect to f , for every hyperedge e < e, (i) f (e ) = 0 or (ii) there exists a subset U ⊂ E, which contains e and does not contain e , is tight at f by Lemma 40. If (i) holds, then g(e ) = f (e ) = 0, clearly, e cannot transfer valence to e with respect to g. Assume that (ii) is true. Since e can not transfer valence to e m with respect to f there exists E 2 ⊂ E, which contains e m and does not contain e , is tight at f by Lemma 18. Take E = U ∪ E 2 . Then E , which contains e m and e and does not contain e , is tight at f by Theorem 7. Note that e∈E f (e) ≤ e∈E g(e). We have that E is tight at g. Then for every hyperedge e < e, e cannot transfer valence to e with respect to g. Thus, e is externally active with respect to g. For the hyperedge e n , we claim e n is externally active with respect to g. Assume that the opposite is true. Then for some hyperedge e < e n , e can transfer valence to e n with respect to g. Then there exists the hypertree f satisfying f (e m ) = f (e m ) + 1 and f (e ) = f (e ) − 1 and f (e) = f (e) for all e ∈ E and e = e m , e n , that is, e can transfer valence to e m with respect to f . This contradicts the choice of the hyperedge e n . For any externally active hyperedge e > e n with respect to f (in fact, e is externally active if e n < e < e m with respect to f ), e is also externally active with respect to g by Lemma 41. For any externally inactive hyperedge e > e m with respect to f , we claim that e is externally inactive with respect to g. Otherwise, e is externally active with respect to f by Lemma 41, a contradiction. It is obvious that e m is externally active with respect to g by the choice of the hypertree f . In fact, if e m is externally inactive with respect to g, then g is a hypertree with externally inactive k and the hyperedge e m is the smallest externally inactive hyperedge with respect to f . Since f (e m ) < g(e m ), it contradicts the choice of the hypertree f . Thus, g is a hypertree with externally inactive k − 1. Theorem 43. The exterior polynomial of any connected bipartite graph is interpolating. Proof. It follows directly from Lemma 42. Coefficient of linear term In this subsection, we will study the coefficient of the linear term of the exterior polynomial by a similar argument in the interior polynomial. Before studying the coefficient of the linear term of the exterior polynomial, we firstly recall the coefficient of constant term of the interior polynomial and the exterior polynomial. Theorem 44 ( [3]). For any connected bipartite graph, the coefficient of the constant term of the interior polynomial and the exterior polynomial are 1. Let G = (V ∪ E, ε) be a connected bipartite graph. Given an arbitrary order on E (e 1 < e 2 < · · · < e \|E\| ), let G k = (E k ∪V k , ε k ) be the bipartite graph formed by E k = {e k , e k+1 , · · · , e \|E\| }, all edges of G incident with some element of E k , and their endpoints in V for each k = 1, 2, · · · , \|E\|. Clearly, G 1 = G and G \|E\| is a star. Let n(G k ) be the nullity of the graph G k , that is, n(G k ) = \|ε k \| − (\|V k \| + \|E k \|) + c k , where c k is the number of connected components of G k . Then n(G 1 ) = n(G) and n(G \|E\| ) = 0. Let n(e k ) = n(G k ) − n(G k+1 ) for any k = 1, 2, · · · , \|E\| − 1 and n(e \|E\| ) = 0. Moreover, a hypertree g of G is defined as g (e) = d G (e) − 1 − n(e) for all e ∈ E. By the proof of Theorem 44 in [3], we know that E k is tight at g for any k = 1, 2, · · · , \|E\|. Moreover, g is the unique hypertree with external inactivity 0, called the exterior greedy hypertree of G. Theorem 45. For a connected bipartite graph G = (V ∪ E, ε), if G − e is connected for each e ∈ E, then the coefficient of the linear term of the exterior polynomial is \|V \| − 1. Proof. After arbitrarily fixing an order on E (e 1 < e 2 < . . . < e m < . . . < e \|E\| ), let g be the exterior greedy hypertree above under this order. Firstly, we show that for any hyperedge e m with g (e m ) = 0 and for any 0 ≤ i ≤ g (e m ), there is an i-element multiset consisting of hyperedges smaller than e m so that the result of reducing g (e m ) to g (e m ) − i, and increasing the g -value to each of its element of the i-multiset by the multiplicity of the element, is also a hypertree of G. Clearly, for any hyperedge e m , the conclusion is true for i = 0. Assume that the opposite is true, that is, there is no such a hypertree for some e m and some 1 ≤ i ≤ g (e m ). Without loss of generality, we assume that there is a hypertree g j satisfying conditions for some 0 ≤ j ≤ g (e m ) − 1 and there is no hypertree satisfying conditions for j + 1. Then the hyperedge e m is internally active with respect to g j . Since g j (e m ) = 0, for any e x < e m , there is a subset S x ⊂ E , which contains e x and does not contain e m , is tight at g j by Lemma 19. Note that g j (e i ) = g (e i ) for all e i > e m . We have that E m+1 = {e m+1 , · · · , e \|E\| } is tight at g j . Put S = ex<em S x ∪ E m+1 = E \ {e m }. Then S is tight at g j by Theorem 7. Let τ be a spanning tree of G inducing g j . Then τ \| S should be a spanning forest of G\| S by Theorem 8. Because G − e is connected for any e ∈ E, we have that G\| S = G − e m is connected. Then τ \| S is a spanning tree of G − e m , which implies that it is τ − e m . Since g (e m ) > 0, d τ (e m ) ≥ 2. This implies that τ − e m is disconnected, a contradiction. In addition, g (e 1 ) must be zero, otherwise, G − e 1 should not be connected. Next we are going to construct \|V \| − 1 such hypertrees as follows. Given the hyperedge e m with g (e m ) = 0 and i (1 ≤ i ≤ g (e m )), we define f em,i : E → N be the hypertree so that its associated multiset M em,i is the largest in reverse order, that is, if f = f em,i is another hypertree (recall that f is obtained by setting f (e m ) = g (e m ) − i and increasing the g -value to each of its element of one i-multiset consisting of hyperedges smaller than e m by the multiplicity of the element) and e = max{e j \|f em,i (e j ) = f (e j )}, then f em,i (e ) > f (e ). Since e∈E g (e) = \|V \| − 1, there are \|V \| − 1 those hypertrees. In the following we show that e m is a unique externally inactive hyperedge with respect to f em,i . (1) If e j > e m , then e j is externally active with respect to f em,i . This is because f em,i (e) = g (e) for any e > e m . E j is tight at g and also at f em,i , one can not transfer valence from E\E j to e j with respect to f em,i . (2) If e j < e m , then e j is externally active with respect to f em,i due to choice of f em,i . Otherwise, there is the hypertree g 1 with g 1 (e j ) = f em,i (e j ) + 1, g 1 (e x ) = f em,i (e x ) − 1 for some e x < e j and g 1 (e ) = f em,i (e ) for e = e j , e x . It contradicts the choice of f em,i . (3) Since E m is tight at g , it is not tight at f em,i . E m+1 is tight both at g and f em,i . So it is possible to transfer valence from E\E m to e m for f em,i and hence e m is externally inactive with respect to f em,i . Finally, we prove that if f is a hypertree with a unique externally inactive hyperedge e m , then f is one of the f em,i 's. Claim 1. For all e x > e m , E x = {e y ∈ E\|e y ≥ e x } is tight at the hypertree f . Assume that Claim 1 is not true, that is, E x = {e y ∈ E\|e y ≥ e x } is not tight at f for some e x > e m . Then f is a hypertree such that it is possible to transfer valent from element of E \ E x to some element of E x by Lemma 16, that is, some element of E x is externally inactive with respect to f , a contradiction. Thus, Claim 1 is true. Claim 1 implies that f (e y ) = g (e y ) for all e y > e m . Since e m is unique externally inactive hyperedge with respect to f we have f (e m ) < g (e m ). Let i = g (e m ) − f (e m ). Then the following claim holds. Claim 2. f = f em,i . Assume that f = f em,i . Note that f em,i (e x ) = g (e x ) for all e x > e m by the construction of f em,i . It implies that f em,i (e x ) = f (e x ) for all e x ≥ e m . Assume that e y = max{e x \|f em,i (e x ) = f (e x )}. There are two cases. Thus, the number of hypertrees with external inactivity 1 is exactly \|V \| − 1. Examples In this section, as examples we shall compute the interior polynomial and the exterior polynomial for several families of bipartite graphs. Computational results in this section are consistent with results in Sections 3 and 4. Observation 46. If G = (V ∪ E, ε) is a tree, then I G (x) = X G (y) = 1. Proof. Since there is only one spanning tree for a tree there is only one hypertree f for G. Furthermore, every hyperedge e ∈ E will be internally active and externally active with respect to the hypertree f since there are no other hypertrees. So I G (x) = x 0 = 1 and X G (y) = y 0 = 1. Theorem 47. If G = (V ∪ E, ε) is a cycle of length 2n, then I G (x) = 1 + x + x 2 + · · · + x n−1 and X G (y) = 1 + (n − 1)y. Proof. Since the two hypergraphs induced by C 2n are the same, it is justified the exterior polynomial of C 2n . Recall that e∈E f (e) = n−1 for any hypertree f of G. Moreover, 0 ≤ f (e) ≤ d G (e) − 1 and d G (e) = 2 for all e ∈ E. Then given an order e 1 < e 2 < · · · < e n on E, there are at most n hypertrees f 1 = (0, 1, · · · , 1), f 2 = (1, 0, · · · , 1),· · · ,f n = (1, 1, · · · , 0) for G. Moreover, let the subgraph τ i be the bipartite graph obtained from G by removing an edge incident with e i for every i = 1, 2, · · · , n. It is obvious that τ i is a spanning tree inducing f i of G for all i = 1, 2, · · · , n. We obtain that there are exactly n hypertrees f i (i = 1, 2, · · · , n) of G. Clearly, f i is a hypertree with internal inactivity n − i in G for any i = 1, 2, · · · , n, f 1 is a hypertree with external inactivity 0, and f i is a hypertree with external inactivity 1 for any i = 2, 3, · · · , n. Thus, I G (x) = 1 + x + x 2 + · · · + x n−1 and X G (y) = 1 + (n − 1)y. Corollary 48. If G = (V ∪ E, ε) is unicyclic and the length of the unique cycle is 2n, then I G (x) = 1 + x + x 2 + · · · + x n−1 and X G (y) = 1 + (n − 1)y. Proof. It follows from Theorem 27 (1), Observation 46 and Theorem 47. Let P n+1 be the path of length n. Let G × H be the Cartesian product of G and H. Corollary 49. If G = (V ∪ E, ε) is P n+1 × P 2 , then I G (x) = (1 + x) n and X G (y) = (1 + y) n . In the following we consider interior and exterior polynomials of some dense graphs. Beforehand, we recall two combinatorial numbers as follows (see [12], p.19-20). Lemma 50 ( [12]). The number of ordered partitions of n into k parts is Proof. If m = 1, then G is a star. It is clear that the conclusion is true from Observation 46. Assume m ≥ 2. By Theorem 13, the interior polynomial and the exterior polynomial are independent of the chosen order of the hyperedges. Without loss of generality, we assume that \|V \| = m and \|E\| = n. Moveover, let v 1 < v 2 < · · · < v m and e 1 < e 2 < · · · < e n . We have that the interior polynomials of abstract dual hypergraphs are equal by Theorem 14. Without loss of generality we consider vertices of E as hyperdeges. Firstly, we prove that if a function f satisfies e∈E f (e) = m − 1 and 0 ≤ f (e) ≤ m − 1 for all e ∈ E, then f is a hypertree of K m,n . We assume that the cardinality of the set {e\|f (e) = 0} is i (note 0 ≤ f (e) ≤ m − 1 and e∈E f (e) = m − 1, so i > 0) and without loss of generality, suppose that f (e 1 ) = 0,· · · ,f (e i ) = 0. We construct the spanning tree T of G inducing f as follows: e 1 is connected to v 1 , v 2 ,· · · , v f (e 1 )+1 ,· · · , and e k is connected to v [ k−1 j=1 f (e j )]+1 , v [ k−1 j=1 f (e j )]+2 ,· · · , v [ k j=1 f (e j )]+1 ,. . . , and e i is connected to v [ i−1 j=1 f (e j )]+1 , v [ i−1 j=1 f (e j )]+2 ,· · · , v [ i j=1 f (e j )]+1 = v m , finally each e ∈ E\{e 1 , e 2 , · · · , e i } is connected to v 1 . Next, we consider hypertrees f with i positions that are not 0 for i = 1, 2, · · · , m − 1. If f (e 1 ) = 0, then the hypertree f is a hypertree with I G (x) = m−1 i=1 n − 1 i m − 2 i − 1 x i + m−1 i=1 n − 1 i − 1 m − 2 i − 1 x i−1 = m−1 i=1 n − 1 i m − 2 i − 1 x i + m−2 i=0 n − 1 i m − 2 i x i = m−1 i=1 n − 1 i m − 2 i − 1 x i + n − 1 0 m − 2 0 − 1 x 0 + m−2 i=1 n − 1 i m − 2 i x i + n − 1 m − 1 m − 2 m − 1 x m−1 = m−1 i=0 n − 1 i m − 2 i − 1 x i + m−1 i=0 n − 1 i m − 2 i x i = m−1 i=0 n − 1 i m − 1 i x i . For the exterior polynomial X G (y) of G, we assume that vertices of E are hyperdeges. For each i = 0, 1, · · · , n − 1, we consider a hypertree f with f (e i+1 ) = 0 and f (e j ) = 0 for all j = 1, · · · , i. Then the hypertree f is a hypertree with external inactivity n−i−1. If f (e i+1 ) = j for j = 1, 2, · · · , m− −q]y n−1 if we regard E as hyperedges. Proof. Without loss of generality, we assume that v 1 < v 2 < · · · < v m and e 1 < e 2 < · · · < e n in V and E, respectively, and M = {v i e i \|i = 1, 2, · · · , q} and E = {e i \|i = 1, 2, · · · , q}. A similar argument of the proof of complete bipartite graphs shows that if a function f satisfies and 0 ≤ f (e) ≤ m − 1 for all e ∈ E \ E , then f is a hypertree of G. We assume that the cardinality of the set {e\|f (e) = 0, e ∈ E } ({e\|f (e) = 0, e ∈ E \ E }) is i (i , resp.) and f (e 1 ) = 0,· · · ,f (e i ) = 0,f (e q+1 ) = 0,· · · ,f (e q+i ) = 0. Then the spanning tree of G inducing f can be constructed as follows: e 1 is connected to v 2 , v 3 ,· · · , v f (e 1 )+2 ,· · · , and e k is connected to v , and e q+1 is connected to v ,· · · , and e q+i is connected to v ,· · · ,v = v m and v 1 . Moreover, e is connected to v 1 for all e ∈ E \ {e 1 , · · · , e i , e q+1 , · · · , e q+i }. It is clear that k−1 j=1 f (e j ) + 2 > k for k = 1, 2, · · · , i. Compared with complete bipartite graphs, G has only no hypertrees f i (1 = 1, 2, · · · , q) with f i (e i ) = m − 1 and f i (e) = 0 for all e = e i . We firstly consider the interior polynomial, it is clear that ι(f 1 ) = 0 and ι(f i ) = 1 (i = 2, · · · , q) for K m,n . Moreover, ι(f ) = 0 in G and ι(f ) = 1 in K m,n for the hypertree f = (m − 2, 1, 0, · · · , 0). And there are same internal inactivity for all f = f, f i (i = 1, 2, · · · , q) in K m,n and G. Thus, I G (x) = 1 + [(n − 1)(m − 1) − q]x + m−1 i=2 n−1 i m−1 i x i . For the exterior polynomial, we assume that vertices of E are regarded as hyperdeges. It is clear that (f i ) = n−i (1 = 1, 2, · · · , q) for K m,n . Moreover, (f ) = n − j − 1 in G and (f ) = n − j in K m,n for the hypertree f with f (e i ) = m − 2, f (e j ) = 1 for some hyperedge e j < e i , and f (e) = 0 for all e = e i , e j for any i = 1, 2, · · · , q. There are same external inactivity for all f = f, f i (i = 1, 2, · · · , q) in K m,n and G. Thus, by simple calculations, we obtain X G (y) = − q]y n−1 . Definition 12 . 12Let H = (V, E) be a connected hypergraph. For some fixed order on E, we denote the interior polynomial I H (x) = f ∈B H x ι(f ) and the exterior polynomial X H (y) = f ∈B H y (f ) . Theorem 13 ([3]). Let H = (V, E) be a connected hypergraph. Then the interior polynomial and the exterior polynomial of H do not depend on the chosen order on E. Theorem 14 ([5]). If H and H are abstract dual hypergraphs, then I H (x) = I H (x). x i is the set supp(f ) = {i\|a i = 0} of indices of the non-zero coefficients. Then there exists the hypertree f satisfying f (e m ) = f (e m ) − 1 and f (e ) = f (e ) + 1 and f (e) = f (e) for all e ∈ E and e = e m , e , that is, e m can transfer valence to e with respect to f . This contradicts the choice of the hyperedge e n . Theorem 28 . 28Let G = (V ∪ E, ε) be a connected bipartite graph with a 2-vertex cut {e 1 , e 2 } in E. Then the degree of the interior polynomial of G is at most min{\|E\| − 1, \|V \| − t + 1}, where t is the number of connected components of G − {e 1 , e 2 }. − t − 1. Then the cardinality of the set {e ∈ E\{e 1 , e 2 }\|f (e) = 0} is at most \|V \| − t − 1. Note that if e is internally inactive with respect to f , then f (e) = 0. Thus, there are at most \|V \| − t − 1 internally inactive hyperedges in E\{e 1 , e 2 } with respect to f and there are at most \|V \| − t + 1 internally inactive hyperedges in E with respect to f . Case 2. f (e 1 ) + f (e 2 ) = t − 1.In this case we have thate∈E\{e 1 ,e 2 }f (e) = \|V \| − t. Then the cardinality of the set {e ∈ E\{e 1 , e 2 }\|f (e) = 0} is at most \|V \| − t. Lemma 40 . 40Let G = (V ∪ E, ε) be a connected bipartite graph and f be a hypertree of G. Given an order on E, then the hyperedge e is externally inactive with respect to f if and only if there exists a hyperedge e < e so that f (e ) = 0 and every subset of E, which contains e and does not contain e , is not tight at f . Proof. The hyperedge e is externally inactive with respect to f , if and only if for some hyperedge e < e, e can transfer valence to e with respect to f . We know that e can transfer valence to e with respect to f if and only if f (e ) = 0, and every subset E ⊂ E, which contains e and does not contain e , is not tight at f by Lemma 18, which completes the proof.Let G = (V ∪E, ε) be a connected bipartite graph, e 1 and e 2 be hyperedges of G and e 1 = e 2 . Let f 1 and f 2 be hypertrees of G with f 1 (e 1 ) > f 2 (e 1 ) and f 1 (e) = f 2 (e) for all e ∈ E and e = e 1 , e 2 . The following property in Lemma 41 is true for two hypertrees f 1 and f 2 . Case 1. f em,i (e y ) > f (e y ).Let S = {e x \|f (e x ) = 0 and e x < e y }. Since e y is externally active with respect to f , there is a subset E x ⊂ E, which contains e y and does not contain e x , is tight at f for any e x ∈ S by Lemma 40.Let E = ex∈S E x . Then E is tight at f by Theorem 7, that is, ex∈E f (e x ) = µ(E ). Note e y ∈ E , E ∩ S = ∅ and f em,i (e y ) > f (e y ). Moreover, f em,i (e x ) = f (e x ) or f (e x ) = 0 for any e x ∈ E . We have that ex∈E f em,i (e x ) >ex∈E f (e x ) = µ(E ), a contradiction. Case 2. f em,i (e y ) < f (e y ). Let S = {e x \|f em,i (e x ) = 0 and e x < e y }. Since e y is externally active with respect to f em,i , there is a subset E x ⊂ E, which contains e y and does not contain e x , is tight at f em,i for any e x ∈ S by Lemma 40. Let E = ex∈S E x . Then E is tight at f em,i by Theorem 7, that is, ex∈E f em,i (e x ) = µ(E ). Note f (e y ) > f em,i (e y ), and f (e x ) = f em,i (e x ) or f em,i (e x ) = 0 for any e x ∈ E . We have that ex∈E f (e x ) > ex∈E f em,i (e x ) = µ(E ), a contradiction. Hence, Claim 2 is true. . Lemma 51 ([12]). The number of m-element multisets of an n-element set is m+n−Let G = (V ∪ E, ε) be the complete bipartite graph K m,n (m ≤ n). if we regard E as hyperedges. hypertrees by Lemma 50. If f (e 1 ) = 0, then hypertree f is a hypertree with internal inactivity i − 1 and there are n Theorem 53 . 53Let G = (V ∪E, ε) be a bipartite graph obtained from K m,n (m ≤ n) by deleting a matching with q (q ≤ m) edges. Then(1) I G (x) = 1 + [(n − 1)(m − 1) − q]x + e∈E f (e) = m − 1, 0 ≤ f (e) ≤ m − 2 for all e ∈ E f (e j )]+2+f (e q+1 ) Theorem 5 ([3]).Let H = (V, E) be a connected hypergraph and BipH be the bipartite graph associated to the hypergraph H. Let f be a hypertree of H. Then Definition 6. Let H = (V, E) be a connected hypergraph. Let f be a hypertree of H. For the subset E ⊂ E, we say that E is tight at f , if It is clear that ∅ and E are always tight at any hypertree, and if E is tight at f and for another hypertree g, then E is also tight at g. Moreover, the next Theorem follows immediately from Theorem 44.2 in[10] since µ is submodular.Theorem 7. Let H = (V, E) be a connected hypergraph. Let f be a hypertree of H. If the subsets A ⊂ E and B ⊂ E are both tight at f , then A ∩ B and A ∪ B are both tight at f . Theorem 8 ([3]). Let H = (V, E) be a connected hypergraph and BipH be the bipartite graph associated to H. Let f be a hypertree of H, and τ be a spanning tree of BipH inducing f . The subset A ⊂ E is tight at f if and only if τ \| A is a spanning forest of BipH\| A , i.e., components of τ \| A are exactly spanning trees of components of BipH\| A .e∈E f (e) = µ(E ) holds. e∈E f (e) = e∈E g(e) AcknowledgementsThis work is supported by NSFC (No. 12171402) and the Fundamental Research Funds for the Central Universities (No. 20720190062). A new polynomial invariant of knots and links. P Freyd, D Yetter, J Hoste, W B R Lickorish, K Millett, A Ocneanu, Bull. Amer. Math. Soc. (N.S.). 122P. Freyd, D. Yetter, J. Hoste, W. B. R. Lickorish, K. Millett and A. Ocneanu, A new polynomial invariant of knots and links, Bull. Amer. Math. Soc. (N.S.) 12(2) (1985) 239-246. A polynomial invariant for knots via von Neumann algebras. V F R Jones, Bull. Amer. Math. Soc. 12V. F. R. Jones, A polynomial invariant for knots via von Neumann alge- bras, Bull. Amer. Math. Soc. 12 (1985) 103-111. A version of Tutte's polynomial for hypergraphs. T Kálmán, Adv. Math. 244T. Kálmán, A version of Tutte's polynomial for hypergraphs, Adv. Math. 244 (2013) 823-873. Root polytopes, parking functions, and the HOMFLY polynomial. T Kálmán, H Murakami, Quantum Topol. 82T. Kálmán and H. Murakami, Root polytopes, parking functions, and the HOMFLY polynomial, Quantum Topol. 8(2) (2017) 205-248. Root polytopes, Tutte polynomial, and a duality theorem for bipartite graphs. T Kálmán, A Postnikov, Proc. Lond. Math. Soc. 1143T. Kálmán and A. Postnikov, Root polytopes, Tutte polynomial, and a duality theorem for bipartite graphs, Proc. Lond. Math. Soc. 114(3) (2017) 561-588. T Kálmán, L Tóthmérész, Hypergraph polynomials and the Bernardi process. T. Kálmán and L. Tóthmérész, Hypergraph polynomials and the Bernardi process, https://arxiv.org/pdf/1810.00812, 2018. Interior polynomial of signed bipartite graphs and the HOMFLY polynomial. K Kato, J. Knot Theory Ramifications. 291241K. Kato, Interior polynomial of signed bipartite graphs and the HOMFLY polynomial, J. Knot Theory Ramifications 29(12) (2020) 2050077, 41 pp. . A Postnikov, Associahedra Permutohedra, Beyond, Int. Math. Res. Not. 6A. Postnikov, Permutohedra, Associahedra, and Beyond, Int. Math. Res. Not. 6 (2009) 1026-1106. Invariants of links of Conway type. J H Przytycki, P Traczyk, Kobe J. Math. 4J. H. Przytycki and P. Traczyk, Invariants of links of Conway type, Kobe J. Math. 4 (1987) 115-139. Combinatorial optimization: Polyhedra and efficiency. A Schrijver, Algorithms and Combinatorics. 24Springer-VerlagA. Schrijver, Combinatorial optimization: Polyhedra and efficiency, Al- gorithms and Combinatorics, 24 (Springer-Verlag, Berlin, 2003). A contribution to the theory of chromatic polynomials. W T Tutte, Canad. J. Math. 6W. T. Tutte, A contribution to the theory of chromatic polynomials, Canad. J. Math. 6 (1954) 80-91. D B West, Combinatorial mathematics. New YorkCambridge University PressD. B. West, Combinatorial mathematics, Cambridge University Press, New York, 2020.	[]
[ "Magnetostriction-driven multiferroicity of MnTe and MnTe/ZnTe epitaxial films", "Magnetostriction-driven multiferroicity of MnTe and MnTe/ZnTe epitaxial films" ]	[ "Helen V Gomonay \nNational Technical University of Ukraine \"KPI\"\nave Peremogy, 3703056KyivUkraine\n", "Ievgeniia G Korniienko \nNational Technical University of Ukraine \"KPI\"\nave Peremogy, 3703056KyivUkraine\n" ]	[ "National Technical University of Ukraine \"KPI\"\nave Peremogy, 3703056KyivUkraine", "National Technical University of Ukraine \"KPI\"\nave Peremogy, 3703056KyivUkraine" ]	[]	Here we demonstrate that MnTe epitaxial films with zinc-blend structure and MnTe/ZnTe multilayers should show ferroelectric polarization in antiferromagnetically (AFM) ordered state and thus belong to multiferroics. Spontaneous ferroelectric polarization results from the bending of highly ionic Mn-Te-Mn bonds induced by magnetostrictive shear strain. Orientation of ferroelectric polarization is coupled with orientation of AFM vector and thus can be controlled by application of the external magnetic field. Due to the clamping of electric and magnetic order parameters, domain structure in MnTe is governed by two mechanisms: depolarizing field produced by electric dipoles and destressing field produced by magnetoelastic dipoles. The values of monodomainization electric and magnetic fields depend upon the sample shape and diminish with the film thickness. Magnetoelectric nature of the domains make it possible to visualize the domain structure by linear and nonlinear optical methods (Kerr effect, second harmonic generation technique). PACS numbers: 75.50.Ee Antiferromagnetics 75.50.Pp Magnetic semiconductors 75.80.+q Magnetomechanical and magnetoelectric effects, magnetostriction 77.80.-e Ferroelectricity and antiferroelectricity	null	[ "https://arxiv.org/pdf/0811.2167v1.pdf" ]	117,769,734	0811.2167	a36bebd8a5f18fc61343007b339fd6ade23269c0	Magnetostriction-driven multiferroicity of MnTe and MnTe/ZnTe epitaxial films 13 Nov 2008 (Dated: November 13, 2008) Helen V Gomonay National Technical University of Ukraine "KPI" ave Peremogy, 3703056KyivUkraine Ievgeniia G Korniienko National Technical University of Ukraine "KPI" ave Peremogy, 3703056KyivUkraine Magnetostriction-driven multiferroicity of MnTe and MnTe/ZnTe epitaxial films 13 Nov 2008 (Dated: November 13, 2008) Here we demonstrate that MnTe epitaxial films with zinc-blend structure and MnTe/ZnTe multilayers should show ferroelectric polarization in antiferromagnetically (AFM) ordered state and thus belong to multiferroics. Spontaneous ferroelectric polarization results from the bending of highly ionic Mn-Te-Mn bonds induced by magnetostrictive shear strain. Orientation of ferroelectric polarization is coupled with orientation of AFM vector and thus can be controlled by application of the external magnetic field. Due to the clamping of electric and magnetic order parameters, domain structure in MnTe is governed by two mechanisms: depolarizing field produced by electric dipoles and destressing field produced by magnetoelastic dipoles. The values of monodomainization electric and magnetic fields depend upon the sample shape and diminish with the film thickness. Magnetoelectric nature of the domains make it possible to visualize the domain structure by linear and nonlinear optical methods (Kerr effect, second harmonic generation technique). PACS numbers: 75.50.Ee Antiferromagnetics 75.50.Pp Magnetic semiconductors 75.80.+q Magnetomechanical and magnetoelectric effects, magnetostriction 77.80.-e Ferroelectricity and antiferroelectricity I. INTRODUCTION Multiferroics are a class of a single phase or composite materials with coexisting magnetic and ferroelectric ordering. High susceptibility of multiferroics to both electric and magnetic fields which enables electrical control of magnetic state and vice versa makes them interesting from fundamental and applied points of view. Some of multiferroics are ferroelectric and ferromagnetic (see, e.g., Refs. 1,2,3) and could be used as a storage media with high information capacity or as functionalized materials for electronic devices. Others (mainly rare earth compounds with Mn ions like RMnO 3 or RMn 2 O 5 , where R = Tb, Dy, Ho, Y, etc.) show ferroelectric and antiferromagnetic ordering 4,5,6,7,8 and may find an application as mediators for indirect electric control of ferromagnetic state of an adjacent layer. 9 Cross-coupling between the magnetic and ferroelectric properties may be caused by different mechanisms. In type-I multiferroics (according to classification proposed in the recent review 10 ) ferroelectricity and magnetism have different origin and coexist only in a certain temperature range. Materials classified as type-II show ferroelectricity only in a magnetically ordered state. Until now such a combined ferroelectric-magnetic ordering was observed in crystals which posses at least small noncompensated magnetization, i.e. in ferro-, ferri-and weak ferromagnets (see, for example, Refs. 4,11,12,13,etc.). Two microscopic mechanisms of the magneticallyinduced ferroelectricity originate from the exchange interactions 14 , namely, i) anisotropic exchange 15,16 (Dzyaloshinskii-Moria interaction, DMI) and ii) exchange striction 17 (magnetoelastic coupling to the lattice). In the first case antisymmetric DMI activated by noncollinear spin ordering breaks inversion symmetry and induces electric polarization through the concomitant lattice and electronic distortion. Some typical examples of the compounds which show DMI are listed in Table I. The second mechanism may play the dominant role in the antiferromagnets with superexchange interactions between the magnetic ions mediated by bridging via nonmagnetic anions, like DyMn 2 O 5 and YMn 2 O 5 , see Table I. In this case the value of exchange integral strongly depends upon an angle between anion-cation bonds. In the crystals with the competing antiferromagnetic (AFM) interactions even small bond-bending may reduce the exchange energy, stabilizes long-range AFM ordering and produce nonzero electric polarization. In the present paper we predict multiferroicity caused by magnetoelastic mechanism in the epitaxially grown MnTe/ZnTe films and heterostructures that belong to the family of II-VI semiconductors 18 well suited to many optoelectronic applications in the infrared and visible range 19 . Our hypothesis is based on the following facts. We argue that the value of spontaneous ferroelectric polarization in MnTe can be as large as 60 nC/cm 2 and thus is comparable with polarization of many other antiferromagnets (see Table I). Moreover, due to the direct coupling between magnetic and ferroelectric ordering, one should expect strong magnetoelectric effects (i.e., induction of polarization by a magnetic field) in this material. Magnetoelastic mechanism is responsible also for the nonlinear optical effects and opens a possibility to visualize antiferromagnetic (and corresponding ferroelectric) domains by use of second harmonic generation (SHG) technique (see, e.g, Refs. 26,27). The paper is organized as follows. In Sec.II we describe crystal and magnetic structure of MnTe/ZnTe films and in Sec.III give some intuitive reasons for appearance of spontaneous electric polarization below the Néel point. In Sec.IV we calculate the value of spontaneous polarization on the basis of phenomenological model. Sec.V is devoted to the discussion of AFM and ferroelectric domains and methods of the domain structure control. In Sec.VI we discuss possible magnetooptical effects that could be helpful in visualization of the magnetic and ferroelectric structure of MnTe. In the last Sec. the summary of the results obtained is given. II. STRUCTURE AND MAGNETIC PROPERTIES OF MNTE Bulk-like films of paramagnetic MnTe grown by MBE technique have zinc-blend structure 35 shown in Fig. 1 (symmetry group is F43m). The d − d exchange interaction between the nearest localized Mn 2+ spins (S = 5/2) is accomplished via Mn-Te-Mn bonds 21 and thus is antiferromagnetic. In the strain-free fcc lattice AFM exchange between the nearest neighbors (NN) is frustrated. Nevertheless, neutron diffraction experiments 23,35 reveal appearance of the long-range AFM type-III structure below the Néel temperature (T N = 65 K). As shown in Fig. 2, this spin arrangement consists of AFM sheets parallel to (001) plane (represented in the plot by the shaded parallelograms). Stabilization of long-range ordering is attributed to the influence of weak NNN coupling 23 and to misfit-induced strain of the magnetic layer 36 which in the case of MnTe/ZnTe corresponds to elongation 23 (c/a = 1.06) in the direction of film growth. It should be noted that in the described AFM-III structure the coupling energy between the spins in the adjacent (001) sheets sums to zero, so, mutual rotation of spins in different layers does not change the energy of exchange interactions 48 . Stabilization of the collinear AFM-III structure can be due to the pronounced magnetostriction 23 [001] It should be also mentioned that the symmetry of AFM state (corresponding group is generated by the rotation 2 [001] and translation [ 1 2 , 1 2 ,0] both combined with time inversion 1 ′ ) allows the existence of macroscopic electric polarization vector oriented along the [001] axis and forbids existence of macrosopic magnetization. The role of magnetoelastic coupling in the formation of AFM-III structure can be traced from the following qualitative considerations. It was already mentioned that the value of Mn-Te-Mn angle φ is the key quantity in determining the exchange coupling constant J d−d between the NN Mn 2+ ions: upon decreasing the φ the strength of AFM interaction increases. For small deflection from an ideal φ 0 = 109.5 • angle peculiar to fcc lattice, this dependence can be approximated as follows (from the results of Bruno and Lascaray 37 ) J d−d (φ) = −3.45 + 0.135(φ − φ 0 ), K.(1) Now, let us turn to Fig. 3 which illustrates the effect of tetragonal strain induced by the presence of ZnTe layers. In a nondeformed cubic lattice all the bonds make an "ideal" angle φ 0 and all NN interactions are equivalent. Tetragonal distortion in [001] direction removes degeneracy between in-plane (001) and interplane NN exchange. In the case of elongation the angles for pairs 12, 34 are smaller than those for the other pairs: φ 12 = φ 34 ≡ φ in < φ 13 = φ 14 = φ 23 = φ 24 ≡ φ out (2) Typical values for in-plane, φ in , and interplane, φ out , angles calculated from geometrical considerations for some MnTe/ZnTe superlattices are listed in Table II. In the last two columns of the Table we give the values of the exchange constants calculated from Eq.(1). It can be easily seen that NN exchange interaction favors AFM ordering for the atoms within (001) plane. At the same time, the bonds between the atoms in neighboring planes are still equivalent and so are frustrated. Further deformation within (001) plane (see Fig. 4 a, b) leads to optimization of the exchange in the interplane FM and AFM bonds by strengthening exchange interactions with the "right" sign and weakening those with the the following difference between the exchange constants J d−d (φ 14 ) − J d−d (φ 13 ) = 0.27a xy a 2 xy + c 2 (u xx − u yy ), K,(3) where a xy and c are in-plane and interplane lattice parameters, correspondingly. Thus, magnetostriction can stabilize an AFM-III structure even in approximation of NN exchange interactions. On the other hand, magnetostriction plays an important role in formation of ferroelectric ordering, as we will show below. In the case of strong spin-orbit coupling some components of magnetostrictive tensor could depend upon the mutual orientation of a localized magnetic moment and a direction of anion-cation bond. This situation in application to MnTe is illustrated in Figs. 4c, 5. Shear strain u xy in (001) plane makes AFM bonds between 1-2 and 3-4 pair inequivalent: φ 12 − φ 34 = √ 2a xy c 3c 2 − 2a 2 xy 2a 2 xy + c 2 u xy ≈ 0.6u xy . (4) The displacement ∆ℓ between the positive (Mn 2+ ) and negative (Te 2− ) ions is thereby equal to ∆ℓ = a xy 9 √ 2 u xy .(5) For the totally ionic bond (with the effective charge 2e) and spontaneous strain u xy = 3 · 10 −3 such a displacement induces a local dipole moment ∝ 0.12 D. Accurate calculations of ferroelectric polarization induced by spontaneous magnetostriction will be given in the next Sec. IV. at Mn sites: In a single-domain sample with the fixed magnetic order parameter (or AFM vector) L the equilibrium values of magnetostriction tensor, u, and electrical polarization, P, can be calculated by minimization of the following expression for the free energy P P (a) (b) (c) (d)M(r j ) = L(q) exp(iqr j ).(6)F = dV {[λ ′ (u xx − u yy ) + 2λ 16 u xy ](L 2 x − L 2 y )θ(z) + 1 2 u ij c ijkl u kl + P 2 2ǫ 0 κ + 2e 14 (z) ǫ 0 κ (P x u yz + P y u zx + P z u xy )} (7) constructed from symmetry considerations. Parameters λ are magnetoelastic constants related to the shear strain, c is a tensor of the elastic modula characteristic to cubic structure. Electrical properties are described by low-frequency dielectric constant ǫ pm = κ + 1 and piezoelectric coefficient e 14 , ǫ 0 is the vacuum permittivity. Periodical alteration of the AFM (MnTe) and nonmagnetic (ZnTe) layers in a heterostructure is described by a formfunction θ(z) which equals 1 in AFM layer and vanishes in a nonmagnetic spacer. In the case of ideal interfaces the strain induced by AFM ordering is homogeneous throughout the AFM and nonmagnetic layers and thus depends on the ratio of AFM layer thickness d to the period of superstructure D. In particular, macroscopic shear strain component u xy which is responsible for polarization effect is given by the following expression u (spon) xy = − λ 16 M 2 0 ρ 2c 44 d D ,(8) where ρ = ±1 distinguish between two orthogonal orientations of L For the multilayered structure the piezoelectric coefficient e 14 (z) should be averaged over the AFM/NM bilayer. It can be easily seen from the Expr. (9) that the direction of vector P correlates with that of AFM vector L (factor ρ) and takes an opposite direction in the domains with L [100] and [010], in accordance with intuitive predictions given above (see Fig. 5). So, coupling between the ferroelectric and antiferromagnetic order is accomplished by the rigid alignment of the antiferromagnetic axis perpendicular to the polarization direction. Characteristic value of the spontaneous ferroelectric polarization in a single-domain state can be evaluated from (9) and available experimental data (see Table VII) as P (spon) = 60 nC/cm 2 . For the thick MnTe film P (spon) can be even higher because Mn atoms cause an enhancement of electromechanical coupling in zinc-blende structures. 39 It is remarkable that among the materials with comparable ferroelectric polarization (see Table I) MnTe shows the highest Néel temperature. The value of the internal electric field E (spon) ∼ 80 kV/cm corresponding to the spontaneous polarization of MnTe is, in turn, close to strain-generated electric field in non-magnetic III-V heterostructures. 40 It seems to be large enough to be detected due to the change in the photoluminescence spectra, like it was done in the Ref. 20. V. SWITCHING OF POLARIZATION BY EXTERNAL MAGNETIC/ELECTRIC FIELD Practical applications of multiferroics in information technology are based on their ability to maintain single ferroelectric domain state for a long time and change it under application of the external field. This can be achieved, for example, by using materials with a high ferroelectric Curie temperature and a robust large polarization (say, RMnO 3 , R=Sc, Y, In, Ho-Lu, see Refs. 41,42) that, in turn, gives rise to a large value of switching field. In the systems with strong coupling between polarization and AFM order a single domain state can be easily fixed by a proper choice of the sample shape, even for small polarization. Moreover, if AFM and ferroelectric domains are intimately related and match up spatially (as, e.g., should be in MnTe), electric polarization of the sample can be switched by either magnetic or electric field. In this section we analyze field dependence of macroscopic polarization for different sample shapes. Existence of equilibrium domain structure in the ferroics is usually attributed to the presence of long-range dipole-dipole interactions between the physically small ordered regions of the sample. Quantitative description of the phenomena is based on the account of shapedependent contribution (stray energy) into free energy of the sample. In the case of ferroelectric (or ferromagnetic) materials this contribution can be written as Φ f−e = V 2 P j N jk P k ,(10) where brackets . . . mean averaging over the sample volume V and N is the second-rank depolarization tensor the components of which depend upon the shape of the sample. Equilibrium domain structure in AFM with nonzero magnetoelastic coupling can be described in a similar way. Long-range (destressing) effects originate from the local internal stress fields σ (mag) jk induced by magnetic ordering. 43,44 Corresponding contribution into free energy of the sample takes a form Φ destr = V 2 σ (mag) jp ℵ jkpt σ (mag) kt ,(11) where ℵ jkpt is the forth-rank destressing tensor which, like N, depends upon the sample shape. In the most practical applications 50 macroscopic properties (such as polarization or elongation) of the multidomain sample depend on the single parameter 0 ≤ α ≤ 1 that can be calculated from minimization of shapedependent (Φ f−e + Φ destr ) and field-dependent contributions into free energy: Φ = V S 1 − E z P (spon) z − χd 2D H 2 y − H 2 x α − 1 2 + 1 2 S 2 α − 1 2 2 .(12) Here spontaneous polarization P (spon) z within a domain is defined by the expression (9), χ is the magnetic susceptibility of MnTe and S 1,2 are shape-dependent coefficients 51 : S 1 = λ ′ M 2 0 σ (mf) (ℵ 11 − ℵ 22 ), S 2 = 4N 3 P (spon) z 2(13)+ M 4 0 [(λ ′ ) 2 (ℵ 11 + ℵ 22 − 2ℵ 12 ) + 16λ 2 16 ℵ 66 ], where σ (mf) ≡ σ (mf) xx = σ (mf) yy are isotropic stresses induced in the film plane due to the lattice mismatch. Analysis of Eq. (12) shows that a single domain state of a sample (α = 1 or 1) can be achieved either by the proper choice of the shape (2S 1 ≥ S 2 ) or application of the external field (E ≥ E MD or H ≥ H MD ) where we've introduced characteristic monodomainization fields as follows: E MD = S 2 2P (spon) z , H MD = S 2 D χd .(14) In the multidomain state macroscopic ferroelectric polarization depends upon the external fields in a following way P z = P (spon) z 2S 1 S 2 − E z E MD − H 2 y − H 2 x H 2 MD .(15) Let us analyze the properties of the coefficients S 1,2 assuming that i) the sample has the shape of an ellipse (with semiaxes a, b and eccentricity k = 1 − b 2 /a 2 ) within the film plane, film thickness is h; ii) elastic properties of the material are isotropic (c 11 − c 12 = 2c 44 ) and are characterized with the shear modulus c 44 and Poisson ratio ν ≡ c 12 /(c 11 + c 12 ). The destressing and depolarizing coefficients can be calculated explicitly in two limiting cases: i) "pillar" with h ≫ a > b: S 1 = σ (mf) dλ ′ M 2 0 2c 44 D k 2 (1 + √ 1 − k 2 ) 2 , S 2 = M 4 0 d 2 2c 44 D 2 (λ ′ ) 2 + 4λ 2 16 + 4λ 2 16 − (λ ′ ) 2 1 − ν k 4 (1 + √ 1 − k 2 ) 4 ;(16) ii) "thin film" with h ≪ b < a: S 1 = h b · σ (mf) λ ′ M 2 0 d 4c 44 (1 − ν)D J 2 (k), S 2 = 4 ǫ 0 κ P (spon) z 2 ,(17) where the dimensionless shape-factor is given by an integral J 2 (k) = π/2 0 (sin 2 φ + cos 2φ/k 2 )dφ 1 − k 2 sin 2 φ → 3πk 2 /16, k → 0 1, k → 1.(18) It is remarkable that coefficient S 2 that favors formation of the domain structure is nonzero in both limiting cases, as can be seen from (13), (16), (17). This fact can be easily extended to any geometry. Really, all the depolarizing effects (stray fields) are related with the flux of the corresponding (ferroelectric, magnetoelastic, etc) dipole moment through the sample surface. In our case the flux of ferroelectric polarization is nonzero only through the surface parallel to the film plane, thus "thin film" shows strong depolarizing ferroelectric effect while "pillar" shows none. In turn, AFM ordering produces stress dipoles within the film plane, so, corresponding flux is maximal through the side faces. As a result, strong destressing effect should be observed for the "pillar", not for the "thin film". For the intermediate case of the sample with the developed side and face surfaces ("ball" or "cube") coefficient S 2 is contributed by both depolarizing mechanisms and so is nonvanishing for any shape. Calculations based on the formulas (16), (17) and experimental data from Table VII show that in the case of MnTe multiferroic the destressing effects are much stronger (S 2 ≈ 10 5 J/m 3 ) that the electric depolarization (S 2 ≈ 5 · 10 3 J/m 3 ), so, monodomainization can be easier achieved in the "thin film". Another interesting feature of the destressing phenomena is existence of the effective internal shape-induced field described by coefficient S 1 . It arises due to crosscorrelation between isotropic (induced either by substrate or by magnetovolume effect) and anisotropic internal stresses and has no analog in ferromagnetic or ferroelectric materials. Coefficient S 1 , as seen from (16), (17), (18), depends upon the eccentricity k and vanishes for the samples isotropic within the film plane (a = b). So, we can deduce that the shape of the sample influences the domain structure in two ways. First, the field of monodomainization (14) strongly depends upon the h/b ratio. Characteristic values of the electric and magnetic monodomainization fields for two limiting cases (h ≫ b and h ≪ b) are given in Table VII. The value of H MD for the thick sample ("pillar") is close to the value of anisotropy field 3.25 T (see Ref.45) and diminishes down to 0.8 T for 'thin film". Second, single domain state is energetically favorable for the samples with the overcritical eccentricity (calculated from the condition 2S 1 ≥ S 2 ). For example, for the thick sample the critical eccentricity is 0.57 (corresponding aspect ratio a/b = 1.22). Shape effects are clearly seen in Fig. 6 which shows field dependence of macroscopic ferroelectric polarization for the "thin film" with h/b = 0.1 and for the "pillar", both samples having the same in-plane eccentricity 0.1 (corresponding aspect ratio a/b = 1.005) In the first case monodomainization field is smaller and biasing effect is more pronounced. Monodomainization of the sample can be also achieved by the combined application of the electric and magnetic field, as illustrated in Fig. 7 for the case of "thin film" with zero eccentricity. Relation between the switching fields in this case is given by the formula Figure 6: (Color online) Macroscopic ferroelectric polarization as a function of external magnetic field calculated according to (15) for "thin film" with h/b = 0.1(curve A, magenta) and "pillar" (curve B, dark cyan). The eccentricity in both cases is k = 0.1. Magnetic field H is switched between x and y directions. Dotted curve (magenta) shows the possible field switching between two opposite polarizations in a single domain sample. E z E MD + H 2 y − H 2 x H 2 MD = 1.(19) VI. VISUALIZATION OF THE DOMAIN STRUCTURE The zinc-blend wide-gap semiconductors are also known to show nonlinear optical properties studied in second harmonic generation (SHG) experiments. 46 In the paramagnetic phase these crystals have the only nonzero SHG tensor component d 14 (−2ω, ω, ω) = d 25 = d 36 which is rather large, e.g. for ZnTe it is 119 pm/V at the fundamental wavelength 1047 nm. 46 AFM ordering may bring to being additional SHG components. To calculate them we analyze the response to the external electric field with account of the nonlinear contribution into free energy F (nonlin) = − 2d 14 ǫ 2 0 κ 3 P x P y P z + η ′ 2ǫ 0 κ (P 2 x − P 2 y )(u xx − u yy ) + 2η 44 ǫ 0 κ (P x P y u xy + P y P z u yz + P z P x u zx ) + θ(z)[ξ 1 (M 2 x + M 2 y )P z (P 2 x − P 2 y ) + ξ 2 (M 2 x − M 2 y )P 3 z ].(20) Coefficients η ′ , η 44 introduced in (20) describe electrostrictive effect and the last term describes high-order magneto-electric coupling. Calculations show that in the AFM phase the crystal becomes biaxial with anisotropic dielectric tensor: ǫ xx − ǫ zz = ǫ zz − ǫ yy = η′λ′ c′ M 2 0 ρ(ǫ pm − 1), ǫ xy = η 44 λ 16 c 44 M 2 0 ρ(ǫ pm − 1),(21) in accordance to symmetry prediction for C 2 point group of AFM phase. The SHG tensor components proportional to AFM vector are of two types. Like dielectric coefficients, the components d 31 + d 32 = 4η 44 d 14 λ 16 c 44 M 2 0 ρ = 2u (spon) xy η 44 d 14 ρ,(22) have opposite signs for the domains with perpendicular orientation of AFM vectors (ρ = ±1). The other coefficients are insensitive to the domain structure: d 24 = −d 15 = 1 2 (d 32 − d 31 ) = M 2 0 ξ 1 ǫ 2 0 (ǫ pm (ω) − 1), d 33 = −M 2 0 ξ 2 ǫ 2 0 (ǫ pm (ω) − 1).(23) The symmetry predicted difference between the d 14 , d 25 and d 36 components is due to the high-order terms in the magnetic order parameter and is neglected. Rotation of the polarization plane which may stem from anisotropy of dielectric tensor and effect of SHG are sensitive to the direction of AFM vector and thus open a possibility to visualize the AFM domain pattern. 26 VII. SUMMARY AND CONCLUSIONS In the present paper we demonstrate for the first time that MnTe epitaxial films and MnTe/ZnTe multilayers grown in a convenient (001)-direction should become electrically polarized below the Néel temperature. Spontaneous ferroelectric polarization results from the bending of highly ionic Mn-Te-Mn bonds induced by magnetostrictive shear strain. In contrast to many other multiferroics, MnTe has no macroscopic magnetic moment and the direction of ferroelectric polarization is coupled with the orientation of AFM order parameter. Thus, the domain structure is formed by the combining action of the electric and magnetoelastic dipole-dipole interactions. Corresponding depolarizing long-range contribution into free energy of the sample is nonzero for any sample shape. In the case of thick film depolarizing effects are governed mainly by magnetoelastic mechanism and corresponding monodomainization field is of the same order as characteristic anisotropy field. In the case of thin film depolarization is due to the presence of the electric dipole interactions and corresponding monodomainization field is much smaller. Due to the clamping between the electrical and AFM domains switching between the opposite direction of ferroelectric polarization can be induced by application of the magnetic field parallel to "easy" AFM axis. Vice versa, one can switch between different (100-oriented and [010]-oriented AFM domains) by application of the electric field. This effect seems to be useful for controlling the state of the adjacent ferromagnetic layer coupled to the multiferroic through exchange interactions like it was proposed in Ref. 9. AFM ordering may also induce additional (anisotropy) components of dielectric and SHG tensors. The sign of the effect is sensitive to orientation of AFM vectors and thus opens a possibility to distinguish different domain types. The spontaneous ferroelectric effect is peculiar to the collinear AFM-III ordering and is not allowed (from symmetry point of view) in the canted AFM-III structure (e.g., such an effect is impossible in MnTe/CdTe multilayers). Figure 1 : 1(Color online) Structure of zinc-blend MnTe. Mn -magenta (large) spheres, Te -blue (small) spheres. Inset (small cube) shows orientation of Mn-Te bonds. Figure 2 : 2(Color online) Collinear AFM-III structure in the tetragonally distorted fcc spin lattice. Film growth direction is parallel to [001] crystallographic axis. Mn 2+ ions with opposite spin direction are shown in different colours (opposite arrows). Figure 3 :Figure 4 : 34(Color online) Effect of the mismatch-induced stresses. Blue (1, 3) and magenta (2, 4) spheres correspond to Mn atoms with opposite directions of magnetization. (a) Nondeformed (cubic) cell, atoms in 1-4 positions are equivalent and angles between all the Mn-Te bonds are the same and equal 109.5 • . (b) Elongation in [001] direction removes degeneracy between angles. "wrong" sign. Namely, elongation in [100] direction (corresponding strain component u xx − u yy > 0) results in (Color online) Magnetostriction in (001) plane. Pairs of antiferromagnetically coupled Mn atoms 1, 2 (hollow) and 3,4 (filled) belong to different atomic planes.(a) Nondeformed state, angles between Mn-Te-Mn bonds take only two different values: φin (pairs 12 and 34) and φout (pairs 13, 14, 23 and 24). (b) Elongation in [100] direction removes degeneracy between the angles φ13 = φ24 > φout > φ23 = φ14. (c) Shear strain uxy removes degeneracy between the angles φ12 and φ34. Figure 5 : 5(Color online) Two types of AFM (and ferroelectric) domains. (a), (c) Mutual orientation of the sublattice magnetization (vectors Mj , j = 1, 4) and shift vector (uj , j = 1, 4) of Mn atom. (b), (d) Arrangement of atoms after the shear strain. Nonequivalence of 1-2 and 3-4 bonds results in appearance of ferroelectric polarization P. Fig. 5 5reveals one interesting feature of the tetragonally distorted MnTe system: AFM structure can be implemented in two types of domains that have different (perpendicular) orientation of local magnetic moments (with respect to crystal axes) and opposite direction of electric polarization. AFM domains are clamped with ferroelectric ones. This means that macroscopic state of a sample can be controlled by application of either magnetic or electric field (or both). Behavior of the domain structure in the presence of the external fields will be discussed in Sec. V.IV. POLARIZATION INDUCED BY AFM ORDERING: MODEL AND CALCULATIONSFormally, AFM type-III structure can be described by the following distribution of magnetization vectors M(r j ) Three equivalent orientations of the structure vector q = (2π/a, 0, π/a) (where a is a lattice constant), and two orientations of L vector (parallel to [100] and [010] axes) generate six types of AFM domains in MnTe films. Misfit strains in the MnTe/ZnTe multilayers reduce de-generacy to two possible domain types with the same q and mutually perpendicular L vectors in the film plane.38 [100] and[010] in different domains, M 0 = \|L\| is sublattice magnetization.Strain-induced contribution 49 into spontaneous polarization calculated from(7), (8) is P (spon) z = −2e 14 u (spon) xy = e 14 λ 16 M 2 0 ρ c 44 d D . It was already mentioned that the domain structure of MnTe/ZnTe multilayers is formed by two (out of six) types of the domains with (i) L [100], P [001], volume fraction α, and (ii) L [010], P [001], volume fraction (1− α). Switching between this two types can be performed by application of the external electric field E parallel to [001] axis or by the magnetic field H directed along one of the "easy" AFM axes ([100] or [010]). Figure 7 : 7(Color online) Phase diagram in E −H variables for the "thin film" (h/b = 0.1) with zero eccentricity. Magnetic field H is switched between x and y directions. 1 . 1Epitaxial layers of MnTe, ZnTe, CdTe and corre-sponding heterostructures posses zinc-blend (ZB) structure consistent with piezoelectric activity. In particular, isomorphous to MnTe nonmagnetic films of ZnTe, CdTe, ZnTe/CdTe(111) can produce a macroscopic electric polarization when stressed or strained 20 . 2. Predominant mechanism of the magnetic interac- tions between Mn 2+ ions is superexchange via Te anions 18,21,22 that favours antiferromagnetic order- ing and noticeably varies with Mn-Te-Mn bond bending. 3. Magnetoelastic coupling in MnTe is rather strong as can be deduced from step-wise variation of lat- tice parameters at the Néel temperature. 23,24 4. MnTe is a wide-gap (band gap is 3.2 eV, as re- ported in Ref. 18) semiconductor with vanish- ingly small concentration of the mobile carriers at low temperatures. 25 Thus, ferroelectric polariza- tion should not be seriously affected by screening by the mobile charge. In MnTe/ZnTe heterostructures such a screening can be avoided by a proper choice of the superlattice parameters and/or by doping. Table I : IValues of spontaneous electric polarization, P (spon) , Néel temperature, TN , type of AFM ordering (MO), microscopic mechanism and type (according to classification Ref.10) of ferroelectricity for some multiferroics (MF).Compound P (spon) (nC/cm 2 ) TN (K) Type of MO Mechanism Type of MF Source BiFeO3 12·10 4 673 cycloid octahedra rot. I Ref.9 BiSrFeO3 10·10 4 643 noncollinear DMI I Ref.28 DyMn2O5 150 39 noncollinear ME II Ref.29,30 YMn2O5 100 45 collinear ME II Ref.17 MnTe (ZB) 60 65 collinear ME II this work TbMnO3 50 27 spiral DMI II Ref.31 TbMn2O5 40 37 noncollinear DMI II Ref.8 Ni3V2O8 12.5 6.3 noncollinear DMI II Ref.32,33 MnWO4 4 12.7 noncollinear DMI II Ref.7 Ca3CoMnO6 - 13 collinear, Ising ME II Ref.34 III. MULTIFERROICITY INDUCED BY MAGNETOSTRICTION: INTUITIVE CONSIDERATIONS Table II : IIAngles between Mn-Te-Mn bonds for in-plane NN (φin) and interplane NN (φout) and corresponding values of exchange integrals J d−d (φin), J d−d (φout) (K) calculated from geometrical considerations for different superlattices. In-plane, axy, and interplane, c, lattice parameters (Å) are taken from the experimental paper Ref.23.Type axy c φin φout J d−d (φin) J d−d (φout) MnTe10/ZnTe18 6.130 6.570 105.7 111.4 -3.20 -3.97 MnTe20/ZnTe18 6.183 6.505 106.7 110.9 -3.27 -3.84 MnTe130/ZnTe330 6.150 6.470 106.7 110.9 -3.28 -3.83 MnTe 6.346 6.346 109.5 109.5 -3.45 -3.45 Table III : IIIValues of constants for MnTe (ZB) used in calculations and calculated (see text for notations).Constant Value Source/formula Rem. 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[ "What can nuclear collisions teach us about the boiling of water or the formation of multi-star systems ? a", "What can nuclear collisions teach us about the boiling of water or the formation of multi-star systems ? a" ]	[ "D H E Gross \nBereich Theoretische Physik\nHahn-Meitner-Institut Berlin\nGlienickerstr.10014109BerlinGermany\n\nFachbereich Physik\nFreie Universität Berlin\nBolognaItalia\n" ]	[ "Bereich Theoretische Physik\nHahn-Meitner-Institut Berlin\nGlienickerstr.10014109BerlinGermany", "Fachbereich Physik\nFreie Universität Berlin\nBolognaItalia" ]	[]	Phase transitions in nuclei, small atomic clusters and self-gravitating systems demand the extension of thermo-statistics to "Small" systems. The main obstacle is the thermodynamic limit. It is shown how the original definition of the entropy by Boltzmann as the volume of the energy-manifold of the N-body phase space allows a geometrical definition of the entropy as function of the conserved quantities. Without invoking the thermodynamic limit the whole "zoo" of phase transitions and critical points/lines can be unambiguously defined. The relation to the Yang-Lee singularities of the grand-canonical partition sum is pointed out. It is shown that just phase transitions in non-extensive systems give the complete set of characteristic parameters of the transition including the surface tension. Nuclear heavy-ion collisions are an experimental playground to explore this extension of thermo-statisticsThermodynamicists will certainly answer our question by: "Nothing" and yet:	10.1142/9789812810939_0002	[ "https://arxiv.org/pdf/cond-mat/0006203v1.pdf" ]	119,513,958	cond-mat/0006203	0f1d9ef9aa15cb2b2c255b93096943308fc5e012	What can nuclear collisions teach us about the boiling of water or the formation of multi-star systems ? a 13 Jun 2000 May 29 -June 3, 2000 D H E Gross Bereich Theoretische Physik Hahn-Meitner-Institut Berlin Glienickerstr.10014109BerlinGermany Fachbereich Physik Freie Universität Berlin BolognaItalia What can nuclear collisions teach us about the boiling of water or the formation of multi-star systems ? a 13 Jun 2000 May 29 -June 3, 2000b Dividing extensive systems into larger pieces, the total energy and entropy are equal to the sum of those of the pieces. Phase transitions in nuclei, small atomic clusters and self-gravitating systems demand the extension of thermo-statistics to "Small" systems. The main obstacle is the thermodynamic limit. It is shown how the original definition of the entropy by Boltzmann as the volume of the energy-manifold of the N-body phase space allows a geometrical definition of the entropy as function of the conserved quantities. Without invoking the thermodynamic limit the whole "zoo" of phase transitions and critical points/lines can be unambiguously defined. The relation to the Yang-Lee singularities of the grand-canonical partition sum is pointed out. It is shown that just phase transitions in non-extensive systems give the complete set of characteristic parameters of the transition including the surface tension. Nuclear heavy-ion collisions are an experimental playground to explore this extension of thermo-statisticsThermodynamicists will certainly answer our question by: "Nothing" and yet: 1 There is a lot to add to classical equilibrium statistics from our experience with "Small" systems: Following Lieb 1 extensivity b and the existence of the thermodynamic limit N → ∞\| N/V =const are essential conditions for conventional (canonical) thermodynamics to apply. Certainly, this implies also the homogeneity of the system. Phase transitions are somehow foreign to this: The essence of first order transitions is that the systems become inhomogeneous and split into different phases separated by interfaces. In general are phase transitions consequently represented by singularities in the grand-canonical partition sum (Yang-Lee singularities). In the following we show that the micro-canonical ensemble gives much more detailed insight. There is a whole group of physical many-body systems called "Small" in the following which cannot be addressed by conventional thermo-statistics: If the range of the force or the thickness of the surface layers is such that the number of surface particles is not negligible compared to the total number of particles, these systems are non-extensive. It is common to all these examples, that the systems are non-extensive. For such systems the thermodynamic limit does not exist or makes no sense. Either the range of the forces (Coulomb, gravitation) are of the order of the linear dimensions of these systems, and/or they are strongly inhomogeneous e.g. at phase-separation. Before inventing any new type of non-extensive thermodynamics or entropy like the one introduced by Tsallis 2 we should realize that Boltzmann's definition: S = k * lnW with W (E, N, V ) = ǫ 0 trδ(E − H N ) (1) trδ(E − H N ) = d 3N p d 3N q N !(2πh) 3N δ(E − H N ). (ǫ 0 a suitable small energy constant) does not invoke the thermodynamic limit, nor additivity, nor extensivity, nor concavity of the entropy S(E, N ) (downwards bending). This was largely forgotten since hundred years. We have to go back to pre Gibbsian times. It is a purely geometrical definition of the entropy and applies as well to "Small" systems. Moreover, the entropy S(E, N ) as defined above is everywhere single-valued and multiple differentiable. There are no singularities in it. This is the most simple access to equilibrium statistics. We will explore its consequences in this contribution. Moreover, we will see that this way we get simultaneously the complete information about the three crucial parameters characterizing a phase transition of first order: transition temperature T tr , latent heat per atom q lat and surface tension σ surf . Boltzmann's famous epitaph above contains everything what can be said about equilibrium thermodynamics in its most condensed form. W is the volume of the sub-manifold at sharp energy in the 6N -dim. phase space. 2 Relation of the topology of S(E, N ) to the Yang-Lee singularities In conventional thermo-statistics phase transitions are indicated by singularities of the grand-canonical partition function Z(T, µ, V ), V is the volume. See more details in 3,4,5,6 Z(T, µ, V ) = ∞ 0 dE ǫ 0 dN e −[E−µN −T S(E)]/T = V 2 ǫ 0 ∞ 0 de dn e −V [e−µn−T s(e,n)]/T .(2) ≈ e const.+lin.+quadr. in the thermodynamic limit V → ∞\| N/V =const . The double Laplace integral (2) can be evaluated asymptotically for large V by expanding the exponent as indicated in the last line to second order in ∆e, ∆n around the "stationary point" e s , n s where the linear term vanish: 1 T = ∂S ∂E s µ T = − ∂S ∂N s P T = ∂S ∂V s(3) the only term remaining to be integrated is the quadratic one. If the two eigen-curvatures λ 1 < 0, λ 2 < 0 this is then a Gaussian integral and yields: Z(T, µ, V ) = V 2 ǫ 0 e −V [es−µns−T s(es,ns)]/T ∞ 0 dv 1 dv 2 e V [λ1v 2 1 +λ2v 2 2 ]/2 (4) Z(T, µ, V ) = e −F (T,µ,V ) (5) F (T, µ, V ) V → e s − µn s − T s s + T ln ( det(e s , n s )) V + o( ln V V ).(6) Here det(e s , n s ) is the determinant of the curvatures of s(e, n). det(e, n) = ∂ 2 s ∂e 2 ∂ 2 s ∂n∂e ∂ 2 s ∂e∂n ∂ 2 s ∂n 2 = s ee s en s ne s nn = λ 1 λ 2 , λ 1 ≥ λ 2(7) In the cases studied here λ 2 < 0 but λ 1 can be positive or negative. If det(e s , n s ) is positive (λ 1 < 0) the last two terms in eq.(6) go to 0, and we If this is the case, Z(T, µ, V ) is analytical and due to Yang and Lee we have a single, stable phase. Or otherwise, the Yang-Lee singularities reflect anomalous points/regions of λ 1 ≥ 0 (det(e, n) ≤ 0). This is crucial. As det(e s , n s ) can be studied for finite or even small systems as well, this is the only proper extension of phase transitions to "Small" systems. 3 The physical origin of the wrong (positive) curvature λ 1 of s(e s , n s ) Before we proceed with the general micro-canonical classification of the various types of phase transitions we will now discuss the physical origin of convex (upwards bending) intruders in the entropy surface. As nuclear matter is not accessible to us we should investigate atomic clusters and compare their thermodynamic behavior with that of the known bulk. In the figure (1) we compare the "liquid-gas" phase transition in sodium clusters of a few hundred atoms with that of the bulk at 1 atm.. In the table (1) we show in comparison with the known bulk values the four important parameters, transition temperature T tr , latent heat per atom q lat , the entropy gain for the evaporation of one atom s boil as proposed by the empirical Trouton's rule (∼ 10) and ∆s surf , the surface entropy per atom as defined above. N Now we can give a systematic and generic classification of phase transitions which applies also to "Small" systems and their relation to the Yang-Lee singularities: • A single stable phase by d(e, n) > 0 (λ 1 < 0). Here s(e, n) is concave (downwards bended) in both directions. Then there is a one to one mapping of the grand-canonical ↔the micro-ensemble. In the two examples on the right the order parameter, the direction v 1 of the eigenvector of largest curvature λ 1 was simply assumed to be the energy. • A transition of first order with phase separation and surface tension is indicated by d(e, n) < 0 (λ 1 > 0). s(e, n) has a convex intruder (upwards bended) in the direction v 1 of the largest curvature. The whole convex area of {e,n} is mapped into a single point in the canonical ensemble.. I.e. if the curvature of S(E, N ) is λ 1 ≥ 0 both ensembles are not equivalent. • A continuous ("second order") transition with vanishing surface tension, where two neighboring phases become indistinguishable. This is at points where the two stationary points move into one another. I.e. where d(e, n) = 0 and v λ=0 · ∇d = 0. These are the catastrophes of the Laplace transform E → T . Here v λ=0 is the eigenvector of d belonging to the largest curvature eigenvalue λ = 0 (→ order parameter). • Finally, a multi-critical point where more than two phases become indistinguishable is at the branching of several lines in the {e, n}-phasediagram with d = 0, ∇d = 0. An example showing all these possible types of phase transitions in a small system is discussed in 3,4,5 On the statistical formation of multi-star systems under rotation Having discussed the micro-canonical description of phase transitions in "Small" systems we sketch here the relevance of our new formulation of thermo-statistics for astrophysical systems. A finite self-gravitating system is controlled by its accessible phase space and its topology: • Due to the long-range gravity the system is non-extensive. • Because there is no heat-, no angular momentum bath a microcanonical treatment under the observation of the conservation laws is neccessary. • Attractive gravity leads to a collaps (phase transition of first order). • Under rotation the collapsed system may multi-fragment and multi-star systems will be formed. • Also the fragmentation of Shoemaker-Levy 9 may be viewed as statistical multifragmentation under the linear stress of Jupiter's gravitation field. With the analogy to the long-range Coulomb force one may study many aspects by the study of collision of heavy nuclei. E.g.: • At higher rotation the system prefers larger moment of inertia. This leads to more symmetric heaviest fragments. For the astro-problem higher angular momentum leads to double or multiple stars instead of a mono star. • For heavy-ion collisions, rotation leads to higher radial energy of fragments, larger variance in the statistical fragmentation which must be distinguished from from radial flow ! Conclusion Instead of using the boiling of water we used the boiling of sodium to demonstrate the physics of first order phase transitions and especially the surface tension. "Small" systems show clearly how to determine all important characteristica of first order transitions: transition temperature, latent heat, and surface tension. This is possible just because these systems are not extensive and their entropy s(e, n) has convex intruders from which the surface entropy can be determined. In the thermodynamic limit this fact becomes more obscured than illuminated. We also mentioned collapsing of self-gravitating and rotating hydrogen clouds towards multi-star systems. As far as the structure of the phase space is concerned there is a striking analogy to the phase space of fragmenting hot nuclei under rotation. The experimental study of the latter is also interesting in view of this analogy. One of the main differences is of course that the latter are experimentally accessible. Another important question is whether there is a kind of statistical equilibrium established in the astrophysical systems and if what the equilibration mechanism might be. Here our learning process has just started. a Invited talk Bologna 2000 -Structure of the Nucleus at the Dawn of the Century, Bologna, Italia May 29 -June 3, 2000 b Dividing extensive systems into larger pieces, the total energy and entropy are equal to the sum of those of the pieces. • first order transitions are distinguished from continuous transitions by the appearance of phase-separations and interfaces with surface tension. Figure 1 : 1MMMC simulation of the entropy s(e) (e in eV per atom) of a system of 1000 sodium atoms with realistic interaction at an external pressure of 1 atm. At the energy per atom e 1 the system is in the pure liquid phase and at e 3 in the pure gas phase, of course with fluctuations. The latent heat per atom is q lat = e 3 − e 1 . Attention: the curve s(e) is artifically sheared by subtracting a linear function 25 + e * 11.5 in order to make the convex intruder visible. s(e) is always a steeply monotonic rising function. We clearly see the global concave (downwards bending) nature of s(e) and its convex intruder. Its depth is the entropy loss due to the additional correlations by the interfaces. From this one can calculate the surface tension σ surf /Ttr = ∆s surf * N/N surf . The double tangent is the concave hull of s(e). Its derivative gives the Maxwell line in the caloric curve T (e) at Ttr obtain the familiar result f (T, µ, V ) = e s − µn s − T s s . I.e. the curvature λ 1 of the entropy surface s(e, n, V ) decides whether the grand-canonical ensemble agrees with the fundamental micro ensemble in the thermodynamic limit. N is the average number of surface atoms of the clusters. σ/T tr is the surface tension over the transition temperature. 4 Phase transitions in the micro-ensemble: The topology of the determinant of curvatures of s(e, n): The physics and mathematics of the second law of thermodynamics. H Elliott, J Lieb, Yngvason, cond-mat/9708200Physics Report. 310Elliott H. Lieb and J. Yngvason. The physics and mathematics of the sec- ond law of thermodynamics. Physics Report,cond-mat/9708200, 310:1- 96, 1999. Possible generalization of Boltzmann-Gibbs statistics. C Tsallis, C. Tsallis. Possible generalization of Boltzmann-Gibbs statistics. . J.Stat.Phys. 52479J.Stat.Phys, 52:479, 1988. Microcanonical thermodynamics: Phase transitions in "Small" systems. D H E Gross, Lecture Notes in Physics. World Scientific. D.H.E. Gross. Microcanonical thermodynamics: Phase transitions in "Small" systems. To be published in Lecture Notes in Physics. World Scientific, Singapore, 2000. Phase transitions in "small" systems. D H E Gross, E Votyakov, Eur.Phys.J.B. 15D.H.E. Gross and E. Votyakov. Phase transitions in "small" sys- tems. Eur.Phys.J.B, 15:115-126, (2000), http://arXiv.org/abs/cond- mat/?9911257. Micro-canonical statistical mechanics of some nonextensive systems. D H E Gross, Invited talk to the International Workshop on Classical and Quantum Complexity and Nonextensive Thermodynamics. Denton-TexasD.H.E. Gross. Micro-canonical statistical mechanics of some non- extensive systems. Invited talk to the International Workshop on Classical and Quantum Complexity and Nonextensive Thermodynamics (Denton-Texas, April 3-6, 2000) http://arXiv.org/abs/astro-ph/?cond- mat/0004268. Phase transitions in "Small" systems -a challenge for thermodynamics. Invited talk to CRIS 2000, 3rd Catania Relativistic Ion Studies. D H E Gross, Phase Transitions in Strong Interactions: Status and Perspectives. Acicastello, ItalyD.H.E. Gross. Phase transitions in "Small" systems -a challenge for thermodynamics. Invited talk to CRIS 2000, 3rd Catania Relativistic Ion Studies, Phase Transitions in Strong Interactions: Status and Per- spectives, Acicastello, Italy, May 22-26, 2000 http://arXiv.org/abs/cond- mat/?0006087.	[]
[ "Pure Monte Carlo Method: a Third Way for Plasma Simulation", "Pure Monte Carlo Method: a Third Way for Plasma Simulation" ]	[ "Hua-Sheng Xie \nInstitute for Fusion Theory and Simulation\nZhejiang University\n310027HangzhouPeople's Republic of China\n" ]	[ "Institute for Fusion Theory and Simulation\nZhejiang University\n310027HangzhouPeople's Republic of China" ]	[]	We bring a totally new concept for plasma simulation, other than the conventional two ways: Fluid/Kinetic Continuum (FKC) method and Particle-in-Cell (PIC) method. This method is based on Pure Monte Carlo (PMC), but far beyond traditional treatments. PMC solves all the equations (kinetic, fluid, field) and treats all the procedures (collisions, others) in the system via MC method. As shown in two paradigms, many advantages have found. It has shown the capability to be the third importance approach for plasma simulation or even completely substitute the other two in the future. It's also suitable for many unsolved problems, then bring plasma simulation to a new era. PACS numbers: 52.65.-y, 52.65.Pp Introduction-The original of Monte Carlo (MC) method for computations can back to hundreds of years ago, e.g., the Buffon's needle experiment. With the emergence of the modern computer in 1940s and pioneer works by John von Neumann, Stanislaw Ulam and Nicholas Metropolis, this method brings amounts of practical applications. The famous and classical [Metropolis1953] paper is said [7] mainly contributed by the hero of plasma physics, M. N. Rosenbluth. However, up to now, MC for physics is mainly on statistical physics, and only in merely special cases for plasma physics.Some people had known that, in fact, the PIC method can be seen a type of MC method.[1] provides a unified MC interpretation of particle simulations, which then can use the MC theory to estimate the errors and develop many types of sampling approaches, e.g., simple MC, importance sampling, the δ-f method and more.While, at most, the conventionally PIC only treats the kinetic equation using MC viewpoint and the field equations are still solved by Deterministic Discretizations (DD, finite differences and so on). FKC is solved totally by DD. Generally, each approach has its advantages and disadvantages. The main disadvantages of DD are the numerical unstability and dissipation and not easy to treat complex geometries and complex procedures which cannot be described easily by differential equations (ODEs, PDEs). To avoid old disadvantages, a way out is to develop new approaches.We find, indeed, we can have a third way for plasma simulation, i.e., we use totally stochastic statistical (Monte Carlo) approach. We need only a change of concept. In Copenhagen interpretation of quantum mechanics, we have known that the intrinsic of the world is uncertainty and all things are probabilities, only the statistic is determined. If we can accept this, we can easily accept the philosophy of this PMC letter. The remaining thing is how we do that, or exactly, how we solve differential equations (in fluid and kinetic) via MC. A branch of mathematic called Stochastic Differential Equations (SDE) has provided us some basic tools.At below, we will firstly describe the PMC method and	null	[ "https://arxiv.org/pdf/1210.2265v2.pdf" ]	118,038,431	1210.2265	fdc5d81ee6e6416d53bf61e00c394c7b4a1b55e6	Pure Monte Carlo Method: a Third Way for Plasma Simulation 10 Oct 2012 (Dated: May 3, 2014) Hua-Sheng Xie Institute for Fusion Theory and Simulation Zhejiang University 310027HangzhouPeople's Republic of China Pure Monte Carlo Method: a Third Way for Plasma Simulation 10 Oct 2012 (Dated: May 3, 2014)To submit for PRL or CPC or ... We bring a totally new concept for plasma simulation, other than the conventional two ways: Fluid/Kinetic Continuum (FKC) method and Particle-in-Cell (PIC) method. This method is based on Pure Monte Carlo (PMC), but far beyond traditional treatments. PMC solves all the equations (kinetic, fluid, field) and treats all the procedures (collisions, others) in the system via MC method. As shown in two paradigms, many advantages have found. It has shown the capability to be the third importance approach for plasma simulation or even completely substitute the other two in the future. It's also suitable for many unsolved problems, then bring plasma simulation to a new era. PACS numbers: 52.65.-y, 52.65.Pp Introduction-The original of Monte Carlo (MC) method for computations can back to hundreds of years ago, e.g., the Buffon's needle experiment. With the emergence of the modern computer in 1940s and pioneer works by John von Neumann, Stanislaw Ulam and Nicholas Metropolis, this method brings amounts of practical applications. The famous and classical [Metropolis1953] paper is said [7] mainly contributed by the hero of plasma physics, M. N. Rosenbluth. However, up to now, MC for physics is mainly on statistical physics, and only in merely special cases for plasma physics.Some people had known that, in fact, the PIC method can be seen a type of MC method.[1] provides a unified MC interpretation of particle simulations, which then can use the MC theory to estimate the errors and develop many types of sampling approaches, e.g., simple MC, importance sampling, the δ-f method and more.While, at most, the conventionally PIC only treats the kinetic equation using MC viewpoint and the field equations are still solved by Deterministic Discretizations (DD, finite differences and so on). FKC is solved totally by DD. Generally, each approach has its advantages and disadvantages. The main disadvantages of DD are the numerical unstability and dissipation and not easy to treat complex geometries and complex procedures which cannot be described easily by differential equations (ODEs, PDEs). To avoid old disadvantages, a way out is to develop new approaches.We find, indeed, we can have a third way for plasma simulation, i.e., we use totally stochastic statistical (Monte Carlo) approach. We need only a change of concept. In Copenhagen interpretation of quantum mechanics, we have known that the intrinsic of the world is uncertainty and all things are probabilities, only the statistic is determined. If we can accept this, we can easily accept the philosophy of this PMC letter. The remaining thing is how we do that, or exactly, how we solve differential equations (in fluid and kinetic) via MC. A branch of mathematic called Stochastic Differential Equations (SDE) has provided us some basic tools.At below, we will firstly describe the PMC method and We bring a totally new concept for plasma simulation, other than the conventional two ways: Fluid/Kinetic Continuum (FKC) method and Particle-in-Cell (PIC) method. This method is based on Pure Monte Carlo (PMC), but far beyond traditional treatments. PMC solves all the equations (kinetic, fluid, field) and treats all the procedures (collisions, others) in the system via MC method. As shown in two paradigms, many advantages have found. It has shown the capability to be the third importance approach for plasma simulation or even completely substitute the other two in the future. It's also suitable for many unsolved problems, then bring plasma simulation to a new era. Introduction-The original of Monte Carlo (MC) method for computations can back to hundreds of years ago, e.g., the Buffon's needle experiment. With the emergence of the modern computer in 1940s and pioneer works by John von Neumann, Stanislaw Ulam and Nicholas Metropolis, this method brings amounts of practical applications. The famous and classical [Metropolis1953] paper is said [7] mainly contributed by the hero of plasma physics, M. N. Rosenbluth. However, up to now, MC for physics is mainly on statistical physics, and only in merely special cases for plasma physics. Some people had known that, in fact, the PIC method can be seen a type of MC method. [1] provides a unified MC interpretation of particle simulations, which then can use the MC theory to estimate the errors and develop many types of sampling approaches, e.g., simple MC, importance sampling, the δ-f method and more. While, at most, the conventionally PIC only treats the kinetic equation using MC viewpoint and the field equations are still solved by Deterministic Discretizations (DD, finite differences and so on). FKC is solved totally by DD. Generally, each approach has its advantages and disadvantages. The main disadvantages of DD are the numerical unstability and dissipation and not easy to treat complex geometries and complex procedures which cannot be described easily by differential equations (ODEs, PDEs). To avoid old disadvantages, a way out is to develop new approaches. We find, indeed, we can have a third way for plasma simulation, i.e., we use totally stochastic statistical (Monte Carlo) approach. We need only a change of concept. In Copenhagen interpretation of quantum mechanics, we have known that the intrinsic of the world is uncertainty and all things are probabilities, only the statistic is determined. If we can accept this, we can easily accept the philosophy of this PMC letter. The remaining thing is how we do that, or exactly, how we solve differential equations (in fluid and kinetic) via MC. A branch of mathematic called Stochastic Differential Equations (SDE) has provided us some basic tools. At below, we will firstly describe the PMC method and then treat a fluid edge pedestal transport model and the kinetic 1D electrostatic problem as paradigms. PMC Method -In plasma simulation, MC has been used to many complex procedures, such as collisions and impurities transport, which has been widely known. So, here, we need not talk too much about that, and all the old MC methods in plasma simulation can be inherited by PMC method naturally. The new steps of PMC are to tell how we solve the fluid and field equations. Using MC to solve PDEs can back to two pivotal papers, [4] and [14], the former uses probabilistic interpretations for linear elliptic and parabolic equations, the latter gives MC methods for solving integro-differential equations occur in various branches of the natural sciences. Feynman-Kac formula-Many types of plasma equations (e.g., transport equations, Vlasov equation) have the form: ∂f ∂t + µ(x, t) ∂f ∂x − 1 2 σ 2 (x, t) ∂ 2 f ∂x 2 − V (x, t)f + p(x, t) = 0. (1) Feynman-Kac formula [6][10] tells us, Eq.(1) is equivalent to SDE when V, p = 0 (V, p =0 case can also find), dX = µ(X, t)dt + σ(X, t)dW t ,(2) where W t is Wiener process (Brownian motion), which is described by normal (Gaussian) distribution. Eqs. (1) and (2) are easily general to high dimensions or more variables, e.g., (x, v, t). The above descriptions are also shown by Itō's lemma [8], a classical work in SDE. Now, we solve the convection-diffusion equation as example to show how to use this formula, which is standard steps and well known by SDE people but may not familiar by plasma community. At this case, a = µ and D = σ 2 /2, are constants. For f (x, 0) = f 0 (x) , we start with a set N samples ξ 0 1 , · · · , ξ 0 N from f 0 (x). A sample for every next step is ξ t+dt i = ξ t i + adt + √ 2Ddtη i , i = 1, · · · , N.(3) where η i is normal distribution with zero mean and unit variance. For a = 0, we get pure diffusion, and for D = 0, pure convection. In fact, integral form explicit exact solution is f (x, t) = G t (x − y)f 0 (y − at)dy = G t * f 0 (x − at),(4)G t (x − y) = 1 (4πDt) 1/2 e − (x−y) 2 4Dt . If we use MC to integral (4) directly, and use N ′ samples for particles and M points in space, the total cost will be O(N ′ × M ) . While using (3), the total cost is O(N ). We write (4) here for benchmark. Using (3), a result is show in FIG.1. We can see that the MC result reproduces the exact solution very well, when we using larger N and M , the result can be even better. Here, M is not for calculation but just for reconstructing the grids for f . The above is a summary of SDE for our usage, which is a key fundament of our PMC method, the applications to plasma examples will be shown at below. Poisson equation-We may meet Poisson equation frequently in plasma physics, for example, in the electrostatic kinetic case or the electromagnetic gyrokinetic case. The equation is as the form ∇ 2 φ(x) = −ρ(x), x ∈ D.(5) If ρ = 0, gives Laplace equation, which can be found solved by MC in many places, e.g., [12]. A good introduction of MC methods for Poisson equation can find in [5]. The method can also take from an extended Feynman-Kac formula. For Dirichlet boundary, φ(x) = f (x), x ∈ ∂D. Standard solution for position x 0 using SDE notation is φ (x 0 ) = 1 2 E τ ∂D 0 ρ (W t ) dt + E [f (W τ ∂D )] ,(6) where, τ ∂D = inf {t : W t ∈ ∂D} is the first-passage time and W τ ∂D is the first-passage location on the boundary, another advantage that we can obtain the solution at a few points directly instead of calculating the whole range, and even if there are steep gradients or irregular boundary. Some updates of MC for Poisson equation may also find in recent literatures, such as for Neumann boundary, improving efficiencies, or reducing errors. Other equations-There are many other types of PDEs in plasma physics. Scalar conservation law equation ∂u/∂t + ∂F (u)/∂x = 0, e.g., Burgers' equation, can find be solved generally in [9]. Navier-Stokes (vector) equation for turbulence can find in [2]. For ODE, we often need just the standard MC integral. Some more MC methods and details for electromagnetic problems are introduced in [16]. One can also refer [11] for some general descriptions. The types of equations mentioned at above have contented many of the main plasma physics equations. Application Paradigms-We use the above methods to solve two practical plasma problems. Simple fluid edge pedestal transport model -The simplest model for calculating pedestal width can find in [18] or [20], the inner range of SOL (Scrape-Off Layer, for divertor or limiter) is mainly slowly perpendicular transport, and the outer range is mainly fast parallel (to the magnetic field line, mainly on toroidal ϕ direction) transport and the particles travel at most L (2πR or 2πqR) then hit the target with the thermal velocity, where R is the major radius, q is edge safety factor, and the thermal velocity is equal to acoustic velocity c s at edge from sheath calculations. We can write the inner range transport equation as ∂n ∂t = 1 r ∂ ∂r rD ∂n ∂r + S(n, r, t) = D r ∂n ∂r + D ∂ 2 n ∂r 2 + S. For simplification, we only use the density diffusion equation and treat the coefficient D as constant, and also ignore the source term S. For outer range, we find quickly that we cannot easily write a 1D equation combine to the inner range equation. Even though we write down the outer range equation, we will quickly meet the singularity problem at the last closed (magnetic) flux surface (LCFS, separatrix). A special treatment for transport barriers can find in [19], which rewrites the transport equations to new form. However, using PMC, we can avoid the above problems automatically. The inner equation (7) can be solved using (2). The outer range needs an extra MC treatment: if a particle comes out to the outer range, we use the same equation as the inner range for perpendicular transport but in the same time let it vanish random in time [0, T ) if it is still in the outer range, with T = L/c s . Quickly, we can find when using (3) to treat (7), the drift velocity a = −D/r → ∞ for the MC particles at r → 0. This problem is also met similarly in PIC [3] when using polar coordinate, also in large scale PIC simulation as in GTC [13]. The conventional method to treat this problem is making a hollow and ignoring a small r → 0 region. While, one can find in [16] other ways of MC implement transport equations and treat the r → 0 problem (e.g., L'Hospital's rule lim ρ→0 ∂f /(ρ∂ρ) = ∂ 2 f /∂ρ 2 ), which means that this won't be a severe problem in PMC. However, in fact, we will find MC can be a natural method to solve this geometry and complex boundary problems. The only reason why people using different coordinates is that we can use the symmetry of the system to make us treat (e.g., analytical calculations) or understand problem easier. People care but the great nature never cares the coordinates. The physics law won't change in any coordinates. So, we can rewrite (7) from (r, θ) coordinate to Cartesian coordinate (x, y) and use x 2 + y 2 = r = r b as boundary. MC is suitable for any complex boundaries, and the circle boundary is just one of them. Then, in use of PMC, people can use Cartesian coordinate always (in fact, as the circle symmetry case in FIG.2, we have already given an example), which will simplify many old complex coordinates (notable: the magnetic surface coordinate) or boundaries problems a lot. A 2D implement of PMC for this edge pedestal transport model is shown in FIG.3. In tokamak experiment, one can see lots of figures as FIG.3, i.e., the above simple model can model the experiment at least qualitatively. Kinetic 1D electrostatic (ES1D) problem-For ES1D, the system is described by kinetic equation ∂f ∂t + v ∂f ∂x + qE m ∂f ∂v = C(f ). (8) where, q is the charge, m is mass, E = −∇φ is the electrical field, C(f ) is collision term. The field equation is (5), with ρ = (f i − f e )dv the charge density. All the variables has normalized in units of eφ/T e , ω −1 pe , λ De . For 1D, (5) can rewrite to a more simple ODE form dE/dx = ρ.(9) Firstly, we can easily find (8) has the form of (1), gives dx i = v i dt, dv i = (qE/m)dt,(10) which is totally the same as in PIC. Eq.(10) tells us that more dimensions won't bring significant new time cost (note: higher dimensions we may need more particles for accuracy, which will bring some new time cost), while (8) will if we use DD. If the collision term has the form C(f ) = αf +β∂f /∂v +γ∂ 2 f /∂v 2 , the kinetic equation is still nothing else but (1). We see here, for PIC simulation of collision, we need an extra MC step, while in PMC method, we can write this step to equation of motion (10) directly. Here, we assume ions immobile for simplify and treat the collisionless (Vlasov) case, i.e., C(f ) = 0 , because that which is easy for benchmark. For f (x, v, t) , x and v can be anywhere. For E(x) or φ(x) , the x can also be anywhere. But, (8) and (9) are coupled. We should find out a way to connect the position x in f and E. A naturally thought is calculating the E(x) directly using MC for every f -particle one by one, but which is too much time consumed, especially when f -particles (N f ) and φ-particles (N φ ) are very large. An adjust way is using grids and then interpolating, which can reduce the time cost by an order of O(N φ ). We find this is just what PIC does, i.e., we re-deduce all the key steps of PIC method by PMC! The interpolating method is easy and has checked by PIC, however, one may also find other ways to implement this PMC step if can make sure they work. Using ODE (9) as field equation, for MC, which is just an integral from one side to another side, and we use random particles to do this, the errors of each step will cancel and average is zero. While, for DD (e.g., Euler, Runge-Kutta or else), the integral is summation and errors will accumulate. Using Poisson equation, the MC method is provided at before, an extra step is needed to calculate E from φ by calculating gradient from the MC φ-particles. Periodic boundary for the MC particles of f is easy to implement, which is the same as PIC. We comment here, for φ, the periodic boundary means φ(0) = φ(L), but when we see E, which is not E(0) = E(L) but average < E(x) >= 0 ( [3] has known this). In fact, this implies (8) and (9), a result is show in FIG. 4. Note that, due to noises, the phase space plot- ting will change for both PIC and PMC, even though under the same parameters (e.g., keep source code unchanged), especially in nonlinear stages. The above figure is a select from typical runs. The main feature of phase space holes is clearly in both PMC and PIC results. PMC is slightly slower than PIC in this case. Summary and Comments-The above two paradigms have shown many new advantages. In fact, PMC is more powerful when problems are complicated. For example, many large scaled codes are possible be completely rewritten via this new method, an example is BOUT ( [21], gyrofluid). For large scaled gyro-kinetic, this method is also possible. Some complicated paradigms are beyond this short letter. A comparison of FKC, PIC and PMC is listed in TABEL. I, which can also shows that why we can call PMC be a third method for plasma simulation. At present, many tools of PMC are brown from other fields. But the concept change is the key important thing. The unified viewpoint by combining everything in one provides us a totally new picture. In this new angle, we can see difficult old problems naturally and problems become simple. We can not only do new problems but also provide new insights or more efficient methods for old problems. For kinetic problems, PMC can be seen as advanced PIC; for fluid problems and complex procedures, PMC is new. Whatever, people may just concern that if an approach is useful and works well. PMC matches this. PMC is not a single fixed approach as PIC, but a combine of varies of old and new MC methods, flexible. As far as the author (HSX) knows, Max-Planck-Institut für Plasmaphysik is developing MC methods for Stellarator edge transport physics, while it seems [17] has already or almost implemented a fully PMC ES2D code though they didn't know this concept when they did that. New applications for resistive MHD tearing mode, Hasegawa-Mima equation, Grad-Shafranov equation and many other problems are on the way by the author or others. This letter has given/built the framework of PMC, which is enough for many practical applications. But, there may are still many new intelligent works need do for some special types of cases. Some unexpected problems may also meet when using to more complicated examples. The author does not know that yet. If also implemented well for the above untested several examples, this new approach is possible completely substitute the conventional two methods, FKC and PIC, and also suitable for solving many old unsolved or hard to solve problems, then bring plasma simulation to a new era. FIG . 1: (Color online) Solve convection-diffusion equation using (3), as example to show PMC method. E is average. Using floating random walk, the results for 2D are show in FIG.2. The MC method here has online) MC for 2D Poisson equation, example. FIG. 3 : 3(Color online) PMC for (Tokamak) edge pedestal transport. FIG. 4 : 4(Color online) Compare with PMC and PIC for 2stream instability, phase space plotting. TABLE I : IComparison of FKC, PIC and PMC for kinetic plasma simulation. PIC treatment and new PMC treatment can be very similar. The implements can be close, but the concepts are totally different. PIC is fixed and nonadjustable. PMC provides more possibilities and can bring us new ways.Using PMC to solvingFKC PIC PMC Approach Continuum Particle + continuum Only particle Equations Only PDEs New method for eqn. of motion, Field eqn. brown from FKC Eqn. of motion brown from PIC, New method for field equation Good Accurate Efficient Easy for complex problems, Effi- cient for parallel, Easy for coding Bad Numerical unstability and dissipation Noise Crude, with error ∝ 1/ √ N Philosophy More mathematical Half math, half first principle More first principle that, we have added two charged slabs (not real 'parti- cles', because this is 1D, or called 'markers' in PIC) at the boundary to cancel the extra E field! Few ES1D PIC researchers have seen this. Thinking in PMC way will bring us many new insights or re-thoughts. For this ES1D problem, we can find, the conventional Acknowledgement-This work is also inspired by Zhichen FENG (IFTS-ZJU), who once (2010) proposed a new method for PIC, though which performances not well in practical application yet. . A Y Aydemir, Physics of Plasmas. 1Aydemir, A. Y., Physics of Plasmas, 1994, 1, 822-831. . O M Belotserkovskii, Y I Khlopkov, Monte Carlo, Methods In Mechanics Of Fluid And Gas. Belotserkovskii, O. M. and Khlopkov, Y. I., Monte Carlo Methods In Mechanics Of Fluid And Gas, 2010. Plasma Physcis via Computer Simulation. C Birdsall, A Langdon, IOPBirdsall, C. and Langdon, A., Plasma Physcis via Com- puter Simulation, IOP, 1991. Or, English translation. R Courant, K Friedrichs, H Lewy, Mathematische Annalen, IBM J. Res. Develop. 100R. Courant, K. Friedrichs, and H. Lewy, Mathematische Annalen, 100: 32-74, 1928. Or, English translation, IBM J. Res. Develop., 11: 215-234, 1967. . 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A Guide to Monte Carlo Simulations in Statistical Physics. D P Landau, K Binder, Cambridge University Press2ndLandau, D. P. and Binder, K., A Guide to Monte Carlo Simulations in Statistical Physics, Cambridge University Press, 2008 (2nd). . Z Lin, T S Hahm, W W Lee, W M Tang, R B White, Science. 281Lin, Z.; Hahm, T. S.; Lee, W. W.; Tang, W. M. and White, R. B., Science, 1998, 281, 1835-1837. . N Metropolis, S Ulam, Journal of the American Statistical Association. 44Metropolis, N. and Ulam, S., Journal of the American Statistical Association, 1949, 44, 335-341. . N Metropolis, A W Rosenbluth, M N Rosenbluth, A H Teller, E Teller, The Journal of Chemical Physics. 21Metropolis, N.; Rosenbluth, A. W.; Rosenbluth, M. N.; Teller, A. H. and Teller, E., The Journal of Chemical Physics, 1953, 21, 1087-1092. Monte Carlo Methods for Electromagnetics. M N O Sadiku, CRC PressSadiku, M. N. O., Monte Carlo Methods for Electromag- netics, CRC Press, 2009. . 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[ "Realisation of Abelian varieties as automorphism groups", "Realisation of Abelian varieties as automorphism groups" ]	[ "Mathieu Florence " ]	[]	[]	Let A be an Abelian variety over a field F . We show that A is isomorphic to the automorphism group scheme of a smooth projective F -variety if, and only if, Aut gp (A) is finite. This result was proved by Lombardo and Maffei [7] in the case F = C , and recently by Blanc and Brion [1] in the case of an algebraically closed F .Résumé.Soit A une variété abélienne sur un corps F . On montre que A est isomorphe au schéma en groupes des automorphismes d'une F -variété projective et lisse, si et seulement si le groupe des F -automorphismes de A est fini. Ce résultat est dûà Lombardo et Maffei [7] lorsque F = C. Il est dûà Blanc et Brion [1] lorsque F = F .	null	[ "https://arxiv.org/pdf/2102.02581v2.pdf" ]	231,801,940	2102.02581	000619c9abb11f243faa4c64e3465ccd36923e26	Realisation of Abelian varieties as automorphism groups 12 May 2022 Mathieu Florence Realisation of Abelian varieties as automorphism groups 12 May 20221 Let A be an Abelian variety over a field F . We show that A is isomorphic to the automorphism group scheme of a smooth projective F -variety if, and only if, Aut gp (A) is finite. This result was proved by Lombardo and Maffei [7] in the case F = C , and recently by Blanc and Brion [1] in the case of an algebraically closed F .Résumé.Soit A une variété abélienne sur un corps F . On montre que A est isomorphe au schéma en groupes des automorphismes d'une F -variété projective et lisse, si et seulement si le groupe des F -automorphismes de A est fini. Ce résultat est dûà Lombardo et Maffei [7] lorsque F = C. Il est dûà Blanc et Brion [1] lorsque F = F . Introduction. Let F be a field, with algebraic closure F . Let X be a projective variety over F . The automorphism group functor Aut(X) is represented by a group scheme, locally of finite type over F ( [8], Theorem 3.7). Conversely, given a group scheme G, of finite type over F , it is natural to ask whether G can be realised as the automorphism group of such an X. When G = A is an abelian variety, this question was recently considered in [7]. When F = C, Lombardo and Maffei prove that A is the automorphism group of a projective smooth complex variety, if and only if Aut gp (A) is finite. They use analytic methods. Their result was extended to F algebraically closed of any characteristic in [1], using algebro-geometric techniques: blowups, Lie algebra computations and modding out actions of finite group schemes. Making a different use of these tools, we provide a generalisation of this result, to the case of all ground fields F . Sketch of our construction. Let A/F be an abelian variety over a field F , such that G := Aut gp (A) is finite. We first introduce an integer n ≥ 1, invertible in F , such that G acts faithfully on the n-torsion subgroup A[n](F ). Then, we pick an abelian variety B 1 /F , enjoying the following properties. (1) The abelian varieties A and B 1 are 'orthogonal', in the sense that We denote by B 2 /F the abelian variety fitting into the diagonal extension 0 −→ A[n] a →(a,ι(a)) −→ A × B 1 π −→ B 2 −→ 0. Using point (1) above, we prove that automorphisms of (the variety) B 2 are diagonal: they come from automorphisms of A × B 1 , respecting orbits under the embedded A[n]. Next, we build an appropriate smooth closed F -subvariety Y 2 ⊂ B 2 , stable by translations by A ≃ π(A × {0}) ⊂ B 2 . We define a smooth F -variety X as the blowup X := Bl Y2 B 2 . The natural arrow A −→ Aut(B 2 ), given by translations, lifts to an arrow τ : A −→ Aut(X). We show that τ is an isomorphism of algebraic groups over F . Notation. 2.1. Geometry over F . Let F be a field, with algebraic closure F , and separable closure F s ⊂ F . We denote by F [ǫ], ǫ 2 = 0, the F -algebra of dual numbers. We use it for differential calculus. By a variety over F , we mean a separated F -scheme of finite type. An algebraic F -group (or simply F -group) is an F -group scheme of finite type. It is often assumed to be reduced, hence smooth over F . Let X be a variety over F . For a field extension E/F , we denote by X E := X × F E the E-variety obtained from X by extending scalars. We put X := X × F F . If X is smooth over F , we denote by T X −→ X the tangent bundle of X. A global section of the tangent bundle is called a vector field on X. We denote by Aut(X) the (abstract) group of automorphisms of the F -variety X, and by Aut(X) the group of automorphisms of the F -variety X. If X/F is a projective variety, we denote by Aut(X) the F -group scheme of automorphisms of X; it is locally of finite type over F . By [2], Lemma 3.1, there is a canonical isomorphism H 0 (X, T X) ∼ −→ Lie(Aut(X)). If an abstract group G acts on a variety X, and if Z ⊂ X is a closed subvariety, we denote by Stab G (Z) ⊂ G, or simply by Stab(Z) ⊂ G when no confusion arises, the subgroup of transformations leaving Z (globally) invariant. Let G/F be a group scheme, locally of finite type. In the situation where G acts on X, we use the notation Stab G (Z) ⊂ G for the closed F -subgroup scheme defined by Stab G (Z)(A) = {g ∈ G(A), g(Z A ) = Z A },Frob X : X −→ X (1) ; it is a morphism of F -varieties, functorial in X. If X/F is an algebraic group, it is a group homomorphism. If X is a commutative algebraic group, there is the Verschiebung homomorphism Ver X : X (1) −→ X, satisfying (Ver X • Frob X ) = pId X . If moreover X/F is a semi-abelian variety, Ver X and Frob X are isogenies. 2.3. Abelian varieties. If A and B are Abelian varieties over F , we denote by Hom gp (A, B) the group of homomorphisms of algebraic F -groups, from A to B. We denote by Hom gp (A, B) the group of homomorphisms of algebraic F -groups, from A to B. These are finite free Z-modules. We adopt the similar notation for endomorphisms (End gp ) and automorphisms (Aut gp ). For an integer n ≥ 1, we denote by A[n] the n-torsion of A, seen as a finite group scheme over F . Barycentric operations. Let A be an abelian variety over F . Then A comes naturally equipped with barycentric operations with integer coefficients. More precisely, for a positive integer n, denote by Z n 1 ⊂ Z n the subset consisting of integers α = (α 1 , . . . , α n ), with α 1 + . . . + α n = 1. For α ∈ Z n 1 , there is a barycentric operation B α : A n −→ A, (x 1 , . . . , x n ) → α 1 x 1 + . . . + α n x n . Associativity of the group law of A, provides natural associativity relations between the B α 's, for various α ′ s. For instance, pick α = (α 1 , α 2 ) ∈ Z 2 1 and γ = (γ 1 , γ 2 ) ∈ Z 2 1 , and set δ := (α 1 γ 1 , α 2 γ 1 , γ 2 ) ∈ Z 3 1 . Then, we have the associativity rule B γ (B α (x 1 , x 2 ), x 3 ) = B δ (x 1 , x 2 , x 3 ).(B α ) \|X n : X n → A factors through the closed immersion X ֒→ A, for every n ≥ 2 and every α ∈ Z n 1 . In this case, we also say that X is barycentric. Note that X is barycentric if and only if it is a translate of an algebraic F -subgroup − → X ⊂ A. Checking this fact is left as an exercise for the reader. Of course, X(F ) might be empty. If X is geometrically reduced and geometrically connected, so is − → X -hence − → X is an abelian subvariety of A. Let A and B be two abelian varieties over F . Recall the essential fact Hom F −var (A, B) = B(F ) × Hom gp (A, B). In particular, morphisms (of varieties) between abelian varieties commute with the barycentric operations B α . If X ⊂ A is a geometrically reduced closed F -subvariety, the smallest geometrically reduced barycentric F -subvariety containing X is called the barycentric envelope of X. We denote it by E(X). Assume now that X is geometrically reduced and geometrically connected. Pick n ≥ 1 and α ∈ Z n 1 . Consider B α (X n ) ⊂ A as a geometrically reduced and geometrically connected closed subvariety of A. Then, if n and α are chosen so that B α (X n ) is of maximal dimension, we have B α (X n ) = E(X). Thus, E(X), being geometrically connected and geometrically reduced, is a translate of an abelian subvariety of A. 3. Statement of the theorem. Theorem 3.1. Let A be an Abelian variety, over a field F . The following are equivalent: 1) The group G := Aut gp (A) is finite. 2) There exists a smooth projective F -variety X, such that A is isomorphic to Aut(X) (as algebraic groups over F ). Note that 2) ⇒ 1) can be checked over F , which follows from [1], Theorem A. Our task in this paper is to prove the converse implication. 4. Auxiliary results. 4.1. Blowups. This section contains two elementary lemmas on automorphisms of blowups, which we provide with short proofs. A good recent reference on this topic, also containing more advanced material, is section 2 of [9]. Lemma 4.1. Let Y ֒→ D be a closed immersion of smooth F -varieties, such that all connected components of Y have codimension ≥ 2 in D. Denote by β : X := Bl Y (D) −→ D the blowup of Y inside D. The F -variety X is smooth. Let f be an automorphism of the F -variety D. Then, f lifts via β to an automor- phism of X, if and only if f (Y ) = Y . Proof. If f (Y ) = Y , then f lifts to an automorphism of X by the universal property of the blowup. Conversely, assume that f lifts to an automorphism φ of X, so that we have a commutative square X φ / / X D f / / D. To check that f (Y ) = Y , can assume that F = F . It then suffices to prove that Y ⊂ D and f (Y ) ⊂ D have the same set of F -rational points. This is clear, since the fiber of β over a point Proof. Denote by i : E ֒→ X the exceptional divisor. The restriction x ∈ D(F ) is either a point if s / ∈ Y (F ), or a projective space of dimension ≥ 1 if x ∈ Y (F ).β \|X−E : X − E −→ D − Y is an isomorphism. We thus have a natural injective F -linear arrow ρ : H 0 (X, T X) −→ H 0 (D − Y, T D) = H 0 (D, T D), σ → σ \|X−E . Note that the equality H 0 (D − Y, T D) = H 0 (D, T D) follows from the fact that Y ⊂ D has codimension ≥ 2. On E, we have a natural extension of vector bundles 0 −→ T E −→ i * (T X) −→ N E/X −→ 0, where N E/X ≃ O E (−1) is the normal bundle of E in X. Since Y has codimension ≥ 2 in D, we have H 0 (E, O E (−1)) = 0. Hypersections in projective space. We could not find a reference in the literature for the following result, so that we provide it with a proof. Proof. Pick a projective embedding S ⊂ P n (everything is over F ). Let d ≥ 1 be an integer. Let H ⊂ P n be a degree d hypersurface, given by h ∈ H 0 (P n , O(d)). By Bertini's theorem, for d large enough and h general, S ∩ H is smooth and geometrically irreductible, of dimension one less than S. This version of Bertini's theorem works over any F -see [10] and [4] for the delicate case where F is finite. Proceeding by induction, we reduce to the case where S is a surface. We then take C := S ∩ H, and show that g(= h 1 (C, O C )) goes to infinity with d. To do so, consider the exact sequence of coherent O P n -modules 0 → O S (−d) ×h − − → O S → O C → 0. Taking Euler characteristics, we get g − 1 = −χ(O C ) = χ(O S (−d)) − χ(O S ). We conclude using the following fact, applied to X = S. For a closed m-dimensional F -subvariety X ⊂ P n , the association d → χ(O X (−d)) is a degree m polynomial function of d. A classical proof is by induction on m ≥ 0. 4.3. (Semi-)abelian varieties. The next Lemma is borrowed from [2], Lemma 5.3. We provide here a different proof. In practice, we will apply it to abelian varieties, in which case it is due to Chow. Proof. We have to show the following. Let E/F be a purely inseparable algebraic extension. Let g : A E −→ B E be a homomorphism of algebraic groups over E. Then g is defined over F . Without loss of generality, we can assume that E/F is finite. By induction, we reduce to the case where E = F ( p √ a)/F is a primitive purely inseparable extension of height one. Note that Frob : E −→ E takes values in F . Hence, g (1) : A (1) E −→ B (1) E is defined over F . The Frobenius homomorphism Frob A : A −→ A (1) presents A (1) as a quotient of A, by a finite (characteristic) sub-F -group µ A ⊂ A. From the relation Ver A • Frob A = pId A , we deduce µ A ⊂ A[p]. Same holds for B. Combining these facts, we get that the E-morphism A/µ A −→ B/µ B induced by g, is defined over F . Modding out further, we get that the E-morphism A/A[p] −→ B/B[p], induced by g, is defined over F . Via the iso A/A[p] ∼ −→ A a → pa, this isomorphism is actually g itself. The Lemma is proved. Proof. Since F s is separably closed, 'simple' is the same as 'absolutely simple', for abelian varieties over F s (use Lemma 4.4). Without loss of generality, we assume that F s is the algebraic closure of its prime subfield. Over Q, we can then use the existence of abelian surfaces with a prescribed CM type. Over F p , we can use Honda-Tate theory. For concrete constructions, and more general results, we refer to [9], Theorem 1 (where F s = Q), and [6], Theorem 2 (where F s = F p ). Proof. Assume first that B is F -simple, in the sense that it has no non-trivial proper abelian F -subvariety. By Proposition 4.3, we can pick a geometrically irreducible smooth F -curve C ⊂ B, of arbitrarily large genus g ≥ 2. The group Aut(C) is finite. Indeed, Lie(Aut(C)) is the space of vector fields on C, which vanishes since g ≥ 2. Let us show that E(C) = B. The barycentric envelope E(C) is a translate of an abelian subvariety B ′ ⊂ B. Since B is F -simple, we get B ′ = B, hence E(C) = B. Now, let g ∈ Aut(B)(F [ǫ]) = B(F [ǫ]) × Aut gp (B) be such that g \|C×F F [ǫ] = Id \|C×F F [ǫ] . Because g commutes to barycentric operations, g acts as the identity on the closed subscheme E(C) × F F [ǫ] ⊂ B × F F [ǫ]. Since E(C) = B, it follows that g = Id. Thus, we get a natural embedding of F -group schemes H := Stab Aut(B) (C) ֒→ Aut(C). In particular, H is finiteétale over F . Let E/F be a finite separable field extension, such that H(E) = H(F ). Denote by We can then adapt the preceding proof, as follows. For each i, let C i ⊂ B i , L i /E i /F and b i ∈ B(L i ) be as in the first part of the proof. We can fulfill the extra requirements that no C i passes through 0, and that the C i 's are of different genus (using Proposition 4.3). In particular, when i = j, C i is not F -isomorphic to C j . We can also assume that L i = L and E i = E are independent of i. Set b := (b 1 , . . . , b n ) ∈ B(L). Φ := h∈H(E),h =e B h ⊂ B Define Y to be the disjoint union of [b], and of the n curves C i ≃ {0} × . . . × {0} × C i × {0} × . . . × {0} ֒→ B. It is not hard to see, that Y enjoys the required property. In general, write B = ( Let A/F be an abelian variety, such that G := Aut(A) is finite. We give a construction of a smooth projective F -variety X, such that A = Aut(X), in several steps. Construction of X. Denote by g the dimension of A. Let n ≥ 1 be an integer, invertible in F , such that the action of G on A[n](F s ) ≃ (Z/n) 2g is faithful. Such an n exists: use that G is finite, and that torsion points of order prime to char(F ) in A(F ) are Zariski-dense in A. Let B s be an abelian variety over F s , of dimension g ′ ≥ g, such that Hom gp (A, B) = Hom gp (B, A) = 0. Since A has a finite number of simple components (up to isogeny), which are all defined over F s by Lemma 4.4, the existence of B s follows from Lemma 4.5. For example, take for B s a product of simple abelian varieties, of dimensions greater than that of the simple components of A. Let E/F , be the finite Galois extension, with group Γ, which is minimal w.r.t. the following properties. Proof. We give two (seemingly) different proofs. The first one uses the perfect duality (.) ∨ := Hom(., Z/n), in the category of (Z/n)[Γ]-modules. Pick a generating set t 1 , . . . , t 2g ′ of the Z/nmodule A[n](E) ∨ -which is free of rank 2g ≤ 2g ′ . Introduce the surjection of (Z/n)[Γ]-modules (Z/n)[Γ] 2g ′ −→ A[n](E) ∨ , e i → t i , where e i denotes the i-th element of the canonical basis. Dualizing it yields an injection of (Z/n)[Γ]-modules ι : A[n](E) −→ (Z/n)[Γ] 2g ′ ≃ B 1 [n], concluding the construction. The second proof is more conceptual. Choose an embedding of constant E-group schemes (Z/n) 2g ≃ A E [n] ֒→ B E [n] ≃ (Z/n) 2g ′ , which exists simply because g ≤ g ′ . Applying R E/F yields an embedding of F -group schemes R E/F (A E )[n] ֒→ R E/F (B E )[n] = B 1 [n]. Composing it with the natural embedding of F -groups A[n] ֒→ R E/F (A E )[n], arising by adjunction from the identity of A E [n], we get the desired ι. Introduce the quotient q : B 2 −→ B 3 := B 2 /A ≃ B 1 /ι(A[n]). Let Y 3 ⊂ B 3 be a smooth F -subvariety, enjoying the properties of Lemma 4.6, where we take B to be our B 3 , and set Y 3 := Y . Put Y 2 := q −1 (Y 3 ). The restriction q \|Y2 : Y 2 −→ Y 3 is an A-torsor. We now define X := Bl Y2 (B 2 ) to be the blowup of Y 2 in B 2 . 5.2. Proof that Aut(X) ≃ A. Translating by elements of A inside B 2 yields a natural arrow A −→ Aut(B 2 ). Since Y 2 ⊂ B 2 is stable by these translations, we get an induced arrow of F -group schemes τ : A −→ Aut(X). It is clear that τ is an embedding. We are going to show that it is an isomorphism. Let us first check that it induces a bijection A(F ) ∼ −→ Aut(X) = Aut(X)(F ). Pick φ ∈ Aut(X). It induces a birational isomorphism f 2 of the F -variety B 2 , which is a regular isomorphism since B 2 is an abelian variety. Thus, we get a commutative diagram X φ / / X B 2 f2 / / B 2 , where the vertical arrows are the structure morphism of the blowup. Using Lemma 4.1, we get f 2 (Y 2 ) = Y 2 . We know that f 2 (x) = g 2 (x) + t 2 , where g ∈ Aut gp (B 2 ), and t 2 ∈ B 2 (F ). We have to show that g 2 = Id and t 2 ∈ A(F ). To do so, we can assume without loss of generality that t 2 ∈ B 1 (F ). We then have to prove g 2 = Id and Therefore g 2 leaves A ⊂ B 2 and B 1 ⊂ B 2 stable. We infer that g 2 lifts, via π, to a diagonal group automorphism δ = (h, g 1 ) of A × B 1 , which automatically leaves the diagonally embedded A[n] stable. Consider the automorphism of B 1 given by f 1 (b 1 ) := g 1 (b 1 ) + t 2 , and the diagonal automorphism of A × B 1 given by f 1 (b 1 )). ∆(a, b 1 ) := (h(a), Since δ leaves A × ι(A[n]) ⊂ A × B 1 stable, there exists f 3 ∈ Aut(B 3 ) such that the diagram A × B 1 ∆ / / π A × B 1 π B 2 f2 / / q B 2 q B 3 f3 / / B 3 commutes. Because f 2 (Y 2 ) = Y 2 , we get f 3 (Y 3 ) = Y 3 . By Lemma 4.6, we conclude that f 3 = Id. Hence, we have t 2 ∈ ι(A[n])(F ) and g 2 = Id. Since δ preserves the diagonally embedded A[n], we get that h, restricted to A[n] ⊂ A, is the identity. Since G acts faithfully on A[n], we conclude that h = Id. Hence, g 2 = Id as well, and our job is done. We have proved that τ induces a bijection on F -points. If F has characteristic zero, this is enough to conclude that τ is an isomorphism of algebraic F -groups. In general, it remains to check that the F -linear map on tangent spaces d e (τ ) : Lie(A) −→ Lie(Aut(X)) is bijective. Recall that Lie(Aut(X) is the space of vector fields on X; that is, global section of the tangent bundle T X −→ X. Let s : X −→ T X be such a section. Restricting s to the complement of the exceptional divisor, we get a global section σ ′ of the tangent bundle of B 2 − Y 2 . Since B 2 is an abelian variety, its tangent bundle is trivial, so that σ ′ is given by an arrow of F -varieties σ ′ : B 2 − Y 2 −→ A(Lie(B 2 )), with target an affine space of dimension dim(B 2 ). Since Z 2 has codimension ≥ 2 in B 2 , σ ′ extends to a morphism σ : B 2 −→ A(Lie(B 2 )), which is constant because B 2 /F is proper. Write σ = t, with t ∈ Lie(B 2 ). To conclude, we have to show t ∈ Lie(A). For y ∈ B 2 (F ), denote by α y : B 2 −→ B 2 the F -morphism given by x → x + y. Recall that the linear isomorphisms Since σ lifts to a section of the tangent bundle of the blowup Bl Y2 (B 2 ), Lemma 4.2 implies, when y ∈ Y 2 (F ), that t belongs to d y α −y (T y (Y 2 )) ⊂ Lie(B 2 ) ⊗ F F . Taking a y lying above (via q) an isolated separable point of Y 3 , we conclude that t ∈ Lie(A), as desired. Hom gp (A, B 1 ) = 0, where homomorphisms are taken over F . (2) There exists an injection (of algebraic F -groups) ι : A[n] ֒→ B 1 . Lemma 4.1 has an infinitesimal analogue, as follows. Lemma 4 . 2 . 42Let Y ֒→ D be a closed immersion of smooth F -varieties, such that all connected components of Y have codimension ≥ 2 in D. Denote by β : X := Bl Y (D) −→ D the blowup of Y inside D. Let s : D −→ T D be a vector field on D. Then, s lifts to a vector field on X, if and only if s \|Y takes values in T Y . This can be checked on the fibers of β over geometric points of Y , which are projective spaces of dimension ≥ 1. Hence, σ \|E takes values in T E. Consequently, ρ(σ) \|Y takes values in T Y . Conversely, let s : D −→ T D be a vector field on D. Then s corresponds to an automorphism ψ of the F [ǫ]-scheme D × F F [ǫ], reducing to the identity at ǫ = 0. Assume that s \|Y takes values in T Y . Then, ψ restricts to an automorphism of the closed subscheme Z × F F [ǫ] ⊂ D × F F [ǫ]. By the universal property (and compatibility with base change) of the blowup, ψ lifts, via β × F F [ǫ], to an automorphism of X × F F [ǫ]. Equivalenty, s lifts, via β, to a vector field on X. Proposition 4. 3 . 3Let S be a geometrically irreductible smooth projective Fvariety, of dimension ≥ 2. Let m ≥ 1 be an integer. Then, S contains a geometrically irreductible smooth projective F -curve, of genus g ≥ m. Lemma 4. 4 . 4Assume that F has characteristic p > 0. Let A, B be semi-abelian varieties over F . Then, all elements of Hom gp(A, B)are defined over the separable closure F s ⊂ F . Lemma 4 . 5 . 45For each n ≥ 2, there exists an (absolutely) simple n-dimensional abelian variety A over F s . Lemma 4. 6 . 6Let B be an abelian variety over F , whose simple factors (over F ) are of dimensions ≥ 2. (Equivalently: all F -homomorphisms from an elliptic curve to B are constant.) Then, there exists a smooth F -subvariety Y ⊂ B, which is a disjoint union of smooth F -curves, and of a separable closed point, such that Stab(Y ) = {Id} ⊂ Aut(B). be the (strict) closed subscheme, consisting of points fixed by at least one nontrivial element h ∈ H(E). It is defined over F by Galois descent. There exists a finite separable field extension L/E, and a point b = 0 ∈ B(L), which does not lie in Φ(L), nor in C(L). We then have a separable zero-cycle [b] in the F -variety B, of degree [L : F ]. Define Y ⊂ B as the disjoint union of [b] and C. We claim that Y has the required property. Indeed, let f ∈ Aut(B)(F [ǫ]) be an automorphism stabilizing Y -or more accurately, Y × F F [ǫ] ⊂ B × F F [ǫ].Then, f permutes the two connected components of the scheme Y × F F [ǫ]. For dimension reasons, it preserves C × F F [ǫ] on the one hand, and [b] × F F [ǫ] on the other hand. From the first fact, we know that f belongs to H(F ); in particular, it is defined over E, hence over L. From the latter fact, we get f (b) = b, hence f = Id. The Lemma is proved in this case. Assume now that B = B 1 × . . . B n , where the B i 's are F -simple abelian varieties. 1 B j )/µ, where S 1 , . . . , S r are F -simple abelian varieties, and where µ is a finite F -subgroup, intersecting trivially each coordinate axis. We can choose Y ֒→ B 1 × . . . B n as in the previous part of the proof, and such that the composite Y ֒→ B 1 × . . . B n can −→ ( r 1 B j )/µ = B is a closed immersion, identifying Y to a smooth closed subvariety of B. An automorphism of B stabilizing Y ⊂ B then lifts, via the quotient can, to an automorphism of B 1 × . . . B n stabilizing Y ⊂ B 1 × . . . B n . We conclude as before. 5. Proof of the implication 1) ⇒ 2). ( 1 ) 1The extension E/F splits the F -group of multiplicative type A[n]. In other words, A[n](E) ≃ (Z/n) 2g .(2) The abelian variety B s is defined over E: there exists an abelian E-varietyB E , such that B E × E F s ≃ B s . (3) Same as (1), for B E : we have B E [n](E) ≃ (Z/n) 2g ′ .Using(1), we view A[n](E) as a (Z/n)[Γ]-module. Introduce the Weil restriction of scalars B 1 := R E/F (B E ). Geometrically, we have B 1 ≃ B m s , where m is the cardinality of Γ. We have B 1 [n] = R E/F ((Z/n) 2g ′ ), so that E/F splits B 1 [n], and B 1 [n](E) is a free (Z/n)[Γ]-module of rank 2g ′ . Lemma 5.1. There exists an embedding of (Z/nZ)[Γ]-modules A[n](E) ֒→ B 1 [n](E); that is to say, an embedding of finiteétale F -group schemes ι : A[n] ֒→ B 1 [n]. B 2 is an abelian variety over F . We have F -embeddings t 2 ∈ 2A(F ) ∩ B 1 (F ) = ι(A[n])(F ). Geometrically, B 1 ≃ B m s . Since Hom gp (A, B s ) = Hom gp (B s , A) = 0, we get Hom gp (A, B 1 ) = Hom gp (B 1 , A) = 0. d y α −y : T y (B 2 ) ∼ −→ Lie(B 2 ) ⊗ F Fare used to trivialize the tangent bundle of B 2 . for all commutative F -algebras A. That it is representable follows from [3], II 1.3.6. 2.2. Frobenius and Verschiebung. If F has characteristic p > 0, we put X (1) := X × Frob F, extension of scalars taken with respect to Frob : Fx →x p −→ F. Recall the Frobenius homomorphism Acknowledgments.We are grateful to the referee for meaningful comments, leading to an improved exposition. We thank Michel Brion for his careful reading, and for helpful suggestions concerning blowups. We thank Frans Oort and Olivier Wittenberg for helping us understand why Lemma 4.5 is true, and Daniel Bertrand for pointing out an existing reference for its proof.Bibliography M Brion, Homomorphisms of algebraic groups: representability and rigidity. preprintM. Brion, Homomorphisms of algebraic groups: representability and rigidity, preprint, available at https://arxiv.org/abs/2101.12460 M Demazure, P Gabriel, Groupes algébriques. ParisMassonM. Demazure, P. Gabriel, Groupes algébriques, 1970, Masson, Paris. Bertini irreducibility theorems over finite fields. F Charles, B Poonen, J. Amer. Math. Soc. 291F. Charles, B. Poonen, Bertini irreducibility theorems over finite fields, 2016, J. Amer. Math. Soc. 29 (1), 81-94. Infinitesimal automorphisms of algebraic varieties and vector fields on elliptic surfaces. G Martin, G. Martin, Infinitesimal automorphisms of algebraic varieties and vector fields on elliptic surfaces, available at https://arxiv.org/abs/2004.07227 On the Existence of Absolutely Simple Abelian Varieties of a Given Dimension over an Arbitrary Field. E W Howe, H J Zhu, J. of Number Theory. 92E. W. Howe, H. J. Zhu, On the Existence of Absolutely Simple Abelian Varieties of a Given Dimension over an Arbitrary Field, 2002, J. of Number Theory 92, 139-163. Abelian varieties as automorphism groups of smooth projective varieties. D Lombardo, A Maffei, 2020D. Lombardo, A. Maffei, Abelian varieties as automorphism groups of smooth projective varieties, 2020, IMRN (7), 1921-1932. Oort, representability of group functors, and automorphisms of algebraic schemes. H Matsumura, F , Inv. Math. 4H. Matsumura, F. Oort, representability of group functors, and automorphisms of algebraic schemes, 1967, Inv. Math. 4, 1-25. The endomorphism rings of some abelian varieties. S Mori, Japan. J. Math. 3S. Mori,The endomorphism rings of some abelian varieties, 1977, Japan. J. Math 3, 105-109. Bertini theorems over finite fields. B Poonen, Ann. of Math. 1603B. Poonen, Bertini theorems over finite fields, 2004, Ann. of Math. 160 (3), 1099-1127. . Mathieu Florence, Equipe De Topologie Et Géométrie Algébriques, ParisInstitut de Mathématiques de Jussieu, Sorbonne UniversitéMathieu Florence, Equipe de Topologie et Géométrie Algébriques, Institut de Mathématiques de Jussieu, Sorbonne Université, Paris. Email address: https://webusers.imj-prg.fr/~mathieu.florence/	[]
[ "The horizon of 2-dichromatic oriented graphs", "The horizon of 2-dichromatic oriented graphs" ]	[ "J Barát ", "Mátyás Czett ", "\nAlfréd Rényi Institute of Mathematics\nDepartment of Mathematics\nUniversity of Pannonia\nEgyetem utca 108200VeszprémHungary\n", "\nEötvös Loránd University\nPázmány Péter sétány 11117BudapestHungary\n" ]	[ "Alfréd Rényi Institute of Mathematics\nDepartment of Mathematics\nUniversity of Pannonia\nEgyetem utca 108200VeszprémHungary", "Eötvös Loránd University\nPázmány Péter sétány 11117BudapestHungary" ]	[]	The dichromatic number of a directed graph is at most 2, if we can 2-color the vertices such that each monochromatic part is acyclic. An oriented graph arises from a graph by orienting its edges in one of the two possible directions. We study oriented graphs, which have dichromatic number more than 2. Such a graph D is 3-dicritical if the removal of any arc of D reduces the dichromatic number to 2. We construct infinitely many 3-dicritical oriented graphs. Neumann-Lara found the four 7-vertex 3-dichromatic tournaments. We determine the 8-vertex 3-dichromatic tournaments, which do not contain any of these, there are 64 of them. We also find all 3-dicritical oriented graphs on 8 vertices, there are 159 of them. We determine the smallest number of arcs that a 3-dicritical oriented graph can have. There is a unique oriented graph with 7 vertices and 20 arcs.	null	[ "https://arxiv.org/pdf/2201.13161v1.pdf" ]	246,430,838	2201.13161	b3173df0a6abc716d34ff3337ada94da1340eee5	The horizon of 2-dichromatic oriented graphs Jan 2022 J Barát Mátyás Czett Alfréd Rényi Institute of Mathematics Department of Mathematics University of Pannonia Egyetem utca 108200VeszprémHungary Eötvös Loránd University Pázmány Péter sétány 11117BudapestHungary The horizon of 2-dichromatic oriented graphs Jan 2022 The dichromatic number of a directed graph is at most 2, if we can 2-color the vertices such that each monochromatic part is acyclic. An oriented graph arises from a graph by orienting its edges in one of the two possible directions. We study oriented graphs, which have dichromatic number more than 2. Such a graph D is 3-dicritical if the removal of any arc of D reduces the dichromatic number to 2. We construct infinitely many 3-dicritical oriented graphs. Neumann-Lara found the four 7-vertex 3-dichromatic tournaments. We determine the 8-vertex 3-dichromatic tournaments, which do not contain any of these, there are 64 of them. We also find all 3-dicritical oriented graphs on 8 vertices, there are 159 of them. We determine the smallest number of arcs that a 3-dicritical oriented graph can have. There is a unique oriented graph with 7 vertices and 20 arcs. Preliminaries The in-degree(out-degree) of a vertex in a digraph is the number of incoming(outgoing) arcs. The degree of a vertex in a digraph is the sum of the in-and out-degrees. A k-coloring of a digraph D is a mapping of the vertices to {1, . . . , k} such that each color class is acyclic. To be more perceptible, in the arguments with 2 colors, we usually refer to the colors as red and blue. In what follows, we concentrate on oriented graphs, that is directed graphs without 2-cycles. We also exclude loops and parallel arcs. The dichromatic number χ d of a digraph D is the minimum integer k such that D admits a k-coloring. A digraph D is k-dicritical if χ d (D) ≥ k, but χ d (D ′ ) < k for every proper subdigraph D ′ of D. Let Crit(k) denote the class of k-dicritical digraphs. Now Crit (2) consists of all circuits (directed cycles). For undirected graphs, 3-critical graphs are characterized as the odd cycles. However, it is widely open to describe Crit(3) for digraphs. Neumann-Lara described the 3-dichromatic tournaments on 7 vertices in [14]. From this, it is easy to find the 3-dicritical oriented graphs on 7 vertices. The most notable conjecture regarding the dichromatic number of oriented graphs was raised by Neumann-Lara in [13] and independently byŠkrekovski [3]. Conjecture 1.1. Every orientation of a planar graph has dichromatic number at most 2. For instance, the dual of the dodecahedron contains two induced paths P 1 and P 2 , that together cover all vertices. If we color the vertices of P 1 red and the vertices of P 2 blue, then there is no monochromatic cycle. Therefore, any orientation of the icosahedron has dichromatic number at most 2. It is tempting to ask whether the same method works and conclusion holds for the dual of fullerenes (and other graph classes). The vertex-arboricity a(G) of a graph G is the minimum number of subsets into which V (G) can be partitioned so that each subset induces a forest. The above coloring argument works for any planar graph with arboricity at most 2. Stein proved that a planar triangulation T satisfies a(T ) ≤ 2 if and only if G * is Hamiltonian [15]. On the other hand, Kardoš proved that 3-connected planar cubic 1 graphs with faces of size at most 6 are Hamiltonian. These premises hold for fullerenes [9]. That is, duals of fullerenes satisfy Conjecture 1.1. Li and Mohar [11] proved that oriented planar graphs of digirth 4 are 2-dichromatic. Harutyunyan and Mohar [6] proved there exist bounded degree digraphs with large dichromatic number and large girth. The very natural generalisation to list-coloring has been addressed in several papers [2,7]. Our main target was to study the landscape of 2-dichromatic oriented graphs. This landscape has a boundary or horizon. We consider the critical 3-dichromatic graphs in the next Sections. These are the graphs just outside of the landscape. In various related questions, the boundary of the landscape can be described by the saturated graphs of the class. A (di)graph D is saturated in the class C if D belongs to C, but if we add any edge/arc to D, then the resulting graph D ′ does not belong to C. In our context of saturation of oriented graphs, we exclude tournaments since there are no arcs to be added. We fix C as the 2-dichromatic graphs. Here we notice the following interesting phenomenon. Proof. Assume to the contrary that D was 2-dichromatic and adding any arc to D results in an oriented graph that is not 2-dichromatic. We know that the vertices of D can be split into two sets R and B such that each of them induces an acyclic digraph. Let a ′ be a missing arc from u to v in D, and let us add a ′ to D to form D ′ . Split the vertices of D ′ to R and B again. By the assumption, one of the classes, R say, induces a directed cycle C ′ , that necessarily contains the arc a ′ . Let P 1 be the directed path C ′ − a ′ from v to u. Add now the reversed arc a ′′ from v to u in D to form D ′′ . Split the vertices of D ′′ to R and B again. By the assumption, one of the classes induces a directed cycle C ′′ , that necessarily contains the arc a ′′ . We know that u and v were in R, the cycle C ′′ lies also in R. Let P 2 be the directed path C ′′ − a ′′ from u to v. Consider now the union of P 1 and P 2 . It necessarily contains a circuit, that lies entirely in R, so monochromatic. This contradicts that D was 2-dichromatic. Among other things, we show several infinite families of 3-dicritical graphs and determine all of them on at most 8 vertices. We notice here that each infinite class collapses under taking butterfly minors. This concept refers to the following operation, that preserves the directed cycles of an oriented graph, and therefore seems relevant to dicoloring. A butterfly minor of a digraph G is a digraph obtained from a subgraph of G by contracting arcs, which are either the only outgoing arc of their tail or the only incoming arc of their head [8]. So it feels plausible to constrain 3-dicoloring to butterfly minor-minimal graphs rather than just critical graphs. However, the situation might be more subtle 2 . During a butterfly minor operation, deleting edges might result in a 2-dicolorable graph. This might be later corrected by a butterfly contraction. We describe such an example in the penultimate paragraph of Section 2. Recall that a(G) ≤ 2 implies that every orientation D of G satisfies χ d (D) ≤ 2. We simply color the partition classes monochromatic. This way we cannot create a monochromatic circuit, independent of the orientations. Kronk and Mitchem [10] proved that a(G) ≤ ⌈ ∆(G) 2 ⌉ unless G is an odd clique or a cycle. In particular, this implies the following Corollary 1.6. For every 4-regular graph G, any orientation D of G satisfies χ d (D) ≤ 2. Dirac [5] proved that any k-critical graph has minimum degree k − 1. Similarly, the minimum in-and outdegree of a k-dicritical graph is at least k − 1. Therefore, any 3-dicritical graph satisfies δ − ≥ 2 and δ + ≥ 2, and need to have more than 2n edges by Corollary 1.6. The degree condition can be sometimes changed to degeneracy. However, Corollary 1.6 does not generalise to 4-degenerate graphs as Lemma 2.1 shows. Infinitely many critical 3-chromatic digraphs We define a simple 5-vertex gadget S. Let x 1 x 2 x 3 be a directed triangle and ab an arc. Let b dominate each x i and let each x i dominate a. Altogether they form an orientation of K 5 . We glue together 3 copies of S at the 3 vertices of a directed triangle abc such that each copy of S -call them S ab , S bc , S ca -contains one of the arcs ab, bc and ca as also shown in Figure 1. The resulting oriented graph D 1 is 3-dicritical as we show next. Proof. Suppose to the contrary that we 2-color the vertices. Consider the directed triangle abc in the middle. One of the arcs, ab say, is monochromatic, red say. The end-vertices a and b are adjacent to three vertices of a directed triangle, xyz say. If any of x, y, z is red, then a, b together with this red vertex form a red directed triangle. Otherwise xyz is monochromatic blue. This shows that the dichromatic number of D 1 is at least 3. We have to show criticality. By symmetry, we have to consider three cases. Either we delete an arc of the triangle in the middle or an arc of an outer triangle or one that connects the two types. The latter can also go in two directions. Suppose the deleted arc is ab in the middle triangle abc. We color a and b red and c blue. Now there is no monochromatic circuit containing any two of a, b, c as consecutive vertices. We use two colors on the three vertices of any outer triangle. Now, there is no monochromatic circuit in any of the copies of the gadget S. Suppose next the deleted arc was x 1 x 2 in the outer triangle of S ab . We color a and b red and c blue. Now there is no monochromatic circuit containing bc or ca. We use two colors on the three vertices of the outer triangle connected to bc and ca. We color the 3 vertices of the outer triangle in S ab blue. Now, there is no monochromatic circuit in any of the copies of the gadget S. Suppose last that the arc bx 1 is missing in S ab . We color a, b and x 1 red and x 2 , x 3 blue. Therefore, this copy of S contains no monochromatic circuit. We color c blue and use two colors in the remaining two outer triangles. Therefore, there is no monochromatic circuit in the other two gadgets either. We notice the essence of the previous proof, and remark the directed triangle in the middle may be replaced by any odd circuit O and the outer circuits might have any length. The connection between the middle and the outer circuits must remain that each arc of O is contained in a gadget similar to S. For each arc ab of O, the vertices of the outer cycle C ab dominate a and vertex b dominates the vertices of C ab . Corollary 2.2. There are infinitely many non-isomorphic oriented graphs in Crit(3). One might say that these infinitely many graphs are very similar and argue that changing/reducing the length of the outer cycle should be allowed. Indeed, with respect to butterfly minors, this infinite class collapses to one graph. However, since the length of the middle circuit is an arbitrary odd number, we still have infinitely many different 3-dicritical graphs. We describe another construction. Let v 1 v 2 v 3 and v 4 v 5 v 6 form two disjoint directed triangles. We construct D 3 as follows. Let all arcs go from {v 1 , v 2 , v 3 } towards v 7 . Let all arcs go from {v 4 , v 5 , v 6 } towards {v 1 , v 2 , v 3 }. Finally let all arcs go from v 7 towards {v 4 , v 5 , v 6 }. It is easy to check that D 3 belongs to Crit(3). We can generalise the above construction to an infinite family to further support Corollary 2.2. Instead of the two disjoint triangles let x 1 , . . . , x k and y 1 , . . . , y l be two circuits of any length greater than 2. Let v be a special vertex such that v dominates each x i and each y j dominates v. We also draw every arc of form x i y j . The oriented graphs O k,l constructed this way are all 3-dicritical. At the same time, we again notice that this class collapses under the butterfly minor operation. We delete an arc vx 2 . Now x 1 x 2 is the only arc entering x 2 . Therefore, we butterfly contract the arc x 1 x 2 and get a construction as before, but the length of the x-circuit is one less 3 . Notice here the following interesting fact: The graph O k,l − vx 2 is 2-dicolorable. Indeed, let v, x 2 , y 2 , . . . y l be blue and all other vertices red. Now the monochromatic triples containing v, x 2 and y j do not form a circuit any more. However, after the butterfly contraction, O k,l−1 is 3-dichromatic again. Let us delete the matching v 4 v 1 , v 5 v 2 and v 6 v 3 from D 3 and add two arcs v 2 v 5 and v 3 v 6 to get the oriented graph D 2 . One can check that D 2 is 3-dicritical. We can generalise the construction of D 2 as follows. Let u 1 . . . u k and v 1 . . . v k be two disjoint circuits for k ≥ 3. Let x be a vertex that dominates {v 1 , . . . , v k } and each of {u 1 , . . . , u k } dominates x. Add all arcs of form v i u j , where i = j. Also add two arcs u 1 v 1 and u 2 v 2 . One can prove that these oriented graphs are also 3-dichromatic. Critical graphs on eight vertices We wrote a computer program to find each 3-dicritical oriented graph on 8 vertices. We represented each oriented graph as a matrix A with 8 rows and 8 columns labelled from 1 to 8 such that • if ij is an arc of the digraph, then the element in row i and column j is +, the element in row j and column i is -. • if there is no arc between two vertices i and j, then both elements a ij and a ji are 0 (there are only 0s in the main diagonal). We first sought tournaments on 8 vertices, which do not contain any of the three 7-vertex 3-dicritical graphs and cannot be colored with two colors. We started with the 6440 tournaments on 8 vertices and removed vertices one at a time in each possible way. We dropped the 8-vertex tournaments, which contained one of the four tournaments on 7 vertices with dichromatic number 3. From the remaining tournaments, we selected those, which contained a monochromatic circuit for any vertex coloring by two colors. See Algorithm 1. We found 64 tournaments that met the above criteria. Next, we started from the pool of 64 tournaments, and we looked for graphs, which cannot be 2-dicolored, but after removing any arc become 2-dichromatic. Notice, we might obtain the same graph in different forms. Therefore, we checked the digraphs that had the same in-degrees and out-degrees and examined for isomorphism. As a result, we obtained 159 different 3-dicritical oriented graphs on 8 vertices. One of them had 21 arcs, see Figure 2, and all others had more arcs: 11 of them had 22 arcs, 84 had 23, 51 had 24 and 12 had 25 arcs. In various further questions, for instance filtering oriented graphs on at least 9 vertices for 2-dicoloring, the following characterization might be useful. What is the list D 1 , . . . D s of 8-vertex oriented graphs that satisfy the following? If an oriented graph O contains a 3-dicritical tournament on 8 vertices as a subgraph, then it contains at least one of D 1 , . . . D s . We determined this list and found that 9 oriented graphs are enough. It is worth mentioning that the containment is distributed very unevenly. There is a graph, which is contained in only 2 tournaments on 8 vertices. On the other hand, there is another graph, which is contained in 34 tournaments on 8 vertices. The 9 oriented graphs are listed here in matrix form: 0+000-+-0-000++-0+000-+--0000++-+0----0+ -0000++-000+-+--0+0+-+--000+-0-+ 00-0++--0+-0++--00-0--++ 00+-0+--0++-0+--00++0+--+----0++ -+---0++ +-0+-0+---+++-0+ -0+++-0+ --+-+-0+ +++++--0 +-+++--0 ++--++-0 0000-++-0000++--000-+0+-00-+0-+-000+-+--00+00-+-0+00-+-+ 000-+-+-0-00-+-+ 0-00+++-0-+00+-+ +000-++-+0+-0+---+-00++--0++0+---+---0++ --+--0++ 0+---0++ --+-+-0+ ++-+--0+ --+-+-0+ ++-++--0 +++-+--0 ++-++--0 0++000--000-+0+-000+-0+--0+000-+ 00+00-+-00-0+-+---0---++ 0-00-++-0+0-0-++ 00+0-++-+000-+-+ -0+0-+-+ 00++0-+--0++0+-- +-0+0--+ 00+-+0+- 0+---0++ 0++-+0-- ++----0+ ---++-0+ ---+++0- +--+++-0 +++-+--0 ++---++0 Algorithm 1 Do two colors suffice? 1: t:=the adjacency matrix of the oriented graph G 2: for each possible 2-coloring of the vertices of G do for each vertex f in graph G do 8: if ContainsCycle(t, a, m, f ) then Discussion It is a natural test to try strengthening Conjecture 1.1. One way to generalise planar graphs is to allow a few crossings in the planar embedding of G. We know an oriented 3-dicritical graph D, whose underlying graph G has crossing number 3. For instance, the 8-vertex 3-dicritical graph with 21 arcs, see Figure 2. Among the 3-critical digraphs, that we found on 7 and 8 vertices, one had 20 arcs, and three had 21 arcs. By the Dirac-type condition on critical digraphs, we know that the in-degree and the out-degree must be at least 2. Therefore, if the number of vertices is at least 10, then any critical digraph must have at least 20 arcs. By Lemma 1.6, we deduce that the underlying graph cannot be 4-regular either. Therefore, there must be at least 21 arcs if n ≥ 10. If we would like to determine the minimum number of arcs in a 3-dicritical graph, then it remains to check the possible 9-vertex graphs. Again, we deduce that there must be at least 19 edges in the underlying graph. We generated all such undirected graphs using nauty by Brendan McKay [12] and oriented the edges to have minimum in-and out-degree 2. We found 33700 such oriented graphs and checked its 2-colorings to find that all of them have dichromatic number at most 2. Similarly, we generated the oriented graphs with 20 arcs and 9 vertices, which have minimum in-and out-degree 2 using nauty. There were 721603 such oriented graphs. Again, checking the 2-colorings, we found that each of them has dichromatic number at most 2. As a consequence, we deduce Corollary 1 . 2 . 12The dual of any fullerene has dichromatic number at most 2. Result 1 . 3 . 13Saturated 2-dichromatic oriented graphs do not exist. Remark 1. 4 . 4The proof generalises to any number k of colors instead of 2.Therefore, we are left with the following Question 1.5. Which oriented graphs have dichromatic number 3? Which are the critical ones? Lemma 2 . 1 . 21The oriented graph D 1 is a member of Crit(3). It has 12 vertices, 9 of them have degree 4, and 3 vertices have degree 8 in the underlying graph. There are altogether 30 arcs. Figure 1 : 1A 3-dicritical oriented graph using many directed triangles. Figure 2 : 2The 8-vertex 3-critical oriented graph with 21 arcs drawn with 3 crossings. Question 4 . 1 . 41Are there 3-dichromatic graphs with crossing number 1 or 2? Corollary 4. 2 . 2There is only one 3-dicritical oriented graph with 20 arcs. Any other member of Crit(3) must have more arcs. end if 17: end for 18: the graph is not 2-dichromatic Algorithm 2 ContainsCycle(t, a, m, f ) 1: m1:=union(m, f ) 2: for each vertex v in a do 3: if t(f, v)==1 then if m contains v then if ContainsCycle(t, a, m1, v) then end if 14: end for 15: ContainsCycle:=false9: b:=false 10: end if 11: end for 12: end for 13: if b then 14: the graph is 2-dichromatic 15: exit Algorithm 16: 4: 5: ContainsCycle:=true 6: exit Algorithm 7: else 8: 9: ContainsCycle:=true 10: exit Algorithm 11: end if 12: end if 13: 3-regular Also the parity of the length of the circuits might be important. See Lemma 2.1. We can delete the parallel arcs. Hajós and Ore constructions for digraphs. J Bang-Jensen, T Bellitto, T Schweser, M Stiebitz, The Electronic J. Combin. 271J. Bang-Jensen, T. Bellitto, T. Schweser, M. Stiebitz. Hajós and Ore constructions for digraphs. The Electronic J. Combin. 27 (1), (2020). List coloring digraphs. J Bensmail, A Harutyunyan, N K Le, J. 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Planar digraphs of digirth four are 2-colourable. Z Li, B Mohar, SIAM J. Disc. Math. 313Z. Li and B. Mohar. Planar digraphs of digirth four are 2-colourable. SIAM J. Disc. Math. 31(3): 2201-2205, (2018). Practical graph isomorphism II. B D Mckay, A Piperno, J. Symbolic Comput. 60B.D. McKay and A. Piperno. Practical graph isomorphism II. J. Symbolic Comput., 60 94-112, (2014). The Dichromatic Number of a Digraph. V Neumann-Lara, JCTB. 33V. Neumann-Lara. The Dichromatic Number of a Digraph. JCTB 33, 265-270. (1982). https://core.ac.uk/download/pdf/82562368.pdf The 3 and 4-dichromatic tournaments of minimum order. V Neumann-Lara, Discrete Mathematics. 1351-3V. Neumann-Lara. The 3 and 4-dichromatic tournaments of minimum order. Discrete Mathematics Volume 135, Issues 1-3, 233-243, (1994). B-set and planar maps. S K Stein, Pacific J. Math. 37S.K. Stein. B-set and planar maps. Pacific J. Math. 37 217-224, (1971). A Note on Graphs of Dichromatic Number 2. 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[ "The Dijkgraaf-Witten invariants of Seifert 3-manifolds with orientable bases", "The Dijkgraaf-Witten invariants of Seifert 3-manifolds with orientable bases" ]	[ "Haimiao Chen \nBeijing Technology and Business University\nBeijingChina\n" ]	[ "Beijing Technology and Business University\nBeijingChina" ]	[]	We derive a formula for the Dijkgraaf-Witten invariants of orientable Seifert 3-manifolds with orientable bases. *	10.1016/j.geomphys.2016.06.006	[ "https://arxiv.org/pdf/1307.0364v4.pdf" ]	117,937,006	1307.0364	3bbca04a766b570a4190a0d74e79a927ff5f4f61	The Dijkgraaf-Witten invariants of Seifert 3-manifolds with orientable bases 6 Apr 2014 Haimiao Chen Beijing Technology and Business University BeijingChina The Dijkgraaf-Witten invariants of Seifert 3-manifolds with orientable bases 6 Apr 2014arXiv:1307.0364v3 [math.GT] We derive a formula for the Dijkgraaf-Witten invariants of orientable Seifert 3-manifolds with orientable bases. * Introduction Let Γ be a finite group and let ω ∈ Z 3 (BΓ; U(1)) be a degree 3 cocycle. where f (Φ) : M → BΓ is a mapping inducing Φ on the fundamental groups whose homotopy class is determined by Φ, [M] is the fundamental class of M, and , is the pairing H 3 (M; U(1)) × H 3 (M) → U(1). DW invariant is the partition function of a topological quantum field theory which was first proposed by Dijkgraaf-Witten [4] and then further developed by Wakui [15], Freed-Quinn [5], Freed [6]; see also [12,16] and the references therein. This TQFT associates to each closed oriented surface Σ an inner product space Z(Σ), and associates to each compact oriented 3-manifold M with non-empty boundary a vector Z(M) ∈ Z(Σ). Freed [6] generalized DW theory to arbitrary dimension. DW invariant encodes information about the fundamental group and the fundamental class of a manifold, so it will not be surprising if there exists some connection between DW invariant and classical invariants. As an evidence, recently Chen [3] used DW invariants to count homotopy classes of mappings from a closed manifold to a topological spherical space form with given degree. But till now, there are few explicit computations seen in the literature. In dimension 2, Turaev [14] derived the formula for the DW invariants of closed surfaces using projective representations of finite groups, and Khoi [11] extended the result to surfaces with boundary. In dimension 3, Chen [2] gave a formula for the untwisted DW invariant of Seifert 3-manifolds. As another contribution, in this article we compute the general DW invariants of Seifert 3-manifolds with orientable bases. It is not a straight generalization of [2], but involves some fine ingredients. Finally we shall point out that, Hansen [8] had derived a formula for the Reshetikhin-Turaev invariants of Seifert 3-manifolds, for any modular tensor category. In fact, as proved by Freed [7], DW theory is a TQFT of Reshetikhin-Turaev type, the corresponding modular tensor category being the representation category of the twisted quantum double. It could be said that our result is essentially not new. However, our approach unfolds many details and highlights the essence; in contrast, it is by no means intuitive to view DW theory as an RT TQFT. Furthermore, our formula is more explicit and elegant. The content is organized as follows. Section 2 consists of an exposition of DW theory. Section 3 is devoted to deriving a formula for the DW invariants of Seifert 3-manifolds with orientable bases; the result is expressed in terms of irreducible characters of the twisted quantum double of Γ. In Section 4, by number-theoretic computations, we give formulas for the DW invariants based on cyclic groups. Exposition of Dijkgraaf-Witten theory This section is a review of the construction of [6] in dimension 3, but we use more technical language in order to make computation more easier. From now on let us fix a finite group Γ and a degree 3 cocycle ω valued in U(1). We can assume that ω is normalized, i.e., ω(x, y, z) = 1 whenever at least one of x, y, z is the identity e ∈ Γ. The following conventions throughout this article. All manifolds are compact and oriented. For a manifold M, let −M denote the manifold obtained by reversing the orientation of M. Let V be the category of finite-dimensional Hermitian spaces. For a Hermitian space V with inner product (·, ·) V , let V denote its conjugate, namely, the same underlying abelian group equipped with the conjugate scalar multiplication (µ, v) → µv and the conjugate inner product (u, v) V = (v, u) V ; if λ is a positive number, then λ·V is the same underlying vector space as V equipped with the inner product (u, v) λ·V = λ · (u, v) V . Finally, by C we mean the standard 1-dimensional Hermitian space. Preparation For a topological space X, let Π 1 (X) denote the fundamental groupoid of X, and let B(X) denote the category whose objects are functors Π 1 (X) → Γ, where Γ is viewed as a groupoid with a single object, and whose morphisms η : φ → φ ′ are functions η : X → Γ such that η(γ(1))φ([γ]) = φ ′ ([γ])η(γ(0)) for any path γ : [0, 1] → X. It is easy to see that B(X) has finitely many connected components if X is a compact manifold. Given φ ∈ B(X). For a singular 3-simplex σ : ∆ 3 → X, set ω(σ, φ) = ω(φ([σ 0 , σ 1 ]), φ([σ 1 , σ 2 ]), φ([σ 2 , σ 3 ]));(2) for a singular 3-chain ζ = i n(i)σ(i), set ω(ζ, φ) = i ω(σ(i), φ) n(i) .(3) Given a morphism µ : φ → φ ′ . For a singular 2-simplex τ : ∆ 2 → X, set ω(τ, φ, µ) = ω(φ([τ 0 , τ 1 ]), φ([τ 1 , τ 2 ]), µ(τ 2 ))ω(µ(τ 0 ), φ ′ ([τ 0 , τ 1 ]), φ ′ ([τ 1 , τ 2 ])) ω(φ([τ 0 , τ 1 ]), µ(τ 1 ), φ ′ ([τ 1 , τ 2 ])) ;(4) for a singular 2-chain ξ = j m(j)τ (j), set ω(ξ, φ, µ) = j ω(τ (j), φ, µ) m(j) .(5) The DW invariant of a surface Suppose Σ is a closed surface. Let φ ∈ B(Σ). For a singular 2-chain ξ homologous to zero, choose ζ ∈ C 3 (Σ; Z) with ∂(ζ) = ξ, and set ω ξ (φ) = ω(ζ, φ).(6) This is independent of the choice of ζ, because if ∂(ζ ′ ) = ξ too, then ζ ′ − ζ = ∂(ν) for some ν ∈ C 4 (Σ; Z), due to H 3 (Σ; Z) = 0, hence ω(ζ ′ , φ)/ω(ζ, φ) = ω(∂(ν), φ) = 1 since ω is a cocycle. Let S(Σ) be the category whose objects are singular 2-cycles ξ ∈ C 2 (Σ; Z) representing the fundamental class [Σ] ∈ H 2 (Σ), and there is a unique morphism from each object to another. For each 2-cycle ξ ∈ S(Σ), and for each connected component [φ] ∈ π 0 (B(Σ)) which can be regarded as a subcategory of B(Σ), define a functor F ξ [φ] : [φ] → V by sending each φ to C and sending each morphism η : φ → φ ′ to the multiplication by ω(ξ, φ, η). Set L ξ [φ] to be the inverse limit of F ξ [φ] , and set Z ω (Σ, ξ) = [φ]∈π 0 (B(Σ)) 1 #Aut(φ) · L ξ [φ] .(7) For any two 2-cycles ξ, ξ ′ ∈ S(Σ), there is a natural transformation F ξ [φ] → F ξ ′ [φ] given by C = F ξ [φ] (φ) → F ξ ′ [φ] (φ) = C, 1 → ω ξ ′ −ξ (φ);(8) the naturalarity follows from the fact that ω(ξ ′ , φ, η)ω ξ ′ −ξ (φ) = ω ξ ′ −ξ (φ ′ )ω(ξ, φ, η). This defines an isometry G ξ ′ ξ : Z ω (Σ, ξ) → Z ω (Σ, ξ ′ ).(9) Definition 2.1. The DW invariant Z ω (Σ) is the inverse limit of the functor S(Σ) → V which sends ξ to Z ω (Σ, ξ) and sends the unique morphism ξ → ξ ′ to G ξ ′ ξ for any ξ, ξ ′ ∈ S(Σ). Remark 2.2. Concretely, each element of Z ω (Σ) is a family of numbers {ϑ(ξ, φ) : ξ ∈ S(Σ), φ ∈ B(Σ)} such that ϑ(ξ ′ , φ ′ ) = ω(ξ ′ , φ, η)ω ξ ′ −ξ (φ)ϑ(ξ, φ)(10) for any ξ, ξ ′ and any morphism η : φ → φ ′ . Remark 2.3. Suppose S 1 (Σ) is a full subcategory of S(Σ). Let Π S 1 1 (Σ) be the full subcategory of Π 1 (Σ) whose objects are 0-cells appearing as the vertices of the singular simplices appearing in objects of S 1 (Σ), and let B S 1 (Σ) be the category of functors Π S 1 1 (Σ) → Γ. Then Z ω (Σ) is isometric to the space of family of numbers {ϑ(ξ, φ) : ξ ∈ S 1 (Σ), φ ∈ B S 1 (Σ)} satisfying the condition (10) for any ξ, ξ ′ ∈ S 1 (Σ) and any morphism η : φ → φ ′ in B S 1 (Σ). The DW invariant of a 3-manifold Suppose M is a 3-manifold with non-empty boundary. For any ξ ∈ S(∂M) and any φ ∈ B(∂M), if possible, take ζ ∈ C 3 (M; Z) with ∂(ζ) = ξ and take Φ ∈ B(M) with Φ\| ∂M = φ, then ω(ζ, Φ) is independent of the choice of ζ, due to the reason similarly as in the previous subsection, noting that H 3 (M; Z) = 0. If η : Φ → Φ ′ is a morphism, then, by the assumption that ω is a cocycle, we have ω(ζ, Φ ′ ) ω(ζ, Φ) = ω(ξ, φ, η\| ∂M ).(11) Let B φ (M) be the subcategory of B(M) having objects the functors Φ : Π 1 (M) → Γ with Φ\| ∂M = φ and having morphisms the maps η : M → Γ with η\| ∂M ≡ e. Then ω(ζ, Φ) depends only on the connected component of Φ in B φ (M); denote it by ω ξ ([Φ]). From definition and (11) it follows that ω ξ ′ ([Φ ′ ]) = ω(ξ ′ , φ, η)ω ξ ′ −ξ (φ)ω ξ ([Φ])(12) for any ξ, ξ ′ and any morphism η : Φ → Φ ′ . Definition 2.4. The DW invariant Z ω (M) is the family of numbers {Z ω (M)(ξ, φ) : ξ ∈ S(∂M), φ ∈ B(∂M)} with Z ω (M)(ξ, φ) = [Φ]∈π 0 (B φ (M )) ω ξ ([Φ]).(13) Remark 2.5. By definition and (12), the family {Z ω (M)(ξ, φ)} indeed satisfies (10) for any ξ, ξ ′ and any η : φ → φ ′ . The TQFT axioms The following is a restatement of Assertion 4.12 of [6]: Theorem 2.6. (a) (Functorality) Each diffeomorphism f : Σ → Σ ′ induces an isometry f * : Z ω (Σ) → Z ω (Σ ′ ), {ϑ(ξ, φ) : ξ ∈ S(Σ), φ ∈ B(Σ)} →{ϑ(f −1 # (ξ ′ ), f * (φ ′ )) : ξ ′ ∈ S(Σ ′ ), φ ′ ∈ B(Σ ′ )},(14) where f # : C 2 (Σ) → C 2 (Σ ′ ) is the induced isomorphism on the group of singular chains, and f * : B(Σ ′ ) → B(Σ) is the pullback induced by f . For each diffeomorphism F : M → M ′ , we have (∂F ) * (Z ω (M)) = Z ω (M ′ ). (15) (b) (Orientation law) There is a canonical isometry Z ω (−Σ) → Z ω (Σ), (16) {ϑ(ξ, φ) : ξ ∈ S(−Σ), φ ∈ B(−Σ)} → {ϑ(−ξ ′ , φ ′ ) : ξ ′ ∈ S(Σ), φ ′ ∈ B(Σ)},(17)through which Z ω (−M) is sent to Z ω (M). (c) (Multiplicavity) There is a canonical isometry Z ω (Σ ⊔ Σ ′ ) ∼ = Z ω (Σ) ⊗ Z ω (Σ ′ ),(18)through which Z ω (M ⊔ M ′ ) is sent to Z ω (M) ⊗ Z ω (M ′ ). (d) (Gluing law) Suppose M is a 3-manifold with ∂M = −Σ ⊔ Σ ′ ⊔ Υ, and f : Σ → Σ ′ is an orientation-preserving diffeomorphism. LetM be the 3-manifold obtained from gluing M via f , i.e.,M = M/{x ∼ f (x), ∀x ∈ Σ}. Then Z ω (M) = tr f (Z ω (M)),(19) where tr f is the composite Z ω (∂M) ∼ = Z ω (Σ) ⊗ Z ω (Σ ′ ) ⊗ Z ω (Υ) → Z ω (Υ) = Z ω (∂M ), (20) u ⊗ v ⊗ w → (v, f * (u)) Z ω (Σ ′ ) · w.(21) 3 Computing the DW invariants of Seifert 3manifolds 3.1 Some fundamental ingredients 3.1.1 The vector space Z ω (Σ 1 ) Let γ ω h (x 1 , x 2 ) = ω(h, x 1 , x 2 )ω(hx 1 h −1 , hx 2 h −1 , h) ω(hx 1 h −1 , h, x 2 ) ,(22)θ ω x (h 1 , h 2 ) = ω(x, h 1 , h 2 )ω(h 1 , h 2 , h 1 h 2 xh −1 2 h −1 1 ) ω(h 1 , h 1 xh −1 1 , h 2 ) .(23) Let π : R 2 → S 1 ×S 1 = Σ 1 be the universal covering map. In the notation of Remark 2.3, take S 1 (Σ 1 ) to be the category whose objects are singular 2chains of the form i n(i)(π •σ(i)) where eachσ(i) is a triangle enclosed by segments connecting integral points of R 2 . We write π •σ as [A, B, C] if σ 0 = A, σ 1 = B, σ 2 = C, and similarly for 3-simplex. Then B S 1 (Σ 1 ) is just the category whose objects are homomorphisms φ : π 1 (Σ 1 , 1 × 1) → Γ and whose morphisms η : φ → φ ′ are elements h such that φ ′ (x) = hφ(x)h −1 for all x ∈ π 1 (Σ 1 , 1 × 1). Note that a homomorphism φ : π 1 (Σ 1 , 1 × 1) → Γ is determined by its values at the meridian mer := S 1 × 1 and the longitude long := 1 × S 1 ; write φ as φ x,h if φ(mer) = x and φ(long) = h. Let ξ 0 denote [(0, 0), (0, 1), (1, 1)] − [(0, 0), (1, 0), (1, 1)] ∈ S 1 (Σ 1 ). Let E be the vector space of functions ϑ : {(x, h) ∈ Γ 2 : xh = hx} → C such that ϑ(h ′ xh ′−1 , h ′ hh ′−1 ) = γ ω h ′ (x, h) γ ω h ′ (h, x) ϑ(x, h) for all h ′ ∈ Γ, and equipped with the inner product (ϑ, ϑ ′ ) E = 1 #Γ · x,h : xh=hx ϑ(x, h)ϑ ′ (x, h).(24) Then Z ω (Σ 1 ) ∼ = E; a canonical isometry is given by sending a family {ϑ(ξ, φ)} to the function which takes the value ϑ(ξ 0 , φ x,h ) at (x, h). There is a canonical orthonormal basis of E which we recall from [1]. Let Conj(Γ) denote the set of conjugacy classes of Γ, for each A ∈ Conj(Γ), choose a representative x A ∈ A. Let N A be the centralizer of x A in Γ, and let R( N A , θ ω x A ) denote the set of irreducible θ ω x A -projective representation of N A . For each ρ ∈ R(N A , θ ω x A ), let χ ρ ∈ E be the function χ ρ (x, h) = δ x∈A θ ω x (h, h ′ ) θ ω x (h ′ , h) tr(ρ(h ′−1 hh ′ )),(25) where h ′ is any element making x = h ′ x A h ′−1 , and define dim χ ρ = x∈A χ ρ (x, e).(26) Let Λ ω = ⊔ A∈Conj(Γ) R(N A , θ ω x A ).(27) Then {χ ρ : ρ ∈ Λ ω } forms an orthonormal basis. We pay special attention to cyclic groups. Let Γ = Z/mZ, and let Z/mZ → {0, 1, · · · , m − 1}, x →x(28) be the obvious map. By Proposition 2.3 of [9] , a complete set of representatives of 3-cocycles can be given by ω l : (Z/mZ) 3 → U(1), (x, y, z) → ζ lx[ỹ +z m ] m , l = 0, · · · , m − 1,(29) where ζ m = exp( 2π √ −1 m ) and [k] mean the largest integer not greater than k. Suppose θ ∈ Z 2 (B(Z/mZ; U(1)) is a 2-cocycle. Since H 2 (B(Z/mZ; U(1)) = 0, there exists a function α : Z/mZ → U(1) such that θ(g, h) = α(g)α(h) α(g + h) . If ρ : Z/mZ → GL(r, C) is an irreducible θ-projective representation of Z/mZ, thenρ := α −1 · ρ is an irreducible ordinary representation of Z/mZ, hence r = 1, and there exists s ∈ Z such thatρ(k) = ζ sk m . Thus ρ(k) = α(k)ζ sk m . For θ = θ l h with θ l h (x, y) = ω l (h, x, y)ω l (x, y, h) ω l (x, h, y) = ζ lh[x +ỹ m ] m ,(30) we can take α = α l h with α l h (x) = ζ lhx m 2 ,(31) so that ω l (h, x, y) = α l h (x)α l h (y) α l h (x + y) .(32) According to [1] Proposition 8, the irreducible characters of D ω l (Z/mZ) are χ l h,s , h ∈ Z/mZ, s = 0, · · · , m − 1, with χ l h,s (x, y) = δ h,x · α l h (y)ζ sỹ m = δ h,x · ζ lhỹ+msỹ m 2 .(33) Algebraic structures on Z ω (Σ 1 ) For any diffeomorphism f : Σ 1 → Σ 1 , by (10) and (14), for each ϑ ∈ Z ω (Σ 1 ), we have (f * (ϑ))(ξ, φ) = ϑ(f −1 # (ξ), f * (φ)) = ω f −1 # (ξ)−ξ (f * (φ)) · ϑ(ξ, f * (φ)).(34) It is well known that the mapping class group of Σ 1 is isomorphic to SL(2, Z). If the isotopy class of f corresponds to a b c d , then f * (φ x,h ) = φ x a h b ,x c h d .(35) Combined with (34), it follows that for each ϑ ∈ E, (f * (ϑ))(x, h) = ω f −1 # (ξ 0 )−ξ 0 (φ x a h b ,x c h d ) · ϑ(x a h b , x c h d ).(36)Let S = 0 −1 1 0 , then S −1 = 0 1 −1 0 . We have (S −1 * ϑ)(x, h) = ω S # (ξ 0 )−ξ 0 (φ h,x −1 ) · ϑ(h, x −1 ) (37) = γ h (x, x −1 ) −1 · ϑ(h, x −1 ).(38) In particular, (S −1 * χ ρ )(x, h) = γ h (x, x −1 ) −1 χ ρ (h, x −1 ) = χ ρ (h, x).(39) Let P be the pair of pants, viewed as a cobordism from S 1 ⊔S 1 to S 1 , then P × S 1 is a cobordism from Σ 1 ⊔ Σ 1 to Σ 1 , hence Z ω (P × S 1 ) is a morphism E ⊗ E → E, which turns out to be the multiplication mul : E ⊗ E → E,(40)mul(ϑ 1 ⊗ ϑ 2 )(x, h) = x 1 x 2 =x γ ω h (x 1 , x 2 )ϑ 1 (x 1 , h)ϑ 2 (x 2 , h).(41) It can be proved that mul(S −1 * χ ρ ⊗ S −1 * χ ρ ′ ) = δ ρ,ρ ′ #Γ dim χ ρ · S −1 * χ ρ .(42) Cycle calculus on torus In order to compute the term ω f −1 # (ξ)−ξ (f * φ) appearing in (36), it is necessary to clarify some rules to which we give the name "cycle calculus on torus". Recall (2) ([A, B, B ′ ] − [A, A ′ , B ′ ]) + ([B, C, C ′ ] − [B, B ′ , C ′ ]) − ([A, C, C ′ ] − [A, A ′ , C ′ ]) =∂([A, B, C, C ′ ] + [A, A ′ , B ′ , C ′ ] − [A, B, B ′ , C ′ ]),(45) and from Figure 1 (b) it follows that ([A, A ′ , B ′ ] − [A, B, B ′ ]) + ([B, B ′ , C ′ ] − [B, C, C ′ ]) − ([A, A ′ , C ′ ] − [A, C, C ′ ]) =∂([A, B, B ′ , C ′ ] − [A, A ′ , B ′ , C ′ ] − [A, B, C, C ′ ]). Formula for Seifert 3-manifolds with orientable bases Recall that (see [13]), a closed orientable Seifert 3-manifold M can be obtained as follows. Take a circle bundle S 1 → F → R where F is orientable as a manifold and R is a surface ∂R = ⊔ n S 1 , so that ∂F = ⊔ n Σ 1 ; glue n copies of solid torus ST := D × S 1 onto F along the boundary Σ 1 's, via diffeomorphisms f j : Σ 1 → Σ 1 , j = 1, · · · , n. The "closure" of R, R = R ∪ (⊔ n D), is the base-surface, and the images of the cores of ST, S 1 ×0, are the exceptional fibers. When f j lies in the mapping class of Σ 1 represented by a j b j a ′ j b ′ j , and R is orientable/non-orientable of genus g, denote M as M O (g; (a 1 , b 1 ), · · · , (a n , b n ))/M N (g; (a 1 , b 1 ), · · · , (a n , b n )). In this article we only pay attention to M O (g; (a 1 , b 1 ), · · · , (a n , b n )). The computation of the DW invariant of M N (g; (a 1 , b 1 ), · · · , (a n , b n )) turns out to be more complicated. U(1)) → H 3 (B z ; U(1)) ∼ = Z/mZ, where ι z : z ֒→ Γ is the inclusion. Put κ ω a,b (z) = ζ ℓ(bǎ−ab) m 2 ,(50) whereǎ ∈ {0, · · · , m − 1} is the residue of a modulo m. Proof. Assume L is horizontal; the case when L is vertical is similar. Without loss of generality we can assume the horizontal coordinate of A is 0, then those of B, C, D are a, a + b, b, respectively. If ab > 0, suppose r i−2 = k i r i−1 +r i , 1 i ν with k 1 0, k 2 , · · · , k ν > 0, and r −1 = b, r 0 = a, r ν = 0. For j = 0, 1, · · · , k 1 + 1, let C j denote the point on L with horizontal coordinate b + (1 − j)a, so C 0 = C, C 1 = D. It is easy to verify that ([A, B, C j−1 ] − [A, C j , C j−1 ]) − ([A, B, C j ] − [A, C j+1 , C j ]) =∂([A, B, C j , C j−1 ]). It follows that [A, B, C] − [A, D, C] = [A, B, C k 1 ] − [A, C k 1 +1 , C k 1 ] + ∂( k 1 j=1 [A, B, C j , C j−1 ]), hence ω [A,B,C]−[A,D,C] (φ z ) = (ω [A,C k 1 +1 ,C k 1 ]−[A,B,C k 1 ] ) −1 · k 1 j=1 ω(z a , z b−ja , z a ). By induction on ν, one can show that ω [A,B,C]−[A,D,C] (φ z ) = ν i=1 k i j=1 ω(z r i−1 , z r i−2 −jr i−1 , z r i−1 ) (−1) i−1 = ν i=1 k i j=1 ω ℓ (r i−1 z, (r i−2 − jr i−1 )z, r i−1 z) (−1) i−1 ,(52) where in the second equation we adopt the additive notation for the cyclic group z . By (32), k i j=1 ω ℓ (r i−1 z, (r i−2 − jr i−1 )z, r i−1 z) = [α ℓ r i−1 z (r i−1 z)] k i α ℓ r i−1 z (r i z) α ℓ r i−1 z (r i−2 z) , hence by (52), ω [A,B,C]−[A,D,C] (φ z ) = ν i=1 [α ℓ r i−1 z (r i−1 z)] k i α ℓ r i−1 z (r i z) α ℓ r i−1 z (r i−2 z) (−1) i = exp 2ℓπ √ −1 m 2 · ν i=1 (−1) i r i−1 z(k i r i−1 z + r i z − r i−2 z) = exp 2ℓπ √ −1 m 2 · ν i=1 (−1) i r i−1 (t i−2 − t i − k i t i−1 )mz (53) = exp 2ℓπ √ −1 m · ν i=1 (−1) i (r i−1 t i−2 − r i−1 t i − r i−2 t i−1 + r i t i−1 )z (54) =ζ ℓ(r −1 t 0 −r 0 t −1 )z m , where in (53) we set r i z = r iz − t i m, in (54) we use the identity r i−2 = k i r i−1 + r i , and in the last equation we use r ν = t ν = 0. By further computation, we obtain ω [A,B,C]−[A,D,C] (φ z ) = ζ ℓz(b az−a bz) m 2 = κ ω ℓ a,b (z).(55) If ab < 0, suppose r i−2 = k i r i−1 +r i , 1 i ν with k 1 0, k 2 , · · · , k ν > 0, and r −1 = b, r 0 = −a, r ν = 0. Applying (47): ω [A,B,C]−[A,D,C] (φ z )ω [B,A,D]−[B,C,D] (φ z ) = ω(z −a , z a , z b−a )ω(z b+a , z −a , z a ) ω(z a , z b , z −a ) , one can reduce this case to the above, replacing (32) by κ ω a,b (z) = ω(z −a , z a , z b−a )ω(z b+a , z −a , z a ) ω(z a , z b , z −a ) · ν i=1 k i j=1 ω(z r i−1 , z r i−2 −jr i−1 , z r i−1 ) (−1) i . Then the same formula (55) can be deduced; we mention that ω l (−az, az, (b − a)z)ω l ((b + a)z, −az, az) ω l (az, bz, −az) = ζ lbz m . Let Σ g;n,1 denote the surface with genus g and n + 1 small disks removed, regarded as a cobordism from ⊔ n S 1 to S 1 . Similarly as in [2], the morphism Z ω (Σ g;n,1 × S 1 ) is the linear map E ⊗n → E, S −1 * χ ρ 1 ⊗ · · · ⊗ S −1 * χ ρn → ( #Γ dim χ ρ 1 ) n+2g−1 δ ρ 1 ,··· ,ρn S −1 * χ ρ 1 .(56) Since Z ω (ST)(x, h) = δ x,e , by (36), (f j ) * (Z ω (ST))(x, h) = ω (f j ) −1 # (ξ 0 )−ξ 0 (φ e,x a ′ j h b ′ j ) · δ x a j h b j ,e(57) In Figure 3, let O = (0, 0), A = (b ′ j , −b j ), C = (−a ′ j , a j ). Then (f j ) −1 # (ξ 0 ) = [O, C, B] − [O, A, B] . By (49), (45), (46), it is easy to see that ω (f j ) −1 # (ξ 0 )−ξ 0 (φ e,x a ′ j h b ′ j ) = ω [E,C,F ]−[E,A ′ ,F ] (φ e,x a ′ j h b ′ j ),(58)which is equal to κ ω a j ,−b j (x a ′ j h b ′ j ) by Lemma 3.2, thus (f j ) * (Z ω (ST)) = 1 #Γ · ρ    xh=hx x a j h b j =e κ ω a j ,−b j (x a ′ j h b ′ j )(S −1 * χ ρ )(x, h)    · S −1 * χ ρ . For integers a, b with (a, b), define η ω ρ (a, b) = z∈Γ κ ω a,−b (z)χ ρ (z a , z −b ).(59) By the proof of Lemma 3.1 of [2], xh=hx x a j h b j =e κ ω a j ,−b j (x a ′ j h b ′ j )(S −1 * χ ρ )(x, h) = η ω ρ (a j , b j ), hence (f j ) * (Z ω (ST )) = 1 #Γ · ρ η ω ρ (a j , b j ) · S −1 * χ ρ . Modifying the proof of Theorem 3.3 of [2], one can establish Theorem 3.3. We have the following formula: Z ω (M O (g; (a 1 , b 1 ), · · · , (a n , b n ))) = ρ∈Λ ω (#Γ) 2g−2 (dim χ ρ ) n+2g−2 · n j=1 η ω ρ (a j , b j ). Explicit computation in the case of cyclic group Continue to use the notations in Section 3.1.1. By the definition (59), η ω l h,s (a, b) = z:az=h ζ labz 2 −(2lh+ms)bz m 2 ;(60) notice that the right-hand side is unchanged if we replacez withz + rm for any r. For each j ∈ {1, · · · , n}, let d j = (a j , m), and let a ′ j = a j /d j , m j = m/d j . Let c j be the unique integer such that 0 c j < m j and a ′ j c j ≡ 1 (mod m j ); suppose a ′ j c j = 1 + t j m j . Let d = [d 1 , · · · , d n ], and leth ′ =h/d, h j =h/d j = h ′ d/d j . We have η ω l h,s (a j , b j ) = d j −1 k=0 ζ b j [la j (km j +c jhj ) 2 −(2lh j d j +ms)(km j +c jhj )] m 2 (61) = ζ −lc jh 2 j mm j · ζ (lt j −s)c jhj m · d j −1 k=0 ζ la ′ j k 2 +(2h j t j −s)k d j .(62) For a prime number p, let ( a p ) denote the Legendre symbol (see Page 51 of [10]), and let g 1 (χ) = Then S p (a) = 1 2 (1 + ( a p )g 1 (χ)) when (a, p) = 1, and S p (a) = p when p\|a. Thus (62) can be computed using the Legendre symbols and Gauss sums. However, the expression is very complicated in general. For simplicity, from now on we assume that m = p is an odd prime number. Suppose p ∤ a j for all j, let c j ∈ {1, · · · , p 2 } be the unique solution of the congruence equation a j c j ≡ 1 (mod p 2 ), then Suppose p\|a j 0 for some j 0 , then and Z ω l (M O (g; (a 1 , b 1 ), · · · , (a n , b n ))) =p 2g−2 ( p−1 s=0 ζ −s 2 n 0 j=1 b j 4la ′ j p ) · n 0 j=1 S p (la ′ j b j ) =p 2g−2 S p (− n 0 j=1 b j 4la ′ j ) · n 0 j=1 S p (la ′ j b j ).(73) Summarizing the above computations, we get Theorem 4.1. Suppose the indices j ∈ {1, · · · , n} are arranged such that p\|a j , p 2 ∤ a j for j ≤ n ′ 0 , p 2 \| a j for n ′ 0 < j ≤ n 0 , and p ∤ a j for j > n 0 ; for j ≤ n 0 , suppose a j = pa ′ j . The DW invariant of M = M O (g; (a 1 , b 1 ), · · · , (a n , b n )) is given as follows: • if n 0 = 0, then • if n ′ 0 = n 0 > 0, then Z ω (M) = p 2g−2 S p (− n 0 j=1 b j 4la ′ j ) · n 0 j=1 S p (la ′ j b j ); • if n ′ 0 < n 0 , then Z ω (M) = p 2g−2 p n 0 −n ′ 0 n ′ 0 j=1 S p (la ′ j b j ). For a closed oriented 3-manifold M, the Dijkgraaf-Witten invariant of M is defined to be Z(M) = 1 #Γ · Φ:π 1 (M )→Γ f (Φ) * [ω], [M] , , (52),(6) and the assumption that ω is normalized. Write[A, B, C] = [A ′ , B ′ , C ′ ] if for any ξ homologous to zero such that [A, B, C] appears in ξ and any φ ∈ B S 1 (Σ 1 ), ω ξ (φ) is unchanged if [A, B, C] is replaced by [A ′ , B ′ , C ′ ]. Now we have ∂[B, A, A, C] = [A, A, C] − [B, A, A], ∂[A, A, B, C] = [A, A, C] − [A, A, B], ∂[C, B, A, A] = [B, A, A] − [C, A, A], hence [A, A, B] = [A, A, C] = [B, A, A] = [C, A, A]. , A, B] = [A ′ , A ′ , B], (44) since [A ′ , A ′ , B ′ ] = [A, A, B ′ − A ′ + A]. Figure 1 :Figure 2 : 12From Figure 1 (a) it follows that B, C, D] − [B, A, D]) + ([A, D, C] − [A, B, C]) =∂([A, B, C, D] − [A, C, D, C] − [B, A, B, D]), (47) ([D, A, B] − [D, C, B]) + ([A, D, C] − [A, B, C]) =∂([A, D, A, B] + [A, B, C, B] − [A, D, C, B]).(48) Figure 3 : 3From Figure 3 we see that [O, C, B] − [O, A, B] =([O, O ′ , B] − [O, B ′ , B]) − ([O, O ′ , F ] − [O, E, F ]) − ([D, G, B] − [D, B ′ , B]) + ([A, G, B] − [O, E, C]) − ([C, F, B] − [O, D, A]) + ([E, C, F ] − [E, A ′ , F ]) + ([O ′ , F, B] − [O, D, B ′ ]) + ∂([O, D, A, B] + [O, C, F, B] − [O, O ′ , F, B] − [O, D, B ′ , B] − [O, E, C, F ] − [D, A, G, B]).(49) Notation 3 . 1 . 31For z ∈ Γ, let m be the order of z and let ℓ ∈ Z/mZ be the image of [ω] under the pull-back homomorphism ι * z : H 3 (BΓ; Lemma 3. 2 . 2Suppose A, B, C, D are integral points on a line L which is horizontal or vertical, and suppose AB = DC = a, BC = AD = b. Let φ z = φ z,e if L is horizontal and φ z = φ e,z if L is vertical. 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[ "Does Flexoelectricity Drive Triboelectricity?", "Does Flexoelectricity Drive Triboelectricity?" ]	[ "C A Mizzi \nDepartment of Materials Science and Engineering\nAngewandte Chemie\nNorthwestern University\n60208Evanston, PhysicalIL\n", "A Y W Lin \nDepartment of Materials Science and Engineering\nAngewandte Chemie\nNorthwestern University\n60208Evanston, PhysicalIL\n", "L D Marks [email protected]. \nDepartment of Materials Science and Engineering\nAngewandte Chemie\nNorthwestern University\n60208Evanston, PhysicalIL\n" ]	[ "Department of Materials Science and Engineering\nAngewandte Chemie\nNorthwestern University\n60208Evanston, PhysicalIL", "Department of Materials Science and Engineering\nAngewandte Chemie\nNorthwestern University\n60208Evanston, PhysicalIL", "Department of Materials Science and Engineering\nAngewandte Chemie\nNorthwestern University\n60208Evanston, PhysicalIL" ]	[]	The triboelectric effect, charge transfer during sliding, is well established but the thermodynamic driver is not well understood. We hypothesize here that flexoelectric potential differences induced by inhomogeneous strains at nanoscale asperities drive tribocharge separation. Modelling single asperity elastic contacts suggests that nanoscale flexoelectric potential differences of ±1-10 V or larger arise during indentation and pull-off. This hypothesis agrees with several experimental observations, including bipolar charging during stick-slip, inhomogeneous tribocharge patterns, charging between similar materials, and surface charge density measurements.	10.1103/physrevlett.123.116103	[ "https://arxiv.org/pdf/1904.10383v1.pdf" ]	128,361,741	1904.10383	fb870c3dea9978b0eaa3092af15137c698174d6b	Does Flexoelectricity Drive Triboelectricity? C A Mizzi Department of Materials Science and Engineering Angewandte Chemie Northwestern University 60208Evanston, PhysicalIL A Y W Lin Department of Materials Science and Engineering Angewandte Chemie Northwestern University 60208Evanston, PhysicalIL L D Marks [email protected]. Department of Materials Science and Engineering Angewandte Chemie Northwestern University 60208Evanston, PhysicalIL Does Flexoelectricity Drive Triboelectricity? 1 * These authors contributed equally to this work Corresponding Author 1 To whom correspondence should be addressed. The triboelectric effect, charge transfer during sliding, is well established but the thermodynamic driver is not well understood. We hypothesize here that flexoelectric potential differences induced by inhomogeneous strains at nanoscale asperities drive tribocharge separation. Modelling single asperity elastic contacts suggests that nanoscale flexoelectric potential differences of ±1-10 V or larger arise during indentation and pull-off. This hypothesis agrees with several experimental observations, including bipolar charging during stick-slip, inhomogeneous tribocharge patterns, charging between similar materials, and surface charge density measurements. The triboelectric effect, the transfer of charge associated with rubbing or contacting two materials, has been known for at least twenty-five centuries [1,2]. The consequences of this transfer are known to be beneficial and detrimental; for instance, tribocharging is widely exploited in technologies such as laser printers but can also cause electrostatic discharges that lead to fires. It is accepted that it involves the transfer of charged species, either electrons [3][4][5], ions [6,7], or charged molecular fragments [8], between two materials. The nature and identification of these charged species has been the focus of considerable research [2,9], but an important unresolved issue is the thermodynamic driver for charge transfer; the process of separating and transferring charge must reduce the free energy of the system. What is the charge transfer driver? In some cases specific drivers are well understood. For instance, when two metals with different work functions are brought into contact charge transfer will occur until the chemical potential of the electrons (Fermi level) is the same everywhere. Triboelectric charge transfer in insulators is less understood; proposed models include local heating [10] and trapped charge tunneling [11][12][13] but these models do not explicitly address the significant mechanical deformations associated with bringing two materials into contact and rubbing them together. Furthermore there is currently little ab-initio or direct numerical connection between experimental measurements and proposed drivers. Since the pioneering work of Bowden and Tabor [14] it has been known that friction and wear at the nanoscale is associated with adhesion between, as well as the elastic and plastic deformation of, a statistical population of asperities. It is also well established that elastic deformation is thermodynamically linked to polarization: the linear coupling between strain and polarization is the piezoelectric effect and the linear coupling between strain gradient and polarization is the flexoelectric effect [15][16][17]. While piezoelectric contributions only occur for materials without an inversion center, flexoelectric contributions occur in all insulators and can be large at the nanoscale due to the intrinsic size scaling of strain gradients [17][18][19]. Quite a few papers have analyzed the implications of these coupling terms in phenomena including nanoindentation [20,21] and fracture [22]. There also exists literature where the consequences of charging on friction have been studied [23][24][25]. However, triboelectricity, flexoelectricity, and friction during sliding are typically considered as three independent phenomena. Are they really uncoupled phenomena? In this paper we hypothesize that the electric fields induced by inhomogeneous deformations at asperities via the flexoelectric effect lead to significant surface potentials differences, which can act as the driver for triboelectric charge separation and transfer. The flexoelectric effect may therefore be a very significant, and perhaps even the dominant, thermodynamic driver underlying triboelectric phenomena in many cases. To investigate this hypothesis in detail we analyze, within the conventional Hertzian [26] and Johnson-Kendall-Roberts (JKR) [27] contact models, the typical surface potential differences around an asperity in contact with a surface during indentation and pull-off. We find that surface potential differences in the range of ±1-10 V or more can be readily induced for typical polymers and ceramics at the nanoscale, and that the intrinsic asymmetry of the inhomogeneous strains during indentation and pull-off changes the sign of the surface potential difference. We argue that our model is consistent with a range of experimental observations, in particular bipolar tribocurrents associated with stick-slip [28], the scaling of tribocurrent with indentation force [29], the phenomenon of tribocharging of similar materials [30][31][32][33], and the inhomogeneous charging of insulators [34,35]. Taking the analysis a step further, our model suggests a suitable upper bound for the triboelectric surface charge density is the flexoelectric polarization that is found to be in semi-quantitative agreement with published experimental data without the need to invoke any empirical parameters. Given the recent ab-initio developments of flexoelectric theory [36][37][38][39], we argue that flexoelectricity can provide an ab-initio understanding of many triboelectric phenomena. Nanoscale asperity contact consists of two main phenomena, indentation and pull-off, which are illustrated in Fig. 1. To investigate the electric fields arising from the strain gradients associated with these two processes, we combine the constitutive flexoelectric equations with the classic Hertzian and JKR models. As discussed further in the Supplemental Material, the normal component of the electric field induced by a flexoelectric coupling in an isotropic non-piezoelectric half plane oriented normal to ̂ is given by: $ = − () ($ * +,, = − -3ϵ 000 + ϵ 022 + ϵ 202 + ϵ 220 + ϵ 033 + ϵ 303 + ϵ 330 4 (1) where $ is the electric field linearly induced by During indentation and pull-off the elastic body will deform, developing a net strain gradient opposite to the direction of the applied force (F). First, we will analyze the indentation case. Because of the axial symmetry of Hertzian indentation, only five strain gradient components in Equation (1) Material) and depicted in Fig. 2(a)-(e) as contour plots. From these plots it is evident that the strain gradient components have complex spatial distributions, the details of which depend on the materials properties of the deformed body (Young's modulus, Poisson's ratio) as well as external parameters (applied force, indenter size). Further insight can be gained by calculating the average effective strain gradient within the indentation volume, which is taken to be the cube of the deformation radius. The average effective strain gradient is negative and scales inversely with indenter size, independent of the materials properties of the deformed body and the applied force. The former is intuitive since a material deformed by an indenter should develop a curvature opposite to the direction of the applied force, and the latter is a consequence of averaging (Supplemental Material). As shown in Fig. 2(f), the average effective strain gradient associated with Hertzian indentation is on the order of -10 8 m -1 in all materials at the nanoscale. Such large strain gradients immediately suggest the importance of flexoelectric couplings [17,18]. For pull-off we use JKR theory, which incorporates adhesion effects between a spherical indenter and an elastic half-space into the Hertz contact model. The tensile force required to separate the indenter from the surface, also known as the pull-off force, can be written as ?@A = − B C(2) where Δγ is the adhesive energy per unit area and R is the radius of the spherical indenter. Replacing the applied force in the Hertzian indentation strain gradient expressions with this force yields pull-off strain gradients immediately before contact is broken. This analysis for the pull-off case yields strain gradient distributions qualitatively similar to those shown in Fig. 2, except with opposite signs because the force is applied in the opposite direction. Importantly, as in the indentation case, the average effective strain gradient within the pull-off volume scales inversely with indenter size, is independent of the materials properties of the deformed body, and is on the order of 10 8 m -1 in all materials at the nanoscale. tends to be larger in magnitude and spatial extent than the indentation surface potential difference. In both cases the magnitude of the maximum surface potential difference is sensitive to the \|E z \| (V/m) (b) R (nm) materials properties of the deformed body (Young's modulus, Poisson's ratio, adhesion energy, flexocoupling voltage) and external parameters (applied force, indenter size). Specifically, the surface potential differences for indentation and pull-off scale as LM@+MN?NLOM,QLM ∝ − S T U V W X Y/B (3) [\99]O,,,Q?< ∝ S^_ U W X Y/B(4) where LM@+MN?NLOM,QLM is the minimum surface potential difference for indentation, [\99]O,,,Q?< is the maximum surface potential difference for pull-off, is the flexocoupling voltage, is the applied force, is the indenter radius, is the Young's modulus, and Δ is the energy of adhesion. The above analysis indicates that large strain gradients arising from deformations by nanoscale asperities yield surface potential differences via a flexoelectric coupling in the ±1-10 V range, as a conservative estimate. The magnitude of this surface potential difference is sufficient to drive charge transfer, suggesting that flexoelectric couplings during indentation and pull-off can be responsible for triboelectric charging. Furthermore, this model implies that the direction of charge transfer is controlled by a combination of the direction of the applied force and local topography (i.e. is the asperity indenting or pulling-off), as well as the sign of the flexocoupling voltage. These features are consistent with and can explain a significant number of previous triboelectric observations without introducing any adjustable parameters. First, it has been observed that tribocurrents exhibit bipolar characteristics associated with stick-slip [28]. This bipolar nature is consistent with the change in the sign of the surface potential difference for indentation and pull-off predicted by our model. Second, the tribocurrent has been shown to scale with the indentation force to the power of Y B [29], which matches the scaling of the indentation surface potential difference with force. Thirdly, charging between similar materials [30][31][32][33] and the formation of non-uniform tribocharge patterns [34,35] can be explained by considering the effect of local surface topography and crystallography on the direction of charge transfer: local variation in surface topography dictates which material locally acts as the asperity, and consequently the direction in which charge transfers. In addition, it is established for crystalline materials that both the magnitude and sign of the flexocoupling voltage can change with crystallographic orientation (Supplemental Table 1). Going beyond these qualitative conclusions, it is relevant to explore whether flexoelectricity can quantitatively explain experimental triboelectric charge transfer measurements. An important quantitative parameter in the triboelectric literature is the magnitude of triboelectric surface charge density which has been measured in a number of systems including spherical particles [33,40] and patterned triboelectric devices [41,42], and normally enters models as an empirical parameter [43,44]. We hypothesize that the upper bound for the triboelectric surface charge density is set by the flexoelectric polarization, i.e. charge will transfer until the flexoelectric polarization is screened (Supplemental Material). As shown in Table 1, this hypothesis agrees with existing tribocharge measurements on a range of length scales to within an order of magnitude without invoking anomalous flexoelectric coefficients. In addition, the formalism we have used is not limited to inorganic materials, but is quite general. As one extension it is known that semi-crystalline layers are formed at the confined spaces during sliding in a lubricant [45], so it is not unreasonable that flexoelectric effects can drive charge separation in lubricants. Another extension is biological materials, as flexoelectric effects in biological membranes are well-established [46]. We also note the magnitude of the flexoelectricity-induced electric fields and surface potential differences at asperities (and crack tips [22]) suggest flexoelectricity can play a role in triboluminescence [47][48][49], triboplasma generation [50] or tribochemical reactions. Such hypotheses merit further work. In summary, using the Hertz and JKR models for indentation and pull-off, we show that Department of Materials Science and Engineering Northwestern University, Evanston, IL 60208 * These authors contributed equally to this work Corresponding Author 1 To whom correspondence should be addressed. Email: [email protected]. S1. General details Flexoelectricity, tribology, and triboelectricity are separate subdisciplines, each with their own terminology and literature. Since this paper discusses the cross-connections between the three some general discussion of flexoelectricity and tribology is provided here for readers who are less familiar with them. The flexoelectric effect describes the linear coupling between polarization and strain gradient. Because flexoelectricity relates strain gradient (a third-rank tensor) and polarization (a first-rank tensor), the flexoelectric effect is a fourth-rank tensor property described by a coupling coefficient known as the flexoelectric coefficient. Flexoelectricity was first observed in solids by Bursian and Zaikovskii [1], but the term flexoelectricity was coined in the field of liquid crystals [2]. Flexoelectric characterization of relaxor ferroelectrics by Cross and Ma in the early 2000s [3] rekindled interest in this subject in the context of oxides, and since then there has been significant experimental [4][5][6][7] and theoretical [8][9][10][11][12][13][14] progress in understanding flexoelectricity. While piezoelectric effects are well established, they only occur for crystallographies which do not have an inversion center. In contrast, flexoelectric effects occur in all insulators independent of the crystallography. The coupling is a fourth-order tensor which can conveniently be scaled by the dielectric coefficient; this is known as the flexocoupling voltage tensor. For many materials flexocoupling voltages are in the range of 1-10 V, although there are cases where it is significantly larger, for instance ~40 V for DyScO3 [15]. Both the sign and the magnitude of the flexocoupling voltage in a given material can depend strongly upon the crystallographic orientation [16], and a few experimental values are given in Table S1 for oxides and Table S2 for polymers. It is known that in some cases the effective flexocoupling voltage can be anomalously large if there is coupling to charged defects [4] in the material or other effects [17]. This is an area of continuing research. Some recent reviews can be found in references [3,16,18]. Table S1. A few examples illustrating how the magnitude and sign of the flexoelectric coefficient change with crystallographic orientation in oxides. All measurements were made in a three-point bending geometry with orientation corresponding to the bending direction. Note, the flexoelectric coefficient reported here is a linear combination of tensor coefficients. For more details, see [5,15,16]. Table S2. Some examples illustrating the magnitude of the flexoelectric coefficient and flexocoupling voltage in polymers. All measurements were made in a cantilever geometry and the sign of the coefficient was not specified. For more details, see [16,19]. While the classic continuum laws of friction have been known for centuries, since the pioneering work of Bowden and Tabor it has been established that they are statistical averages over many asperity contacts. In many cases, single asperities do not follow the statistical macroscopic laws of friction, as reviewed in [20]. While there remains some debate about the exact mechanisms of energy dissipation in sliding contact, for instance the importance of electron coupling [21] versus movement of misfit dislocations [22,23], collective motion of dislocations [24,25] or local chemistry [26,27], the general nature is well understood. The consequences of tribocharging on increasing friction has also been explored in the literature [28][29][30]. S2. Flexoelectricity in an isotropic non-piezoelectric material The constitutive equation for flexoelectricity in a non-piezoelectric dielectric material is L = L789 () :; (< = + L7 7 (S1) where L is the dielectric displacement, L789 is the flexoelectric coefficient, S3. Hertzian strain gradients from Hertzian stresses For a non-adhesive single asperity contact, the Hertzian contact model [31] is used to describe the deformation mechanics of a rigid sphere on an elastic flat half-space. The Hertzian model assumes the materials in contact to be homogeneous and isotropic, and deformations are perfectly elastic and governed by classical continuum mechanics (Hooke's law). Furthermore, the Young's modulus Y and Poisson's ratio ν are also assumed to be constant under load. The stress fields given by the Hertzian contact model with a spherical indenter have been thoroughly analyzed elsewhere [32], so we will merely use the results of this model to derive analytical expressions for the Hertzian strain gradient fields. In cylindrical coordinates, stresses in the bulk of a Hertzian deformed elastic half-space are given by: rr Q = 3 2 u 1 − 2 3 C C z1 − S Y/C X B \| + S Y/C X B C C + C C + Y/C } 1 − C + + (1 + ) Y/C tan ]Y S Y/C X − 2ƒ" (S7) ‡ ‡ Q = − 3 2ˆ1 − 2 3 C C ‰1 − Š Y C ‹ B OE + Y C ‰2 + 1 − C + + (1 + ) Y C tan ]Y Š Y C ‹OE• (S8) $$ Q = − 3 2ˆŠ Y C ‹ B C C + C C • (S9) r$ Q = − 3 2 } C C + C C ƒ ‰ C Y C C + OE (S10) where = 1 2 •( C + C − C ) + (( C + C − C ) C + 4 C C ) Y/C ' (S11) Q = C (S12) = } 3 4 (1 − C )ƒ Y/B (S13) In these expressions, is the applied force, is the deformation radius, is the Young's modulus, is the Poisson's ratio, and and are cylindrical coordinates. There is no dependence is these formulas because of radial symmetry. These stresses were related to strain using the isotropic Hooke's law in cylindrical coordinates: rr = 1 rr − ( ‡ ‡ + $$ )4 (S14) ‡ ‡ = 1 - ‡ ‡ − ( rr + $$ )4 (S15) $$ = 1 -$$ − ( ‡ ‡ + rr )4 (S16) r$ = 2(1 + ) r$ (S17) Then, these cylindrical strains were transformed into Cartesian strains using the transformation matrix = Š cos (θ) −sin (θ) 0 sin (θ) cos(θ) 0 0 0 1 ‹ (S18) Lastly, these expressions were differentiated to determine expressions for relevant strain gradient components. The strain gradients that can couple to the normal component of the electric field as derived in S1 are $$$ , $<< , <<$ , <$< , $oeoe , oeoe$ , and oe$oe . Each of these strain gradients are functions of , , , , , and . Additionally, because of the axial symmetry of this problem, oe$oe = $oeoe and <$< = $<< . S4. Pull-off Model The adhesion between contacts has been extensively studied by two groups: Johnson, Kendall, and Roberts (JKR) [33] and Derjaguin, Muller, and Toporov (DMT) [34]. The JKR theory, which incorporates adhesion effects due to the increased contact area caused by the elastic surface into the Hertzian contact model, is used to model pull-off. In the JKR theory, long-range adhesive interactions outside the contact area are neglected so it has been shown that this model is appropriate for systems with compliant materials that have high adhesion and large indenter radii [35]. Additionally, the JKR theory will also predict a non-zero contact area even in the absence of applied loads, resulting in a tensile load required to separate the two adhered surfaces. The pulloff force is the maximum magnitude of this tensile load. In contrast, the DMT theory includes adhesion interactions outside the contact area by considering long-range adhesive interactions such as van der Waals forces. Consequently, the DMT theory is better for approximating systems with stiff materials that have weak, long-range adhesion and small indenter radii. In reality, most physical systems will fall between the JKR and DMT limits, which are described quantitatively by Tabor's parameter • [35]: • = } (1 − C ) C C O B ƒ Y B (S19) where R is the radius of the spherical indenter, ∆ is the adhesive energy per unit area, ν is the • is calculated to be 0.53, showing that the JKR theory may not yield the best approximation. However, since we are not using the full JKR theory to model deformations and their induced polarizations and only using the pull-off force expression, this analysis demonstrates that the adhesive tensile loads predicted by the JKR theory is sufficient for the purposes of this Letter. S5. Average strain gradient for indentation and pull-off Recall the effective strain gradient, defined as +,, = 3ϵ 000 + ϵ 022 + ϵ 202 + ϵ 220 + ϵ 033 + ϵ 303 + ϵ 330 (S20) in the main text, linearly induces an electric field via the flexoelectric effect. It is comprised of a number of strain gradient components, each with complex spatial distributions. Therefore, to get a sense of the overall magnitude and impact of the effective strain gradient, it is convenient to average it. A natural choice of integration volume is the deformation volume defined as B , where is the deformation radius. III ¡ +,, = 1 B ¢ +,,(S21) This is particularly convenient because for indentation, is a function of materials parameters via . Therefore, averaging over the deformation volume effectively removes all dependences except for the indenter radius. This is confirmed numerically. Moreover, since $ = − +,,(S22) and is a constant, it follows that an average electric field can be defined as $ III = − III ¡ +,,(S23) which is also independent of materials properties and applied parameters except the indenter size. S6. Comparison between indentation and pull-off flexoelectric responses To The surface potential differences above are also roughly linear with (1 − C ), but this proportionality is not exact. S8. Surface charge density and flexoelectric polarization This model was developed under the assumption there is no free charge present to screen the polarization/electric field arising from strain gradients via a flexoelectric coupling. In reality, free charge will be present (e.g. from bulk defects, surface defects, or nearby air and water) and is only a function of indenter size making $ a function of indenter size and the flexoelectric coefficient . Unfortunately, the flexoelectric coefficient is not a wellcharacterized materials property, so for this analysis a typical value of = 10 -9 C/m is assumed [16]. strain gradient. The proportionality constant is the flexocoupling voltage (i.e., the flexoelectric coefficient divided by the dielectric constant) and the effective strain gradient is the sum of the symmetry-allowed strain gradient components (where 789 = Fig. 1 . 1Schematic of asperity contact between a rigid sphere (blue) and an elastic body (red). Fig. 2 .) 2(a) -(e) Symmetrically inequivalent strain gradients arising from Hertzian indentation of an elastic half-space that can flexoelectrically couple to the normal component of the electric field. Lines indicate constant strain gradient contours in units of 10 6 m -1 , is the direction normal to the surface with positive values going into the bulk, is an in-plane direction, and the origin is the central point of contact. Data corresponds to 1 nN of force (a conservatively small number) applied to an elastic half-space with a Young's modulus of 3 GPa and a Poisson's ratio of 0.3 (typical polymer) by a 10 nm rigid indenter. (f) The magnitude of the average effective strain as a function of indenter radius ( ). The average effective strain gradient corresponds to a sum of the strain gradient components shown in (a) -(e) averaged over the indentation/pull-off volumes. We now turn to the flexoelectric response to these deformations. Obtaining analytical expressions for the normal component of the electric field in the deformed body induced by indentation and pull-off involves substituting the strain gradient components shown in Fig. 2 into Equation (1). This electric field component is shown in Fig. 3 for the indentation case with a positive flexocoupling voltage. The pull-off case is similar, but the signs of the electric fields are reversed. Because the electric field induced by the flexoelectric effect is the effective strain by the flexocoupling voltage, its magnitude is linearly proportional to the flexocoupling voltage and inversely proportional to the indenter size. The average electric field in the indentation/pull-off volume is on the order of 10 8 -10 9 V/m for all materials at the nanoscale assuming a conservative flexocoupling voltage of 1 V.Fig. 3. (a) Normal component of the electric field induced by Hertzian indentation via a flexoelectric coupling. Lines indicate constant electric field contours in units of MV/m, is the direction normal to the surface with positive values going into the bulk, is an in-plane direction, and the origin is the central point of contact. Data corresponds to 1 nN of force applied to an elastic half-space with a Young's modulus of 3 GPa and a Poisson's ratio of 0.3 (typical polymer) by a 10 nm indenter. A flexocoupling voltage of 1 V is assumed. (b) Magnitude of the average electric field (\| $ III \|) in the indentation/pull-off volumes as a function of indenter radius ( ) assuming a flexocoupling voltage of 1 V (dashed) and 10 V (solid). The electric fields induced by the flexoelectric effect in the bulk of the deformed body will generate a potential on its surface. Fig. 4 depicts the surface potential difference calculated from the normal component of the electric field (Supplemental Material) along the deformed surface of a typical polymer with a flexocoupling voltage of 10 V. The pull-off surface potential difference Fig. 4 . 4Electric potential difference along the surface of the deformed body for indentation (solid) and pull-off (dashed). is an in-plane direction and the origin is the central point of contact. Data corresponds to 1 nN of force applied to an elastic half-space with a Young's modulus of 3 GPa, a Poisson's ratio of 0.3, adhesion energy of 0.06 N/m (typical polymer), and flexocoupling voltage of 10 V by a 10 nm indenter. 1 . 1Comparison between measured triboelectric surface charge (σtribo) and calculated flexoelectric polarization (PFxE) for feature sizes in the mm to µm range assuming a flexoelectric coefficient of 1 nC/m. These results make a strong case that the flexoelectric effect drives triboelectric charge separation and transfer, and that nanoscale friction, flexoelectricity, and triboelectricity occur simultaneously and are intimately linked: macroscopic forces during sliding on insulators cause local inhomogeneous strains at contacting asperities which induce significant local electric fields which in turn drive charge separation. This analysis does not depend upon the details of the charge species, they may be electrons, polymeric ions, charged point defects in oxides, or some combination. Hence our model does not contradict any of the existing literature on the nature of the charge species, instead it provides a thermodynamic rationale for the charge separation to occur. Our analysis also suggests one can optimize charge separation: assuming pull-off dominates, based upon equation (4) one wants a relatively soft material with a high flexocoupling voltage, large adhesion, and many small asperities. deformations by nanoscale asperities yield surface potential differences via a flexoelectric coupling in the ±1-10 V range or more, large enough to drive charge separation and transfer. The direction and magnitude of the surface potential differences depend on the applied force, asperity size, local topography, and material properties. These findings explain some previous tribocharging observations and we argue are the first steps towards an ab-initio understanding of triboelectric phenomena. Supplemental Material: Does Flexoelectricity Drive Triboelectricity? C. A. Mizzi * , A. Y. W. Lin * and L. D. Marks 1 ) strain gradient, L7 is the dielectric constant, 7 is the electric field, and subscripts are Cartesian directions using the Einstein convention. In this work, we will assume the material is isotropic which greatly reduces the number of non-trivial components of L789 and L7 . we assume the non-trivial components of the isotropic flexoelectric coefficient tensor are approximately the same soL789 = -L7 89 + L8 79 + L9 78 4 (S4) In the absence of surface charge the normal component of the dielectric displacement vanishes. = j ⋅ mm⃗ = 0 (S5) Taking j = q and combining the surface charge condition with the constitutive equation for a nonpiezoelectric, isotropic dielectric material yields an expression for the normal component of the electric field induced by a flexoelectric coupling. expression, the symmetry-allowed strain gradients are denoted explicitly ( coefficient normalized by the dielectric constant has been replaced by the flexocoupling voltage . Poisson's ratio, Y is the Young's modulus, and O is the equilibrium separation distance. Using the materials properties for a typical polymer (Y = 3 GPa, ν = 0.3, ∆ = 0.06 N/m) and setting zo to the bond length of a C-C bond (1.54 Å), • is calculated as 0.97, which is in between the JKR and DMT limits but closer to the JKR limit. For typical ceramics (Y = 250 GPa, ν = 0.3, ∆ = 2 N/m), a function of materials parameters and the applied force via . Similarly, for pull-off model the pull-off case, the Hertz expressions for the deformation radius and pressure are replaced with JKR expressions. Namely, The net effect on the induced electric field is demonstrated below in Fig. S1 for a typical polymer ( = 3 GPa, = 0.3, ∆ = 0.06 N/m, and = 10 V) contacted by a rigid sphere with radius = 10 nm and an indentation force = 1 nN. This plot depicts the magnitude of the normal component of the electric field at the central point of contact ( = 0, = 0) as a function of depth into the bulk of the deformed body ( ). From this plot, it is apparent that besides the change in the sign, the main difference between the pull-off and indentation electric fields is their spatial distribution. Fig. S1 . S1Comparison between normal component of the electric field in the indentation and pulloff cases. Data corresponds to the normal component of the electric field at the central point of contact ( = 0, = 0) as a function of depth into the bulk of the deformed body ( ) for typical polymer ( = 3 GPa, = 0.3, ∆ = 0.06 N/m, and = 10 V) contacted by a rigid sphere with radius = 10 nm and an indentation force = 1 nN.S7. Surface potential difference: calculation and scaling relationshipsThe electric fields induced by the flexoelectric effect in the bulk of the deformed body will also generate a potential on its surface. This flexoelectric surface potential difference can becalculated from the normal component of the electric field via ©\r,?ª+ ( , ) way to characterize the size of this surface potential difference is the magnitude of the minimum surface potential difference during indentation and maximum surface potential difference during pull-off. These values correspond to ©\r,?ª+ ( = 0, = 0). The set of figures below demonstrate how the indentation and pull-off surface potential differences scale with materials properties and external parameters. They were obtained by calculating the indentation and pull-off surface potential differences while varying one property/parameter with all other terms held constant. Power-law fits used to determine the scaling behavior are shown in red in Fig. S2 and S3. The end results are summarized in the expressions LM@+MN?NLOM,QLM ∝ − Fig. S2 .Fig. S3 . S2S3Scaling of the magnitude of the minimum surface potential difference during indentation with applied force (F), indenter radius (R), Young's modulus (Y), flexocoupling voltage (f), and Poisson ratio ( ). Surface potential differences are calculated numerically (blue squares) by varying one quantity while keeping all other parameters constant (constant values are black text in each plot). Red lines show fits to the calculated values and the equation of fit is in red text. Scaling of the magnitude of the maximum surface potential difference during pull-off with adhesion energy (∆ ), indenter radius (R), Young's modulus (Y), flexocoupling voltage (f), and Poisson ratio ( ). Surface potential differences are calculated numerically (blue squares) by varying one quantity while keeping all other parameters constant (constant values are black text in each plot). Red lines show fits to the calculated values and the equation of fit is in red text. tend to accumulate on the surface of the deformed body to screen the polarization developed via the flexoelectric effect. Therefore, an estimate for upper bound of the surface charge density is the value of the flexoelectric polarization. In both the indentation and pull-off cases, the average polarization in the deformation volume is related to the average effective strain gradient in Fig. S4 . S4Average flexoelectric polarization (\| $ \| )as a function of indenter size (R) assuming a flexoelectric coefficient of 1 nC/m. in units of C/m 2 and e/nm 2 . are symmetrically inequivalent.Expressions for these components are derived from classic Hertzian stresses (see SupplementalIndentation Pull-off Pre-contact F F Table W R Harper, Contact and Frictional Electrification. OxfordOxford University PressW. R. Harper, Contact and Frictional Electrification (Oxford University Press, Oxford, 1967). . 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[ "p-ADIC GKZ HYPERGEOMETRIC COMPLEX", "p-ADIC GKZ HYPERGEOMETRIC COMPLEX" ]	[ "Lei Fu ", "DAQINGPeigen Li ", "Hao Zhang " ]	[]	[]	To a torus action on a complex vector space, Gelfand, Kapranov and Zelevinsky introduce a system of differential equations, which are now called the GKZ hypergeometric system. Its solutions are GKZ hypergeometric functions. We study the p-adic counterpart of the GKZ hypergeometric system. The p-adic GKZ hypergeometric complex is a twisted relative de Rham complex of over-convergent differential forms with logarithmic poles. It is an overholonomic object in the derived category of arithmetic D-modules with Frobenius structures. Traces of Frobenius on fibers at Techmüller points of the GKZ hypergeometric complex define the hypergeometric function over the finite field introduced by Gelfand and Graev. Over the nondegenerate locus, the GKZ hypergeometric complex defines an over-convergent F -isocrystal. It is the crystalline companion of the ℓ-adic GKZ hypergeometric sheaf that we constructed before. Our method is a combination of Dwork's theory and the theory of arithmetic D-modules of Berthelot.	null	[ "https://arxiv.org/pdf/1804.05297v2.pdf" ]	119,571,577	1804.05297	863defc702e7070d2438785a98f6f552cbe6396c	p-ADIC GKZ HYPERGEOMETRIC COMPLEX Sep 2021 Lei Fu DAQINGPeigen Li Hao Zhang p-ADIC GKZ HYPERGEOMETRIC COMPLEX Sep 2021GKZ hypergeometric F -isocrystalarithmetic D-moduleDwork trace formula Mathematics Subject Classification: Primary 14F30; Secondary 11T2314G1533C70 To a torus action on a complex vector space, Gelfand, Kapranov and Zelevinsky introduce a system of differential equations, which are now called the GKZ hypergeometric system. Its solutions are GKZ hypergeometric functions. We study the p-adic counterpart of the GKZ hypergeometric system. The p-adic GKZ hypergeometric complex is a twisted relative de Rham complex of over-convergent differential forms with logarithmic poles. It is an overholonomic object in the derived category of arithmetic D-modules with Frobenius structures. Traces of Frobenius on fibers at Techmüller points of the GKZ hypergeometric complex define the hypergeometric function over the finite field introduced by Gelfand and Graev. Over the nondegenerate locus, the GKZ hypergeometric complex defines an over-convergent F -isocrystal. It is the crystalline companion of the ℓ-adic GKZ hypergeometric sheaf that we constructed before. Our method is a combination of Dwork's theory and the theory of arithmetic D-modules of Berthelot. Introduction 0.1. The GKZ hypergeometric system. Let A =    w 11 · · · w 1N . . . . . . w n1 · · · w nN    be an (n×N )-matrix of rank n with integer entries. Denote the column vectors of A by w 1 , . . . , w N ∈ Z n . It defines an action of the n-dimensional torus T n Z = Spec Z[t ±1 1 , . . . , t ±1 n ] on the N -dimensional affine space A N Z = Spec Z[x 1 , . . . , x N ]: T n Z × A N Z → A N Z , (t 1 , . . . , t n ), (x 1 , . . . , x N ) → (t w11 1 · · · t wn1 n x 1 , . . . , t w1N 1 · · · t wnN n x N ). Let γ 1 , . . . , γ n ∈ C. In [27], Gelfand, Kapranov and Zelevinsky define the A-hypergeometric system to be the system of differential equations where for the second system of equations, (λ 1 , . . . , λ N ) ∈ Z N goes over the family of integral linear relations N j=1 λ j w j = 0 We would like to thank Jiangxue Fang for helpful discussions. The research is supported by the NSFC.. 1 among w 1 , . . . , w N . We now call the A-hypergeometric system the GKZ hypergeometric system. An integral representation of a solution of the GKZ hypergeometric system is given by f (x 1 , . . . , x N ) = Σ t γ1 1 · · · t γn n e N j=1 xj t w 1j 1 ···t w nj n dt 1 t 1 · · · dt n t n (0. 1.2) where Σ is a real n-dimensional cycle in T n . Confer [1, equation (2.6)], [21, section 3] and [25, Corollary 2 in §4.2]. 0.2. The GKZ hypergeometric function over a finite field. Let p be a prime number, q a power of p, F q the finite field with q elements, ψ : F q → Q * a nontrivial additive character, and χ 1 , . . . , χ n : F * q → Q * multiplicative characters. In [24] and [26], Gelfand and Graev define the hypergeometric function over the finite field F q to be the function defined by the family of twisted exponential sums Hyp(x 1 , . . . , x N ) = t1,...,tn∈F * q χ 1 (t 1 ) · · · χ n (t n )ψ N j=1 x j t where (x 1 , . . . , x N ) varies in A N (F q ). It is an arithmetic analogue of the expression (0.1.2). In [22], we introduce the ℓ-adic GKZ hypergeometric sheaf Hyp which is a perverse sheaf on A N Fq such that for any rational point x = (x 1 , . . . , x N ) ∈ A N (F q ), we have Hyp(x 1 , . . . , x N ) = (−1) n+N Tr(Frob x , Hypx), (0. 2.2) where Frob x is the geometric Frobenius at x. In this paper, we study the crystalline companion of the ℓ-adic GKZ hypergeometric sheaf. 0.3. The p-adic GKZ hypergeometric complex. For any v = (v 1 , . . . , v N ) ∈ Z N ≥0 and w = (w 1 , . . . , w n ) ∈ Z n , write x v = x v1 1 · · · x vN N , t w = t w1 1 · · · t wn n , \|v\| = v 1 + · · · + v N , \|w\| = \|w 1 \| + · · · + \|w n \|. Fix an algebraic closure Q p of Q p . Throughout this paper, all finite extensions of Q p are taken inside Q p . Denote by \| · \| the p-adic norm on Q p defined by \|a\| = p −ordp(a) . Let π ∈ Q p be an element satisfying π p−1 + p = 0, K a finite extension of Q p containing π, R the discrete valuation ring of K, and κ = F q the residue field of R. For each real number r > 0, consider the algebras K{r −1 x} = { v∈Z N ≥0 a v x v : a v ∈ K, \|a v \|r \|v\| are bounded}, K r −1 x = { v∈Z N ≥0 a v x v : a v ∈ K, lim \|v\|→∞ \|a v \|r \|v\| = 0}. They are Banach K-algebras with respect to the norm v∈Z N ≥0 a v x v r = sup \|a v \|r \|v\| . We have K r −1 x ⊂ K{r −1 x}. Elements in K r −1 x are exactly those power series converging in the closed polydisc D(0, r + ) N = {(x 1 , . . . , x N ) : x i ∈ Q p , \|x i \| ≤ r}. Moreover, for any r < r ′ , we have K{r ′−1 x} ⊂ K r −1 x ⊂ K{r −1 x}. Let K x † = r>1 K{r −1 x} = r>1 K r −1 x . K x † is the ring of over-convergent power series, that is, series converging in closed polydiscs of radii > 1. The main goal of this paper is to construct the GKZ hypergeometric complex C · (L † ) of K x † -modules with connections ∇, and a Frobenius structure given by a horizontal morphism F : Fr * (C · (L † ), ∇) → (C · (L † ), ∇), where Fr : D(0, 1 + ) N → D(0, 1 + ) N is the morphism (x 1 , . . . , x N ) → (x q 1 , . . . , x q N ). See Propositions 0.4 and 0.5. For anyā = (ā 1 , . . . ,ā N ) ∈ A N (F q ), let a = (a 1 , . . . , a N ) ∈ D(0, 1 + ) N be its Techmüller lifting. In Theorem 0.15, we prove the following p-adic analogue of the formula (0.2.2): Hyp(ā 1 , . . . ,ā N ) = Tr q n F −1 a , C · (L † ) ⊗ L K x † K(a) , where K(a) = K is regarded as a K x † -algebra via the homomorphism K x † → K, x i → a i . Over the non-degenerate locus U which we will make precise later, we prove in Theorem 0.11 that H i (C · (L † ))\| U = 0 for i = n, and (H n (C · (L † ))\| U , ∇) is an over-convergent F -crystal. The connection ∇ makes C · (L † ) a complex of arithmetic D-modules defined by Berthelot. In Proposition 0.9, we prove C · (L † ) is over-holonomic in the sense of Caro. We now give the detailed construction. Let δ be a rational convex polyhedral cone in R n containing w 1 , . . . , w N . We often work either with the the maximal case where δ = R n , or the minimal case where δ = {λ 1 w 1 + · · · + λ N w N : λ j ≥ 0} is the rational convex polyhedral cone in R n generated by {w 1 , . . . , w N }. For any real numbers r > 1 and s > 1, define L(r, s) = { w∈Z n ∩δ a w (x)t w : a w (x) ∈ K{r −1 x}, a w (x) r s \|w\| are bounded} = { v∈Z N ≥0 , w∈Z n ∩δ a vw x v t w : a vw ∈ K, \|a vw \|r \|v\| s \|w\| are bounded}, L † = r>1, s>1 L(r, s) Note that L(r, s) and L † are rings. Let γ = (γ 1 , . . . , γ n ) be an n-tuple of rational numbers, and let F (x, t) = N j=1 x j t w1j 1 · · · t wnj n , t γ = t γ1 1 · · · t γn n . Consider the logarithmic twisted de Rham complex C · (L † ) defined as follows: We set C k (L † ) = { 1≤i1<···<i k ≤n f i1...i k (x, t) dt i1 t i1 ∧ · · · ∧ dt i k t i k : f i1...i k (x, t) ∈ L † } with differential d : C k (L † ) → C k+1 (L † ) given by d(ω) = t γ exp(πF (x, t)) −1 • d t • t γ exp(πF (x, t)) (ω) = d t ω + n i=1 γ i + N j=1 πw ij x j t wj dt i t i ∧ ω for any ω ∈ C k (L † ), where d t is the exterior differentiation with respect to the t variable. Let Ω 1 K x † be the free K x † -module with basis dx 1 , . . . , dx N . We have an integrable connection ∇ : C · (L † ) → C · (L † ) ⊗ K x † Ω 1 K x † defined by ∇(ω) = t γ exp(πF (x, t)) −1 • d x • t γ exp(πF (x, t)) (ω) = d x ω + πd x F ∧ ω = d x ω + N j=1 πt wj dx j ∧ ω, where d x is the exterior differentiation with respect to the x variable. Since d x commutes with d t , ∇ commutes with d : C k (L † ) → C k+1 (L † ), and hence ∇ induces connections on cohomology groups of C · (L † ). Consider the Frobenius map in the variable t defined by Φ(f (x, t)) = f (x, t q ). One verifies directly that Φ(L(r, s)) ⊂ L(r, q √ s) and hence Φ(L † ) ⊂ L † . It induces maps Φ : C k (L † ) → C k (L † ) on differential forms commuting with d t : Φ 1≤i1<···<i k ≤n f i1...i k (x, t) dt i1 t i1 ∧ · · · ∧ dt i k t i k = 1≤i1<···<i k ≤n q k f i1...i k (x, t q ) dt i1 t i1 ∧ · · · ∧ dt i k t i k Suppose furthermore that (q − 1)γ ∈ Z n ∩ δ. Consider the maps F : C k (L † ) → C k (L † ) defined by F = t γ exp(πF (x, t)) −1 • Φ • t γ exp(πF (x q , t)) (0.3.1) = t (q−1)γ exp πF (x q , t q ) − πF (x, t) • Φ. (0.3.2) Even though t γ exp(πF (x, t)) does not lie in L † and multiplication by it does not define an endomorphism on C · (L † ), Proposition 0.4 (i) below shows that exp πF (x q , t q ) − πF (x, t) lies in L † , and hence the expression (0.3.2) shows that F defines an endomorphism on each C k (L † ). Proposition 0.4. (i) Let w = max(\|w 1 \|, . . . , \|w N \|). Both exp πF (x q , t q )−πF (x, t) and exp πF (x, t)−πF (x q , t q ) lie in L(r, r − 1 w p p−1 pqw ) for any 1 < r ≤ p p−1 pq . (ii) Suppose (q − 1)γ ∈ Z n ∩ δ. Consider the K-algebra homomorphism Fr ♮ : K x → K x , x j → x q j . Let Fr * (C · (L † ), ∇) = (C · (L † ), ∇) ⊗ K x † ,Fr ♮ K x † . Then F defines a horizontal morphism of complexes of K x † -modules with connections F : Fr * (C · (L † ), ∇) → (C · (L † ), ∇). Let Ψ ′ be the substitution t → t 1 q : Ψ ′ w a w (x)t w = w a w (x)t w q , and let pr : K x † 1 q Z n ∩ δ → K x † [[Z n ∩ δ]] be the K x † -linear map defined by pr(t w ) = t w if w ∈ Z n ∩ δ, 0 otherwise. Consider the operator Ψ : L † → L † defined by Ψ = pr • Ψ ′ . We extend Ψ ′ and Ψ to differential forms by Ψ ′ 1≤i1<···<i k ≤n f i1...i k (x, t) dt i1 t i1 ∧ · · · ∧ dt i k t i k = 1≤i1<···<i k ≤n q −k f i1...ij (x, t 1 q ) dt i1 t i1 ∧ · · · ∧ dt i k t i k , Ψ 1≤i1<···<i k ≤n f i1...i k (x, t) dt i1 t i1 ∧ · · · ∧ dt i k t i k = 1≤i1<···<i k ≤n q −k Ψ(f i1...ij (x, t)) dt i1 t i1 ∧ · · · ∧ dt i k t i k . They commute with d t . Suppose (1 − q)γ ∈ Z n ∩ δ. Define G : C · (L † ) → C · (L † ) by G = pr • t γ exp(πF (x q , t)) −1 • Ψ ′ • t γ exp(πF (x, t)) = Ψ • t (1−q)γ exp πF (x, t) − πF (x q , t q ) . By Proposition 0.4 (i), exp πF (x, t) − πF (x q , t q ) lies in L † and hence G defines an operator on C · (L † ). Proposition 0.5. (i) Suppose (1 − q)γ ∈ Z n ∩ δ. Then G defines a horizontal morphism of complexes of K x †modules with connections G : (C · (L † ), ∇) → Fr * (C · (L † ), ∇). (ii) Suppose ±(1 − q)γ ∈ Z n ∩ δ. We have G • F = id. Moreover, F and G induce isomorphisms on cohomology groups inverse to each other. Definition 0.6. Suppose ±(1 − q)γ ∈ Z n ∩ δ. The p-adic GKZ hypergeometric complex is defined to be the tuple (C · (L † ), ∇, F ) consisting of the complex C · (L † ) of K x † -modules with the connection ∇ and the Frobenius structure defined by the horizontal morphism F : Fr * (C · (L † ), ∇) → (C · (L † ), ∇). In the rest of this paper, we assume δ = {λ 1 w 1 + · · · + λ N w N : λ j ≥ 0}, and we assume either γ = 0 or 0 lies in the interior of the convex hull of {0, w 1 , . . . , w N } so that δ = R n and ±(1 − q)γ ∈ Z n ∩ δ. 0.7. The GKZ hypergeometric D † -module. We study the GKZ hypergeometric complex using the arithmetic D-module theory. Let P N R be the formal projective space obtained by taking the formal completion of the projective space P N R along the special fiber. We often omit the subscript R in P N R for convenience. Let D † P N ,Q (∞) be the sheaf of differential operators of finite levels and possibly of infinite orders with over-convergent poles along the ∞ divisor. For the definition of this sheaf, see [8] or 4.6. By [29], we have Γ(P N , D † P N ,Q (∞)) = D † , where D † = r>1, s>1 { v∈Z N ≥0 f v (x) ∂ v v ! : f v (x) ∈ K{r −1 x}, f v (x) r s \|v\| are bounded}, ∂ v = ∂ v 1 +···+v N ∂x v 1 1 ···∂x v N N and v! = v 1 ! · · · v N ! for any v = (v 1 , . . . , v N ) ∈ Z N ≥0 . By the result in [29], D † is a coherent ring. Let ∂ ∂xj ∈ D † act on L † via ∇ ∂ ∂x j = t γ exp(πF (x, t)) −1 • ∂ ∂x j • t γ exp(πF (x, t)) = ∂ ∂x j + πt wj . Proposition 0.8. L † is a coherent left D † -modules. Thus C · (L † ) is a complex of coherent D † -modules. Denote by D b coh (D † P N ,Q (∞)) the derive category of complexes of D † P N ,Q (∞)-(L † ) defines an object in D b coh (D † P N ,Q (∞)), which we denote by C · (L † ). Caro [15, Définition 3.1] defines D b ovhol (D † P N ,Q (∞)) as the subcategory of D b coh (D † P N ,Q (∞)) con- sisting of over-holonomic complexes. Denote the full subcategory of D b ovhol (D † P N ,Q (∞)) consisting of those objects with Frobenius structures by F -D b ovhol (D † P N ,Q (∞)). (Confer [10, Définition 5.1.1]) Proposition 0.9. C · (L † ) is an object in F -D b ovhol (D † P N ,Q (∞)). 0.10. The GKZ hypergeometric F -isocrystal. Let F i,γ = t γ exp(πF (x, t)) −1 • t i ∂ ∂t i • t γ exp(πF (x, t)) = t i ∂ ∂t i + γ i + π N j=1 w ij x j t wj . It follows from the definition of the logarithmic twisted de Rham complex that the homomorphism L † → C n (L † ), f (x, t) → f (x, t) dt 1 t 1 ∧ · · · ∧ dt n t n induces an isomorphism L † / n i=1 F i,γ L † ∼ = H n (C · (L † )). ∇ defines a connection on H n (C · (L † )), and F defines a horizontal isomorphism F : Fr * (H n (C · (L † )), ∇) → (H n (C · (L † )), ∇). Let ∆ be the convex hull of {0, w 1 , . . . , w N } in R n , κ = F q the residue field of the integer ring of K, and U 0 the Zariski open subset of A n κ consisting of those pointsā = (ā 1 , . . . ,ā N ) so that F (ā, t) = N j=1ā j t wj is non-degenerate in the sense that for any face τ of ∆ not containing the origin, the system of equations ∂ ∂t 1 F τ (ā, t) = · · · = ∂ ∂t n F τ (ā, t) = 0 has no solution in (κ * ) n , where F τ (ā, t) = wj ∈τā j t wj . Let U = sp −1 (U 0 ), where sp : D(0, 1 + ) N → A N κ is the specialization morphism. Let X be an open subset of P N κ such that its complement T is a divisor. Let F -isoc † (X) be the category of F -isocrystals on X over-convergent along T as defined in [9,Définition 2.3.6], and let F -Coh(D † P N ,Q ( † T )) be the category of coherent D † P N ,Q ( † T )-modules with Frobenius structures. We have a fully faithful functor induced by the specialization morphism sp * : F -isoc † (X) → F -Coh(D † P N ,Q ( † T )). By [16, Théorème 2.2.12] the essential image consists of coherent D † P N ,Q ( † T )-modules with Frobenius structures whose restriction to P N \T is (O P N ,Q )\| P N \T -coherent, where O P N ,Q = O P N ⊗ Z Q. We also call an object in the essential image of sp * a convergent F -isocrystal on X over-convergent along T , or simply an over-convergent F -isocrystal on X. Theorem 0.11. (i) H k (C · (L † ))\| U0 = 0 for k = n. (ii) H n (C · (L † ))\| U0 is an over-convergent F -isocrystal. More generally we will prove that for any divisor T containing P N k \U 0 , ( † T )(H n (C · (L † ))) is a convergent F -isocrystal on P N 0 \T overconvergent along T . Definition 0.12. We define the GKZ hypergeometric F-isocrystal to be H n (C · (L † ))\| U0 , and denote it by Hyp U . 0.13. Fibers of the GKZ hypergeometric complex. Let a = (a 1 , . . . , a N ) be a point in the closed unit polydisc D(0, 1 + ) N , where a i ∈ K ′ for some finite extension K ′ of K, and a i lie in the discrete valuation ring R ′ of K ′ . This point defines a closed immersion i a : Spf R ′ → P N of formal schemes. Let L † 0 = s>1 { w∈Z n ∩δ a w t w : a w ∈ K ′ , \|a w \|s \|w\| are bounded}. Consider the logarithmic twisted de Rham complex C · (L † 0 ) such that C k (L † 0 ) = { 1≤i1<···<i k ≤n f i1...i k (t) dt i1 t i1 ∧ · · · ∧ dt i k t i k : f i1...i k (t) ∈ L † 0 } and the differential d : C k (L † 0 ) → C k+1 (L † 0 ) is given by d(ω) = t γ exp(πF (a, t)) −1 • d t • t γ exp(πF (a, t)) (ω) = d t ω + n i=1 γ i + π N j=1 w ij a j t wj dt i t i ∧ ω for any ω ∈ C k (L † 0 ). Proposition 0.14. (i) For any a = (a 1 , . . . , a N ) ∈ D(0, 1 + ) N , H k (C · (L † 0 )) are finite dimensional. If a = (a 1 , . . . , a N ) lies in U , then dim H k (C · (L † 0 )) ∼ = 0 if k = n, n!vol(∆) if k = n. (ii) For any a in D(0, 1 + ) N , we have isomorphisms i ! a C · (L † )[N ] ∼ = C · (L † ) ⊗ L K x † K ′ ∼ = C · (L † 0 ), where K ′ is regarded as a K x † -algebra via the homomorphism K x † → K ′ , x j → a j . Let Φ a : C k (L † ) → C k (L † ) be the morphism Φ a 1≤i1<···<i k ≤n f i1...i k (t) dt i1 t i1 ∧ · · · ∧ dt i k t i k = 1≤i1<···<i k ≤n q k f i1...ij (t q ) dt i1 t i1 ∧ · · · ∧ dt i k t i k . It commutes with d t . Define F a = t γ exp(πF (a, t)) −1 • Φ a • t γ exp(πF (a q , t)) = t (q−1)γ exp πF (a q , t q ) − πF (a, t) • Φ a . By Proposition 0.4 (i), exp πF (a q , t q )−πF (a, t) lie in L † 0 , and hence F a defines an endomorphism on each C k (L † 0 ). Define Ψ a : L † 0 → L † 0 by Ψ a ( w∈Z n ∩δ c w t w ) = w∈Z n c qw t w . On differential forms, define Ψ a 1≤i1<···<i k ≤n f i1...i k (t) dt i1 t i1 ∧ · · · ∧ dt i k t i k = 1≤i1<···<i k ≤n q −k Ψ a (f i1...ij (t)) dt i1 t i1 ∧ · · · ∧ dt i k t i k . Define G a = t γ exp(πF (a q , t)) −1 • Ψ a • t γ exp(πF (a, t)) = Ψ a • t (1−q)γ exp πF (a, t) − πF (a q , t q ) . By Proposition 0.4 (i), exp πF (a, t) − πF (a q , t q ) lies in L † 0 and hence G a defines an operator on C · (L † 0 ). From now on, we assume that a is a Techmüller point, that is, a q j = a j (j = 1, . . . , N ). Then a is a fixed point of Fr : D(0, 1 + ) N → D(0, 1 + ) N , (x 1 , . . . , x N ) → (x q 1 , . . . , x q N ). In this case both F a and G a commute with d : C k (L † 0 ) → C k+1 (L † 0 ) and hence are chain maps. By Proposition 0.5, we have G a • F a = id and F a and G a induce isomorphisms on H · (C · (L † 0 )) inverse to each other. We will show that each G a : C k (L † 0 ) → C k (L † 0 ) is a nuclear operator and hence the homomorphism on each H k (C · (L † 0 )) induced by G a is also nuclear. We can talk about their traces and characteristic power series. But F a may not be nuclear. Let Tr G a , C · (L † 0 ) = n k=0 (−1) k Tr G a , C k (L † 0 ) = n k=0 (−1) k Tr G a , H k (C · (L † 0 )) , det I − T G a , C · (L † 0 ) = n k=0 det I − T G a , C k (L † 0 ) (−1) k = n k=0 det I − T G a , H k (C · (L † 0 )) (−1) k . Let χ : F * q → Q * p be the Techmüller character which maps each u in F * q to its Techmüller lifting. By [36,Theorems 4.1 and 4.3], the formal power series θ(z) = exp(πz − πz p ) converges in a disc of radius > 1, and its value θ(1) at z = 1 is a primitive p-th root of unity in K. Let ψ : F q → K * be the additive character defined by ψ(ā) = θ(1) Tr Fq /Fp (ā) for anyā ∈ F q . Letā j ∈ F q be the residue class of a j , let S m (F (ā, t)) = ū1,...,ūn∈F * q m χ 1 (Norm F m q /Fq (ū 1 )) · · · χ n (Norm F m q /Fq (ū n ))ψ Tr F q m /Fq N j=1ā jū w1j 1 · · ·ū wnj n be the twisted exponential sum for the multiplicative characters χ i = χ (1−q)γi , the nontrivial additive character ψ : F q → Q * p , and the polynomial F (ā, t), and let L(F (ā, t), T ) = exp ∞ m=1 S m (F (ā, t)) T m m be the L-function for the twisted exponential sums. Note that for m = 1, we have S m (F (ā, t)) = Hyp(ā 1 , . . . ,ā n ), where Hyp(x 1 , . . . , x N ) is given by (0.2.1). Theorem 0.15. Suppose (1 − q)γ ∈ Z n ∩ δ, and suppose K ′ contains all (q − 1)-th root of unity. Let a = (a 1 , . . . , a n ) be a Techmüller point, that is, a q j = a j . Then each G a : C k (L † 0 ) → C k (L † 0 ) is nuclear. Moreover, we have S m (F (ā, t)) = Tr (q n G a ) m , C · (L † 0 ) , L(F (ā, t), T ) = det I − T q n G a , C · (L † 0 ) −1 , 0.16. Parallel transport and the L-function. As an application of our results, let's study the behavior of the L-function L(F (x, t), T ) with respect to the parameter x. Let a = (a 1 , . . . , a N ) be a point in U with coordinates a i ∈ K ′ , and let Hyp(a) = i ! a Hyp U [N ] be the fiber of Hyp U at a. By Theorem 0.11 (i) and Proposition 0.14, we have i ! a H n (C · (L † ))[N ] ∼ = i ! a C · (L † )[n][N ] ∼ = C · (L † 0 )[n] ∼ = H n (C · (L † 0 )). So we have Hyp(a) ∼ = L † 0 / n i=1 F i,γ,a L † 0 , where F i,γ,a = t i ∂ ∂ti + γ i + π N j=1 w ij a j t wj . If a is a Techmüller point, then we have S m (F (ā, t)) = (−1) n Tr (q n F −1 a ) m , Hyp(a) , L(F (ā, t), T ) = det I − T q n F −1 a , Hyp(a) (−1) n+1 . Let a = (a 1 , . . . , a N ) and b = (b 1 , . . . , b N ) be points in U with coordinates in K ′ , and let T a,b : Hyp(a) ∼ = → Hyp(b) be the parallel transport for Hyp U . It is well-defined if \|b i − a i \| < 1 for all i. It can be described as follows: For any formal power series f (t) ∈ Q p [[Z n ∩ δ]], we have ∇ exp(−πF (x, t))f (t) = t γ exp(πF (x, t)) −1 • d x • t γ exp(πF (x, t)) exp(−πF (x, t))f (t) = 0. So exp(−πF (x, t))f (t) is horizontal with respect to ∇. But it is only a formal horizontal section since it may not lie in L † . Formally, T a,b maps exp(−πF (a, t))f (t) to exp(−πF (b, t))f (t). We have exp(−πF (b, t))f (t) = exp πF (a, t) − πF (b, t) (exp(−πF (a, t))f (t)). So T a,b : Hyp(a) ∼ = → Hyp(b) can be identified with the isomorphism T a,b : L † 0 / n i=1 F i,γ,a L † 0 → L † 0 / n i=1 F i,γ,b L † 0 , g(t) → exp πF (a, t) − πF (b, t) g(t). This is well-defined if \|b i − a i \| < 1 for all i since we then have exp πF (a, t) − πF (b, t) ∈ L † 0 . Since F : Fr * (Hyp, ∇) → (Hyp, ∇) is a horizontal morphism, we have a commutative diagram Hyp(a q ) T a q ,x q → Hyp(x q ) Fa ↓ ↓ Fx Hyp(a) Ta,x → Hyp(x). The commutativity of this diagram can also be checked directly using the explicit formulas for T a,x , F a and F x . Let {e 1 (x), . . . , e M (x)} be a local basis for Hyp U . Write where P (x) and Q(x) are matrices of power series. By the commutativity of the above diagram, we have Q(x) = P (x q )Q(a)P (x) −1 . As ∇ ∂ ∂x j (T a,x (e k (a))) = 0 for all k, P (x) satisfies the system of differential equations (q n F −1 x ) e 1 (x), . . . , e M (x) = (e 1 (x q ), . . . , e M (x q ))Q(x), So we have ∂ ∂x j (P (x)) + A j (x)P (x) = 0. (0.16.3) Equations (0.16.1)-(0. 16.3) give formulas for calculating the exponential sums and the L-function using a solution of a system of differential equations. Note that since Hyp U is an over-convergent F -crystal, the matrix Q(x) = P (x q )Q(a)P (x) −1 is over-convergent even though P (x) may be convergent only in the unit open polydisc centered at a. In particular, L(F (x, t), T ) (−1) n+1 is over-convergent in x. The paper is organized as follows. In section 1, we prove Propositions 0.4 and 0.5 on basic properties of the GKZ hypergeometric complex. In Section 2, we prove Proposition 0.8 about the D † -module structure on C · (L † ). In Section 3, we prove the Dwork trace formula and Theorem 0.15 relating the twisted exponential sum with the trace of the Frobenius on C · (L † 0 ). Section 4 is the most technical part of the paper. By relating C · (L † 0 ) with the rigid cohomology of a crystal, we prove Proposition 0.14 saying that the cohomology groups of C · (L † 0 ) are finite dimensional. Combining this finiteness result with a theorem of Caro, we deduce Proposition 0.9 which says C · (L † ) is over holonomic. Finally we use Berthelot's theory of arithmetic D-modules and results of Caro and Ogus to prove Theorems 0.11 which claims that H n (C · (L † )) defines an over-convergent F -isocrystal on the non-degenerate locus U . This proof is inspired by [35] which treats the onevariable hypergeometric systems. In Section 5, we state some open questions. 1. Proof of some basic propositions 1.1. Proof of Proposition 0.4. (i) Write exp(πz − πz q ) = 1 + ∞ i=1 c i z i . We have \|c i \| ≤ p − p−1 pq i by [36, Theorem 4.1]. Write exp(πz q − πz) = 1 − ( ∞ i=1 c i z i ) + ( ∞ i=1 c i z i ) 2 − · · · = ∞ i=0 c ′ i z i . Then we also have \|c ′ i \| ≤ p − p−1 pq i . For the monomial x j t wj , we have exp π(x j t wj ) q − πx j t wj = ∞ i=0 c ′ i x i j t iwj , c ′ i x i j r ≤ p − p−1 pq i r i = r −1 p p−1 pq −i ≤ r − 1 w p p−1 pqw −\|iwj \| Here for the last inequality, we use the fact that \|iwj \| w ≤ i and the assumption that r ≤ p p−1 pq . So we have exp π(x j t wj ) q − πx j t wj ∈ L(r, r − 1 w p p−1 pqw ). We have exp πF (x q , t q ) − πF (x, t) = N j=1 exp π(x j t wj ) q − πx j t wj . Since r − 1 w p p−1 pqw ≥ 1, the space L(r, r − 1 w p p−1 pqw ) is a ring. So exp πF (x q , t q ) − πF (x, t) lies in L(r, r − 1 w p p−1 pqw ). Similarly exp πF (x, t) − πF (x q , t q ) lies in L(r, r − 1 w p p−1 pqw ). (ii) Let C (1)· (L † ) be the logarithmic twisted de Rham complex so that C (1),k (L † ) = C k (L † ) for each k, and d (1) : C (1),k (L † ) → C (1),k+1 (L † ) is given by d (1) = t γ exp(πF (x q , t)) −1 • d t • t γ exp(πF (x q , t) = d t + n i=1 γ i + N j=1 πw ij x q j t wj dt i t i . Let ∇ (1) be the connection on C (1)· (L † ) defined by ∇ (1) (ω) = t γ exp(πF (x q , t)) −1 • d x • t γ exp(πF (x q , t)) (ω) = d x ω + N j=1 qπx q−1 j t wj dx j ∧ ω, We first prove Fr * (C · (L † ), ∇) ∼ = (C (1)· (L † ), ∇ (1) ). Consider the K-algebra homomorphism K y 1 , . . . , y N † → K x 1 , . . . , x N † , y j → x q j . This makes K x † a finite K y † -algebra. We have a canonical isomorphism L † ⊗ K y † K x † ∼ = → L † , whereL † is defined in the same way as L † except that we change the variables from x j to y j . The connection ∇ onL † defines a connection onL † ⊗ K y † K x † by the Leibniz rule. Via the above isomorphism, it defines the connection Fr * ∇ on L † . Let's verify that it coincides with the connection ∇ (1) on L † . Any element in L † can be written as a finite sum of elements of the form f (x)g(x q , t) with f (x) ∈ K[x] and g(y, t) ∈L † . By the Leibniz rule, we have (Fr * ∇)(f (x)g(x q , t)) = d x (f (x))g(x q , t) + f (x)(∇g(y, t))\| y=x q = d x (f (x))g(x q , t) + f (x) d y g(y, t) + πg(y, t)d y F (y, t) \| y=x q = d x f (x)g(x q , t) + πf (x)g(x q , t)d x F (x q , t). This proves our assertion. Similarly, one verifies that the connection Fr * ∇ on Fr * C k (L † ) can be identified with the connection ∇ (1) on C (1),k (L † ). Using the fact that Φ • d t = d t • Φ and Φ • d x = d x • Φ, one checks that F • d (1) = d • F and F • ∇ (1) = ∇ • F. So F defines a horizontal morphism of complexes of K x † -modules with connections F : (C (1)· (L † ), ∇ (1) ) → (C · (L † ), ∇). 1.2. Proof of Proposition 0.5. That G : (C · (L † ), ∇) → (C (1)· (L † ), ∇ (1) ) is a horizontal morphism can be proved in the same way as the proof of Proposition 0.4 (ii). To prove the assertions in 0.5 (ii), we first work with the usual logarithmic de Rham complex and later with the twisted logarithmic de Rham complex. We have Ψ • Φ = id on C · (L † ). By enlarging K, we may assume K contains all q-th roots of unity. Let µ q be the group of q-th roots of unity in K. For any ζ = (ζ 1 , . . . , ζ n ) ∈ µ n q , we use the notation ζt = (ζ 1 t 1 , . . . , ζ n t n ), ζ w = ζ w1 1 · · · ζ wn n . We have ζ∈µ n q ζ w = q n if q\|w, 0 otherwise. So we have Φ • Ψ( w a w (x)t w ) = w a qw (x)t qw = 1 q n ζ∈µ n q w a w (x)(ζt) w . Let Θ ζ be the endomorphism on differential forms induced by the substitution t → ζt: Θ ζ 1≤i1<···<i k ≤n f i1...i k (x, t) dt i1 t i1 ∧ · · · ∧ dt i k t i k = 1≤i1<···<i k ≤n f i1...i k (x, ζt) dt i1 t i1 ∧ · · · ∧ dt i k t i k . It commutes with d t . We have Φ • Ψ = 1 q n ζ∈µ n q Θ ζ . Let's show that Φ • Ψ is homotopic to id. It suffices to that Θ ζ is homotopic to id for each ζ ∈ µ n q . Let L † T = r>1, s>1 { k∈Z ≥0 , w∈Z n ∩δ a (k,w) (x)T k t w : a (k,w) (x) ∈ K{r −1 x}, a (k,w) (x) r s k+\|w\| are bounded}. Consider the de Rham complex (C · (L † T ), d (T,t) ) so that C k (L † T ) is the space of k-forms which can be written as sums of products of dT, dt1 t1 , . . . , dtn tn and functions in L † T , and d (T,t) : C k (L † T ) → C k+1 (L † T ) is the usual exterior diffferentiation of differential forms in the variables T, t. The substitution t i → (1 + (ζ i − 1)T )t i (i = 1, . . . , n) induces a chain map ι : (C · (L † ), d t ) → (C · (L † T ), d (T,t) ). Here we use the fact that ζ i ≡ 1 mod p so that each 1 + (ζ i − 1)T is a unit in L † T . In particular, d (1+(ζi−1)T )ti (1+(ζi−1)T )ti lies in C · (L † T ) . The evaluation at T = 0 (resp. T = 1) induces a chain map ev 0 : (C · (L † T ), d) → (C · (L † ), d t ) (resp. ev 1 : (C · (L † T ), d) → (C · (L † ), d t )). We have ev 1 • ι = Θ ζ , ev 0 • ι = id. To prove Θ ζ is homotopic to identity, it suffices to show ev 1 is homotopic to ev 0 . Note that 1 0 g(x, T, t)dT lies in L † for any g(x, T, t) ∈ L † T . Define Ξ : C k (L † T ) → C k−1 (L † ) by Ξ f (x, T, t) dt i1 t i1 ∧ · · · ∧ dt i k t i k = 0, Ξ g(x, T, t)dT ∧ dt j1 t j1 ∧ · · · ∧ dt j k−1 t j k−1 = 1 0 g(x, T, t)dT dt j1 t j1 ∧ · · · ∧ dt j k−1 t j k−1 . Then we have d t Ξ + Ξd (T,t) = ev 1 − ev 0 . We now consider the logarithmic twisted de Rham complexes. Let T ζ , L, E 0 , E 1 , H be the conjugates of Θ ζ , ι, ev 0 , ev 1 , Ξ by t γ exp(πF (x, t)) , respectively. One verifies that they are defined on C · (L † ) or C · (L † T ). Actually we have E 0 = ev 0 , E 1 = ev 1 , H = Ξ, T ζ = ζ γ exp πF (x, ζt) − πF (x, t) Θ ζ , L = 1 + (ζ − 1)T γ exp πF x, (1 + (ζ − 1)T )t − πF (x, t) ι, where for any q-th root of unity ζ i and any γ i with (1 − q)γ i ∈ Z, we define ζ γi i to be the unique q-th root of unity such that (ζ γi i ) 1−q = ζ (1−q)γi i . This makes sense since taking (1 − q)-th power defines an isomorphism on the group of q-th roots of unity. But taking (1 − q)-th power on the group of q-th roots of unity is the identity map. So we actually have ζ γi i = ζ (1−q)γi i . Define 1 + (ζ i − 1)T γi by 1 + (ζ i − 1)T γi = 1 + γ i (ζ i − 1)T + γ i (γ i − 1) 2! (ζ i − 1)T 2 + · · · . Then we have 1 + (ζ i − 1)T γi \| T =1 = ζ γi i . One can verifty that GF = id, F G = 1 q n ζ∈µ n q T ζ , E 1 • L = T ζ , E 0 • L = id, dH + Hd = E 1 − E 1 . It follows that each T ζ is homotopic to identity and hence F G is also homotopic to identity. D † -modules Recall that D † = Γ(P N , D † P N ,Q (∞)) = r>1, s>1 { v∈Z N ≥0 f v (x) ∂ v v ! : f v (x) ∈ K{r −1 x}, f v (x) r s \|v\| are bounded}. In this section, we mainly use the following description for D † . Proposition 2.1. We have D † = r>1, s>1 { v∈Z N ≥0 f v (x) ∂ v π \|v\| : f v (x) ∈ K{r −1 x}, f v (x) r s \|v\| are bounded}. Proof. Let m be a positive integer and let m = a 0 + a 1 p + a 2 p 2 + · · · + a l p l be its p-expansion, where 0 ≤ a i ≤ p − 1 for all i and a l = 0. Set σ(m) = a 0 + a 1 + a 2 + · · · + a l . We have σ(m) ≤ (p − 1)(l + 1) ≤ (p − 1)(log p m + 1). It is a standard fact that ord p (m!) = m − σ(m) p − 1 . So we have ord p π m m! = σ(m) p − 1 , 0 ≤ ord p π m m! ≤ log p m + 1. For any nonzero v = (v 1 , . . . , v N ) ∈ Z N ≥0 , let v i1 , . . . , v im be those nonzero components. We have 0 ≤ ord p π \|v\| v! ≤ m j=1 (log p v ij + 1) = log p m j=1 v ij + m ≤ log p m j=1 v ij m m + m ≤ N log p \|v\| + N, that is, 0 ≤ ord p π \|v\| v! ≤ N log p \|v\| + N. Set B † = r>1, s>1 { v∈Z N ≥0 f v (x) ∂ v v ! : f v (x) ∈ K{r −1 x}, f v (x) r s \|v\| are bounded}. Let's prove B † = D † . Given v∈Z N ≥0 f v (x) ∂ v v ! in B † , choose real numbers r > 1, s > 1 and C > 0 such that f v (x) r s \|v\| ≤ C. We have v∈Z N ≥0 f v (x) ∂ v v ! = v∈Z N ≥0 f v (x) π \|v\| v! ∂ v π \|v\| . For any nonzero v ∈ Z N ≥0 , we have ord p π \|v\| v! ≥ 0. So we have f v (x) π \|v\| v! r s \|v\| ≤ f v (x) r s \|v\| ≤ C. Hence v∈Z N ≥0 f v (x) ∂ v v ! lies in D † . Conversely, given v∈Z N ≥0 f v (x) ∂ v π \|v\| in D † , choose real numbers r > 1, s > 1 and C ′ > 0 such that f v (x) r s \|v\| ≤ C ′ . We have v∈Z N ≥0 f v (x) ∂ v π \|v\| = v∈Z N ≥0 f v (x) v! π \|v\| ∂ v v ! . For any nonzero v = (v 1 , . . . , v N ) ∈ Z N ≥0 , we have ord p π \|v\| v! ≤ N log p \|v\| + N . So we have v! π \|v\| ≤ p N \|v\| N For any 1 < s ′ < s, we have f v (x) v! π \|v\| r s ′\|v\| ≤ f v (x) r s \|v\| p N \|v\| N (s ′ /s) \|v\| ≤ p N C ′ \|v\| N (s ′ /s) \|v\| . Note that \|v\| N (s ′ /s) \|v\| are bounded since 0 < s ′ /s < 1. So v∈Z N ≥0 f v (x) ∂ v v ! lies in B † . Lemma 2.2. Let S be any subset of Z n ≥0 . There exists a finite subset S 0 of S such that S ⊂ v∈S0 (v + Z n ≥0 ). Proof. We use induction on n. When n = 1, we have S ⊂ v + Z ≥0 , where v is the minimal element in S ⊂ Z ≥0 . Suppose the assertion holds for any subset of Z n ≥0 , and let S be a subset of Z n+1 ≥0 . If S is empty, our assertion holds trivially. Otherwise, we fix an element a = (a 1 , . . . , a n+1 ) in S. For any i ∈ {1, . . . , n + 1} and any 0 ≤ b i ≤ a i , let S i,bi = {(c 1 , . . . , c n+1 ) ∈ S : c i = b i }. By the induction hypothesis, there exists a finite subset T i,bi ⊂ S i,bi such that S i,bi ⊂ v∈T i,b i (v + Z n+1 ≥0 ). We have S ⊂ 1≤i≤n+1 0≤bi≤ai S i,bi (a + Z n+1 ≥0 ) ⊂ 1≤i≤n+1 0≤bi≤ai v∈T i,b i (v + Z n+1 ≥0 ) (a + Z n+1 ≥0 ). We can take S 0 = 1≤i≤n+1 0≤bi≤ai T i,bi {a}. Let C(A) = {k 1 w 1 + · · · + k N w N : k i ∈ Z ≥0 }, L †′ = r>1, s>1 { w∈C(A) a w (x)t w : a w (x) ∈ K{r −1 x}, a w (x) r s \|w\| are bounded}. C(A) is a submonoid of Z n ∩ δ, and L †′ is a subring of L † . Lemma 2.3. (i) The ring homomorphism φ : K x, y † → L †′ , v∈Z N ≥0 f v (x)y v → v∈Z N ≥0 f v (x)t v1w1+···+vN wN is surjective, where y = (y 1 , . . . , y N ) and K x, y † = r>1, s>1 { v∈Z N ≥0 f v (x)y v : f v (x) ∈ K{r −1 x} and f v (x) r s \|v\| are bounded}. (ii) Let ∂ ∂xj acts on L †′ via ∇ ∂ ∂x j = exp(πF (x, t)) −1 • ∂ ∂x j • exp(πF (x, t)) = ∂ ∂x j + πt wj . Then L †′ is a D † -module, and the map ϕ : D † → L †′ , v∈Z N ≥0 f v (x) ∂ v π \|v\| → ( v∈Z N ≥0 f v (x) ∂ v π \|v\| ) · 1 = v∈Z N ≥0 f v (x)t v1w1+···+vN wN is an epimorphism of left D † -modules. Proof. (i) Let ∆ be the convex hull of {0, w 1 , . . . , w N }. Decompose ∆ into a finite union τ τ so that each τ is a simplex of dimension n with vertices {0, w i1 , . . . , w in } for some subset {i 1 , . . . , i n } ⊂ {1, . . . , N }. For each τ , let δ(τ ) be the cone generated by τ , and let B(τ ) = Z n ∩ {c 1 w i1 + · · · + c n w in : 0 ≤ c i < 1}, C(τ ) = {k 1 w i1 + · · · + k n w in : k i ∈ Z ≥0 }. Being a discrete bounded set, B(τ ) is finite. Every element w ∈ Z n ∩ δ(τ ) can be written uniquely as w = b(w) + c(w) with b(w) ∈ B(τ ) and c(w) ∈ C(τ ). So we have Z n ∩ δ(τ ) = w∈B(τ ) (w + C(τ )), and hence C(A) = τ (C(A) ∩ δ(τ )) = τ w∈B(τ ) C(A) ∩ (w + C(τ )) . For each C(A) ∩ (w + C(τ )), the map Z n ≥0 → w + C(τ ), (k 1 , . . . , k n ) → w + k 1 w i1 + · · · + k n w in is a bijection. Applying Lemma 2.2 to the inverse image of C(A) ∩ (w + C(τ )), we can find finitely many u 1 , . . . , u m ∈ C(A) ∩ (w + C(τ )) such that C(A) ∩ (w + C(τ )) = m i=1 (u i + C(τ )). We thus decompose C(A) into a finite union of subsets of the form u + C(τ ) such that τ is a simplex of dimension n with vertices {0, w i1 , . . . , w in } for some subset {i 1 , . . . , i n } ⊂ {1, . . . , N }, and u ∈ C(A) ∩ (w + C(τ )) for some w ∈ B(τ ). Elements in L †′ is a sum of elements of the form w∈u+C(τ ) a w (x)t w , where a w (x) ∈ K{r −1 x} and a w (x) r s \|w\| are bounded for some r, s > 1. To prove φ : K x, y † → L †′ is surjective, it suffices to show w∈u+C(τ ) a w (x)t w lies in the image of φ. Write u = c 1 w 1 + · · · + c N w N , where c i ∈ Z ≥0 . A preimage for w∈u+C(τ ) a w (x)t w is v1,...,vn≥0 a u+v1wi 1 +···+vnwi n (x)y ci 1 +v1 i1 · · · y ci n +vn in j∈{1,...,N }−{i1,...,in} y cj j . Here to verify this element lies in K x, y † , we use the fact that \|v\| ≤ C\|v 1 w i1 + · · · + v n w in \| for some constant C since w i1 , . . . , w in form a basis of R n . So we have a u+v1wi 1 +···+vnwi n (x) r ( C √ s) c1+···+cN +v1+···+vn ≤ ( C √ s) c1+···+cN a u+v1wi 1 +···+vnwi n (x) r s \|v1wi 1 +···+vnwi n \| ≤ s \|−u\| ( C √ s) c1+···+cN a u+v1wi 1 +···+vnwi n (x) r s \|u+v1wi 1 +···+vnwi n \| . Hence a u+v1wi 1 +···+vnwi n (x) r ( C √ s) c1+···+cN +v1+···+vn are bounded. This proves that φ : K x, y † → L †′ is surjective. (ii) By (i), the map ϕ : D † → L †′ is surjective. For any f ∈ L †′ , choose Q ∈ D † such that ϕ(Q) = f . For any P ∈ D † , we have P f = P ϕ(Q) = ϕ(P Q) ∈ L †′ . So L †′ is a left D † -module. Then ϕ is a homomorphism of left D † -modules. 2.4. Let C k (L †′ ) = { 1≤i1<···<i k ≤n f i1...i k (x, t) dt i1 t i1 ∧ · · · ∧ dt i k t i k : f i1...i k (x, t) ∈ L †′ }. Note that d : C k (L † ) → C k+1 (L † ) (resp. ∇ ∂ ∂x j ) maps C k (L †′ ) to C k+1 (L †′ ) (resp. C k (L †′ )). So C · (L †′ ) is a subcomplex of C · (L † ). Let F i,γ = t γ exp(πF (x, t)) −1 • t i ∂ ∂t i • t γ exp(πF (x, t)) = t i ∂ ∂t i + γ i + π N j=1 w ij x j t wj . It follows from the definition of the logarithmic twisted de Rham complex that the homomorphism L †′ → C n (L †′ ), f (x, t) → f (x, t) dt 1 t 1 ∧ · · · ∧ dt n t n induces an isomorphism L †′ / n i=1 F i,γ L †′ ∼ = H n (C · (L †′ )). Let Λ = {λ = (λ 1 , . . . , λ N ) ∈ Z N : N j=1 λ j w j = 0}, λ = λj >0 1 π ∂ ∂x j λj − λj <0 1 π ∂ ∂x j −λj (λ ∈ Λ), E i,γ = N j=1 w ij x j ∂ ∂x j + γ i (i = 1, . . . , n), Proposition 2.5. The homomorphism of D † -modules ϕ : D † → L †′ , v∈Z N ≥0 f v (x) ∂ v π \|v\| → v∈Z N ≥0 f v (x) ∂ v π \|v\| · 1 = v∈Z N ≥0 f v (x)t v1w1+···+vN wN induces isomorphisms D † / λ∈Λ D † λ ∼ = → L †′ , D † /( n i=1 D † E i,γ + λ∈Λ D † λ ) ∼ = → L †′ / n i=1 F i,γ L †′ ∼ = H n (C · (L †′ )). Moreover, there exist finitely many µ 1 , . . . , µ m ∈ Λ such that m i=1 D † µm = λ∈Λ D † λ . Proof. We have shown that ϕ is surjective in Lemma 2.3. One can verify that ϕ( λ ) = 0 for all λ ∈ Λ. Suppose v f v (x) ∂ v π \|v\| lies in the kernel of ϕ, that is, v f v (x)t v1w1+···+vN wN = 0, where f v (x) ∈ K{r −1 x} and f v (x) r s \|v\| are bounded for some r, s > 1. For each w ∈ C(A), let S w = {v ∈ Z n ≥0 : w = v 1 w 1 + · · · + v N w N }. Then we have v∈Sw f v (x) = 0. For each nonempty S w , choose an element v (0) = (v (0) 1 , . . . , v (0) N ) ∈ S w such that \|v (0) \| is minimal in S w . For any v ∈ S w , let λ v = v − v (0) . We have λ v ∈ Λ and ∂ v π \|v\| − ∂ v (0) π \|v (0) \| = ∂ min(v,v (0) ) π j min(vj ,v (0) j ) vj >v (0) j 1 π ∂ ∂x j vj −v (0) j − vj <v (0) j 1 π ∂ ∂x j v (0) j −vj = ∂ min(v,v (0) ) π j min(vj ,v (0) j ) λv . So ∂ v π \|v\| − ∂ v (0) π \|v (0) \| lies in the ideal λ∈Λ D λ . Let D = K 1 π ∂ ∂x1 , . . . , 1 π ∂ ∂xN . We have an isomorphism of rings N ). φ : D → K[z 1 , . . . , z N ], 1 π ∂ ∂x i → z i (i = 1, . . . ,Let˜ λ = φ( λ ) (λ ∈ Λ), and let f v = φ ∂ v π \|v\| − ∂ v (0) π \|v (0) \| = z v − z v (0) . Then we have f v = z min(v,v (0) )˜ λv . Let µ 1 , . . . , µ m be as in Lemma 2.6 below. We can writẽ λv = b (1)˜ µ1 + · · · + b (m)˜ µm where each b (i) is a polynomial with integer coefficients whose total degree does not exceed that of λv . We can thus write f v = a (1) v˜ µ1 + · · · + a (m) v˜ µm , where each a (i) is a polynomials with integer coefficients whose total degree does not exceed totdeg f v = \|v\|. So we can write a (i) v = u a (i) vu z u such that a (i) vu are integers and a (i) vu = 0 for \|u\| > \|v\|. We then have ∂ v π \|v\| − ∂ v (0) π \|v (0) \| = u a (1) vu ∂ u π \|u\| µ1 + · · · + u a (m) vu ∂ u π \|u\| µm . So v f v (x) ∂ v π \|v\| = w∈C(A) v∈Sw f v (x) ∂ v π \|v\| = w∈C(A) v∈Sw f v (x) ∂ v π \|v\| − ∂ v (0) π \|v (0) \| = m k=1 u v a (k) vu f v (x) ∂ u π \|u\| µ k . Hence v f v (x) ∂ v π \|v\| lies in the left ideal of D † generated by µ k , and hence we have ker ϕ = m k=1 D † µ k = λ∈Λ D † λ . Here we need to check u v a (k) vu f v (x) ∂ u π \|u\| is a well-defined element in D † . First note that v a (k) vu f v (x) converges in K r −1 x since f v (x) r s \|v\| are bounded and \|a (k) vu \| ≤ 1. Moreover, we have v a (k) vu f v (x) r s \|u\| ≤ max v, \|u\|≤\|v\| a (k) vu f v (x) r s \|u\| ≤ max v f v (x) r s \|v\| . So u v a (k) vu f v (x) ∂ u π \|u\| does define an element in D † . For any g i ∈ L †′ (i = 1, . . . , n), choose P i ∈ D † such that ϕ(P i ) = g i . One can check directly that E i,γ (1) = F i,γ (1). Moreover, F i,γ commutes with each ∇ ∂ ∂x j and hence with P i . So we have ϕ( i P i E i,γ ) = i P i E i,γ (1) = i P i F i,γ (1) = i F i,γ P i (1) = i F i,γ ϕ(P i ) = i F i,γ g i . Therefore ϕ( i D † E i,γ ) = i F i,γ L †′ . Together with the fact that ϕ is surjective and ker ϕ = λ∈Λ D † λ , we get D † / λ∈Λ D † λ ∼ = L †′ , D † /( n i=1 D † E i,γ + λ∈Λ D † λ ) ∼ = L †′ / n i=1 F i,γ L †′ . Lemma 2.6. For any λ = (λ 1 , . . . , λ N ) ∈ Λ, let λ + = (max(λ 1 , 0), . . . , max(λ N , 0)), λ − = (max(−λ 1 , 0), . . . , max(−λ N , 0)),˜ λ = z λ + − z λ − . There exist finitely many µ 1 , . . . , µ m ∈ Λ such that for any λ ∈ Λ, we can writẽ λ = a (1)˜ µ1 + · · · + a (m)˜ µm , where a (i) are polynomials with integer coefficients such that their total degrees do not exceed the total degree max(\|λ + \|, \|λ − \|) of˜ λ . Proof. Consider the set S = {(λ + , λ − ) : λ ∈ Λ − {0}} ⊂ Z 2N ≥0 . By Lemma 2.2, there exists finitely many µ 1 , . . . , µ m ∈ Λ − {0} such that S ⊂ m i=1 (µ + i , µ − i ) + Z 2N ≥0 . We prove the lemma by induction on \|λ + \| + \|λ − \|. For any λ ∈ Λ, we can find some µ i such that (λ + , λ − ) ∈ (µ + i , µ − i ) + Z 2N ≥0 . Then λ − µ i ∈ Λ and (λ − µ i ) + = λ + − µ + i , (λ − µ i ) − = λ − − µ − i . In the case where \|µ + i \| ≥ \|µ − i \|, we havẽ λ = z λ + − z λ − = z λ + −µ + i (z µ + i − z µ − i ) + z µ − i (z λ + −µ + i − z λ − −µ − i ) = z λ + −µ + i˜ µi + z µ − i˜ λ−µi Note that \|(λ − µ i ) + \| + \|(λ − µ i ) − \| < \|λ + \| + \|λ − \|. By the induction hypothesis, we can writẽ λ−µi = a (1)˜ µ1 + · · · + a (m)˜ µm such that a (j) are polynomials with integer coefficients of total degree ≤ max(\|λ + − µ + i \|, \|λ − − µ − i \|). Then we have˜ λ = z λ + −µ + i˜ µi + m j=1 z µ − i a (j)˜ µj Since we are in the case \|µ + i \| ≥ \|µ − i \|, the total degrees of z µ − i a (j) are ≤ max(\|λ + \|, \|λ − \|) . This gives the required expression for˜ λ . In the case where \|µ + i \| ≤ \|µ − i \|, we havẽ λ = z λ + − z λ − = z λ − −µ − i (z µ + i − z µ − i ) + z µ + i (z λ + −µ + i − z λ − −µ − i ) = z λ − −µ + i˜ µi + z µ + i˜ λ−µi We conclude as before using the induction hypothesis. Remark 2.7. The left D † -module H n (C · (L †′ )) ∼ = D † /( n i=1 D † E i,γ + λ∈Λ D † λ ) is the p-adic analogue of the (complex) hypergeometric D-module ([1]) associated to the GKZ hypergeometric system of differential equations (0.1.1). In general, the action of G on C · (L † ) does not preserve C · (L †′ ), and the above D † -module does not have a Frobenius structure with interesting arithmetic properties. 2.8. Proof of Proposition 0.8. We have shown that L †′ is a D † -module in Lemma 2.3. It is known that D † is coherent ( [29]). So by Proposition 2.5, L †′ is a coherent D † -module. Keep the notation in the proof of Lemma 2.3. Decompose ∆ into a finite union τ τ so that each τ is a simplex of dimension n with vertices {0, w i1 , . . . , w in } for some subset {i 1 , . . . , i n } ⊂ {1, . . . , N }. Let B be the finite set τ B(τ ). Consider the map ψ : β∈B L †′ → L † , (f β ) → β∈B f β t β . We will show ψ is surjective. As in the proof of Lemma 2.3 (ii), this implies that L † is a left D † -module. It is then clear that ψ is a homomorphism of D † -modules. We will prove ker ψ is a finitely generated D † -module. Combined with the fact that L †′ is a coherent D † -module, this implies that L † is a coherent D † -module. We have Z n ∩ δ = τ (Z n ∩ δ(τ )). To prove ψ is surjective, it suffices to show every element in L † of the form w∈Z n ∩δ(τ ) a w (x)t w lies in the image of ψ, where a w (x) ∈ K{r −1 x} and a w (x) r s \|w\| are bounded for some r, s > 1. Every element w ∈ Z n ∩ δ(τ ) can be written uniquely as w = b(w) + c(w) with b(w) ∈ B(τ ) and c(w) ∈ C(τ ). We have w∈Z n ∩δ(τ ) a w (x)t w = β∈B(τ ) w∈Z n ∩δ(τ ), b(w)=β a w (x)t c(w) t β . Note that w∈Z n ∩δ(τ ), b(w)=β a w (x)t c(w) lie in L †′ since a w (x) r s \|c(w)\| ≤ ( a w (x) r s \|w\| )s \|−b(w)\| are bounded. Thus ψ is surjective. Given β ′ , β ′′ ∈ B, set L β ′ ,β ′′ = {f ∈ L †′ : f t β ′ −β ′′ ∈ L †′ }, S β ′ ,β ′′ = {w ∈ C(A) : w + β ′ − β ′′ ∈ C(A)}. We have L β ′ ,β ′′ = r>1, s>1 { w∈S β ′ ,β ′′ a w (x)t w : a w (x) ∈ K{r −1 x}, a w (x) r s \|w\| are bounded for some r, s > 1}. We have S β ′ ,β ′′ + w j ⊂ S β ′ ,β ′′ for all j, and L β ′ ,β ′′ is a D † -submodule of L †′ . For any f ∈ L β ′ ,β ′′ and β ∈ B, let ι β ′ ,β ′′ (f ) β =    f if β = β ′ , −f t β ′ −β ′′ if β = β ′′ , 0 if β ∈ B\{β ′ , β ′′ }. Then the map ι β ′ ,β ′′ : L β ′ ,β ′′ → β∈B L †′ , f → (ι β ′ ,β ′′ (f ) β ) β∈B is a homomorphism of D † -modules and its image is contained in ker ψ. We will prove each L β ′ ,β ′′ is a finitely generated D † -module, and ker ψ = β ′ ,β ′′ ι β ′ ,β ′′ (L β ′ ,β ′′ ). It follows that ker ψ is a finitely generated D † -module. We have S β ′ ,β ′′ = τ (S β ′ ,β ′′ ∩ δ(τ )) = τ w∈B(τ ) (S β ′ ,β ′′ ∩ (w + C(τ ))). By Lemma 2.2, for each S β ′ ,β ′′ ∩ (w + C(τ )), we can find finitely many u 1 , . . . , u m ∈ S β ′ ,β ′′ ∩ (w + C(τ )) such that S β ′ ,β ′′ ∩ (w + C(τ )) = m i=1 (u i + C(τ )). We thus decompose S β ′ ,β ′′ into a finite union of subsets of the form u + C(τ ) such that τ is a simplicial complex of dimension n with vertices {0, w i1 , . . . , w in } for some subset {i 1 , . . . , i n } ⊂ {1, . . . , N }, and u ∈ S β ′ ,β ′′ ∩(w+C(τ )) for some w ∈ B(τ ). Then L β ′ ,β ′′ is generated by these t u as a D † -module. Indeed, every element in L β ′ ,β ′′ is a sum of elements of the form w∈u+C(τ ) a w (x)t w . We have w∈u+C(τ ) a w (x)t w = v1,...,vn≥0 a u+v1wi 1 +···+vnwi n (x) 1 π ∂ ∂x i1 v1 · · · 1 π ∂ ∂x im vn · t u . Suppose (f (0) β ) ∈ β∈B L †′ is an element in ker ψ. We then have β∈B f (0) β t β = 0. Suppose B = {β 1 , . . . , β k }, and write f (0) β = w∈C(A) a βw (x)t w . Define f (1) β = w∈C(A), w+(β−β1) ∈C(A) a βw (x)t w , g (1) β = w∈C(A), w+(β−β1)∈C(A) a βw (x)t w . In particular, f (1) β1 is 0 since it is a sum over the empty set. We have g a βw (x)t w . We have g (i) β ∈ L β,βi and (f (i−1) β ) − α∈B\{βi} ι α,βi (g (i) α ) = (f (i) β ) . We have f (k) β = 0 for all β ∈ B = {β 1 , . . . , β k }. So we have (f (0) β ) = k i=1 α∈B\{βi} ι α,βi (g (i) α ). Hence ker ψ = β ′ ,β ′′ ι β ′ ,β ′′ (L β ′ ,β ′′ ). This finishes the proof of Proposition 0.8. The results in this section can be used to prove the following. Lemma 2.9. L † is flat over K x † . Let a = (a 1 , . . . , a N ) be a point in the closed polydisc D(0, 1 + ) N such that a i ∈ K ′ , where K ′ is a finite extension of K. Regard K ′ as a K x † -algebra via the homomorphism K x † → K ′ , x j → a j . We have L † ⊗ K x † K ′ ∼ = L † 0 . Proof. Let R be the integer ring of K, and let R x † = r>1 { v∈Z N ≥0 a v x v : a v ∈ R, \|a v \|r \|v\| are bounded }, R x, y † = r>1, s>1 { u,v∈Z N ≥0 a uv x u y v : a uv ∈ R, \|a uv \|r \|u\| s \|v\| are bounded }, L † R = r>1,s>1 { v∈Z N ≥0 , w∈Z n ∩δ a vw x v t w : a vw ∈ R, \|a vw \|r \|v\| s \|w\| are bounded}, L †′ R = r>1,s>1 { v∈Z N ≥0 , w∈C(A) a vw x v t w : a vw ∈ R, \|a vw \|r \|v\| s \|w\| are bounded}. We have K x † ∼ = R x † ⊗ R K, L † ∼ = L † R ⊗ R K. To prove L † is flat over K x † , it suffices to show L † R is flat over R x † . Keep the notation in the proof of Lemma 2.3 and 2.8. The same proof shows that the following homomorphisms v1wN +···+vN wN are surjective. By [23], R x, y † is a noetherian ring. It follows that L † R is also noetherian. We have β∈B L †′ R → L † R , (f β ) → β∈B f β t β , R x, y † → L †′ R , v∈Z N ≥0 f v (x)y v → v∈Z N ≥0 f v (x)tL † R /π k L † R ∼ = (R/π k )[x][Z n ∩ δ], R x † /π k R x † ∼ = (R/π k )[x]. So L † R /π k L † R is flat over R x † /π k R x † for all k. By [28, IV Théorème 5.6], L † R is flat over R x † . Finally let's prove L † ⊗ K x † K ′ ∼ = L † 0 . One can verify directly that in the case where K ′ = K, the homomorphism L † → L † 0 , w∈Z n ∩δ a w (x)t w → w∈Z n ∩δ a w (0)t w is surjective with kernel (x 1 , . . . , x N )L † . This proves our assertion in the case where K = K ′ and a = (0, . . . , 0). In general, we have an isomorphism L † ⊗ K K ′ ∼ = L † K ′ , where L † K ′ = r>1,s>1 { v∈Z N ≥0 , w∈Z n ∩δ a vw x v t w : a vw ∈ K ′ , \|a vw \|r \|v\| s \|w\| are bounded}. By base change from K to K ′ and using this isomorphism, we can reduce to the case where K ′ = K. Then using the automorphism K ′ x † → K ′ x † , x i → x i − a i , we can reduce to the case where a = (0, . . . , 0). Dwork's theory Let θ(z) = exp(πz − πz p ), θ m (z) = exp(πz − πz p m ) = m−1 i=0 θ(z p i ). Then θ m (z) converges in a disc of radius > 1, and the value θ(1) = θ(z)\| z=1 of the power series θ(z) at z = 1 is a primitive p-th root of unity in K ([36, Theorems 4.1 and 4.3]). Letū ∈ F p m and let u ∈ Q p be its Techmüller lifting so that u p m = u. θ m (z)\| z=u = θ(1) Tr F p m /Fp (ū) . From now on, we denote elements in finite fields by letters with bars such asū,ā j ,ū i ... and denote their Techmüller liftings by the same letters without bars such as u, a j , u i ... Let ψ m : F q m → K * be the additive character defined by ψ m (ū) = θ(1) Tr F q m /Fp (ū) . Then we have ψ m (ū) = exp(πz − πz q m )\| z=u . Denote ψ 1 by ψ. We have ψ m = ψ • Tr F q m /Fq . Letā 1 , . . . ,ā N ∈ F q . For anyū 1 , . . . ,ū n ∈ F * q m , we have Let χ : F * q → Q * p be the Techmüller character, that is, χ(ū) = u is the Techmüller lifting of u ∈ F q . It is a generator for the group of multiplicative characters on F q . Any multiplicative character F * q → Q * p is of the form χ γ = χ γ(1−q) for some rational number γ ∈ 1 1−q Z. Moreover, for anyū ∈ F q m , we have χ γ (Norm F m q /Fq (ū)) = (u 1+q+···+q m−1 ) γ(1−q) = u γ(1−q m ) , (3.1.2) Let γ 1 , . . . , γ n ∈ 1 1−q Z. Set χ i = χ γi(1−q) (i = 1, . . . , n). Consider the twisted exponential sum S m (F (ā, t)) = ū1,...,ūn∈F * q m χ 1 (Norm F m q /Fq (ū 1 )) · · · χ n (Norm F m q /Fq (ū n ))ψ Tr F q m /Fq = u q m −1 i =1 u γ1(1−q m ) 1 · · · u γn(1−q m ) n N j=1 exp(πz − πz q m )\| z=aj u w 1j 1 ···u w nj n = u q m −1 i =1 u γ1(1−q m ) 1 · · · u γn(1−q m ) n N j=1 ∞ i=1 c i (a j u w1j 1 · · · u wnj n ) i = u q m −1 i =1   t γ1(1−q m ) 1 · · · t γn(1−q m ) n N j=1 ∞ i=1 c i (a j t w1j 1 · · · t wnj n ) i   \| ti=ui = u q m −1 i =1   t γ1(1−q m ) 1 · · · t γn(1−q m ) n N j=1 exp πa j t w1j 1 · · · t wnj n − πa j t q m w1j 1 · · · t q m wnj n   \| ti=ui = u q m −1 i =1 t γ1(1−q m ) 1 · · · t γn(1−q m ) n exp πF (a, t) − πF (a, t q m ) \| ti=ui . We thus have S m (F (ā, t)) = u q m −1 i =1 t γ1(1−q m ) 1 · · · t γn(1−q m ) n exp πF (a, t) − πF (a, t q m ) \| ti=ui .(q m − 1) n Tr(G m a , L † 0 ) = u q m −1 i =1 t γ1(1−q m ) 1 · · · t γn(1−q m ) n exp πF (a, t) − πF (a, t q m ) \| ti=ui . Proof. For any real number s > 1, definẽ So i S converges to i as S goes over all finite subsets of {t w } w∈Z n ∩δ . Moreover i S has finite ranks. So i is completely continuous. L(s) 0 = { w∈Z n ∩δ a w t w : a w ∈ K ′ , lim \|w\|→∞ \|a w \|s \|w\| = 0}. Let H(t) = t (1−q)γ exp πF (a, t) − πF (a, t q ) . By Proposition 0.4, we have H q (t) ∈ L(p p−1 pqw ) 0 . For any s > 1, we have Ψ a (L(s) 0 ) ⊂ L(s q ) 0 . Consider the operator G a = t γ exp(πF (a, t)) −1 • Ψ a • t γ exp(πF (a, t)) = Ψ a • H(t). If 1 < s < p For any u ∈ Z n ∩ δ, we have G a (t u ) = Ψ a ( w∈Z n ∩δ c w t w+u ) = Ψ a ( w∈u+Z n ∩δ c w−u t w ) = w∈u+Z n ∩δ c qw−u t w . The coefficient of t u on the righthand side is c (q−1)u . Taking the sum of diagonal entries of the matrix of G a onL(s) 0 with respect to {t w } w∈Z n ∩δ , we get Tr(G a ,L(s) 0 ) = u∈Z n ∩δ c (q−1)u .(s) 0 = N (s) f W (s) f , where N (s) f and W (s) f are G a -invariant spaces, N (s) f is finite dimensional over K ′ , f (G a ) is nilpotent on N (s) f and bijective on W (s) f . We have N (s) f = ∞ m=1 ker (f (G a )) m , W (s) f = ∞ m=1 im (f (G a )) m . For any pair s < s ′ , we havẽ pw . This shows that G a : L † 0 → L † 0 is nuclear and L(s ′ ) 0 ⊂L(s) 0 , N (s ′ ) f ⊂ N (s) f , W (s ′ ) f ⊂ W (s) f . Let N f = 1<s<p p−1 pw N (s) f and W f = 1<s<p p−1 pw W (s) f . Then L † 0 = N f W f , N f and W f are G a -invariant, f (G a )Tr(G a , L † 0 ) = u∈Z n ∩δ c (q−1)u . On the other hand, we have u q−1 =1 u w = q − 1 if q − 1\|w, 0 otherwise. So we have u q−1 i =1 t γ1(1−q) 1 · · · t γn(1−q) n exp πF (a, t) − πF (a, t q ) \| ti=ui = w∈Z n ∩δ u q−1 i =1 c w u w1 1 · · · u wn n = (q − 1) n u∈Z n ∩δ c (q−1)u . We thus get (q − 1) n Tr(G a , L † 0 ) = u q−1 i =1 t γ1(1−q) 1 · · · t γn(1−q) n exp πF (a, t) − πF (a, t q ) \| ti=ui . This proves the theorem for m = 1. We have G m a = exp(πF (a, t)) −1 • Ψ m a • exp(πF (a, t)) = Ψ m a • exp πF (a, t) − πF (a, t q m ) . So the assertion for general m follows from the case m = 1. Proof of Theorem 0.15. By the equation (3.1.3) and the Dwork trace formula 3.3, we have S m (F (ā, t)) = (q m − 1) n Tr(G m a , L † 0 ) = n k=0 n k (−1) k (q m ) n−k Tr(G m a , L † 0 ) = n k=0 (−1) k Tr (q n−k G a ) m , L †( n k ) 0 . For the L-function, we have L(F (ā, t), T ) = exp ∞ m=1 S m (F (ā, t)) T m m = exp ∞ m=1 n k=0 (−1) k Tr (q n−k G a ) m , L †( n k ) 0 T m m = n k=0 exp (−1) k ∞ m=1 Tr (q n−k G a ) m , L †( n k ) 0 T m m = n k=0 det I − T q n−k G a , L †( n k ) 0 (−1) k+1 This proves Theorem 0.15. Overholonomic D † -Modules and F -isocrystals 4.1. In this section, we take δ to be the cone generated by w 1 , . . . , w N . We first compare our twisted logarithmic de Rham complex with the rigid cohomology of a certain isocrystal. As in the proof of Lemma 2.3, we decompose ∆ into a finite union τ τ so that each τ is a simplex of dimension n with vertices {0, w i1 , . . . , w in } for some subset {i 1 , . . . , i n } ⊂ {1, . . . , N }. For each τ , let δ(τ ) be the cone generated by τ , and let B(τ ) = Z n ∩ {c 1 w i1 + · · · + c n w in : 0 ≤ c i < 1}, C(τ ) = {k 1 w i1 + · · · + k n w in : k i ∈ Z ≥0 }. B(τ ) is finite. Every element w ∈ Z n ∩ δ(τ ) can be written uniquely as w = b(w) + c(w) with b(w) ∈ B(τ ) and c(w) ∈ C(τ ). We have Z n ∩ δ = τ (Z n ∩ δ(τ )) = τ w∈B(τ ) (w + C(τ )) . The finite set B = {w 1 , . . . , w N } τ B(τ ) generates the monoid Z n ∩ δ. Denote elements in B by b 1 , . . . , b M with b i = w i for 1 ≤ i ≤ N . Let Σ be the fan of all faces of dual coneδ of δ, and let X R = X R (Σ) = Spec R[Z n ∩ δ] be the affine toric R-scheme defined by Σ. We have a closed immersion X R → A M R defined by the R-algebra homomorphism R[y 1 , . . . , y M ] → R[Z n ∩ δ], y i → t bi . Let X R be the closure of X R in P M R , let X R be the formal completion of X R with respect to the adic topology defined by the maximal ideal of R, and let X κ = X R ⊗ R κ (resp. X κ = X R ⊗ R κ) be the special fiber of X R (resp. X R ). The analytification X an K of the generic fiber of X R coincides with the rigid analytic space associate to the formal scheme X R . We have a specialization map sp : X an K → X R . The tube ]X κ [ is defined to be ]X κ [= sp −1 (X κ ). We have a commutative diagram ]X κ [ / / X an K = / / X an K sp X κ / / X κ / / X R . The analytification X an K of the generic fiber of X R is a strict neighborhood of ]X κ [ in X an K . Denote by j :]X κ [→ X an K the inclusion. Lemma 4.2. We have an isomorphism Γ(X an K , j † O X an K ) ∼ − → L † 0 . Proof. By [9, 1.2.4 (iii)], ]X κ [ is the intersection of X an K with the closed unit polydisc D(0, 1 + ) M in A M,an K , and the intersections V λ of X an K with the closed polydiscs D(0, λ + ) M of radii λ form a fundamental system of strict neighborhoods when λ → 1 + . We have Γ(X an K , j † O X an K ) = lim −→ λ→1 + Γ(V λ , O V λ ) = K y 1 , . . . , y M † /IK y 1 , . . . , y M † , where I is the ideal of K[y 1 , . . . , y M ] defining the closed subscheme X K of A M K . Let A = {λ = (λ 1 , . . . , λ M ) ∈ Z M : M i=1 λ i b i = 0}, λ = λi>0 y λi i − λi<0 y λi i (λ ∈ A), and let J be the ideal of K[y 1 , . . . , y M ] generated by˜ λ (λ ∈ A). We have J ⊂ I. Choose finitely many µ 1 , . . . , µ m ∈ A so that for any λ ∈ A, we have˜ λ = a 1˜ µ1 + · · · + a m˜ µm , where a i are polynomials with integer coefficients, and the total degrees of a i do not exceed that of˜ λ . Such a choice is possible by Lemma 2.6. Consider the homomorphism φ : K y 1 , . . . , y M † → L † 0 , y i → t bi . It is well-defined. In fact, given v a v y v ∈ K y 1 , . . . , y M † such that \|a v \|s \|v\| are bounded for some s > 1, v a v y v is mapped to u ( i vibi=u a v )t u . Let c = max 1≤i≤M \|b i \|. We have \|u\| ≤ c\|v\| whenever u = v i b i . Thus i vibi=u a v s c −1 \|u\| ≤ max v \|a v \|s \|v\| . Suppose v a v y v lies in ker φ. For any u, let S u = {v ∈ Z M ≥0 : i v i b i = u}. Then we have v∈Su a v = 0 for any u. For any nonempty S u , choose v (0) = (v (0) 1 , . . . , v (0) M ) ∈ S u such that \|v (0) \| is minimal in S u . Then v − v (0) ∈ A. We have y v − y v (0) = y min(v,v (0) ) vj >v (0) j y vj −v (0) j j − vj <v (0) j y v (0) j −vj j = y min(v,v (0) )˜ v−v (0) ∈ J. As in the proof of Proposition 2.5, we can write y v − y v (0) = w a (1) vw y w ˜ µ1 + · · · + w a (s) vw y w ˜ µm for some a (k) vw ∈ Z such that a (k) vw = 0 if \|w\| > \|v\|. We have v a v y v = u∈Z n ∩δ v∈Su a v y v = u∈Z n ∩δ v∈Su a v (y v − y v (0) ) = m k=1 w v a (k) vw a v y w ˜ µ k As a (k) vw ∈ Z and \|a v \|s \|v\| are bounded for some s > 1, v a (k) vw a v converges in R. We have v a (k) vw a v s \|w\| ≤ max v, \|w\|≤\|v\| \|a (k) vw a v \|s \|w\| ≤ max v \|a v \|s \|v\| . So w v a (k) vw a v y w ∈ K y 1 , . . . , y M † , and v a v y v ∈ JK y 1 , . . . , y M † . Hence ker φ ⊂ JK y 1 , . . . , y M † ⊂ IK y 1 , . . . , y M † . It is clear that IK y 1 , . . . , y M † ⊂ kerφ. We thus have ker φ = IK y 1 , . . . , y M † . Let's prove φ is surjective. It suffices to show w∈bi+C(τ ) a w t w ∈ L † 0 lies in the image of φ, where τ is a simplex with vertices {0, w i1 , . . . , w in } and b i ∈ B(τ ). A preimage for w∈bi+C(τ ) a w (x)t w is v1,...,vn≥0 a bi+v1wi 1 +···+vnwi n y i y v1 i1 · · · y vn in . Let's prove it lies in K y † . Suppose \|a w \|s \|w\| are bounded for some s > 1. Since w i1 , . . . , w in are linearly independent, there exist c 1 , c 2 > 0 such that c 1 \|v\| ≤ \|v 1 w i1 + · · · + v n w in \| ≤ c 2 \|v\| for any v = (v 1 , . . . , v n ). We have \|a bi+v1wi 1 +···+vnwi n \| · (s c1 ) 1+v1+···+vn ≤ \|a bi+v1wi 1 +···+vnwi n \| · s c1+\|v1wi 1 +···+vnwi n \| ≤ \|a bi+v1wi 1 +···+vnwi n \| · s \|bi+v1wi 1 +···+vnwi n \| s c1+\|−bi\| . Thus \|a bi+v1wi 1 +···+vnwi n \|·(s c1 ) 1+v1+···+vn are bounded and v1,...,vn≥0 a bi+v1wi 1 +···+vnwi n y i y v1 i1 · · · y vn in lies in K y † . Therefore φ induces an isomorphism Γ(X an K , j † O X an K ) ∼ = L † 0 . Let Σ ′ be a regular refinement of Σ, X ′ R the toric R-scheme associate to Σ ′ , and X ′ R a compactification of X ′ R . Replacing X ′ R by the scheme theoretic image of the immersion X ′ R → X R × R X ′ R , we may assume that we have a commutative diagram X ′ R → X ′ R ↓ ↓ X R → X R . It gives rise to a commutative diagram X ′ R → X R × XR X ′ R → X ′ R ց ↓ ↓ X R → X R . The morphism X ′ R → X R × X R X ′ R is an open immersion since X ′ R → X ′ R and the projection X R × X R X ′ R → X ′ R are open immersions. It is proper since X ′ R → X R is proper and the projection X R × X R X ′ R → X R is separated. Since X ′ R is dense in X ′ R , we have X ′ R ∼ = X R × XR X ′ R . Consider the Cartesian squares X ′ κ → X ′ κ → X ′ R ↓ ↓ ↓ X κ → X κ → X R , (4.2.1) where X ′ κ (resp. X ′ κ ) is the special fibre of X ′ R (resp. X ′ R ), and X ′ R is the formal completion of X ′ R . Let ]X ′ κ [ be the tube of X ′ κ in the analytification X ′an K of generic fibre X ′ K of X ′ , let j ′ :]X ′ κ [→ X ′an K be the inclusion, and let φ : X ′an K → X an K be the canonical morphism. We have a commutative diagram ]X ′ κ [ j ′ → X ′an K → X ′an K ↓ ↓ φ ↓ ]X κ [ j → X an K → X an K . (4.2.2) Lemma 4.3. We have R k φ * j ′ † O X ′an K ∼ = j † O X an K if k = 0, 0 if k ≥ 1, H k (X an K , j † O X an K ) ∼ = L † 0 if k = 0, 0 if k ≥ 1. Proof. For any λ > 1, let V λ be the intersection of X an K with the closed polydisc D(0, λ + ) M in P M,an K . Then V λ (λ > 1) form a fundamental system of affinoid strict neighborhoods of ]X κ [ in X an K and j λ : V λ → X an K are affine morphisms. Since squares in (4.2.1) are Cartesian, V ′ λ = φ −1 (V λ ) (λ > 1) form a fundamental system of strict neighborhoods of ]X ′ κ [ in X ′an K , and j ′ λ : V ′ λ → X ′an K are affine. Consider the commutative diagram ]X ′ κ [ → V ′ λ j ′ λ → X ′an K ↓ ↓ φ λ ↓ φ ]X κ [ → V λ j λ → X an K . We have R k φ * (j † O X ′ an K ) = R k φ * lim − →λ→1 + j ′ λ * O V ′ λ ∼ = lim − →λ→1 + R k φ * j ′ λ * O V ′ λ (4.3.1) ∼ = lim − →λ→1 + j λ * R k φ λ * O V ′ λ (4.3.2) ∼ = lim − →λ→1 + j λ * j * λ R k φ * O X ′an K ∼ = lim − →λ→1 + j λ * j * λ (R k φ K * O X ′ K ) an (4.3.3) = j † (R k φ K * O X ′ K ) an . HereR k φ K * O X ′ K = 0 for k ≥ 1 and φ K * O X ′ K = O XK . The first assertion of the lemma follows. We have H k (X an K , j † O X an K ) ∼ = H k (X an K , lim − →λ→1 + j λ * O V λ ) ∼ = lim − →λ→1 + H k (X an K , Rj λ * O V λ ) ∼ = lim − →λ→1 + H k (V λ , O V λ ). For k ≥ 1, we thus get H k (X an K , j † O X an K ) = 0. The case k = 0 is given by Lemma 4.2. Regard K y † as a free K y † -module with the connection ∇(1) = πdy. It defines the Dwork isocrystal, which is j † 0 O A 1,∇(1) = e −πF (a,t) • d t • e πF (a,t) (1) = n i=1 N j=1 πw ij a j t wj dt i t i . Let T K be the maximal torus in the toric scheme X ′ K , and let D ′ = X ′ K − T n K . As X ′ K is a smooth toric scheme and D ′ is a normal crossing divisor, the sheaf j ′ † Ω k X ′an K (log D ′ ) of forms with logarithmic poles along D ′ is a free j ′ † O X ′an K -module with basis dt i1 t i1 ∧ · · · · ∧ dt i k t i k (1 ≤ i 1 < · · · < i k ≤ n) for each k. Lemma 4.4. We have RΓ(X ′an K , j ′ † Ω · X ′an K (log D ′ ) ⊗ j ′ † O X ′an K L ψ,Fa ) ∼ = RΓ rig (T n κ , L ψ,Fa•i ), where L ψ,Fa•i is the pull-back of the Dwork crystal L ψ by the composite T n K i → X ′ K Fa → A 1 K . Proof. Choose a regular fan Σ ′′ containing Σ ′ such that \|Σ ′′ \| = R n . The toric R-scheme X ′′ R associated to Σ ′′ is proper over R. Let X ′′ R (resp. X ′′ κ and X ′′ K ) be its formal completion (resp. special fiber, resp. generic fibre), and let D ′′ = X ′′ R − T n R . Then D ′′ is a divisor with normal crossing in X ′′ R . We have open immersions T n κ ֒→ X ′ κ ֒→ X ′′ κ . Note that X ′an K is a strict neighborhood of ]X ′ k [ in X ′′an K . In [6, Corollary A.4], taking X = X ′′ R , Z = D ′′ , U = T n κ , V = X ′ κ and W = X ′an K , we get RΓ(X ′an K , j ′ † Ω · X ′an K (logD ′ ) ⊗ j ′ † O X ′an K L ψ,Fa ) ∼ = RΓ rig (T n κ , L ψ,Fa•i ). 4.5. Proof of Proposition 0.14 (i). Combining Lemmas 4.3 and 4.4 and the fact that j ′ † Ω k X ′an K (log D ′ ) is a free j ′ † O X ′an K -module, we get RΓ rig (T n κ , L ψ,Fa•i ) ∼ = RΓ X an K , Rφ * j ′ † Ω · X ′an K (log D ′ ) ⊗ j ′ † O X ′an K L ψ,Fa ∼ = Γ X an K , φ * j ′ † Ω · X ′an K (log D ′ ) ⊗ j ′ † O X ′an K L ψ,Fa ∼ = C · (L † 0 ) . In the case where γ = 0, we have δ = R n by our assumption. Let K χi be Kummer isocrystal, that is, K y, y −1 considered as a K y, y −1 -module with the connection ∇ i (1) = γ i dy y . Let p i : T n k → T 1 k be the projection to the i-th factor. Define K χ = ⊗ i p * i K χi . The same argument as above shows that RΓ rig (T n k , K χ ⊗ L ψ,Fa ) ∼ = C · (L † 0 ). Rigid cohomology groups of the tensor product of pull-backs of the Kummer isocrystal and the Dwork isocrystal are finite dimensional by [12, 3.10]. So H k (C · (L † 0 )) are finite dimensional for arbitrary a = (a 1 , . . . , a N ) ∈ D(0, 1 + ) N . Next we prove the second part of Proposition 0.14 (i) using the method in [14,Corollary 1.3]. For any w ∈ Z n ∩ δ, define ω(w) = inf{r : w ∈ r∆}. One can show that there exist positive real numbers C 1 and C 2 such that for any w ∈ Z n ∩ δ, we have C 1 \|w\| ≤ ω(w) ≤ C 2 \|w\|. For any real numbers b > 0 and c, define L(b, c) = c { w∈Z n ∩δ a w t w : ord p (a w ) ≥ bω(w) + c}, L(b) = c L(b, c), B = { w∈Z n ∩δ a w π ω(w) t w a w → 0 as \|w\| → ∞}. 4.6. Arithmetic D-modules. The reader may consult [6, §2], [8] and [15, §1] for details. Let f : X ′ → X be a morphism of smooth formal R-schemes, a a uniformizer of R, f i : X ′ i → X i the reduction of f modulo a i+1 , f 0 : X ′ 0 → X 0 the induced morphism on special fibers, and T (resp. T ′ ) a divisor on X 0 (resp. X ′ 0 ) such that f −1 0 (T ) ⊂ T ′ . Let U ⊂ X be an affine open subset and let h ∈ Γ(U , O X ) so that the divisor T of X 0 is given by h ≡ 0 mod a. Let U i and h i ∈ Γ(U i , O Ui ) be the reduction of U and h modulo a i+1 . For any m ≥ 0, define B (m) Xi (T )\| Ui = O Ui [t]/(h p m+1 i t − p), B (m) X (T )\| U = lim ← − i B (m) Xi (T )\| Ui = O U {t}/(h p m+1 t − p). B (m) X (T ) is a p-adically complete O X -algebra depending only on X and T . Define the sheaf of functions on X with overconvergent singularities along T by O X ,Q ( † T ) = lim − → m B (m) X (T ) ⊗ Z Q. Denote by D (m) Xi the sheaf differential operators of level m on X i . If x 1 , . . . , x N are local coordinates on U i and ∂ j = ∂/∂x j for 1 ≤ j ≤ N , we have where d X (resp. d X ′ ) is the dimension of X (resp. X ′ ). Let T 1 be another divisor of X 0 containing T . We have a functor [8, 4.3.10-11].) ( † T 1 , T ) : D b coh (D † X ,Q ( † T )) → D b coh (D † X ,Q ( † T 1 )), ( † T 1 , T )(E) := D † X ,Q ( † T 1 ) ⊗ L D † X ,Q ( † T ) E. Here we could omit the symbol L since the morphism D † X ,Q ( † T ) → D † X ,Q ( † T 1 ) is flat. (cf. In the following, we work with X = P N and the divisor ∞ = P N − A N . For simplicity, when there is no confusion on the divisors T = ∞ and T ′ , we use the notation O X ,Q (∞), D † X ,Q (∞), D † X ′ ,Q (∞), D (m) X ′ →X (∞), D † X ′ →X ,Q (∞) instead of O X ,Q ( † T ), D † X ,Q ( † T ), D † X ′ ,Q ( † T ′ ), D (m) X ′ →X (T ′ , T ), D † X ′ →X ,Q ( † T ′ , T ) and use ( † T 1 ) instead of ( † T 1 , T ). 4.7. Proof of Propositions 0.9 and 0.14 (ii). For a closed point a = (a 1 , . . . , a N ) in the formal affine space A N , let i a : a → P N be the closed immersion and let J be its ideal sheaf. By [17,Théorème 2.1.4] and Proposition 0.14 (i) which we have shown, we only have to check i ! a C · (L † ) ∼ = C · (L † 0 )[−N ]. By [16, Lemma 2.2.3], we have D † a→P N ,Q = i −1 a (D † P N ,Q /J D † P N ,Q ). Applying [16, Proposition 1.1.10] to D † P N ,Q (∞), we have D † a→P N ,Q (∞) ∼ = D † a→P N ,Q (∞) ⊗ L i −1 a D † P N ,Q (∞) i −1 a D † P N ,Q (∞) ∼ = D † a→P N ,Q ⊗ L i −1 a D † P N ,Q i −1 a D † P N ,Q (∞) ∼ = i −1 a (D † P N ,Q /J D † P N ,Q ) ⊗ L i −1 a D † P N ,Q i −1 a D † P N ,Q (∞) ∼ = i −1 a D † P N ,Q /J D † P N ,Q ⊗ L D † P N ,Q D † P N ,Q (∞) ∼ = i −1 a D † P N ,Q (∞)/J D † P N ,Q (∞) ∼ = i −1 a O P N ,Q (∞)/J O P N ,Q (∞) ⊗ L O P N ,Q (∞) D † P N ,Q (∞) , where for the last two isomorphisms, we use the fact that D † i ! a C · (L † ) = D † a→P N ,Q (∞) ⊗ L i −1 a D † P N ,Q (∞) i −1 a C · (L † )[−N ] ∼ = i −1 a O P N ,Q (∞)/J O P N ,Q (∞) ⊗ L O P N ,Q (∞) D † P N ,Q (∞) ⊗ L i −1 a D † P N ,Q (∞) i −1 a C · (L † )[−N ] ∼ = i −1 a O P N ,Q (∞)/J O P N ,Q (∞) ⊗ L O P N ,Q (∞) D † P N ,Q (∞) ⊗ L D † P N ,Q (∞) C · (L † ) [−N ] ∼ = i −1 a O P N ,Q (∞)/J O P N ,Q (∞) ⊗ L O P N ,Q (∞) C · (L † ) [−N ] ∼ = K ′ ⊗ L K x † C · (L † )[−N ] ∼ = C · (L † 0 )[−N ]. Here K ′ denotes the residue field of a, and the last isomorphism follows from Lemma 2.9. This finishes the proof. Recall that D † = Γ(P N , D † P N ,Q (∞)) = r>1, s>1 { v∈Z N ≥0 f v (x) ∂ v π \|v\| : f v (x) ∈ K{r −1 x}, f v (x) r s \|v\| are bounded}. L † = r>1, s>1 { w∈Z n ∩δ a w (x)t w : a w (x) ∈ K{r −1 x}, a w (x) r s \|w\| are bounded}, L †′ = r>1, s>1 { w∈C(A) a w (x)t w : a w (x) ∈ K{r −1 x}, a w (x) r s \|w\| are bounded}. where C(A) = {k 1 w 1 + · · · + k N w N : k i ∈ Z ≥0 }. Consider their algebraic counterpart D = { v∈Z N ≥0 f v (x)∂ v : f v (x) ∈ K[x] , f v (x) = 0 for finitely many v}, L = { w∈Z n ∩δ a w (x)t w : a w (x) ∈ K[x] , a w (x) = 0 for finitely many w}, L ′ = { w∈C(A) a w (x)t w : a w (x) ∈ K[x] , a w (x) = 0 for finitely many w}. Then L ′ , L and L/ n i=1 F i,γ L are D-module. By the proof of [1, theorem 4.4], we have D/ λ∈Λ D λ ∼ = L ′ . Combined with Proposition 2.5, we have D † ⊗ D L ′ ∼ = L †′ . In 2.8 where we prove Proposition 0.8, we construct an exact sequence B ′ L †′ → B L †′ → L † → 0, where the direct sums are taken over certain finite sets B ′ and B. The same proof shows that we have an exact sequence B ′ L ′ → B L ′ → L → 0. We have a commutative diagram (4.8.1) B ′ L ′ / / B L ′ / / L / / 0 B ′ L †′ / / B L †′ / / L † / / 0. It follow that D † ⊗ D L ∼ = L † . By [8, Proposition 3.1.1], D is coherent. So L ′ is a coherent D-module. This implies that L and L/ n i=1 F i,γ L are coherent. Let D P N be the sheaf of the usual differential operators on the formal scheme P N , and let D P N ,Q = D P N ⊗ Z Q, D P N ,Q (∞) = O P N ,Q (∞) ⊗ O P N ,Q D P N ,Q . They are coherent. By [17, 1.2.1], the functor Γ(P N , −) induces an equivalence of between the category of coherent D P N ,Q (∞)-modules and the category of coherent Γ(P N , D P N ,Q (∞))-modules. Let D an = { v∈Z N ≥0 f v (x)∂ v : f v (x) ∈ K x † , f v (x) = 0 for finitely many v}. We have Γ(P N , D P N ,Q (∞)) ∼ = D an . Denote the coherent D P N ,Q (∞)-module corresponding to D an ⊗ D L by L. We then have D † P N ,Q (∞) ⊗ D P N ,Q (∞) L ∼ = L † . It follows that D † P N ,Q (∞) ⊗ D P N ,Q (∞) L i F i,γ L ∼ = L † / i F i,γ L † . (4.8.2) Fix a ∈ U . Then F (a, x) is non-degenerate. Fix a monomial basis {t u1 , . . . , t u d } of L 0 / n i=1 F i,γ,a L 0 , where L 0 = K ′ [t u : u ∈ Z n ∩ δ] and d = n!vol(∆). By [2, Proposition 6.6], there exists a polynomial b(x) depending on a such that for any a ′ ∈ A N K satisfying b(a ′ ) = 0, F (a ′ , t) is non-degenerate and {t u1 , . . . , t u d } form a basis of L 0 / n i=1 F i,γ,a ′ L 0 . Lemma 4.9. (L/ n i=1 F i,γ L) b is a free K[x] b module with basis {t u1 , . . . , t u d }. Proof. Let M be a positive integer so that ω(u) ∈ 1 M Z for any u ∈ Z n ∩ δ. Let L m/M be the K[x]-A j t uj + ζ + n i=1 t i ∂ ∂t i F (x, t)η i = ω(uj )=m/M A j t uj + ζ ′ + n i=1 F i,γ η ′ i . with A j ∈ K[x] b , ζ, ζ ′ ∈ (L (m−1)/M ) b and η i , η ′ i ∈ (L (m/M−1) ) b . Applying the same procedure to ζ ′ and after finitely many steps, we see t u1 , . . . , ) n →L (m/M) . If a is a lifting of a ∈ U 0 , then we have a ∈ U . By the same argument in [2], we can chooseb(x) ∈ κ[x] so that b is a lifting ofb, and (L 0 / n i=1 F i,γ,āL0 )b is free with basis {t u1 , . . . , t u d }. t u d generate (L/ n i=1 F i,γ L) b . Suppose f 1 , . . . , f d ∈ K[x] b and f 1 t u1 + · · · + f d t u d = n i=1 F i,γ η i for some η i ∈ L. Then for any a ′ ∈ A N with b(a ′ ) = 0, we have f 1 (a ′ )t u1 + · · · + f d (a ′ )t u d = n i=1 F i,γ,a ′ η i (a ′ ). As {t u1 , . . . , t u d } is a basis of L 0 / n i=1 F i,γ,a ′ L 0 , we have f 1 (a ′ ) = · · · = f d (a ′ ) = 0. Thus f 1 = · · · = f d = 0 in K[x] b .Lemma 4.11. L/ n i=1 F i,γ L \| U0 is a locally free O P N ,Q \| U0 -module. Proof. Let H = L/ n i=1 F i,γ L and H = L/ n i=1 F i,γ L. Then H is the coherent D P N ,Q (∞)module corresponding to the coherent D an -module D an ⊗ D H. By [17, 1.2.1], H is the sheaf associated to the presheaf V → D P N ,Q (∞)(V ) ⊗ D an D an ⊗ D H ∼ = D P N ,Q (∞)(V ) ⊗ D H for any open subset V of P N . For any closed pointā in U 0 , chooseb(x) and b(x) as in Remark 4.10 and Lemma 4.9. Let U ′ 0 be the open subset of U 0 consisting of those pointsā ′ ∈ U 0 such that b(a ′ ) = 0, and let U ′ = sp −1 (U ′ 0 ). Then H\| U ′ 0 is the sheaf associated to the presheaf V → D P N ,Q (∞)(V ) ⊗ D H ∼ = D P N ,Q (∞)(V ) ⊗ Γ(U ′ 0 ,D P N ,Q (∞)) Γ(U ′ 0 , D P N ,Q (∞)) ⊗ D H for any open subset V of U ′ 0 . We have Γ(U ′ 0 , D P N ,Q (∞)) ∼ = O P N,an K (U ′ ) ⊗ K[x] D. So Γ(U ′ 0 , D P N ,Q (∞)) ⊗ D H ∼ = O P N,an K (U ′ ) ⊗ K[x] H. By Lemma 4.9, O P N,an K (U ′ ) ⊗ K[x] H is a free O P N,an K (U ′ )-module. Thus H\| U ′ 0 is a free O P N ,Q \| U ′ 0 - module. Asān i=1 F i,γ L)\| U0∩V0 ∼ = → (L † / n i=1 F i,γ L † )\| U0∩V0 . Proof. Let φ : (L/ n i=1 F i,γ L)\| U0∩V0 → (L † / n i=1 F i,γ L † )\| U0∩V0 be the homomorphism induced by the inclusion L ֒→ L † . Both (L/ n i=1 F i,γ L)\| U0∩V0 and (L † / n i=1 F i,γ L † )\| U0∩V0 are locally free O P N ,Q \| U0∩V0 -modules of the same rank d = n!vol(∆) by Lemma 4.11 and Proposition 0.14. So we only need to show φ is surjective. Cover U 0 ∩ V 0 by finitely many affine open subsets W i so that (L/ n i=1 F i,γ L)\| Wi and (L † / n i=1 F i,γ L † )\| Wi are free O P N ,Q \| Wi -modules. In 2.2-2.4, we construct an epimorphism D † → L †′ whose kernel is λ∈Λ D † λ , and we construct an epimorphism B L †′ → L † . We thus get an epimorphism B D † → L † / n i=1 F i,γ L † and hence an epimorphism of sheaves f † : B D † P N ,Q (∞) → L † / n i=1 F i,γ L † . Similarly, we can construct an epimorphism of sheaves B D P N ,Q (∞) → L/ n i=1 F i,γ L. Note that D P N ,Q (∞)\| A N ∼ = D P N ,Q \| A N . The following diagram commutes: B D P N ,Q (W i ) f → (L/ n i=1 F i,γ L)(W i ) ↓ ↓ B D (m) P N ,Q (W i ) f (m) → (L † / n i=1 F i,γ L † )(W i ) ↓ φ(Wi) B D † P N ,Q (∞)(W i ) f † → (L † / n i=1 F i,γ L † )(W i ). We have D † P N ,Q (∞)(W i ) = m D (m) P N ,Q (W i ). By [8,Lemma 4.1.2], the Banach norm on (L † / n i=1 F i,γ L † )(W i ) inherited from O P N ,Q (W i ) and D (m) P N ,Q (W i ) are equivalent. Since Γ(W i , O P N ,Q (W i )) is a Noetherian Banach K-algebra, the image of φ(W i ) : (L/ n i=1 F i,γ L)(W i ) → (L † / n i=1 F i,γ L † )(W i ) is closed by [13, Proposition 3.7.2/2]. Let's prove the image is dense, which implies that φ is surjective. For any element ξ ∈ (L † / n i=1 F i,γ L † )(W i ), we can find a large m and η ∈ B D (m) P N ,Q (W i ) such that ξ = f (m) (η). Since B D P N ,Q (W i ) is dense in B D (m) P N ,Q (W i ), we can find a sequence {η n } ⊂ B D P N ,Q (W i ) with limit η. Then the sequence {φ(f (η n )} has limit ξ. n i=1 F i,γ L)\| U0 is a crystal. Since (L † / n i=1 F i,γ L † )\| U0∩V0 is a convergent isocrystal, (L/ n i=1 F i,γ L)\| U0∩V0 is(L/ n i=1 F i,γ L)\| U0 ∼ = → D † P N ,Q (∞)\| U0 ⊗ D P N ,Q (∞)\|U 0 (L/ n i=1 F i,γ L)\| U0 . Combined with the isomorphism (4.8.2), we get an isomorphism L/ n i=1 F i,γ L \| U0 ∼ = → L † / n i=1 F i,γ L † \| U0 . Together with Lemma 4.11, we deduce that L † / n i=1 F i,γ L † \| U0 is O P N ,Q \| U0 -coherent. So H n (C · (L † ))\| U0 is O P N ,Q \| U0 -coherent. For any divisor T in P N κ containing P N κ \U 0 , we have ( † T ) H n (C · (L † )) = D † P N ,Q ( † T ) ⊗ D † P N ,Q (∞) H n (C · (L † )). Hence ( † T ) H n (C · (L †· )) is a coherent D † P N ,Q ( † T )-module whose restriction to P N \T is O P N ,Q \| P N \Tcoherent. By [16, Théorème 2.2.12], ( † T ) H n (C · (L † )) is a convergent F -isocrystal on P N \T overconvergent along T. This proves Theorem 0.11 (ii). Consider the complexes C · : = 0 → C 0 (L † ) → · · · → C n (L † ) → H n C · (L † ) → 0 , C · : = 0 → C 0 (L † ) → · · · → C n (L † ) → H n C · (L † ) → 0 , C · 0 : = 0 → C 0 (L † 0 ) → · · · → C n (L † 0 ) → H n C · (L † 0 ) → 0 . For any closed point a in U , as in 4.7, using the right-exactness of the tensor functor and the flatness of H n C · (L † ) \| U0 over O P N ,Q \| U0 proved in Theorem 0.11 (ii), we have i ! a C · = D † a→P N ,Q (∞) ⊗ L i −1 a D † P N ,Q (∞) i −1 a C · [−N ] ∼ = i −1 a O P N ,Q (∞)/J O P N ,Q (∞) ⊗ L O P N ,Q (∞) D † P N ,Q (∞) ⊗ L D P N ,Q (∞) C · [−N ] ∼ = i −1 a O P N ,Q (∞)/J O P N ,Q (∞) ⊗ L O P N ,Q (∞) C · [−N ] ∼ = K ′ ⊗ L K x † C · [−N ] ∼ = K ′ ⊗ K x † C · [−N ] ∼ = C · 0 [−N ] . By Proposition 0.14, C · 0 is acyclic. So i ! a C · is acyclic for any closed point a of U . By [3, Lemma 1.3.11], this implies that C · \| U0 is acyclic. So H k (C · )\| U0 = 0 for k = n. This proves Theorem 0.11 (i). Open questions In this final section, we state a few open questions on the arithmetic and geometry of the overconvergent F -isocrystal Hyp U . For any geometric point a of U such that its specialization is rational over some finite extension F q k , the q k -adic Newton polygon of the action q nk G a on the fiber Hyp(a) = i ! a Hyp U [N ] is independent of the choice of k. This polygon is called the Newton polygon of the fibre Hyp(a) and is denoted by NP (Hyp(a)). It is an important geometric and arithmetic invariant depending on the point a. A basic question is how the Newton polygon NP(Hyp(a)) varies as a varies over U . The well known Grothendieck specialization theorem says that the Newton polygon goes up under specialization. In particular, there is a generic Newton polygon as a varies over U . The generic Newton polygon is the lowest possible Newton polygon as a varies over U . It is attained for all points a whose specialization lies in U 0 − T 0 , where T 0 is a hypersurface in U 0 . This generic Newton polygon depends not only on the data w 1 , . . . , w N and γ 1 , . . . , γ n , but also on p. Denote the generic Newton polygon by GNP(p). The Newton polygon NP(Hyp(a)) is equal to GNP(p) for all a ∈ U − T . Question 5.1. Explicitly determine the generic Newton polygon GNP(p). A good lower bound is obtained by Adolphson-Sperber [4]. For simplicity, we assume that γ = 0 below. In this case, the lower bound depends only the convex hull ∆ of {0, w 1 , . . . , w N } in R n and is independent of p. It is called the Hodge polygon and is denoted by HP(∆). It has slopes 0, 1/D, 2/D, ..., (nD)/D with multiplicities h 0 (∆), h 1 (∆), ..., h nD (∆), where D is some explicitly determined positive integer depending only on ∆, and the Hodge numbers h i (∆) are also explicitly determined by ∆. We have nD i=0 h i (∆) = n!vol(∆). The theorem of Adolphson-Sperber [4] says for every point a ∈ U , NP(Hyp(a)) ≥ HP(∆). In particular, GNP(p) ≥ HP(∆). If this inequality becomes an equality, we say that Hyp is generically ordinary at p. In such an ordinary case, GNP(p) is explicitly determined. Since GNP(p) depends on p but its lower bound HP(∆) does not depend on p, one cannot expect that GNP(p) = HP(∆) in general. In fact, a simple ramification argument shows that a necessary condition for Hyp U to be generically ordinary at p is p ≡ 1 mod D. Adolphson-Sperber [4] conjectured that the converse is also true. It is true in many interesting cases, but the full form is false in general ( [38], [41]). The ideal congruence condition p ≡ 1 mod D needs to be replaced by a slightly weaker congruence condition p ≡ 1 mod D * for some effectively computable positive integer D * . That is, a slightly weaker version of the Adolpson-Sperber conjecture is true. The optimal (smallest) D * can be quite subtle. Suppose that GNP(p) has slopes s 0 < s 1 < . . . with multiplicities h 0 , h 1 , . . .. By Katz's isogeny theorem [31] and the Newton-Hodge decomposition, there is a decreasing filtration of Hyp U \| U−T by convergent sub-F -isocrystals Hyp U \| U−T = M 0 ⊃ M 1 ⊃ M 2 ⊃ ... ⊃ 0, such that each M i /M i−1 is pure of slope s i and rank h i . Twisting by q −si , the convergent Fisocrystal M i /M i−1 becomes a unit root F -isocrytal H i on U − T , which is convergent but may not be over-convergent. This unit root crystal H i corresponds to a lisse p-adic étale sheaf of rank h i on U 0 − T 0 and hence a p-adic representation of π 1 (U 0 − T 0 ). It is a transcendental p-adic object and has no ℓ-adic counterpart. As explained above, in the cases p ≡ 1 mod D * , it is known that the slopes are {0, 1/D, · · · , (nD)/D} and the rank h i is equal to the Hodge number h i (∆) introduced by Adolphson-Sperber [4] for all 0 ≤ i ≤ nD. Question 5.2. Determine the image of the p-adic representation of π 1 (U 0 − T 0 ) → GL(h i ) corresponding to H i . In the classical case of Kloosterman sums over the n-dimensional torus, one has D = D * = 1, the slopes are {0, 1, · · · , n} and the ranks are h 0 = h 1 = · · · = h n = 1. The question on images of the rank one p-adic representations corresponding to H i is asked by Katz [32]. The image lies inside GL(h i , R), where R is a finite ramified extension of Z p . This creates a major new difficulty. For instance, the image will be far from being of finite index in GL(h i , R). In the geometric case of the universal family of ordinary elliptic curves or abelian varieties, this image is the full group GL(h i , Z p ) by Igusa [30] and Chai [19]. A further problem is to study the L-function of the p-adic representation corresponding to H i . By Dwork's conjecture proven in Wan ([39], [40]), the unit root L-function L(H i , t) of H i is a p-adic meromorphic function in t. It is not rational in t in general. See Liu-Wan [34] for the geometric example of universal ordinary elliptic curves, and its close connection to overconvergent p-adic modular forms. The L-function L(H i , t) will have infinitely many zeros and infinitely many poles. Question 5.3. Determine the order of L(H i , t) as a p-adic meromorphic function, that is, determine approximately how many zeros and poles in a large disk \|t\| p < r as r goes to infinity. The answer depends at least on the rank h i , the dimension of U , and possibly on finer geometry of Hyp U \| U−T . The above questions are arithmetic in nature. A geometric problem is the following. For i = 0, H 0 always has rank 1 and nonzero horizontal sections (unique up to constant) of H 0 is of the same form as the integral representation solution (0.1.2) of the GKZ hypergeometric system, as shown by Adolphson-Sperber [5]. It would be interesting to understand the higher slope case when i > 0. T a,x (e 1 1(a), . . . , e M (a)) = (e 1 (x), . . . , e M (x))P (x) S m (F (x, t)) = (−1) n Tr (P (x q )Q(a)P (x) −1 ) m , (0.16.1) L(F (x, t), T ) = det I − T P (x q )Q(a)P (x) −1 (−1) n+1 (0.16.(x), . . . , e M (x) = (e 1 (x), . . . , e M (x))A j (x). this equation, we show it holds componentwisely. The equation clearly holds for those component β = β 1 . Collecting the linear combination of t w with w ∈ C(A) on both sides of This is exactly the β 1 component of the equation (2.8.1). In general, for i = 1, . . . , k, we define f (i) β = w∈C(A), w+(β−β1) ∈C(A),..., w+(β−βi) ∈C(A) A), w+(β−β1) ∈C(A),..., w+(β−βi−1) ∈C(A),w+(β−βi)∈C(A) (πz − πz q m )\| z=aj u . Let K ′ be a finite extension of K containing all q-th roots of unity. Set L(s) 0 = { w∈Z n ∩δ a w t w : a w ∈ K ′ , \|a w \|s \|w\| are bounded}.We have L † 0 = s>1 L(s) 0 . Note that L(s) 0 (s > 1) and L † 0 are rings. Each L(s) 0 is a Banach space with respect to the norm w∈Z n ∩δ a w t w = sup w∈Z n ∩δ \|a w \|s \|w\| . Theorem 3 . 3 ( 33Dwork trace formula). The operator G a : L † 0 → L † 0 is nuclear, and we have For any s < s ′ , we have L(s ′ ) 0 ⊂L(s) 0 ⊂ L(s) 0 , and L † 0 = s>1L (s) 0 . EndowL(s) 0 with the norm w∈Z n ∩δ a w t w = sup w∈Z n ∩δ \|a w \|s \|w\| .ThenL(s) 0 is a Banach space. The inclusion L(s ′ ) 0 ֒→L(s) 0 is completely continuous. Indeed, choose s < s ′′ < s ′ . We can factorize this inclusion as the compositeL(s ′ ) 0 ֒→L(s ′′ ) 0 ֒→L(s) 0 .It suffices to verify the inclusion i :L(s ′′ ) 0 ֒→L(s) 0 is completely continuous. Let L S be the finite dimensional K ′ -vector space spanned by a finite subset S of {t w } w∈Z n ∩δ , and leti S :L(s ′′ ) 0 →L(s) 0be the composite of the projectionL(s ′′ ) 0 → L S and the inclusion L S ֒→L(s) 0 . One can verify thati S − i ≤ sup then G a induces a map G a :L(s) 0 →L(s) 0 . Indeed it is the compositẽ L(s) 0 ֒→ L(s) :L(s) 0 →L(s) 0 is completely continuous since the last inclusion in the above composite is completely continuous. In particular, it is nuclear ([36, Theorem 6.9]). Write t (1−q)γ exp πF (a, t) − πF (a, t q ) = w∈Z n ∩δ c w t w . In particular, Tr(G a ,L(s) 0 ) is independent of s. Similarly, Tr(G m a ,L(s) 0 ) and det(I − T G a ,L(s) of s. For any monic irreducible polynomial f (T ) ∈ K ′ [T ] with nonzero constant term, write ([36, Theorem 6.9])L is nilpotent on N f and bijective on W f . Since det(I − T G a ,L(s) 0 ) is independent of s, all N (s) f have the same dimension, and hence we have N f = N (s) f for all 1 < s < p p−1 (4.3.1) follows from that φ is quasi-compact and quasi-separated. (4.3.2) follows from that j ′ λ and j λ are affine. (4.3.3) follows from [7, Proposition 3.4.9], where φ K : X ′ K → X K is the canonical morphism. By [20, Theorem 9.2.5], we have ∇, where j 0 : D(0, 1 + ) N → A 1,an K is the inclusion. Let L ψ,Fa be the pull-back of L ψ by the morphism F a : X ′an K → A 1,an K defined by y → F (a, t). Then L ψ,Fa is j ′ † O X ′ an K considered as a free j ′ † O X ′ an K -module with the connection P N ,Q (∞) is flat over D † P N ,Q ([8, 4.3.11]), and D † P N ,Q (∞) is flat over O P N ,Q (∞) ([8, 3.5.2]). So we have submodule of L generated by all t u with ω(u) ≤ m/M , and let L (m/M) be the K[x]-submodule generated by t u with ω(u) = m/M. We have b(x) = M(d+1) m=0 b m/M (x), where b m/M (x) are defined in [2, §6]. By [2, Proposition 6.3], for any h ∈ L (m/M) , we have h = ω(uj )=m/M Remark 4 . 10 . 410LetL 0 = κ[x][t u : u ∈ Z ∩ δ], and letL 0,m/M andL (m/M) 0 be defined similarly as in proof of Lemma 4.9. The matrix of the homomorphism φ m : (L (m/M−1) ) n → L (m/M) defined in [2, §6] is a lifting of a similar homomorphismφ m : (L (m/M−1) goes over closed points of U 0 , U ′ 0 form an open covering of U 0 . So H\| U0 is a locally free O P N ,Q \| U0 -module.On the other hand, by Proposition 0.9 and [18, 1.3.4], there exists a Zariski openV 0 ⊂ P N such that H n (C · (L † )) ∼ = L † / n i=1 F i,γ L † defines a convergent isocrystal on V 0 . Lemma 4 . 12 . 412We have an isomorphism (L/ 4. 13 . 13Proof of Theorem 0.11. By Lemma 4.11 and [37, Proposition 1.20], (L/ Question 5 . 4 . 54Explicitly determine the p-adic horizontal sections of the connection for the unit root F -isocrystal H i . modules with coherent cohomology. By[29, Théorème 5.3.3] and Proposition 0.8, C · also a convergent isocrystal by Lemma 4.12. By [37, Theorem 2.16], (L/ n i=1 F i,γ L)\| U0 is a convergent isocrystal. By [11, Proposition 3.1.4], we have an isomorphism X ,Q ( † T ) f −1 E[d X ′ − d X ], We have L † 0 = b>0 L(b). Since ord p (π) = 1 p−1 , we haveLet H(t) = t (1−q)γ exp πF (a, t) − πF (a q , t q ) . By the proof of Proposition 0.4 (i) in 1.1, we have H(t) ∈ L( p−1 pq ). The operator G a = Ψ a • H(t) can be considered as the compositeSo for any 0 < b ≤ p−1 p , G a defines an endomorphism of L(b), and for any 0 < b ≤ p−1 pq , G a induces a morphism L(b) → L(qb). In the case p > 2, we have p−1 p > 1 p−1 . So G a induces a morphism on B:Using L(b) and B instead of L † 0 , we can construct twisted logarithmic de Rham complexes C · (L(b))Here i 0 , i l , i,ĩ are induced by inclusions of complexes. Since i 0 is surjective and G a :by [2, 3.8, 3.9]. The proof for the case p = 2 is similar. We replace π satisfying π + π p p = 0 by π ′ satisfying ∞ i=0 π ′p i p i = 0, and replace the function exp(π(z − z p )) = exp πz + (πz) p p by E(π ′ z),is the Artin-Hasse exponential. These replacements give rise to a proof which works for all p. See[33]for details.k,j < p m for all j. The sheaf of differential operators of level m on X is defined byWe have D (m)where O X ,Q = O X ⊗ Z Q. 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[ "Local well-posedness for the Sixth-Order Boussinesq Equation", "Local well-posedness for the Sixth-Order Boussinesq Equation" ]	[ "Amin Esfahani [email protected] ", "Gustavo Luiz ", "Farah [email protected] ", "\nSchool of Mathematics and Computer Science\nPostal Code\nDamghan University Damghan\n36716-41167Iran\n", "\nUniversidade Federal de Minas Gerais Av. Antônio Carlos\n30123-970 Belo Horizonte-MG6627, 702Caixa PostalBrazil\n" ]	[ "School of Mathematics and Computer Science\nPostal Code\nDamghan University Damghan\n36716-41167Iran", "Universidade Federal de Minas Gerais Av. Antônio Carlos\n30123-970 Belo Horizonte-MG6627, 702Caixa PostalBrazil" ]	[]	This work studies the local well-posedness of the initial-value problem for the nonlinear sixthorder Boussinesq equation utt = uxx + βuxxxx + uxxxxxx + (u 2 )xx, where β = ±1. We prove local well-posedness with initial data in non-homogeneous Sobolev spaces H s (R) for negative indices of s ∈ R. Mathematical subject classification: 35B30, 35Q55, 35Q72. * Supported by FAPEMIG-Brazil and CNPq-Brazil.	10.1016/j.jmaa.2011.06.038	[ "https://arxiv.org/pdf/1012.5088v1.pdf" ]	119,303,906	1012.5088	cf43db7d392896b2c2eedc2a636021497fadfadf	Local well-posedness for the Sixth-Order Boussinesq Equation 22 Dec 2010 Amin Esfahani [email protected] Gustavo Luiz Farah [email protected] School of Mathematics and Computer Science Postal Code Damghan University Damghan 36716-41167Iran Universidade Federal de Minas Gerais Av. Antônio Carlos 30123-970 Belo Horizonte-MG6627, 702Caixa PostalBrazil Local well-posedness for the Sixth-Order Boussinesq Equation 22 Dec 2010 This work studies the local well-posedness of the initial-value problem for the nonlinear sixthorder Boussinesq equation utt = uxx + βuxxxx + uxxxxxx + (u 2 )xx, where β = ±1. We prove local well-posedness with initial data in non-homogeneous Sobolev spaces H s (R) for negative indices of s ∈ R. Mathematical subject classification: 35B30, 35Q55, 35Q72. * Supported by FAPEMIG-Brazil and CNPq-Brazil. Introduction The study of wave propagation on the surface of water has been a subject of considerable theoretical and practical importance during the past decades. In 1872, Joseph Boussinesq [6] derived a model equation for propagation of water waves from Euler's equations of motion for two-dimensional potential flow beneath a free surface by introducing appropriate approximations for small amplitude long waves. Later the Boussinesq equation u tt = u xx + βu xxxx + (f (u)) xx , β = ±1,(1.1) appeared not only in the study of the dynamics of thin inviscid layers with free surface but also in the study of the nonlinear string, the shape-memory alloys, the propagation of waves in elastic rods and in the continuum limit of lattice dynamics or coupled electrical circuits (see [11] and the references therein). Our principal aim here is to study the local well-posedness for the initial value problem associated to the sixth-order Boussinesq equation with quadratic nonlinearity [7,8]: u tt = u xx + βu xxxx + u xxxxxx + (u 2 ) xx , x ∈ R, t ≥ 0, u(0, x) = ϕ(x); u t (0, x) = ψ x (x), (1.2) where β = ±1. It is worth noting that the stationary propagating localized solutions of equation (1.2) have been investigated numerically and the two classes of subsonic solutions corresponding to the sign of β have been obtained, more precisely, the monotone shapes and the shapes with oscillatory tails [7]. Natural spaces to study the initial value problem above are the classical Sobolev spaces H s (R), s ∈ R, which are defined via the spacial Fourier transform g(ξ) = R e −ixξ g(x) dx, as the completion of the Schwarz class S(R) with respect to the norm g H s (R) = ξ s g(ξ) L 2 (R) , where ξ = 1 + \|ξ\|. Given initial data (φ, (ψ) x ) ∈ H s (R) × H s−1 (R) and a positive time T > 0, we say that a function u : R × [0, T ] → R is a real solution of (1.2) if u ∈ C([0, T ]; H s (R)) and u satisfies the integral equation u(t) = V c (t)ϕ + V s (t)ψ x + t 0 V s (t − t ′ )(u 2 ) xx (t ′ )dt ′ . (1.3) where the two operators that constitute the free evolution are defined via Fourier transform by the formulas V c (t)ϕ = e it √ ξ 2 −βξ 4 +ξ 6 + e −it √ ξ 2 −βξ 4 +ξ 6 2 ϕ(ξ) ∨ (1.4) V s (t)ψ x = e it √ ξ 2 −βξ 4 +ξ 6 − e −it √ ξ 2 −βξ 4 +ξ 6 2i ξ 2 − βξ 4 + ξ 6 ψ x (ξ) ∨ . (1. 5) In the case that T can be taken arbitrarily large, we shall say the solution is global-in-time. Here, we focus our attention only in local solutions. Concerning the local well-posedness question, when β = −1, several results are obtained for equation (1.1) (so-called "good" Boussinesq equation) [3,12,13,18,23]. On the other hand while equation (1.1) with β = 1 (so-called the "bad" Boussinesq equation) only soliton type solutions are known. Moreover, taking Fourier transform, we can see that the solution of the linearized equation u grows as e ±ξ 2 t . The same occurs for the nonlinear problem. Therefore, to study well-posedness the component proportional to e ξ 2 t has to be vanished. We refer the reader to [9] for results concerning this "bad" version using the inverse scattering approach. The local well-posedness for dispersive equations with quadratic nonlinearities has been extensively studied in Sobolev spaces with negative indices. The proof of these results are based on the Fourier restriction norm approach introduced by Bourgain [4,5] in his study of the nonlinear Schrödinger (NLS) equation iu t + u xx + \|u\| p u = 0 and the Korteweg-de Vries (KdV) equation u t + u xxx + uu x = 0. This method was further developed by Kenig et al. [16] for the KdV Equation and [17] for the quadratics nonlinear Schrödinger equations. The original Bourgain method makes extensive use of the Strichartz inequalities in order to derive the bilinear estimates corresponding to the nonlinearity. On the other hand, Kenig et al. simplified Bourgain's proof and improved the bilinear estimates using only elementary techniques, such as Cauchy-Schwartz inequality and simple calculus inequalities (see also [15,22]). In this paper, we prove local well-posedness in H s (R) with s > −1/2 for (1.2) using the idea introduced in [13]. Indeed, we modify the Bourgain-type space observing that the dispersion for this equation, given by the symbol ξ 2 − βξ 4 + ξ 6 , is in some sense related with the symbol of the KdVtype equation. This modification allow us to obtain bilinear estimates using the same techniques as in [16,17]. To describe our results we define next the X s,b spaces related to our problem. Definition 1.1 For s, b ∈ R, X s,b denotes the completion of the Schwartz class S (R 2 ) with respect to the norm u X s,b = \|τ \| − γ(ξ) b ξ s u(τ, ξ) L 2 τ,ξ (R 2 ) (1.6) where γ(ξ) ≡ ξ 2 − βξ 4 + ξ 6 and "∧" denotes the time-space Fourier transform. As a consequence of this definition, we immediately have for b > 1/2, that X s,b is embedded in C(R; H s (R)). We will also need the localized X s,b spaces defined as follows. Definition 1.2 For s, b ∈ R and T ≥ 0, X s,b T denotes the space endowed with the norm u X s,b T = inf w∈X s,b { w X s,b : w(t) = u(t) on [0, T ]} . The main result of this paper reads as follows. Theorem 1.1 Let s > −1/2, then for all ϕ ∈ H s (R) and ψ ∈ H s−1 (R), there exist T = T ( ϕ H s (R) , ψ H s−1 (R) ) and a unique solution u of the initial value problem associated to equation (1.2) with initial data u(0) = ϕ and u t (0) = ψ x such that u ∈ C([0, T ]; H s (R)) ∩ X s,b T . Moreover, given T ′ ∈ (0, T ) there exists R = R(T ′ ) > 0 such that giving the set In some sense the previous theorem is quite surprising. There is no difference in the local theory when one considers the signs ± in front of the forth derivative term in equation (1.2). However, despising the sixth order term in (1.2), we obtain the Boussinesq equation (1.1), where the "good" and "bad" models are very distinct. W = {(φ,ψ) ∈ H s (R) × H s−1 (R) : φ − ϕ 2 H s (R) + ψ − ψ 2 H s−1 (R) < R} the map solution S : W −→ C([0, T ′ ] : H s (R)) ∩ X s,b T , (φ,ψ) −→ u(t) is Lipschitz. In addition, if (ϕ, ψ) ∈ H s ′ (R) × H s ′ −1 (R) with s ′ > s, We should remark that because of lack of a scaling argument for the Boussinesq-type equations, it is not clear what is the lower index s where one has local well-posedness for equation (1.2) with initial data u(0) = ϕ and u t (0) = ψ x , where (ϕ, ψ) ∈ H s (R) × H s−1 (R). Here we answer, partially, this question. In fact, our main result is a negative one; it concerns in particular a kind of ill-posedness. We prove that the flow map for the Cauchy problem associated to equation (1.2) is not smooth (more precisely C 2 ) at the origin for initial data in H s (R) × H s−1 (R), with s < −3 (cf. Theorems 4.1 and 4.2). Therefore any iterative method applied to the integral formulation of the Boussinesq equation (1.2) always fails in this functional setting. In other words, if one can apply the contraction mapping principle to solve the integral equation corresponding to (1.2) thus, by the implicit function Theorem, the flow-map data solution is smooth, which is a contradiction (cf. Theorem 4.2). Indeed our ideas are based on an argument similar to Tzvetkov [24] (see also Bourgain [5]) who established a similar result for the KdV equation. The same question was studied by Molinet, Saut and Tzvetkov [19,20], for the Benjamin-Ono (BO) equation u t + Hu xx + uu x = 0 (1.7) and for the Kadomtsev-Petviashvili-I (KPI) equation (u t + uu x + u xxx ) x − u yy = 0,(1.8) respectively (see also [10]). In all the mentioned ill-posedness results it is, in fact, proved that for a fixed t > 0 the flow map ϕ → u(t) is not C 2 differentiable at zero. This, of course, implies that the flow map is not smooth (C 2 ) at the origin. Unfortunately, in our case we cannot fix t > 0 since we do not have good cancelations on the symbol ξ 2 − βξ 4 + ξ 6 . To overcome this difficulty, we allow the variable t to move. Therefore, choosing suitable characteristics functions and sending t to zero to obtain our results (cf. Theorems 4.1 and 4.2). We should remark that this kind of argument also appears in the ill-posed result of Bejenaru and Tao [1]. The plan of this paper is as follows: in Section 2, we state some linear estimates for the integral equation in the X s,b space introduced above. Bilinear estimates and the relevant counterexamples are proved in Section 3. Finally, the ill-posedness question is treated in Section 4. The Cauchy Problem Let us start this section by introducing the notation used throughout the paper. We use c to denote various constants that may vary line by line. Given any positive numbers a and b, the notation a b means that there exists a positive constant c such that a ≤ cb. Also, we denote a ∼ b when, a b and b a. Given θ be a cutoff function satisfying θ ∈ C ∞ 0 (R), 0 ≤ θ ≤ 1, θ ≡ 1 in [−1, 1], supp(θ) ⊆ [−2, 2] and for 0 < T < 1 define θ T (t) = θ(t/T ). In fact, to work in the X s,b spaces we consider another version of (1.3), that is u(t) = θ(t) (V c (t)ϕ + V s (t)ψ x ) + θ T (t) t 0 V s (t − t ′ )(u 2 ) xx (t ′ )dt ′ . (2.1) Note that the integral equation (2.1) is defined for all (t, x) ∈ R 2 . Moreover if u is a solution of (2.1) thanũ = u\| [0,T ] will be a solution of (1.3) in [0, T ]. In the next two lemmas, we estimate the linear and integral part of (2.1). We refer the reader to [13] for the proofs (see also [14] and [15]). Lemma 2.1 Let u(t) the solution of the linear equation u tt = u xx + βu xxxx + u xxxxxx , u(0, x) = ϕ(x); u t (0, x) = ψ x (x) with ϕ ∈ H s (R) and ψ ∈ H s−1 (R). Then there exists c > 0 depending only on θ, s, b such that θu X s,b ≤ c ϕ H s (R) + ψ H s−1 (R) . (2.2) Lemma 2.2 Let − 1 2 < b ′ ≤ 0 ≤ b ≤ b ′ + 1 and 0 < T ≤ 1 then (i) θ T (t) t 0 g(t ′ )dt ′ H b t ≤ T 1−(b−b ′ ) g H b ′ t ; (ii) θ T (t) t 0 V s (t − t ′ )f (u)(t ′ )dt ′ X s,b ≤ T 1−(b−b ′ ) f (u)(τ, ξ) 2iγ(ξ) ∨ X s,b ′ . Bilinear Estimates As it is standard in the Fourier restriction method, the linear estimates given in Lemmas 2.1-2.2 immediately yields Theorem 1.1 (see [13] for details) once we prove the following crucial nonlinear estimate. Theorem 3.1 Let s > −1/2 and u, v ∈ X s,−a . Then, there exists c > 0 such that \|ξ\| 2 uv(τ, ξ) 2iγ(ξ) ∨ X s,−a ≤ c u X s,b v X s,b , (3.1) where ∨ denotes the inverse time-space Fourier transform, holds in the following cases (i) s ≥ 0, b > 1/2 and 1/6 < a < 1/2, (ii) −1/2 < s < 0, b > 1/2 and 1/6 < a < 1/2 such that \|s\| < a. Moreover, the constant c > 0 that appears in (3.1) depends only on a, b, s. Before proceed to the proof of Theorem 3.1, we state some elementary calculus inequalities that will be useful later. Lemma 3.1 For λ, µ ∈ R, p, q > 0 and r = min{p, q, p + q − 1} with p + q > 1, we have R dx x − λ p x − µ q 1 λ − µ r . (3.2) Moreover, for a i ∈ R, i = 0, 1, 2, 3, and q > 1/3 R dx a 0 + a 1 \|x\| + a 2 x 2 + a 3 \|x\| 3 q 1. (3.3) Proof. See Lemma 4.2 in [15] and Lemma 2.5 in [2]. Lemma 3.2 There exists c > 0 such that 1 c ≤ sup x,y≥0 1 + x − y 3 + β 2 √ y 1 + x − y − βy 2 + y 3 ≤ c. (3.4) Proof. Since y 3 − β 2 √ y ≤ y − βy 2 + y 3 ≤ y 3 − β 2 √ y + 1 2 , for all y ≥ 0 a simple computation shows the desired inequalities. Remark 3.1 We should note that by using the previous lemma, we have an equivalent way to compute the X s,b -norm, that is u X s,b ∼ \|τ \| − \|ξ\| 3 + β 2 \|ξ\| b ξ s u(τ, ξ) L 2 τ,ξ (R 2 ) . (3.5) This equivalence will be important in the proof of Theorem 3.1, because the symbol ξ 2 − βξ 4 + ξ 6 of equation ( ξ 2 − βξ 4 + ξ 6 ≤ 1, for all ξ = 0. (3.6) We will prove the theorem for the case β = 1, the case β = −1 can be analogously proved. Let u, v ∈ X s,b and define f (τ, ξ) = \|τ \| − \|ξ\| 3 + \|ξ\|/2 b ξ s u(τ, ξ) and g(τ, ξ) = \|τ \| − \|ξ\| 3 + \|ξ\|/2 b ξ s v(τ, ξ). Using Remark 3.1, inequity (3.6) and a duality argument the desired inequality is equivalent to \|W(f, g, ϕ)\| ≤ c f L 2 ξ,τ g L 2 ξ,τ h L 2 ξ,τ ,(3.7) where W(f, g, h) = R 4 ξ s g(τ 1 , ξ 1 )f (τ 2 , ξ 2 )h(τ, ξ) ξ 1 s ξ 2 s \|τ \| − \|ξ\| 3 + \|ξ\|/2 a \|τ 1 \| − \|ξ 1 \| 3 + \|ξ 1 \|/2 b \|τ 2 \| − \|ξ 2 \| 3 + \|ξ 2 \|/2 b dξdτ dξ 1 dτ 1 , where τ 2 = τ − τ 1 and ξ 2 = ξ − ξ 1 . Therefore to perform the desired estimate we need to analyze all the possible cases for the sign of τ , τ 1 and τ 2 . To do this we split R 4 into the following regions: Γ 1 = (ξ, τ, ξ 1 , τ 1 ) ∈ R 4 : τ 1 , τ 2 < 0 , Γ 2 = (ξ, τ, ξ 1 , τ 1 ) ∈ R 4 : τ 1 , τ ≥ 0, τ 2 < 0 , Γ 3 = (ξ, τ, ξ 1 , τ 1 ) ∈ R 4 : τ 1 ≥ 0, τ 2 , τ < 0 , Γ 4 = (ξ, τ, ξ 1 , τ 1 ) ∈ R 4 : τ 1 < 0, τ 2 , τ ≥ 0 , Γ 5 = (ξ, τ, ξ 1 , τ 1 ) ∈ R 4 : τ 1 , τ < 0, τ 2 ≥ 0 , Γ 6 = (ξ, τ, ξ 1 , τ 1 ) ∈ R 4 : τ 1 , τ 2 ≥ 0 . Thus, it is suffices to prove inequality (3.7) with D(f, g, h) instead of W(f, g, h), where D(f, g, h) = R 4 ξ s ξ 1 s ξ 2 s g(τ 1 , ξ 1 )f (τ 2 , ξ 2 )h(τ, ξ) σ a σ 1 b σ 2 b dξdτ dξ 1 dτ 1 , with ξ 2 , τ 2 and σ, σ 1 , σ 2 belonging to one of the following cases (I) σ = τ + \|ξ\| 3 − 1 2 \|ξ\|, σ 1 = τ 1 + \|ξ 1 \| 3 − 1 2 \|ξ 1 \|, σ 2 = τ 2 + \|ξ 2 \| 3 − 1 2 \|ξ 2 \|, (II) σ = τ − \|ξ\| 3 + 1 2 \|ξ\|, σ 1 = τ 1 − \|ξ 1 \| 3 + 1 2 \|ξ 1 \|, σ 2 = τ 2 + \|ξ 2 \| 3 − 1 2 \|ξ 2 \|, (III) σ = τ + \|ξ\| 3 − 1 2 \|ξ\|, σ 1 = τ 1 − \|ξ 1 \| 3 + 1 2 \|ξ 1 \|, σ 2 = τ 2 + \|ξ 2 \| 3 − 1 2 \|ξ 2 \|, (IV) σ = τ − \|ξ\| 3 + 1 2 \|ξ\|, σ 1 = τ 1 + \|ξ 1 \| 3 − 1 2 \|ξ 1 \|, σ 2 = τ 2 − \|ξ 2 \| 3 + 1 2 \|ξ 2 \|, (V) σ = τ + \|ξ\| 3 − 1 2 \|ξ\|, σ 1 = τ 1 + \|ξ 1 \| 3 − 1 2 \|ξ 1 \|, σ 2 = τ 2 − \|ξ 2 \| 3 + 1 2 \|ξ 2 \|, (VI) σ = τ − \|ξ\| 3 + 1 2 \|ξ\|, σ 1 = τ 1 − \|ξ 1 \| 3 + 1 2 \|ξ 1 \|, σ 2 = τ 2 − \|ξ 2 \| 3 + 1 2 \|ξ 2 \|. First we note that the cases σ = τ + \|ξ\| 3 − \|ξ\|/2, σ 1 = τ 1 − \|ξ 1 \| 3 + \|ξ 2 \|/2, σ 2 = τ 2 − \|ξ 2 \| 3 + \|ξ 2 \|/2 and σ = τ − \|ξ\| 3 + \|ξ\|/2, σ 1 = τ 1 + \|ξ 1 \| 3 − \|ξ 1 \|/2, σ 2 = τ 2 + \|ξ 2 \| 3 − \|ξ 2 \|/2 cannot occur, since τ 1 < 0, τ 2 < 0 implies τ < 0, and τ 1 ≥ 0, τ 2 ≥ 0 implies τ ≥ 0. On the other hand, by applying the change of variables (ξ, τ, ξ 1 , τ 1 ) → −(ξ, τ, ξ 1 , τ 1 ) and observing that the L 2 -norm is preserved under the reflection operation, the cases (IV), (V), (VI) can be easily reduced, respectively, to (III), (II), (I). Moreover, making the change of variables τ 2 = τ − τ 1 , ξ 2 = ξ − ξ 1 and then (ξ, τ, ξ 2 , τ 2 ) → −(ξ, τ, ξ 2 , τ 2 ) the case (II) can be reduced (III). Therefore we need only establish cases (I) and (III). Now we first treat the inequality (3.7) with D(f, g, h) in the case (I). By symmetry we can restrict ourselves to the set A = (ξ, τ, ξ 1 , τ 1 ) ∈ R 4 : \|σ 2 \| ≤ \|σ 1 \| . We divide A into the following four subregions: A 1 = {(ξ, τ, ξ 1 , τ 1 ) ∈ A : \|ξ 1 \| ≤ 10}, A 2 = {(ξ, τ, ξ 1 , τ 1 ) ∈ A : \|ξ 1 \| ≥ 10, \|2ξ 1 − ξ\| ≥ \|ξ 1 \|/2}, A 3 = {(ξ, τ, ξ 1 , τ 1 ) ∈ A : \|ξ 1 \| ≥ 10, \|ξ 1 − ξ\| ≥ \|ξ 1 \|/2, \|σ 1 \| ≤ \|σ\|}, A 4 = {(ξ, τ, ξ 1 , τ 1 ) ∈ A : \|ξ 1 \| ≥ 10, \|ξ 1 − ξ\| ≥ \|ξ 1 \|/2, \|σ 1 \| ≥ \|σ\|}. We have A = A 1 ∪ A 2 ∪ A 3 ∪ A 4 . Indeed \|ξ 1 \| > \|2ξ 1 − ξ\| + \|ξ 1 − ξ\| ≥ \|(2ξ 1 − ξ) − (ξ 1 − ξ)\| = \|ξ 1 \|. Using the Cauchy-Schwarz and Hölder inequalities it is easy to see that \|Z\| ≤ f L 2 ξ,τ (R 2 ) g L 2 ξ,τ (R 2 ) h L 2 ξ,τ (R 2 ) ξ 2s σ 2a R 2 χ A1∪A2∪A3 dξ 1 dτ 1 ξ 1 2s ξ 2 2s σ 1 2b σ 2 2b 1 2 L ∞ ξ,τ (R 2 ) + f L 2 ξ,τ (R 2 ) g L 2 ξ,τ (R 2 ) h L 2 ξ,τ (R 2 ) 1 ξ 1 2s σ 1 2b R 2 χ A4 ξ 2s dξdτ ξ 2 2s σ 2a σ 2 2b 1 2 L ∞ ξ 1 ,τ 1 (R 2 ) . (3.8) Noting that ξ 2s ≤ ξ 1 2\|s\| ξ 2 2s , for s ≥ 0, and ξ 2 −2s ≤ ξ 1 2\|s\| ξ −2s , for s < 0 we have ξ 2s ξ 1 2s ξ 2 2s ≤ ξ 1 ϑ(s) (3.9) where ϑ(s) = 0, if s ≥ 0 4\|s\|, if s ≤ 0 . By employing Lemma 3.1, it sufficient to get bounds for J 1 (ξ 1 , τ 1 ) = 1 σ 2a A1∪A2∪A3 ξ 1 ϑ(s) τ + \|ξ 2 \| 3 − \|ξ 2 \|/2 + \|ξ 1 \| 3 − \|ξ 1 \|/2 2b dξ 1 and J 2 (ξ, τ ) = ξ 1 ϑ(s) σ 1 2b A4 dξ τ 1 − \|ξ 2 \| 3 + \|ξ 2 \|/2 + \|ξ\| 3 − \|ξ\|/2 2a . Case 1. Contribution of A 1 to J 1 . In region A 1 we have ξ 1 ϑ(s) 1. Therefore for a > 0 and b > 1/2 we obtain J 1 (ξ, τ ) \|ξ1\|≤10 dξ 1 1. Case 2. Contribution of A 2 to J 1 . In this region, we use a change of variable η = τ + \|ξ − ξ 1 \| 3 − \|ξ − ξ 1 \|/2 + \|ξ 1 \| 3 − \|ξ 1 \|/2. If ξ 1 ξ 2 ≥ 0, and without loss of generality ξ 1 , ξ 2 ≥ 0, then since ξ ≥ ξ 1 ≥ 10, J 1 1 σ 2a ξ 1 ϑ(s) \|ξ\|\|ξ − 2ξ 1 \| η 2b dξ 1 1 σ 2a ξ 1 ϑ(s)−2 η 2b dη 1. If ξ 1 ξ 2 ≤ 0, and without loss of generality ξ 1 ≥ 0 and ξ 2 ≤ 0. Therefore ξ 1 ≥ ξ and moreover ξ 2 + 2ξ 1 (ξ 1 − ξ) − 1 3 = ξ 2 1 + (ξ 1 − ξ) 2 − 1 3 ≥ 1 2 ξ 2 1 . (3.10) Using this relation, we obtain J 1 1 σ 2a ξ 1 ϑ(s) \|ξ 2 + 2ξ 1 (ξ 1 − ξ) − 1/3\| η 2b dξ 1 1 σ 2a ξ 1 ϑ(s)−2 η 2b dη 1, for a > 0, b > 1/2 and ϑ(s) ≤ 2. Case 3. Contribution of A 3 to J 1 . If ξ 1 ξ 2 ≥ 0, and without loss of generality ξ 1 , ξ 2 ≥ 0, then σ − σ 1 − σ 2 = −3ξξ 1 ξ 2 . Moreover, since ξ ≥ ξ 1 ≥ 10, we conclude σ \|ξξ 1 ξ 2 \| \|ξ 1 \| 3 ξ 3 1 , so that Lemma 3.1 implies that J 1 ∞ 0 ξ 1 ϑ(s)−6a τ + (ξ − ξ 1 ) 3 − (ξ − ξ 1 )/2 + ξ 3 1 − ξ 1 /2 2b dξ 1 1, for b > 1/2 and ϑ(s) ≤ 6a. If ξ 1 ≥ 0 and ξ 2 ≤ 0, then ξ 1 ≥ ξ. Therefore, by a change of variable η = τ − ξ 3 2 + ξ 2 /2 + ξ 3 1 − ξ 1 /2 and using (3.10), we have J 1 1 σ 2a ξ 1 ϑ(s) \|ξ 2 + 2ξ 1 (ξ 1 − ξ) − 1/3\| η 2b dη 1 σ 2a ξ 1 ϑ(s)−2 η 2b dη 1, for a > 0, b > 1/2 and ϑ(s) ≤ 2. Case 4. Now we estimate J 2 (ξ 1 , τ 1 ). We use a change of variable η = τ 1 +\|ξ\| 3 −\|ξ\|/2−\|ξ−ξ 1 \| 3 +\|ξ−ξ 1 \|/2. Hence we have \|η\| \|σ 1 \| + \|σ\| σ 1 . If ξ, ξ 2 ≥ 0, then \|2ξ − ξ 1 \| = 2ξ − ξ 1 ≥ ξ − ξ 1 = \|ξ − ξ 1 \| ≥ \|ξ 1 \|/2 so that J 2 ξ 1 ϑ(s) σ 1 2b \|η\| σ1 dη \|ξ 1 \|\|2ξ − ξ 1 \| η 2a ξ 1 ϑ(s)−2 σ 1 2b+2a−1 1, for b > 1/2, 0 < a < 1/2 and ϑ(s) ≤ 2. If ξ ≥ 0 and ξ − ξ 1 ≤ 0, then one has ξ 2 + (ξ − ξ 1 ) 2 − 1/3 = ξ 2 1 /2 + (2ξ − ξ 1 ) 2 /2 − 1/3 ≥ ξ 2 1 /2 − 1/3 ≥ ξ 2 1 /4, so that J 2 ξ 1 ϑ(s) σ 1 2b \|η\| σ1 dη \|ξ 2 + (ξ − ξ 1 ) 2 − 1/3\| η 2a ξ 1 ϑ(s)−2 σ 1 2b+2a−1 1, for b > 1/2, 0 < a < 1/2 and ϑ(s) ≤ 2. Now we are going to prove the case (III). First we split R 4 into the following six regions: B 1 = (ξ, τ, ξ 1 , τ 1 ) ∈ R 4 : \|ξ 1 \| ≤ 10 , B 2 = (ξ, τ, ξ 1 , τ 1 ) ∈ R 4 : \|ξ 1 \| ≥ 10, \|ξ\| ≤ 1 , B 3 = (ξ, τ, ξ 1 , τ 1 ) ∈ R 4 : \|ξ 1 \| ≥ 10, \|ξ\| ≥ 1, ; \|ξ\| ≥ \|ξ 1 \|/4 , B 4 = (ξ, τ, ξ 1 , τ 1 ) ∈ R 4 : \|ξ 1 \| ≥ 10, \|ξ\| ≥ 1, \|ξ\| ≤ \|ξ 1 \|/4, max{\|σ 1 \|, \|σ 2 \|, \|σ\|} = \|σ\| , B 5 = (ξ, τ, ξ 1 , τ 1 ) ∈ R 4 : \|ξ 1 \| ≥ 10, \|ξ\| ≥ 1, \|ξ\| ≤ \|ξ 1 \|/4, max{\|σ 1 \|, \|σ 2 \|, \|σ\|} = \|σ 1 \| , B 6 = (ξ, τ, ξ 1 , τ 1 ) ∈ R 4 : \|ξ 1 \| ≥ 10, \|ξ\| ≥ 1, \|ξ\| ≤ \|ξ 1 \|/4, max{\|σ 1 \|, \|σ 2 \|, \|σ\|} = \|σ 2 \| . Using the Cauchy-Schwarz and Hölder inequalities and duality it is easy to see that \|Z\| ≤ f L 2 ξ,τ (R 2 ) g L 2 ξ,τ (R 2 ) h L 2 ξ,τ (R 2 ) ξ 2s σ 2a R 2 χ B1∪B3∪B4 dξ 1 dτ 1 ξ 1 2s ξ 2 2s σ 1 2b σ 2 2b 1 2 L ∞ ξ,τ (R 2 ) + f L 2 ξ,τ (R 2 ) g L 2 ξ,τ (R 2 ) h L 2 ξ,τ (R 2 ) 1 ξ 1 2s σ 1 2b R 2 χ B2∪B5 ξ 2s dξdτ ξ 2 2s σ 2a σ 2 2b 1 2 L ∞ ξ 1 ,τ 1 (R 2 ) + f L 2 ξ,τ (R 2 ) g L 2 ξ,τ (R 2 ) h L 2 ξ,τ (R 2 ) 1 ξ 2 2s σ 2 2b R 2 χ B6 ξ 1 + ξ 2 2s dξ 1 dτ 1 ξ 1 2s σ 1 2a σ 2b 1 2 L ∞ ξ 2 ,τ 2 (R 2 ) . where σ, σ 1 , σ 2 were given in the condition (III) and B 6 ⊂ (ξ 2 , τ 2 , ξ 1 , τ 1 ) ∈ R 4 : \|ξ 1 \| ≥ 10, \|ξ 1 + ξ 2 \| ≥ 1 \|ξ 1 + ξ 2 \| ≤ \|ξ 1 \|/4, max{\|σ 1 \|, \|σ 2 \|, \|σ\|} = \|σ 2 \| . Therefore from Lemma 3.1, it sufficient to get bounds for K 1 (ξ, τ ) = 1 σ 2a B1∪B3∪B4 ξ 1 ϑ(s) τ + \|ξ 2 \| 3 − \|ξ 2 \|/2 − \|ξ 1 \| 3 + \|ξ 1 \|/2 2b dξ 1 , K 2 (ξ 1 , τ ) = ξ 1 ϑ(s) σ 1 2b B2∪B5 dξ τ 1 − \|ξ 2 \| 3 + \|ξ 2 \|/2 + \|ξ\| 3 − \|ξ\|/2 2a , K 3 (ξ 2 , τ 2 ) = 1 σ 2 2b B6 ξ 1 ϑ(s) τ 2 + \|ξ 1 + ξ 2 \| 3 − \|ξ 1 + ξ 2 \|/2 + \|ξ 1 \| 3 − \|ξ 1 \|/2 2a dξ 1 , Case 1. Contribution of B 1 to K 1 . In region B 1 we have ξ 1 ϑ(s) 1. Therefore for a > 0 and b > 1/2 we obtain K 1 (ξ, τ ) \|ξ1\|≤10 dξ 1 1. Case 2. Contribution of B 3 to K 1 . In region B 3 , we use a change of variable η = τ + \|ξ 2 \| 3 − \|ξ 2 \|/2 − \|ξ 1 \| 3 + \|ξ 1 \|/2. In the case ξ 1 , ξ − ξ 1 ≥ 0, by (3.10), we have K 1 1 σ 2a ξ 1 ϑ(s) dη \|ξ 2 + 2ξ 1 (ξ 1 − ξ) − 1/3\| η 2b dη 1 σ 2a ξ 1 ϑ(s)−2 η 2b dη 1, for a > 0, b > 1/2 and ϑ(s) ≤ 2. If ξ 1 ≥ 0 and ξ − ξ 1 ≤ 0, then one has \|ξ − 2ξ 1 \| = 2ξ 1 − ξ ≥ ξ 1 , so that K 1 1 σ 2a ξ 1 ϑ(s) \|ξ(ξ − 2ξ 1 )\| η 2b dη 1 σ 2a ξ 1 ϑ(s)−2 η 2b dη 1, for a > 0, b > 1/2 and ϑ(s) ≤ 2. Case 3. Contribution of B 4 to K 1 . In this region, if ξ 1 , ξ − ξ 1 ≥ 0, then by definition of the set B 4 , we have 0 ≤ ξ 1 ≤ ξ ≤ ξ 1 /4, which is a contradiction. Therefore, this case can not happen. Now if ξ 1 ≥ 0 and ξ − ξ 1 ≤ 0, then one has \|ξ − 2ξ 1 \| = 2ξ 1 − ξ ≥ ξ 1 . We consider two cases. If ξ ≤ 0, then σ 1 + σ 2 − σ = 3ξξ 1 (ξ − ξ 1 ). If ξ ≥ 0, then σ 1 + σ 2 − σ = −2ξ 3 + ξ + 3ξξ 1 (ξ − ξ 1 ). Since \|ξ − ξ 1 \| ≥ 3\|ξ 1 \|/4, we have \|ξ\| ≤ \|ξξ 1 ξ 2 \| and 2\|ξ 3 \| ≤ \|ξξ 1 (ξ − ξ 1 )\|/6. Therefore in both cases, we have σ \|ξξ 1 (ξ − ξ 1 )\| \|ξξ 2 1 \|. Thus, by a change of variable η = τ − ξ 3 + 3ξξ 1 (ξ − ξ 1 ) + ξ/2, one gets K 1 1 σ 2a ξ 1 ϑ(s) \|ξ(ξ − 2ξ 1 )\| η 2b dη 1 \|ξξ 2 1 \| 2a ξ 1 ϑ(s) \|ξ(ξ − 2ξ 1 )\| η 2b ddη ξ 1 ϑ(s)−4a−1 \|ξ\| 2a+1 dη η 2b 1, for a > 1/4, b > 1/2 and ϑ(s) ≤ 2. Case 4. Contribution of B 2 to K 2 . First if ξ ≥ 0 and ξ 2 ≤ 0, we use a change of variable η = τ 1 + ξ 3 + ξ 3 2 + ξ 1 /2 and we get \|η\| \|τ 1 − ξ 3 1 + ξ 1 /2\| + \|2ξ 3 − 3ξξ 1 ξ 2 \| σ 1 + \|ξ 1 \|\|ξ 2 \| σ 1 + ξ 2 1 . Since \|ξ 1 \| ≥ 10 and \|ξ\| ≤ 1 we have \|2ξ 2 − ξ 1 (2ξ − ξ 1 )\| \|ξ 1 \| 2 . Thus we derive K 2 ξ 1 ϑ(s) σ 1 2b \|η\| σ1 +\|ξ1ξ2\| dη \|2ξ 2 − ξ 1 (2ξ − ξ 1 )\| η 2a ξ 1 ϑ(s)−2 σ 1 2b \|η\| σ1 +\|ξ1\| 2 dη η 2a ξ 1 ϑ(s)−2 σ 1 2b+2a−1 + ξ 1 ϑ(s)−4a σ 1 2b 1, for ϑ(s) ≤ min{2, 4a}, 0 < a < 1/2, b > 1/2 and 2a + 2b − 1 > 0. If ξ ≥ 0 and ξ − ξ 1 ≥ 0, we consider two cases. If ξ 1 ≥ 0, then 0 ≤ ξ ≤ ξ 1 /10, which is a contradiction with ξ − ξ 1 ≥ 0. So the only possible case is ξ 1 ≤ 0. We use a change of variable η = τ 1 −(ξ−ξ 1 ) 3 +(ξ−ξ 1 )/2+ξ 3 −ξ/2 to get \|η\| \|τ 1 + ξ 3 1 \| + \|3ξξ 1 (ξ − ξ 1 )\| σ 1 + \|ξ 1 \|\|ξ 2 \| σ 1 + ξ 2 1 . Since \|ξ 1 (2ξ − ξ 1 )\| \|ξ 1 \| 2 , we conclude K 2 ξ 1 ϑ(s) σ 1 2b \|η\| σ1 +\|ξ1\| 2 dη \|ξ 1 (2ξ − ξ 1 )\| η 2a ξ 1 ϑ(s)−2 σ 1 2b \|η\| σ1 +\|ξ1\| 2 dη η 2a ξ 1 ϑ(s)−2 σ 1 2b+2a−1 + ξ 1 ϑ(s)−4a σ 1 2b 1, for ϑ(s) ≤ min{2, 4a}, 0 < a < 1/2, b > 1/2 and 2a + 2b − 1 > 0. Case 5. Contribution of B 5 to K 2 . In this region, we use a change of variable η = τ 1 + \|ξ\| 3 − \|ξ\|/2 − \|ξ 2 \| 3 + \|ξ 2 \|/2. If ξ, ξ 2 ≥ 0, then we consider two cases. If ξ 1 ≥ 0, then 0 < ξ 1 ≤ ξ ≤ ξ 1 /4, which is a contradiction. Therefore, this case can not happen. If ξ 1 ≤ 0, then 3\|ξξ 1 ξ 2 \| = \|σ 1 + σ 2 − σ\| σ 1 and \|η\| \|τ 1 + ξ 3 1 + ξ 1 /2\| + \|ξξ 1 ξ 2 \| σ 1 . Thus, since \|ξ 1 (2ξ − ξ 1 )\| \|ξ 1 \| 2 , we have K 2 ξ 1 ϑ(s) σ 1 2b dη \|ξ 1 (2ξ − ξ 1 )\| η 2a ξ 1 ϑ(s)−2 σ 1 2b \|η\| σ1 dη η 2a \|ξ 1 \| ϑ(s)−2 σ 1 2a+2b−1 1, for ϑ(s) ≤ 2, 0 < a < 1/2, b > 1/2 and 2a + 2b − 1 ≥ 0. If ξ ≥ 0 and ξ 2 ≤ 0, then ξ 1 ≥ 0, ξ 2 1 \|ξ 1 (ξ 1 − 2ξ)\| and σ 1 \|σ 1 + σ 2 − σ\| = \| − 2ξ 3 + 3ξξ 1 ξ 2 + ξ\| \|ξξ 1 ξ 2 \|. Hence we have \|η\| \|τ 1 − ξ 3 1 + ξ 1 /2\| + \|ξ + 3ξξ 1 ξ 2 \| σ 1 ; and thus K 2 ξ 1 ϑ(s) σ 1 2b dη \|ξ 1 (ξ 1 − 2ξ) − 1/3\| η 2a ξ 1 ϑ(s)−2 σ 1 2b \|η\| σ1 dη η 2a \|ξ 1 \| ϑ(s)−2 σ 1 2a+2b−1 1, for ϑ(s) ≤ 2, 0 < a < 1/2, b > 1/2 and 2a + 2b − 1 ≥ 0. Case 6. Contribution of B 6 to K 3 . Finally in the region B 6 , we have \|ξ 2 \| \|ξ 1 \|, therefore \|ξ 1 \| 2 \|ξξ 1 ξ 2 \| σ 2 , hence K 3 1 σ 2 2b ξ 1 ϑ(s) \|τ 2 \| + \|ξ\| 3 − \|ξ\|/2 + \|ξ 3 1 \| − \|ξ 1 \|/2 2a dξ 1 σ 2 ϑ(s)/2−2b dξ 1 \|τ 2 \| + \|ξ\| 3 − \|ξ\|/2 + \|ξ 3 1 \| − \|ξ 1 \|/2 2a 1, for a > 1/6, b > 1/2 and ϑ(s) ≤ 4b. Next we show that the bilinear estimate (3.1) does not hold if s ≤ −1/2. More precisely, Theorem 3.2 For any s < −1/2 and any a, b ∈ R, with a < 1/2 the estimate (3.1) fails. The above theorem has an important consequence. It shows that our local result stated in Theorem 1.1 is sharp, in the sense that it cannot be improved by means of the X s,b -spaces given in Definition 1.1. Proof of Theorem 3.2. Recall that γ(ξ) = ξ 2 − βξ 4 + ξ 6 and let ̺(ξ) = ξ 3 − βξ/2, N ≫ 1 and define A N = {(ξ, τ ) ∈ R 2 \|N ≤ ξ ≤ N + N −α , \|τ − ̺(ξ)\| ≤ 1}, where 0 < α < 1 will be choose later. It is easy to see that A N contains a rectangle with (N, 3N 2 − β/2) as a vertex, with dimensions cN −2 × N 2−α and longest side pointing in the (1, 3N 2 − β/2) direction. Define f N (τ, ξ) = χ AN and g N (τ, ξ) = χ −AN , then f N L 2 τ,ξ ∼ N −α/2 , and g N L 2 τ,ξ ∼ N −α/2 . Let u N , v N ∈ X s,b such that f N (τ, ξ) ≡ \|τ \| − ̺(ξ) b ξ s u N (τ, ξ) and g N (τ, ξ) ≡ \|τ \| − ̺(ξ) b ξ s v N (τ, ξ). Therefore, from Lemma 3.2-(3.4) and the fact that \|\|τ \| − ̺(ξ)\| ≤ min{\|τ − ̺(ξ)\|, \|τ + ̺(ξ)\|}, we obtain ξ 2 u N v N (τ, ξ) 2iγ(ξ) ∨ X s,−a ξ 2 ξ s γ(ξ) \|τ \| − γ(ξ) a R 2 f N (τ 1 , ξ 1 )g(τ 2 , ξ 2 ) ξ 1 −s ξ 2 −s dτ 1 dξ 1 \|τ 2 \| − γ(ξ 2 ) b \|τ 1 \| − γ(ξ 1 ) b L 2 τ,ξ (R 2 ) B N , where B N ≡ ξ 2 ξ s γ(ξ) τ − ̺(ξ) a R 2 f N (τ 1 , ξ 1 )g(τ 2 , ξ 2 ) ξ 1 −s ξ 2 −s dτ 1 dξ 1 τ 2 − ̺(ξ 2 ) b τ 1 − ̺(ξ 1 ) b L 2 τ,ξ (R 2 ) . From the definition of A N we have (i) If (τ 1 , ξ 1 ) ∈ supp(f N ) and (τ 2 , ξ 2 ) ∈ supp(g N ) then \|τ 1 − ̺(ξ 1 )\| ≤ 1 and \|τ 2 − ̺(ξ 2 )\| ≤ 1. (ii) f * g(τ, ξ) ≥ χ RN (τ, ξ), where R N is the rectangle of dimensions cN −2 × N 2−α with one of the vertices at the origin and the longest side pointing in the (1, 3N 2 − β/2) direction. (iii) \|ξ 1 \| ∼ N , \|ξ 2 \| ∼ N and \|ξ\| ≤ N −α . Moreover, combining (i) and (iii) we obtain \|τ − ̺(ξ)\| N 2−α , for all \|ξ\| ≥ N −α /2. (3.11) Therefore (i), (ii), (iii), (3.11) and the inequality ξ 2 /γ(ξ) ≥ ξ yields N −α B N N −(2+α)s N (2−α)a \|ξ\| 2 γ(ξ) χ RN L 2 τ,ξ (R 2 ) N −(2+α)s N (2−α)a N −α R {\|ξ\|≥N −α /2} χ 2 RN (τ, ξ)dξdτ 1/2 N −(2+α)s N (2−α)a N −α N −α N −α/2 . Taking N ≫ 1, this inequality is possible only when s ≥ − 3α/2 + (2 − α)a α + 2 . (3.12) Now fix a < 1/2 and choose α = 1 − 2a 1 − a . Then α ∈ (0, 1) and plug it into (3.12) we conclude that the estimate (3.1) must fail for s < −1/2. Ill-posedness Before stating the main results let us define the flow-map data solution as S : H s (R) × H s−1 (R) −→ C([0, T ] : H s (R)) (ϕ, ψ) −→ u(t) (4.1) where u(t) is given in (1.3) below. Our ill-posedness results read as follows. Theorem 4.1 Let s < −3 and any T > 0. Then there does not exist any space X T such that u C([0,T ]:H s (R)) ≤ c u X T , (4.2) for all u ∈ X T V c (t)ϕ + V s (t)ψ x X T ≤ c ϕ H s (R) + ψ H s−1 (R) , (4.3) for all ϕ ∈ H s (R), ψ ∈ H s−1 (R) and t 0 V s (t − t ′ )(uv) xx (t ′ )dt ′ X T ≤ c u XT v X T ,(4. 4) for all u, v ∈ X T . Theorem 4.2 Let s < −3. If there exists some T > 0 such that the initial value problem associated to (1.2) with initial data u(0) = ϕ and u t (0) = ψ x is locally well-posed, then the flow-map data solution S defined in (4.1) is not C 2 at zero. Proof of Theorem 4.1 Suppose that there exists a space X T satisfying the conditions of the theorem for s < −3 and T > 0. Let ϕ, ψ ∈ H s (R) and define u(t) = V c (t)ϕ, v(t) = V c (t)ρ. In view of (4.2), (4.3), (4.4) it is easy to see that the following inequality must hold sup 1≤t≤T t 0 V s (t − t ′ )(V c (t ′ )ϕV c (t ′ )ψ) xx (t ′ )dt ′ H s (R) ≤ c ϕ H s (R) ψ H s (R) . (4.5) We will see that (4.5) fails for an appropriate choice of ϕ, ρ, which would lead to a contradiction. where A ξ = ξ 1 : ξ 1 ∈ supp( ψ) and ξ 2 ∈ supp( ϕ) and K(t, ξ, ξ 1 ) ≡ t 0 sin((t − t ′ )γ(ξ)) cos(t ′ γ(ξ 2 )) cos(t ′ γ(ξ 1 )) dt ′ . Note that for all ξ 1 ∈ supp( ψ) and ξ 2 ∈ supp( ϕ) we have γ(ξ 2 ), γ(ξ 1 ) ∼ N 3 and 1 ≤ ξ ≤ 3. On the other hand, since s < −3, we can choose ε > 0 such that − 2s − 6 − 2ε > 0. (4.6) Let t = 1 N 3+ε , then for N sufficiently large we have cos(t ′ γ(ξ 2 )), cos(t ′ γ(ξ 1 )) ≥ 1/2 and sin((t − t ′ )γ(ξ)) ≥ c(t − t ′ )γ(ξ), for all 0 ≤ t ′ ≤ t, 1 ≤ ξ ≤ 3 and ξ 1 ∈ supp( η). Therefore Proof of Theorem 4.2 Let s < −3 and suppose that there exists T > 0 such that the flow-map S defined in (4.1) is C 2 . When (ϕ, ψ) ∈ H s (R) × H s−1 (R), we denote by u (ϕ,ψ) ≡ S(ϕ, ψ) the solution of the IVP (1.2) with initial data u(0) = ϕ and u t (0) = ψ x , that is K(t, ξ, ξ 1 ) t 0 (t − t ′ )γ(ξ)dt ′ γ(ξ) 1 N 6+2ε .u (ϕ,ψ) (t) = V c (t)ϕ + V s (t)ψ x + t 0 V s (t − t ′ )(u 2 (ϕ,ψ) ) xx (t ′ )dt ′ . The Fréchet derivative of S at (ω, ζ) in the direction (ϕ,φ) is given by d (ϕ,φ) S(ω, ζ) = V c (t)ϕ + V s (t)φ x + 2 t 0 V s (t − t ′ )(u (ϕ,ψ) (t ′ )d (ϕ,φ) S(ω, ζ)(t ′ )) xx dt ′ . (4.7) Using the well-posedness assumption we know that the only solution for initial data (0, 0) is u (0,0) ≡ S(0, 0) = 0. Therefore, (4.7) yields d (ϕ,φ) S(0, 0) = V c (t)ϕ + V s (t)φ x . Computing the second Fréchet derivative at the origin in the direction ((ϕ,φ), (ν,ν)), we obtain d 2 (ϕ,φ),(ν,ν) S(0, 0) = 2 t 0 V s (t − t ′ ) [(V c (t ′ )ϕ + V s (t ′ )φ x )(V c (t ′ )ν + V s (t ′ )ν x )] xx dt ′ . Takingφ,ν = 0, the assumption of C 2 regularity of S yields sup 1≤t≤T t 0 V s (t − t ′ )(V c (t ′ )ϕV c (t ′ )ν) xx (t ′ )dt ′ H s (R) ≤ c ϕ H s (R) ν H s (R) which has been shown to fail in the proof of Theorem 4.1. then the above results hold with s ′ instead of s in the same interval [0, T ] with T = T ( ϕ H s (R) , ψ H s−1 (R) ). Define ϕ(ξ) = N −s χ [−N,−N +1] and ψ(ξ) = N −s χ [N +1,N +2] ,where χ A (·) denotes the characteristic function of the set A. We have ϕ H s (R) ∼ 1 and ψ H s (R) ∼ 1.By the definitions of V c , V s and Fubini's Theorem, we have t 0 V 0s (t − t ′ )(V c (t ′ )ϕV c (t ′ )ψ) xx (t ′ )dt ′ ∧x (ξ) = R − \|ξ\| 2 8iγ(ξ) ϕ(ξ 2 ) ψ(ξ 1 )K(t, ξ, ξ 1 ) dξ 1 = A ξ − \|ξ\| 2 8iγ(ξ)N −2s K(t, ξ, ξ 1 ) dξ 1 VN s (t − t ′ )(V c (t ′ )ϕV c (t ′ )ψ) xx (t ′ )dt ′ −2s−6−2ε , for all N ≫ 1which is in contradiction with (4.6). 1.2) does not have good cancelations to make use of Lemma 3.1. 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[ "Correcting for Reporting Delays in Cyber Incidents", "Correcting for Reporting Delays in Cyber Incidents", "Correcting for Reporting Delays in Cyber Incidents", "Correcting for Reporting Delays in Cyber Incidents" ]	[ "Seema Sangari ", "DrEric Dallal ", "Seema Sangari ", "DrEric Dallal " ]	[]	[]	With an ever evolving cyber domain, delays in reporting incidents are a well-known problem in the cyber insurance industry. Addressing this problem is a requisite to obtaining the true picture of cyber incident rates and to model it appropriately. The proposed algorithm addresses this problem by creating a model of the distribution of reporting delays and using the model to correct reported incident counts to account for the expected proportion of incidents that have occurred but not yet been reported. In particular, this correction shows an increase in the number of cyber events in recent months rather than the decline suggested by reported counts.	null	[ "https://arxiv.org/pdf/2201.10348v1.pdf" ]	246,275,683	2201.10348	105abd255b505269dcf2d3b97d751da42e181333	Correcting for Reporting Delays in Cyber Incidents Seema Sangari DrEric Dallal Correcting for Reporting Delays in Cyber Incidents Delay DistributionOptimizationReporting Delays With an ever evolving cyber domain, delays in reporting incidents are a well-known problem in the cyber insurance industry. Addressing this problem is a requisite to obtaining the true picture of cyber incident rates and to model it appropriately. The proposed algorithm addresses this problem by creating a model of the distribution of reporting delays and using the model to correct reported incident counts to account for the expected proportion of incidents that have occurred but not yet been reported. In particular, this correction shows an increase in the number of cyber events in recent months rather than the decline suggested by reported counts. Introduction The cyber security domain is evolving rapidly, with new attack vectors emerging regularly, and even the most up-to-date data cannot be considered complete. Cyber incidents take a long time to become known and even longer to appear in online databases. While some cyber events are known immediately after they occur, most events are often reported many months or years after the event actually occurred, resulting in biased data. Marriott's major cyber incident occurred in 2014 but was only reported in 2018. Major cyber events become headlines in leading newspapers when reported publicly rather than when they happened. Smaller cyber events may never be reported at all, or have extreme delays, as only public companies and those with personally identifiable information may be obligated to report. Sometimes reporting can take 5-10 years, for various intentional or unintentional reasonsfailing to realize that a cyber incident happened, failing to immediately determine the extent of accessed or stolen data, or deciding not to publicize the incident for fear of reputation risk and consequent financial impacts. As a result, reporting delays are often observed in historical cyber event databases. These databases show a decrease in cyber incidents in the recent few years. Coleman et al. (2021) raised the concern that cyber incidents would remain undetected due to advanced threat techniques. Cyber risk modeling firms rely upon historical data to build their models, which are in turn relied upon by cyber insurers for underwriting, portfolio management, and risk transfer. To build robust loss estimation models for today's evolving cyber world, the most recent and updated information is required, with as little bias as possible. Correcting reporting delays in these databases is therefore a key requirement to have trustworthy cyber insurance models. With the necessary corrections, one can then properly examine temporal trends in the targeting of industries or in attacker tactics. Terminology followed in this paper: i: Incident I: Set of all incidents delay, δ i : Time between the incident date and the reporting date age, A i : Time between the incident date and the last incident reporting date in the data f ∆ : Probability density function of the delay distribution F ∆ : Cumulative distribution function of the delay distribution Literature Review Most of the literature on reporting delays addresses the medical space. We are not aware of any existing papers that propose a methodology for correcting cyber incident counts to account for reporting delays. However, Coleman et al. (2021) does examine both the distribution of the number of days to discover cyber incidents and the number of days to disclose them. Harris (1987) described reporting delays as a statistical problem for the first time. Heisterkamp et al. (1988aHeisterkamp et al. ( ,b, 1989 and Brookmeyer and Damiano (1989) made distributional assumptions and built linear/quadratic models whereas Rosenberg (1990) and Cheng and Ford (1991) suggested Poisson models. These are easy to implement but assume stationary reporting delays and don't capture trends. Morgan and Curran (1986), Downs et al. (1987Downs et al. ( , 1988), Healy and Tillett (1988) and Heisterkamp et al. (1988aHeisterkamp et al. ( ,b, 1989 fitted exponential, integrated logistic and log-linear models to capture trends. Gail and Brookmeyer (1988), Brookmeyer and Liao (1990), Kalbfleisch and Lawless (1991) and Esbjerg et al. (1999) applied conditional probabilities to capture trends but this resulted in over-fitting. Lawless (1994) proposed a multinomial model with distributed random effects based on Dirichlet/Poisson/Gamma distributions to capture trends in a timely fashion but failed to handle longer delays. Wang (1992) suggested maximum likelihood estimation (MLE) based non-parametric and semi-parametric approaches but with complete 1 data. Harris (2020) suggested correcting COVID cases with an expectation-maximization (EM) algorithm and trained the model with complete data to correct test data. White et al. (2009) andWeinberger et al. (2020) proposed a simpler method based on proportions but also require complete data to train. Wang (1992), Keiding and Moeschberger (1992) and Midthune et al. (2005) applied truncated models to avoid random effects but require stable reporting delays. Höhle and An Der Heiden (2014), Bastos et al. (2019) and Chitwood et al. (2020) suggested a Bayesian and hierarchical approach with Poisson and negative binomial distributions. It is easy to implement but makes distributional assumptions. Noufaily et al. (2015Noufaily et al. ( , 2016) suggested a log-likelihood approach with a truncation model that is data driven but sensitive to the choice of three reporting time steps. Bastos et al. (2019) suggested a chain-ladder approach but it is sensitive to outliers. Jewell (1989), Zhao et al. (2009), Zhao andZhou (2010), and Avanzi et al. (2016) investigated cyber claims data to account for reporting delays from a capital reserving perspective. This problem is different from the one being investigated, since reporting delays in claims are due only to detection delays. As stated in Brookmeyer and Liao (1990), none of these approaches deal with delays longer than any previously observed. Data The data used is a proprietary data set constructed by merging multiple source cyber event sets together. The source cyber event sets contained differing fields that, at minimum, provided information as to the "what", "when" and "who" of an event. Concretely, the data sets included a description of the event, the name of the company to which the cyber event occurred (N.B.: aggregation events separately list each company known to be impacted), along with the date that the event occurred and the date that the event was reported. Because company names frequently differ from one data set to another, the data sets could not be de-duplicated based on direct string matching. Instead, the event data sets were matched to a firmographic data set containing approximately 55 million businesses in the US. This was done via a previously developed matching algorithm that examined company name, industry classification (e.g., via NAICS codes), address information, and any other fields common to both the cyber event data set and the firmographic data set. Events in distinct cyber event data sets were identified as the same when: 1. They were matched to the same company in the firmographic data set; and 2. They were listed as having occurred within 1 week of each other. Limitations: Some events do not have an occurrence date listed, and were therefore excluded from this analysis. There are also large spikes in event counts listed as having occurred on January 1st in most years (Fig. 1a), which was assumed to be a default value when only the year of the event was known. These events were therefore re-distributed proportionally throughout the year as shown in Fig. 1b but excluded while developing the approach. Proposed Approach The proposed approach consists of estimating the reporting delay distribution f ∆ from the empirical data. With this distribution, a corrected count of events with age a can be determined by dividing the raw counts by F ∆ (a), as the latter expression represents the proportion of events that are reported within a delay, δ, of less than a or, equivalently, the proportion of events that are reported as of the present. There are four complications with estimating the delay distribution in the "obvious" way: • The nature of reporting delays means that direct estimates from empirical data will be biased towards shorter delays, since recent events could only appear in the data set in the first place if the reporting delay is small. • The estimates assume zero probability of any delay longer than the longest in the data set. • The estimates for long reporting delays are based on few data points. • The reporting delay distribution may not be stationary. Mathematically, the first problem is that, for incidents with age a, the delay δ i can only be observed if δ i <= a. This problem can be resolved if one assumes that f ∆ is stationary, in which case the distribution can be estimated for delays, δ, larger than a from older events. This is the approach taken in Algorithm 1. To test for stationary, the algorithm was run on two year windows of the data set centered at different times. (Note that the two year windows include all events that occur within the window, irrespective of when they are reported.) This showed that the delay distribution is non-stationary, as can be seen in Fig. 2. To deal with this non-stationarity, the delay distribution was modeled via parametric distributions where parameters were estimated over monthly rolling two year windows. For each two year window, optimal parameters were determined based on matching cumulative distributions. Specifically, Algorithm 1 was run on each of the monthly two year windows to obtain an estimate of the delay distribution for that time window. For a given two year window and modeled distribution, an optimization algorithm was used to determine the parameters that gave the best match between: • the delay distribution obtained by Algorithm 1; and • the parametric distribution restricted to the domain where the maximum delay is less than the window's maximum reporting delay. The optimal distribution parameters for each window were computed using a derivative free optimizer. In what follows, the above two distributions will be referred to as the "debiased empirical delay distribution" and the "modeled delay distribution", respectively. Generating the Debiased Empirical Delay Distribution Inspired from Brookmeyer and Liao (1990), the proposed algorithm works on the limitation that the delay, δ, is not considered beyond age and hence only a distribution conditioned on the delay being less than or equal to age can be estimated. The algorithm corrects the incident counts with F ∆ estimated from empirical data. The algorithm applies a top down approach to estimate the distribution "from the outside in", accounting for the estimated proportion of events that have occurred but not yet been reported as the distribution itself is being computed. Let A max := max i∈I A i be the maximal age of any event in the data set. Also let h A (a) be the number of incidents of age a, and let h ∆ (δ) be the number of incidents with delay δ. Formally, h A (a) = \|{i ∈ I : A i = a}\| (1) h ∆ (δ) = \|{i ∈ I : ∆ i = δ}\|(2) Then the equation to estimate the delay distribution is: f ∆ (δ) = h ∆ (δ) Amax a=δ hA(a) /F∆(a)(3) Intuitively, the distribution is generated based on the ratio of the number of events with the given delay period, h ∆ (δ), to the best estimate of the true number of events whose age is old enough to be seen in the incident data set (i.e., where the age is larger than the delay under consideration), Amax a=δ hA(a) /F∆(a). Algorithm 1 shows the algorithmic implementation to generate the distribution. Algorithm 1 Algorithm for computing the debiased empirical delay distribution Input: The histograms, h A and h ∆ , computed as in Eqs. (1) and (2), respectively. Output: The distribution f ∆ . 1: function COMPUTEDELAYDISTRIBUTION(h A , h ∆ ) 2: A max ← max i∈I A i 3: F ∆ (A max ) ← 1 4: δ max ← max i∈I δ i 5: for δ ← A max to δ = 0 do 6: den ← 0 7: for a ← δ to δ max do 8: den ← den + h A (a)/F ∆ (a) Computes denominator 9: end for 10: return f ∆ 15: end function f ∆ (δ) ← h ∆ (δ)/den Computes PDF 11: F ∆ (δ − 1) ← F ∆ (δ) − f ∆ (δ)Updates Generating the Modeled Delay Distribution As previously described, the modeled delay distribution is determined by an optimization that minimizes the difference in the cumulative distribution functions of the debiased empirical delay distribution and the modeled delay distribution, restricted to the domain [0, δ max ] (N.B.: δ max is the maximum observed delay). The PDF and CDF plots shown in Fig. 2 suggest that a single distribution will not provide a good fit, as the delay distribution is bi-modal. Rather, a mixture of distributions would be required. Fig. 2 suggests a mix of exponential and normal distributions. Mathematically, F θ (δ) = αF Exp (δ, Scale) + (1 − α)F N (δ, µ, σ)(4) where F Exp = Exponential CDF with parameter, Scale = 1 /λ F N = Normal CDF with parameters, µ and σ A possible interpretation of the bi-modal nature of the reporting delay distribution is that there are two distributions: one for events that are discovered almost immediately (modeled by the exponential) and one for events which have a delay due to both discovery time and public disclosure time (modeled by the normal). The parameter α can therefore be interpreted as the proportion of events that are discovered right away by the organization. Since the normal distribution is defined on (−∞, ∞) and reporting delays cannot be negative, the modeled delay distribution needs to be adjusted. The CDF in Eq. 4 truncated to [0, ∞) (and re-normalized) can be expressed as F θ (δ) = α(F Exp (δ, Scale)) + (1 − α) Truncated Normal Distribution until δ (F N (δ, µ, σ) − F N (0, µ, σ) α + (1 − α) (1 − F N (0, µ, σ) Truncated Normal Distribution over [0, ∞)(5) where 0 ≤ δ ≤ ∞ Since the debiased empirical delay distribution is only defined on [0, δ max ], which is the domain on which it is compared to the modeled delay distribution, the truncation of the modeled distribution to this domain is also defined, denoted by F θ as below. F θ (δ) = α(F Exp (δ, Scale)) + (1 − α) Truncated Normal Distribution until δ (F N (δ, µ, σ) − F N (0, µ, σ) α(F Exp (δ max , Scale)) + (1 − α) (F N (δ max , µ, σ) − F N (0, µ, σ) Truncated Normal Distribution over [0, δmax] (6) where 0 ≤ δ ≤ δ max Defining the Optimization Function Defining the optimization function was challenging due to two factors in particular: • There are many combinations of parameters that give approximately the same distribution when restricted to the domain [0, δ max ], but which differ substantially in how much of the total distribution's weight is contained in this domain. • As two year windows closer to the present are considered, the quantity of data shrinks, resulting in increasingly unstable parameter estimates. The optimization function used is shown in Eq. 7 below. The first term 2 log 10 F θ − log 10 F ∆ 2 reduces the CDF difference between the debiased empirical delay distribution and the modeled delay distribution over the domain [0, δ max ]. The purpose of applying log 10 weights is to place more emphasis on a good fit for the initial months 3 . As mentioned above, there are many different combinations of parameters that would result in comparable errors in the first optimization term, but which nevertheless differ substantially over the domain [0, ∞). This problem is relatively minor when the range [0, δ max ] contains the bulk of the distribution, which it does when δ max is substantially greater than the second peak of the debiased empirical delay distribution. But as two year windows closer to the present are considered, this ceases to be the case. In order to avoid this problem, the optimization function must consider modeled delay distribution values beyond δ max . The second term log 10 S θ − log 10 S θ 2 penalizes large differences between the CDFs of consecutive modeled distributions beyond δ max . In order to further minimize parameter instability for recent two year windows, another set of weights is assigned to the first two terms, which effectively diminishes the importance of a good fit with the empirical data and increases the importance of parameter stability as the amount of available data diminishes. The third and fourth terms are penalization terms -F 2 N (0, µ, σ) term penalizes negative delays introduced by the normal distribution (defined over (−∞, +∞)) whereas S 2 θ (10Y ) term penalizes delays beyond 10 years. Mathematically, the optimization function is defined as θ Opt = argmin θ=(α,Scale,µ,σ) δ max δ F ix log 10 F θ − log 10 F ∆ 2 δ ∈ [0, δmax] + 1 − δ max δ F ix log 10 S θ − log 10 S θ 2 δ ∈ (δmax, δ F ix ] + F 2 N (0, µ, σ) δ < 0 + S 2 θ (10Y ) δ > 10Y ears where δ F ix is the Maximum value of δ in the dataset. F θ is as defined in Eq. 6 whereas S θ is the survival function, defined as the complement of F θ of Eq. 5: S θ = 1 − F θ(8) Finally, θ refers to the previous two year window's optimal parameters. Since no previous parameters are available for the first two year window, the second term is taken to be zero for that window. log 10 S θ − log 10 S θ 2 = 0 (9) θ refers to optimal parameters at previous step. The covariance matrix adaptation evolution strategy (CMA-ES) was applied to compute the modeled distribution parameters. It is a derivative free optimization algorithm, a type typically used when derivatives are difficult or costly to compute (Hansen (2006(Hansen ( , 2016(Hansen ( , 2019). Computing the Corrected Counts Once the modeled delay distributions (one for each two year window) have been obtained by optimization, event counts can be corrected based on the cumulative distribution function of the full modeled distribution (i.e., defined over [0, ∞)) defined by Eq. 5. Specifically, Corrected Count = Reported Counts for month, 'm' F θ (a)(10) where a is age of the given month, 'm'. From each modeled distribution, the correction factor is computed at only one point, age, of the modeled distribution for the given month's correction. Results Fig . 3 shows two examples of plots comparing PDFs of the fitted parametric modeled distribution, its truncation to the domain [0, δ max ] -where δ max is computed for each window individually, and the debiased empirical delay distribution. Fig. 3a shows this comparison for the two year window starting from July 2012 until June 2016 and Fig. 3b shows this comparison for the most recent window starting from January, 2017 until December 2018. Parameters and Interpretation Fig . 4 shows the parameter plots of the delay distribution generated for each monthly two year rolling window. The alpha plot (Fig. 4a) suggests that organizations discover 8-18% of cyber events right away (8% ≤ α ≤ 18%). The scale plot (Fig. 4b) suggests that the short delays modeled by the exponential distribution had a mean of less than 60 days delay until early 2016 but increased rapidly to around 140 days in early 2018. The normal distribution mean, µ, (Fig. 4c) and standard deviation, σ, (Fig. 4d) parameter plots suggest that the longer delays modeled by the normal distribution remained consistent over time. The period of longer delays remain consistent varying within ±10% range. Year ahead F θ (a, a + 1 Y ear) = F θ (a) F θ (a + 1 Y ear) Corrections and Validation where a is the age of the event counts being corrected. Whereas the 2017 year ahead corrections (Fig. 5a) initially show close agreement with the 2018 counts, more recent year ahead corrections underestimate the 2018 counts. On the other hand, the 2018 year ahead corrections (Fig. 5b) generally overestimate the 2019 counts, except for the most recent months, which show close agreement. As stated in section 4.3, the debiased empirical delay distribution has fewer data points for more recent two year windows so weights of ( δmax /δFix) and (1 − δmax /δFix) are used in the optimization function to dynamically adjust the weight given to the CDF before and after δ max respectively. By removing these weights, better estimates for recent months might be obtained but would come at the cost of more parameter instability and worse validation plots (overfitting). In either Fig. 5a or Fig. 5b, the corrected counts (dashed line) show a trend of increasing incident counts since 2016, which is contrary to the diminishing trend seen in the raw counts. The trend in corrected counts is therefore much more in line with reports from insurers and other organizations that release reports on cyber risk. Conclusion This work examined the long known problem of reporting delays in historical cyber events databases and proposed an algorithm to correct for these delays. Interestingly, the true distribution of reporting delays appears to be bi-modal, which we have interpreted as a mixture of two distributions: one for incidents that are discovered immediately, modeled by an exponential distribution, and one for incidents that are not immediately discovered, modeled by a normal distribution. With this form of reporting delay distribution, we obtained non-stationary modeled delay distributions via optimization. These modeled delay distributions were used to estimate the total number of cyber incidents that will eventually be reported from the current counts. The approach was validated by estimating year ahead corrections. To understand the current cyber threat landscape and to create robust cyber risk models, one needs accurate historical data. While it is not possible to get the exact count of cyber events, the proposed algorithm aims to correct for reporting delays approximately. The reported cyber incident counts in recent times show a decreasing trend simply because incidents have not been reported yet, even though they have actually already occurred. However, in reality, the rate of cyber incidents is increasing and that is what the algorithm reveals. The general approach can be applied to any form of data with reporting delays including other long tailed insurance claims (such as liability), and COVID-19 pandemic data. The current study corrects the overall counts for US cyber events, and the industry specific correction of cyber event counts is a future direction of this work. Figure 1 : 1Cyber Event Counts until December 2018 Figure 2 : 2PDF and CDF of Delay Distribution generated for Dec.,'12 and Dec.,'16 Fig. 5 5shows the corrected incident counts based on the proposed methodology. Figs. 5a and 5b show the corrected counts for the events reported by Dec. 2017 and by Dec. 2018, respectively. Although the corrections follow similar trends in both, the correction factors vary substantially. To validate the proposed algorithm, the counts reported until December 2017 (2018) were corrected for a year ahead and compared against the counts reported as of December 2018 (2019). The year ahead correction factor is computed as Figure 3 :Figure 4 : 34Comparing PDFs of Debiased Delay Distribution with Parametric Modeled DisPlots of Modeled Distribution Parameters based on Empirical Debiased Delay Distribution 11 Figure 5 : 115Vs 2018 cumulative counts (b) 2018 corrections Vs 2019 cumulative counts Until 201X -Counts reported as of 201X adjusted for the default date of January 1, proportionally 201X Corrected -"Counts until 201X" corrected based on Eq. 10 201X Corrected Year Ahead -"Counts until 201X" corrected based on Eq. Validation Plots Complete data -No further events are expected to be reported with delays. 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[ "Cluster Frustration in the Breathing Pyrochlore Magnet LiGaCr 4 S 8", "Cluster Frustration in the Breathing Pyrochlore Magnet LiGaCr 4 S 8" ]	[ "Ganesh Pokharel \nDepartment of Physics & Astronomy\nUniversity of Tennessee\n37996KnoxvilleTNUSA\n\nMaterials Science & Technology Division\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA\n", "Hasitha Suriya Arachchige \nDepartment of Physics & Astronomy\nUniversity of Tennessee\n37996KnoxvilleTNUSA\n\nMaterials Science & Technology Division\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA\n", "Travis J Williams \nNeutron Scattering Division\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA\n", "Andrew F May \nMaterials Science & Technology Division\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA\n", "Randy S Fishman \nMaterials Science & Technology Division\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA\n", "Gabriele Sala \nNeutron Scattering Division\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA\n", "Stuart Calder \nNeutron Scattering Division\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA\n", "Georg Ehlers \nNeutron Technologies Division\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA\n", "David S Parker \nMaterials Science & Technology Division\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA\n", "Tao Hong \nNeutron Scattering Division\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA\n", "Andrew Wildes \nInstitut Laue-Langevin\n20156, 38042, Cdex 9GrenobleCSFrance\n", "David Mandrus \nDepartment of Physics & Astronomy\nUniversity of Tennessee\n37996KnoxvilleTNUSA\n\nMaterials Science & Technology Division\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA\n\nDepartment of Materials Science & Engineering\nUniversity of Tennessee\n37996KnoxvilleTNUSA\n", "Joseph A M Paddison \nMaterials Science & Technology Division\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA\n", "Andrew D Christianson \nMaterials Science & Technology Division\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA\n" ]	[ "Department of Physics & Astronomy\nUniversity of Tennessee\n37996KnoxvilleTNUSA", "Materials Science & Technology Division\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA", "Department of Physics & Astronomy\nUniversity of Tennessee\n37996KnoxvilleTNUSA", "Materials Science & Technology Division\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA", "Neutron Scattering Division\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA", "Materials Science & Technology Division\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA", "Materials Science & Technology Division\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA", "Neutron Scattering Division\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA", "Neutron Scattering Division\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA", "Neutron Technologies Division\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA", "Materials Science & Technology Division\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA", "Neutron Scattering Division\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA", "Institut Laue-Langevin\n20156, 38042, Cdex 9GrenobleCSFrance", "Department of Physics & Astronomy\nUniversity of Tennessee\n37996KnoxvilleTNUSA", "Materials Science & Technology Division\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA", "Department of Materials Science & Engineering\nUniversity of Tennessee\n37996KnoxvilleTNUSA", "Materials Science & Technology Division\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA", "Materials Science & Technology Division\nOak Ridge National Laboratory\n37831Oak RidgeTNUSA" ]	[]	We present a comprehensive neutron scattering study of the breathing pyrochlore magnet LiGaCr4S8. We observe an unconventional magnetic excitation spectrum with a separation of high and low-energy spin dynamics in the correlated paramagnetic regime above a spin-freezing transition at 12(2) K. By fitting to magnetic diffusescattering data, we parameterize the spin Hamiltonian. We find that interactions are ferromagnetic within the large and small tetrahedra of the breathing pyrochlore lattice, but antiferromagnetic further-neighbor interactions are also essential to explain our data, in qualitative agreement with density-functional theory predictions [Ghosh et al., npj Quantum Mater. 4, 63 (2019)]. We explain the origin of geometrical frustration in LiGaCr4S8 in terms of net antiferromagnetic coupling between emergent tetrahedral spin clusters that occupy a face-centered lattice. Our results provide insight into the emergence of frustration in the presence of strong further-neighbor couplings, and a blueprint for the determination of magnetic interactions in classical spin liquids. arXiv:2002.09749v1 [cond-mat.str-el]	10.1103/physrevlett.125.167201	[ "https://arxiv.org/pdf/2002.09749v1.pdf" ]	211,258,560	2002.09749	1becf4e8dea47003e6956f1257c34565c92f53df	Cluster Frustration in the Breathing Pyrochlore Magnet LiGaCr 4 S 8 22 Feb 2020 Ganesh Pokharel Department of Physics & Astronomy University of Tennessee 37996KnoxvilleTNUSA Materials Science & Technology Division Oak Ridge National Laboratory 37831Oak RidgeTNUSA Hasitha Suriya Arachchige Department of Physics & Astronomy University of Tennessee 37996KnoxvilleTNUSA Materials Science & Technology Division Oak Ridge National Laboratory 37831Oak RidgeTNUSA Travis J Williams Neutron Scattering Division Oak Ridge National Laboratory 37831Oak RidgeTNUSA Andrew F May Materials Science & Technology Division Oak Ridge National Laboratory 37831Oak RidgeTNUSA Randy S Fishman Materials Science & Technology Division Oak Ridge National Laboratory 37831Oak RidgeTNUSA Gabriele Sala Neutron Scattering Division Oak Ridge National Laboratory 37831Oak RidgeTNUSA Stuart Calder Neutron Scattering Division Oak Ridge National Laboratory 37831Oak RidgeTNUSA Georg Ehlers Neutron Technologies Division Oak Ridge National Laboratory 37831Oak RidgeTNUSA David S Parker Materials Science & Technology Division Oak Ridge National Laboratory 37831Oak RidgeTNUSA Tao Hong Neutron Scattering Division Oak Ridge National Laboratory 37831Oak RidgeTNUSA Andrew Wildes Institut Laue-Langevin 20156, 38042, Cdex 9GrenobleCSFrance David Mandrus Department of Physics & Astronomy University of Tennessee 37996KnoxvilleTNUSA Materials Science & Technology Division Oak Ridge National Laboratory 37831Oak RidgeTNUSA Department of Materials Science & Engineering University of Tennessee 37996KnoxvilleTNUSA Joseph A M Paddison Materials Science & Technology Division Oak Ridge National Laboratory 37831Oak RidgeTNUSA Andrew D Christianson Materials Science & Technology Division Oak Ridge National Laboratory 37831Oak RidgeTNUSA Cluster Frustration in the Breathing Pyrochlore Magnet LiGaCr 4 S 8 22 Feb 2020(Dated: February 25, 2020) We present a comprehensive neutron scattering study of the breathing pyrochlore magnet LiGaCr4S8. We observe an unconventional magnetic excitation spectrum with a separation of high and low-energy spin dynamics in the correlated paramagnetic regime above a spin-freezing transition at 12(2) K. By fitting to magnetic diffusescattering data, we parameterize the spin Hamiltonian. We find that interactions are ferromagnetic within the large and small tetrahedra of the breathing pyrochlore lattice, but antiferromagnetic further-neighbor interactions are also essential to explain our data, in qualitative agreement with density-functional theory predictions [Ghosh et al., npj Quantum Mater. 4, 63 (2019)]. We explain the origin of geometrical frustration in LiGaCr4S8 in terms of net antiferromagnetic coupling between emergent tetrahedral spin clusters that occupy a face-centered lattice. Our results provide insight into the emergence of frustration in the presence of strong further-neighbor couplings, and a blueprint for the determination of magnetic interactions in classical spin liquids. arXiv:2002.09749v1 [cond-mat.str-el] Geometrical frustration-the inability to satisfy all interactions simultaneously due to geometrical constraints-can generate unusual magnetic states in which long-range magnetic ordering is suppressed but strong short-range spin correlations endure [1]. Canonical models of frustrated magnetism often consider spins coupled by antiferromagnetic nearest-neighbor (NN) interactions, which generate a macroscopic degeneracy of magnetic ground states on lattices such as the pyrocholore network of corner-sharing tetrahedra [2][3][4]. This groundstate degeneracy is not symmetry-protected, and in general is expected to be broken by perturbations such as furtherneighbor interactions or spin-lattice coupling. Remarkably, however, some materials exhibit highly-frustrated behavior, despite having complex magnetic interactions that deviate strongly from canonical frustrated models [5][6][7]. These states are of fundamental interest because they can reveal novel frustration mechanisms. A modification of the pyrochlore lattice with potential to realize such states is an alternating array of small and large tetrahedra [ Fig. 1(a)]. This lattice is conventionally called a "breathing pyrochlore", although the size alternation is static, and corresponds to a symmetry lowering from F d3m to F43m [8]. Different exchange interactions can occur within the small and large tetrahedra (J and J , respectively; see Fig. 1(a)), increasing the richness of the phase diagram [9]. Neglecting further-neighbor interactions, conventional ordering is expected only if both J and J are ferromagnetic. If J and J are both antiferromagnetic, the ground state is a classical spin liquid, whereas if J and J are of opposite sign, the ground-state manifold is dimensionally reduced [9]. Furtherneighbor interactions (J 2 , J 3a , and J 3b ; see Fig. 1(a)) can generate further exotic phases. Perhaps the most intriguing of these is predicted [10] to occur when J or J is large and showing large (small) tetrahedra (colored green (grey)), and the connectivity of the exchange interactions J, J , J2, J3a, and J 3b . J3a and J 3b span the same distance but have different symmetry. (b) Emergent tetrahedral clusters generated by strong ferromagnetic J interactions, coupled by a net antiferromagnetic interaction JAFM ∝ J + 4J2 + 2J3a + 2J 3b (> 0). ferromagnetic, and further-neighbor interactions are antiferromagnetic. The dominant ferromagnetic interactions drive the formation of ferromagnetic tetrahedral clusters, and intercluster interactions are frustrated because these clusters occupy a face-centered cubic (FCC) lattice [ Fig. 1(b)] [10]. This model provides a notable example of the concept of emergent frustration-the frustration of multi-spin degrees of freedom that occupy a different lattice to the spins themselves [6,7]. Experimental realizations of the breathing pyrochlore model include the spinel derivatives AA Cr 4 X 8 , in which the A-site is occupied by an ordered arrangement of Li + and In 3+ /Ga 3+ ; X = O, S or Se; and the Cr 3+ ions occupy a breathing pyrochlore lattice [8]. Since J ∼ J in these materials, collective magnetic behavior is expected, in contrast to the breathing pyrochlore material Ba 3 Yb 2 Zn 5 O 11 in which tetra- hedra are decoupled [11][12][13][14][15]. Series members with X = O have antiferromagnetic J and J and exhibit magnetostructural phase transitions and nematic spin ordering [8,[16][17][18][19]. Replacement of O with S or Se ligands is predicted to cause two key differences: suppression of direct exchange relative to superexchange, which is expected to be ferromagnetic because the Cr-X-Cr bond angles are near to 90 • [20]; and enhancement of further-neighbor interactions [10]. Hence, series members with S or Se ligands [20][21][22][23][24] are promising candidates to realize models of frustration driven by furtherneighbor interactions. However, no experimental determination of the magnetic interactions in such systems exists. Here, we use neutron scattering measurements to study the breathing pyrochlore LiGaCr 4 S 8 . While the Weiss constant of LiGaCr 4 S 8 is relatively small, θ CW ≈ 20 K [20,23,25], its bulk magnetic susceptibility χ shows strong deviations from Curie-Weiss behavior below ∼100 K, suggesting the development of strong spin correlations above its spin-freezing transition at T f = 12(2) K [20]. Spin freezing is probably driven by a small amount of off-stoichiometry, as approximately 4% of Li sites are occupied by Ga [20]. Our three key results explain the nature and origin of spin correlations in LiGaCr 4 S 8 : we experimentally parameterize the spin Hamiltonian to reveal the importance of further-neighbor couplings; we confirm recent theoretical predictions (Ref. 10) of cluster frustration; and we observe a direct signature of cluster formation in its magnetic excitation spectrum. These results show that LiGaCr 4 S 8 realizes the frustration of tetrahedral clusters on an emergent FCC lattice. [26]) using two neutron spectrometers: Fig. 2(a) shows high-energy data measured using the ARCS spectrometer with incident energy E i = 25 meV, and Fig. 2(b) shows low-energy data measured using the CNCS spectrometer with E i = 3.32 meV. All INS data have been corrected for detailed balance, and CNCS data are background-subtracted. The dependence of the scattering on Q and temperature suggests that it is of magnetic origin. The bandwidth of the spectrum is about 15 meV, which is larger than θ CW ≈ 20 K (2 meV), suggesting that both ferromagnetic and antiferromagnetic exchange interactions are significant. Above 100 K the spectrum is broad, as expected for a paramagnet. In contrast, between 20 K and 100 K, a band at 12 meV and low-energy quasielastic excitations are observed. The low-energy scattering is much more intense than the highenergy scattering, and has a pronounced wavevector dependence. On cooling, the quasielastic scattering moves towards low energy; however, analysis of the dynamical susceptibility using a damped-harmonic-oscillator model indicates that the excitations are overdamped and gapless at all measured temperatures [26]. Below T f , most of the quasielastic spectral weight shifts to the elastic line [26], consistent with the expected dramatic slowing-down of spin dynamics associated with spin freezing [27]. Interestingly, the intensity of the highenergy band does not change appreciably compared to 20 Ka point to which we return below. Additional evidence of spin freezing is provided by our muon spin relaxation (µSR) measurements, described in detail in the SM [26]. Zero-field µSR measurements down to 1.8 K showed no evidence of static magnetic order or a canonical spin glass state, however the relaxation rate increased at the same temperature as seen with neutron scattering suggesting a slowing down of the spin fluctuations towards a frozen magnetic state. Longitudinal-field µSR does not show evidence of dynamic spin fluctuations, but rather agrees with the emergence of spin freezing at low temperature in LiGaCr 4 S 8 . We now obtain an estimate of the magnetic interactions in Model J (K) J (K) J2 (K) J3a (K) J 3b (K) DFT (Ref. 10) −7.7(1) −12.2(1) 1.2(1) 6.1(1) 3.0(1) J-J 3.07(3) −29.9(4) 0* 0* 0* J-J -J2-J3a-J 3b −7.8(6) −22.1(3) −1. 6(4) 9.6(1) 0.8 (4) successfully to Cr 3+ -based spinels [28,29]. Here, J ij ∈ {J, J , J 2 , J 3a , J 3b } denotes an interaction as shown in Fig. 1, and S denotes a classical vector of magnitude S(S + 1) with S = 3/2. Because LiGaCr 4 S 8 does not exhibit longrange magnetic order, it is not possible to employ the conventional approach of fitting interactions to spin-wave spectra in an ordered state. Therefore, we consider instead the magnetic diffuse scattering intensity, I(Q) = I(Q, E) dE, which we obtain from background-corrected powder-diffraction data collected using the HB-2A diffractometer at ORNL (see SM [26]). For a given set of interaction parameters, we calculate I(Q) andχT using Onsager reaction field theory [30][31][32], which is equivalent to the self-consistent Gaussian approximation used elsewhere [9,28,33] and gives accurate results for frustrated Heisenberg pyrochlore models [34]. We tested three models against our I(Q) data and the χT data from Ref. 20 [ Fig. 3]. Values of the interaction parameters for each model are given in Table I. First, we considered the five-parameter "DFT model" obtained using densityfunctional theory (DFT) in Ref. 10. Calculations of I(Q) and χT for this model show partial agreement with experiment; however, the calculated position of the main diffuse-scattering peak disagrees with the data [ Fig. 3(a)]. Second, we fitted a simpler model to I(Q) data that included J and J interactions only ("J-J model"). These fits also do not agree with the I(Q) data, and are inconsistent with the χT data [ Fig. 3(b)]. Crucially, this result indicates that longer-ranged interactions beyond J and J are essential to account for our experimental data. Finally, we fitted all five interaction parameters to our I(Q) and χT data ("J-J -J 2 -J 3a -J 3b model"). Our data robustly determine a unique optimal solution (see SM [26]) that gives a good fit to I(Q) and χT [ Fig. 3(c)]. We find J is the largest interaction, J, J are ferromagnetic, J 3a is antiferromagnetic, and J 2 and J 3b are small. The DFT model [10] shows the same trends. The consistency between the results derived by the two methods suggests that the trends determined by the modeling are physically reasonable. With an interaction model in hand, we consider the origin of frustration in LiGaCr 4 S 8 . We hypothesize that, at low temperature, spins coupled by dominant ferromagnetic J are essentially aligned within the large tetrahedra, forming S ≈ 6 clusters. The lattice occupied by these clusters is FCC [ Fig. 1(b)], and the net interaction between clusters for our parameters is given by J AFM = (J + 4J 2 + 2J 3a + 2J 3b )/16 = 0.43 K [10]; i.e., it is antiferromagnetic. We therefore also hypothesize the suppression of T f compared to J occurs because of the frustration of antiferromagnetic inter-cluster interactions on the FCC lattice, as proposed theoretically in Ref. [10]. To test the hypothesis of ferromagnetic cluster formation, we performed classical Monte Carlo simulations driven by our fitted interaction parameters (see SM [26]). Fig. 4(a) shows that, at 20 K, the simulated spin correlation function S(0) · S(r) is close to unity at the distance r within large tetrahedra. This result shows that large tetrahedral clusters are essentially ferromagnetic at 20 K. Fig. 4(b) shows the calculated temperature dependence of S(0) · S(r = r ) , and reveals that the clusters develop below 100 K. As described in the SM [26], our own all-electron first principles calculations support the ferromagnetic cluster picture presented here, along with the counterintuitive distance dependence of the exchange interactions. To test the hypothesis of antiferromagnetic frustration of S ≈ 6 cluster spins, we calculated the Fourier transform of the 3D spin correlation function I(Q) ∝ r S(0) · S(r) exp(iQ · r) from our Monte Carlo model using the program Scatty [36]. Fig. 4(c) shows the calculated I(Q) for LiGaCr 4 S 8 at 20 K. Fig. 4(d) shows the calculated I(Q) from the FCC lattice of cluster spins, defined on each tetrahedron of the breathing-pyrochlore lattice as S = The cluster model helps to explain our INS data. Our 20 K data are shown on a linear scale in Fig. 4(f). From ferromagnetic-cluster spin-wave theory [35,37], we calculate that the excitation spectrum of an isolated tetrahedron with interaction J contains a single flat mode at E = 4J S, whose intensity shows a broad peak centered at a Q of approximately 1.1Å −1 [ Fig. 4(g)]. Despite the simplicity of this calculation, it is in qualitative agreement with both the energy and wavevector dependence of the high-energy excitation in our INS data. The single-cluster approximation neglects the effect of coupling between the tetrahedra and consequently contains no low-energy excitations. A different approximation is obtained by optimizing an ordered magnetic ground state using the SpinW software [35]: this state again has ferromagnetic spins within large tetrahedra, but are ordered with propagation vector k = [0, 0, 1]. The assumption of an ordered ground state proximate to the 20 K state allows the spectrum to be calculated from linear spin-wave theory, but overestimates the effect of coupling between tetrahedra [ Fig. 4(h)]. Our determination of the magnetic interactions of the breathing-pyrochlore magnet LiGaCr 4 S 8 sets a benchmark for quantitative interpretation of neutron data from polycrystalline samples. Our results show that further-neighbor interactions are large, in agreement with DFT predictions [10] but in sharp contrast to oxide spinels [28]. The origin of frustration in LiGaCr 4 S 8 is the formation of tetrahedral clusters due to a dominant ferromagnetic J interaction, and the frustration of net antiferromagnetic inter-cluster interactions. We directly observe cluster formation via the development of an essentially intra-cluster high-energy mode in INS data. Such modes may potentially be present in other materials where emergent clusters are coupled by frustrated interactions, such as the quantum-spin-liquid candidate Ca 10 Cr 7 O 28 [5] and the metallic frustrated magnet β-Mn 0.8 Co 0.2 [6]. Intriguingly, on traversing T f , the high-energy mode in our INS data remains unchanged, whereas the low-energy excitations shift to the elastic line. Hence, the timescale of inter-cluster dynamics is enhanced below T f , while that of the intra-cluster dynamics is unchanged. From the frequency dependence of ac susceptibility data [20], we obtained the Mydosh parameter δT f ∼ 0.012 (see SM [26]). This value is an order of magnitude larger than that of canonical spin-glass systems such as AuMn [38] and CuMn [39], but is compatible with cluster-glass systems such as Cr 0.5 Fe 0.5 Ga [40] and Zn 3 V 3 O 8 [41], suggesting that the ground state of LiGaCr 4 S 8 is cluster-glass-like. It would therefore be interesting to investigate whether traditional cluster-glass materials-in which strong structural disorder typically generates clusters with a broad size distribution-exhibit distinct high-energy excitations similar to LiGaCr 4 S 8 . We thank C. Batista for useful discussions and Gerald Morris for technical support with muon spin resonance measurements. This work was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division. G.P. and H.S.A were partially supported by the Gordon and Betty Moore Foundation's EPiQS Initiative through Grant GBMF4416. JAMP acknowledges financial support from Churchill College, University of Cambridge, during early stages of this project. This research used resources at the Spallation Neutron Source and the High Flux Isotope Reactor, a Department of Energy (DOE) Office of Science User Facility operated by Oak Ridge National Laboratory (ORNL). FIG. 1 . 1(a) Breathing pyrochlore lattice of S = 3/2 Cr 3+ ions (black circles) in LiGaCr4S8, FIG. 2 . 2Inelastic neutron scattering spectra of LiGaCr4S8 measured at temperatures indicated in the panels. (a) High energy excitation spectra measured with Ei = 25 meV. (b) Low energy spin excitations measured with Ei = 3.32 meV. The regions of Q-E space enclosed by dotted lines in (a) indicate the regions shown in (b). Intensity is corrected for detailed balance and shown by color on a logarithmic scale. The intensity scale in (b) is a factor of 10 larger than in (a). Fig. 2 2presents the temperature dependence of our inelastic neutron scattering (INS) data as a function of wavevector transfer Q = \|Q\| and energy transfer E. Data were collected on a ∼2 g polycrystalline sample (see supplementary material (SM) FIG. 3 . 3LiGaCr 4 S 8 . Our starting point is a Heisenberg spin Hamiltonian, H = 1 Data (black circles), model fits (red lines), and data-fit (blue lines) for (a) the DFT model of Ref. [10], (b) the J-J model, and (c) the J-J -J2-J3a-J 3b model discussed in the text. The left-hand panels of (a), (b) and (c) represent the neutron scattering data at temperatures indicated in each panel, and the right-hand panel represents χ. Fits were performed for T ≥ 20 K. FIG. 4 . 4(a) Spin-pair correlation S(0) · S(r) as a function of distance r between spins. The normalization is such that S(0)·S(0) = 1. Results were obtained from classical Monte Carlo simulations driven by interaction parameters optimized to our neutron data. (b) Calculated temperature dependence of the spin-spin correlation function at r = r , the distance between neighboring spins in large tetrahedra.(c) Calculated single-crystal diffuse-scattering pattern I(Q) in the (hk0) plane at 20 K. (d) Calculated I(Q) from the emergent FCC lattice of cluster spins, defined as S = 4 i=1 Si on each tetrahedron. (e) Calculated I(Q) for spins on the FCC lattice with antiferromagnetic NN exchange interactions JAFM = 0.43 K. (f) Experimentally-measured spin excitation spectrum at 20 K. (g) Calculated spin excitation spectrum of an isolated tetrahedral cluster. (h) Spin excitation spectrum calculated assuming a proximate ordered ground state with propagation vector k = [0, 0, 1] [35]. 4 i=1 4S i . Figs. 4(c) and (d) are different because the former includes the structure factor of the tetrahedral cluster, whereas the latter does not. Fig. 4(e) shows the calculated I(Q) for spins on the FCC lattice coupled by NN interactions J AFM . The strong similarity between Figs. 4(d) and (e) demonstrates that antiferromagnetic interactions between cluster spins in LiGaCr 4 S 8 are frustrated in the same way as individual spins on the FCC lattice. TABLE I . 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[ "Automated Translation of Rebar Information from GPR Data into As-Built BIM: A Deep Learning-based Approach", "Automated Translation of Rebar Information from GPR Data into As-Built BIM: A Deep Learning-based Approach" ]	[ "Ph.DZhongming Xiang [email protected] ", "Candidate \nDept. of Civil and Environmental Engineering\nUniv. of Utah\n84112Salt Lake CityUT\n", "Assistant ProfessorGe Ou [email protected] ", "Assistant ProfessorAbbas Rashidi [email protected] \nDept. of Civil and Environmental Engineering\nUniv. of Utah\n84112Salt Lake CityUT\n\nDept. of Civil and Environmental Engineering\nUniv. of Utah\n84112Salt Lake CityUT\n" ]	[ "Dept. of Civil and Environmental Engineering\nUniv. of Utah\n84112Salt Lake CityUT", "Dept. of Civil and Environmental Engineering\nUniv. of Utah\n84112Salt Lake CityUT", "Dept. of Civil and Environmental Engineering\nUniv. of Utah\n84112Salt Lake CityUT" ]	[]	Building Information Modeling (BIM) is increasingly used in the construction industry, but existing studies often ignore embedded rebars. Ground Penetrating Radar (GPR) provides a potential solution to develop as-built BIM with surface elements and rebars. However, automatically translating rebars from GPR into BIM is challenging since GPR cannot provide any information about the scanned element. Thus, we propose an approach to link GPR data and BIM according to Faster R-CNN. A label is attached to each element scanned by GPR for capturing the labeled images, which are used with other images to build a 3D model. Meanwhile, Faster R-CNN is introduced to identify the labels, and the projection relationship between images and the model is used to localize the scanned elements in the 3D model. Two concrete buildings is selected to evaluate the proposed approach, and the results reveal that our method could accurately translate the rebars from GPR data into corresponding elements in BIM with correct distributions.	10.1061/9780784483893.047	[ "https://arxiv.org/pdf/2110.15448v1.pdf" ]	240,288,706	2110.15448	90915dbb50a9b781f11fc874cdf058688095576f	Automated Translation of Rebar Information from GPR Data into As-Built BIM: A Deep Learning-based Approach Ph.DZhongming Xiang [email protected] Candidate Dept. of Civil and Environmental Engineering Univ. of Utah 84112Salt Lake CityUT Assistant ProfessorGe Ou [email protected] Assistant ProfessorAbbas Rashidi [email protected] Dept. of Civil and Environmental Engineering Univ. of Utah 84112Salt Lake CityUT Dept. of Civil and Environmental Engineering Univ. of Utah 84112Salt Lake CityUT Automated Translation of Rebar Information from GPR Data into As-Built BIM: A Deep Learning-based Approach 1As-built BIMGPRRebarDeep Learning Building Information Modeling (BIM) is increasingly used in the construction industry, but existing studies often ignore embedded rebars. Ground Penetrating Radar (GPR) provides a potential solution to develop as-built BIM with surface elements and rebars. However, automatically translating rebars from GPR into BIM is challenging since GPR cannot provide any information about the scanned element. Thus, we propose an approach to link GPR data and BIM according to Faster R-CNN. A label is attached to each element scanned by GPR for capturing the labeled images, which are used with other images to build a 3D model. Meanwhile, Faster R-CNN is introduced to identify the labels, and the projection relationship between images and the model is used to localize the scanned elements in the 3D model. Two concrete buildings is selected to evaluate the proposed approach, and the results reveal that our method could accurately translate the rebars from GPR data into corresponding elements in BIM with correct distributions. INTRODUCTION Due to the significant technological improvement in the visualization, automation, efficiency, and productivity in all lift cycles of building management, building information modeling (BIM) has been widely applied to construction projects (Zhao, 2017). As a result, the investment in BIM-related products is sustainably growing in recent years, with an annual increase rate of 15.5% (ValuatesReports, 2020). To generate as-built BIMs, vendors have launched various products (i.e., Revit®, SketchUp®, MicroStation®, etc.), which utilize the geometrical information of building components as the inputs (Wang et al., 2015). For a building with sufficient drawings, it is efficient to adopt these products to develop as-built BIMs as the geometries are available from the drawings. However, for some buildings lacking drawings, the commercial tools might be inefficient due to the implementation of capturing component sizes, which are time-consuming and error-prone. Thus, finding a reliable solution to generate BIMs for as-built buildings is still an ongoing academic exploration. BIM is typically indicated as the 3D model with identified categories of building elements, and most of the studies are focused on how to generate as-built BIMs for visible elements, including shear walls, slabs, columns, doors, and so on. However, one of the embedded components, rebar, is usually ignored in the process of BIM development due to the inaccessibility of rebar. This phenomenon has limited the BIM applications, such as structural health monitoring maintenance and building renovation. Thus, it is necessary to develop a complete BIM with surface elements and embedded rebars for as-built buildings. To accomplish this goal, three major challenges should be tackled: 1) developing an accurate BIM model for surface elements, 2) localizing rebars from concrete elements, and 3) translating rebars into corresponding elements into BIM. To address the first challenge in BIM development for surface elements, two sub-steps are involved. One is 3D data (point cloud) acquisition from an infrastructure. Researchers have proposed various approaches, which can be categorized as image-based methods (Rashidi and Karan, 2018) and time-of-flight (ToF)-based methods (Turkan et al., 2013). Another sub-step is to identify elements from a point cloud. To address this issue, several kinds of methods have been explored based on the projection relationship between 2D images and 3D models (Braun and Borrmann, 2019), placement rules of structural elements (e.g., columns are vertically placed.) (Lu et al., 2019), or geometrical relationships between different elements (e.g., windows are normally placed on the walls and higher than doors) (Wang et al., 2015). For rebar localization, several non-destructive testing (NDT) tools [i.e., ground penetrating radar (GPR) (Lai et al., 2018), impact echo testing (Hong et al., 2015), and ultrasonic testing (Algernon et al., 2011)], are available. GPR is one of the most popular NDT tools and has been widely used to localize rebar due to the advantages of portable size, ease of operation, and high accuracy (Xiang et al., 2019a). Since rebar signals are recorded as hyperbolic patterns in GPR data, developing an efficient method to explain the hyperbolas is the major task for rebar localization. As a fundamental step, rebar recognition, or indicated as hyperbolic pattern recognition, has been well studied, and researchers have proposed various methods, which can be classified as patternbased methods (Xiang et al., 2021) and machine learning-based methods (Lei et al., 2019). The subsequent step is rebar depth and size determination, which involves 1) modifying GPR device itself (Agred et al., 2018), 2) combining GPR device with other NDT tools (Wiwatrojanagul et al., 2018), and 3) utilizing advanced image processing techniques (Xiang et al., 2020). For the rebar insertion in the as-built BIM, however, there is not any effective study available yet. The limited publications are about placing rebars into BIM based on the information from 2D drawings (Choi et al., 2014). On the commercial side, some vendors have designed GPR devices that can generate 3D rebars, but there are two obvious drawbacks: manual calibration during scanning and no guaranteed compatibility with BIM software (Lai et al., 2018). Therefore, inserting identified rebars into the corresponding element in an as-built BIM is almost a blank area that needs a systematic solution. To solve this issue, this research proposes a deep learning approach to automatically translate rebars from GPR data into an as-built BIM based on the photogrammetry method of 3D reconstruction. Since rebars in GPR data and the concrete elements in BIM are generated from two systems, to translate rebars into BIM, mapping information should be introduced to integrate these two domains. Thus, after scanning each element with the GPR device and capturing images of the elements from different angles and positions, this research designs a label that is attached to each element one by one, and one image (called labeled image) is captured for each element with the attached label. Then, these labeled images are used together with other images to reconstruct a 3D model. After that, this research introduces a deep learning method, Faster RCNN, to implement the automated localization of the labels among all 2D images. The label locations in the 2D images are then mapped to the 3D model based on the projection relationship between 2D images and the 3D model. Subsequently, the scanned elements can be identified in a 3D model, which means the localized rebars in the GPR data are linked to the corresponding element in the BIM model. More details about the proposed approach will be discussed in the following sections. Meanwhile, two concrete buildings are selected as testbeds to demonstrate the feasibility of the proposed approach. RESEARCH METHODOLOGY The framework of the proposed approach is presented in Figure 1, which depicts the components of developing a BIM model based on images, recognizing labels through Faster RCNN, projecting identified labels from 2D images to the 3D model, localizing rebars in GPR data, and linking rebars in GPR data to corresponding elements in the 3D model. In this framework, there are two fundamental steps: BIM development and rebar localization, which, however, will not be discussed in this research since there are already mature solutions for these steps. Figure 1. The framework of the proposed approach On the one hand, as a well-established technique, photogrammetry has shown significant efficiency to reconstruct a 3D model with low cost but high quality (Golparvar-Fard et al., 2011). Several mature commercial programs can implement this task. In this research, the ContextCapture, a reliable software produced by Bentley, is used to reconstruct a 3D model to save program settings. Meanwhile, this research does not explore new techniques for element identification from point clouds and directly assumes that the elements (i.e., columns, shear walls, and slabs) are identified. On the other hand, rebar localization in GPR data includes two sub-steps: rebar recognition and rebar depth and size determination. For rebar recognition, the authors have proposed a method that utilized the distinguish features of rebar signals in the frequency domain to differentiate the rebar from the background based on a series of frequency filters (Xiang et al., 2021). For rebar depth determination, the inverse correlation between the maximum intensity in GPR data and the travel time of electromagnetic waves is intrinsic and has been used to determine rebar depth in several studies (Dinh et al., 2018;Kaur et al., 2015). For rebar size determination, it is the least reliable application among all kinds of applications of GPR (Bungey 2004). Thus, this research combines the frequency filter method and inverse correlation method to recognize rebar and determine the rebar depth, respectively, while the rebar size is predefined as a uniform diameter of 12.7mm. Therefore, two residual steps, Scanned Element Recognition and Rebar Translation from GPR data into BIM, are critical for completed BIM development and will be the focus of this manuscript. Deep Learning-based Scanned Element Recognition This research designs a label attached to each element which is scanned by the GPR device (Figure 2). Then, an image is captured for each labeled element used with other images to reconstruct a 3D model. The label is first detected in images and then projected to the 3D model. In this way, the scanned elements can be identified in BIM to map the GPR data. The procedure is elaborated in the following two sub-steps: A. Label Recognition in 2D: The Faster R-CNN, designed for objection detection, is introduced here to recognize labels in 2D. Compared with traditional CNN, Faster R-CNN has several technical features, including (1) using selective search to extract thousands of region proposals, (2) converting these region proposals as the feature map of the last convolutional layer, and (3) rescaling all region proposals into a uniform size through a region of interest pooling layer. The Faster RCNN is trained by 300 labeled images and subsequently applied to detect the labels among the images used for 3D reconstruction. 2), where is the calibration matrix, is the rotation matrix, and is the translation matrix. , , and are camera parameters. As shown in Figure 3, the identified label in the wall is mapped into the point cloud at the corresponding location based on the projection relationship. Rebar Translation from GPR data into BIM The following step aims to translate rebars from GPR data into the BIM model. Figure 4a is an example of GPR data, which contains rebar signals and noise but has no information that can indicate which element the data is scanned from. This phenomenon leads to a problem that there is no discriminative feature to determine which element the rebar is scanned from if there are more than two elements involved in the project. To solve this issue, this research introduces time stamps as the new coordinate to map the GPR data and the scanned elements. Thus, chronological sequences of the GPR data and the labeled images are recorded. The timestamps of the GPR data are {( 1 ℎ , 1 ), ( 2 ℎ , 2 ), … }, where ℎ and regard to the two directions of element . Meanwhile, the timestamps of the labeled images are { 1 , 2 , … } . Then, we allocate the timestamps as {( 1 ℎ , 1 ) → 1 , ( 2 ℎ , 2 ) → 2 , … } to match each GPR data and labeled image. In this way, the relationship between GPR data and the scanned element is built. Then, rebars localized from the GPR data should be placed to BIM in corresponding locations. The ratio of rebar locations to the length of GPR data is = / , where is the location of the rebar in the unit of pixel, and is the pixel length of GPR data. Thus, the rebar location in the wall is * (Figure 4b), where is the length of wall. Similarly, rebar depth is determined by = / * , where is the depth of rebar in the GPR data in the unit of pixel, is the pixel height of the GPR data, and is the maximum detection depth of the GPR device. It is worth noting that the GPR data scanned in the vertical direction of the element records rebars in the horizontal direction and vice versa. EXPERIMENTAL SETUP AND CASE STUDY For validating the performance of the proposed approach, two concrete buildings located at the campus of the University of Utah are selected as the testbeds in this research. The experimental elements of the two buildings are illustrated in Figure 5, in which Case 1 involves one shear wall, one column, and one slab, and Case 2 involves two shear walls and one slab. D1 and D2 in Figure 5 denote the two scan directions. To scan rebars, this research adopts the GPR device, Handy Search NJJ-105, which has the maximum detection depth of 30 cm and GPR data height of 625 pixels. It should be noted that only the performance of rebar translation will be discussed since the rebar localization and BIM development are not within the scope of this research. Industry Foundation Classes (IFC) is introduced in this research to convert the 3D models into files that are readable in the common BIM platforms (i.e., Revit®). Eventually, 231 images for Case 1 and 296 images for Case 2 are captured to reconstruct 3D models. Among these images, there are three labeled images for Case 1 and three labeled images for Case 2. Similarly, each case has three GPR scans corresponding to each element. After processing these images and GPR data with the proposed approach, the BIM models of the two selected buildings are illustrated in Figure 6. Overall, all rebars are translated from GPR data into the BIMs, which indicates the labels are correctly detected and projected into 3D models, and the mapping between the scanned elements and the rebars in the GPR data is built successfully. a b Figure 6. BIMs with surface elements and rebars: (a) Building #1; and (b) Building #2 The performance of label recognition using Faster RCNN is evaluated first. After training the network, Faster RCNN is utilized to recognize labels among the images that are used to reconstruct 3D models. The accuracy, which denotes if the label is recognized or not, reached 100% for both cases. All labeled images are detected by the Faster RCNN, and the locations of the labels in the images are marked as well. It demonstrates that the deep learning method is applicable for label recognition and feasible to identify the scanned elements. For assessing the performance of the rebar placement, three elements in Case 1 are selected and evaluated. As shown in Table 1, all rebars in the GPR data are successfully translated into corresponding elements in the BIM model in both directions, which proves that the proposed method is effective in placing rebars from GPR data into BIMs. CONCLUSION AND FUTURE WORK In order to fill the gap of inserting rebars into as-built BIM, this research provided an automated approach that utilized Faster RCNN to translate rebars from GPR data into corresponding elements in BIMs. A predefined image label was attached to the elements that were scanned by the GPR device, and the label was then recognized by the trained Faster RCNN. With timestamps of the GPR data and labeled images, the link between the GPR data and corresponding elements in the BIMs were built, and the rebars were subsequently translated from GPR data into elements. At last, two concrete buildings were selected to validate the feasibility of the proposed approach. According to the results, three conclusions can be summarized: • The Faster RCNN is a promising method for label recognition as it could detect the existences of all labeled images and localize labels in these images. • The proposed method could successfully project the labels from 2D images to a 3D model to mark the scanned elements in BIMs. • The rebars were eventually translated from GPR data into BIMs and correctly placed into corresponding elements. In this research, a predefined label has been introduced as an extra coordinate to link GPR data and the scanned elements. However, this research has a limitation in that the proposed approach only considered rebars in the second layers. Besides solving this limitation, the authors also plan to explore deep learning methods to build the link based on the pattern features of GPR data and the scanned elements in 2D images. Meanwhile, a comparison discussion of rebar placement will be conducted between the 3D models and the drawings. Figure 2 .Figure 3 . 23Labeled image to indicate the element scanned by GPR device Identified label in the 3D model B. Label Projection from 2D to 3D: The recognized labels in 2D images are then mapped to the 3D model based on the projection relationship between images and the 3D model. 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[ "Coronal activity from the ASAS eclipsing binaries", "Coronal activity from the ASAS eclipsing binaries" ]	[ "Acta ", "Vol " ]	[]	[]	We combine the catalogue of eclipsing binaries from the All Sky Automated Survey (ASAS) with the ROSAT All Sky Survey (RASS). The combination results in 836 eclipsing binaries that display coronal activity and is the largest sample of active binary stars assembled to date. By using the (V-I ) colors of the ASAS eclipsing binary catalogue, we are able to determine the distances and thus bolometric luminosities for the majority of eclipsing binaries that display significant stellar activity. A typical value for the ratio of soft X-ray to bolometric luminosity is L X /L bol ∼ a few ×10 −4 , similar to the ratio of soft X-ray to bolometric flux F X /F bol in the most active regions of the Sun. Unlike rapidly rotating isolated late-type dwarfs -stars with significant outer convection zones -a tight correlation between Rossby number and activity of eclipsing binaries is absent. We find evidence for the saturation effect and marginal evidence for the so-called "super-saturation" phenomena. Our work shows that wide-field stellar variability searches can produce a high yield of binary stars with strong coronal activity.The combined ASAS and RASS catalogue, as well as the results of this work are available for download in a form of a file.	null	[ "https://arxiv.org/pdf/0812.3909v1.pdf" ]	15,240,164	0812.3909	77ec8b3a424f955a4c721cd808d34ac836b88e49	Coronal activity from the ASAS eclipsing binaries 2008 Acta Vol Coronal activity from the ASAS eclipsing binaries 582008Received Month Day, Yearstars: eclipsing -stars: binary -stars: evolution -stars: X-rays We combine the catalogue of eclipsing binaries from the All Sky Automated Survey (ASAS) with the ROSAT All Sky Survey (RASS). The combination results in 836 eclipsing binaries that display coronal activity and is the largest sample of active binary stars assembled to date. By using the (V-I ) colors of the ASAS eclipsing binary catalogue, we are able to determine the distances and thus bolometric luminosities for the majority of eclipsing binaries that display significant stellar activity. A typical value for the ratio of soft X-ray to bolometric luminosity is L X /L bol ∼ a few ×10 −4 , similar to the ratio of soft X-ray to bolometric flux F X /F bol in the most active regions of the Sun. Unlike rapidly rotating isolated late-type dwarfs -stars with significant outer convection zones -a tight correlation between Rossby number and activity of eclipsing binaries is absent. We find evidence for the saturation effect and marginal evidence for the so-called "super-saturation" phenomena. Our work shows that wide-field stellar variability searches can produce a high yield of binary stars with strong coronal activity.The combined ASAS and RASS catalogue, as well as the results of this work are available for download in a form of a file. Introduction In the last few decades, it has become clear that chromospheric and coronal activity of main sequence dwarfs is intimately tied to stellar rotation rate. Main sequence stars are born rapidly rotating and quickly lose their angular momentum by some combination of stellar winds and magnetic torques. It follows that stars with rapid rotation -ideal for studies of stellar activity -are quite rare. In comparison to isolated main sequence stars, close eclipsing binaries are rare as well. However, they are dramatically variable in comparison to rapidly rotating single stars. Close binaries that consist of two late-type dwarfs are thought to be synchronized as long as their orbital periods are shorter than 10 days or so (Duquennoy & Mayor, 1991). This is true for the majority of close binaries, which then may be fruitfully employed in a study of stellar activity. Contact binary stars were discovered to be strong X-ray emitters almost 30 years ago (Carroll et al. 1980). Shortly afterward Cruddace and Dupree (1984) identified 14 X-ray active W UMa systems and noticed a correlation between X-ray luminosity and rotational period, suggesting that for very fast rotators this relation may be reversed. Since then, the study of W UMa activity has broadened -Xray flux variations were discovered (McGale et al. 1996, Brickhouse andDupree 1998), the color-activity and period-activity dependencies were studied on a bigger sample of 57 W UMa binaries (Stȩpień et al. 2001). But there is still no final theory describing the mechanism of X-ray emission. Recently large databases of contact binaries were searched for coronal activity, offering still larger binary samples and thus better statistics. Geske et al. (2006) combined the Northern Sky Variability Survey (NSVS) catalogue with the ROSAT All-Sky Survey X-Ray catalogue and found 140 new binaries with active corona. A part of the All Sky Automated Survey (ASAS) catalogue (southern hemisphere) was also investigated (Chen et al. 2006), and 34 objects more were identified. In 2005 ASAS released the complete catalogue of variable stars south of declination +28deg (Pojmański et al. 2005), among which there are over 11,000 eclipsing binaries, thus providing the largest sample of galactic field binary stars. In this short observational paper we combine the complete ASAS catalogue of variable stars (ACVS) with the ROSAT All-Sky Survey X-ray catalogue (RASS). We obtain the largest sample of X-ray active contact binary stars up to date, which constitutes 379 sources. Furthermore, we widen the study of stellar activity to include all types of eclipsing binaries i.e., including semi-detached and detached as well. The overall number of coincident sources amounts to 836. The overwhelming majority of the objects in our sample are close binaries that are expected to be in tidal synchronization. Therefore, we take the orbital period to be equal to the rotation period of each individual star. Sections 2 and 3 describe both the ASAS and ROSAT catalogues, the process of preparing the combined sample and incidence of X-ray activity. In Section 4 we discuss the method of determining the bolometric and X-ray luminosities. The complete catalogue is presented in Section 5. A short analysis of the evolution of X-ray, or coronal, activity with rotation and colour is given in Section 6. We summarize the paper in Section 7. A. A. The ASAS eclipsing binary catalogue The All Sky Automated Survey (ASAS) is a blind optical survey that covers approximately 3/4 of the night sky. The scientific goal of ASAS is to study variability at the bright end (8 < V < 14 mag), and at a moderate cadence (1-3 days). For the last 7 years, ASAS has been collecting data in both the V -and I -band. At this sampling rate and limiting magnitude, ASAS is optimized for studies of stellar variability. To date, ASAS has generated a rich V-band catalogue of over 50,000 classified variable stars. 1 Roughly 20% (11,076) of the ASAS variable stars have been identified as eclipsing binaries (Pojmański 2002;Paczyński et al. 2006). The ASAS eclipsing binaries are divided into three classes, which are referred to as Eclipsing Contact (EC), Eclipsing Semi-Detached (ESD) and Eclipsing Detached (ED). Roughly 50% of the ASAS eclipsing binaries from the catalogue belong to the EC group while the other half is approximately evenly split between ESD and ED. Note that the classification scheme of Pojmański (2002), that we employ here, differs from the more traditional convention of dividing eclipsing binaries into EW, EB and EA types. The (EC, ESD, ED) scheme discriminates between eclipsing binaries based upon the photometric smoothness of the eclipse. In the case of EC binaries, each eclipse is relatively smooth, while each eclipse is relatively sharp for the ED binaries. Therefore, the ratio of primary radius to orbital separation is relatively large for the EC systems while for ED binaries, this ratio is relatively small. A strong motivation for utilizing the Pojmański (2002) convention is the abundance of photometric data from ASAS as well as the (initial) lack of spectroscopic information for the ASAS variable star catalogue. In this work, we take advantage of the I-band filter of ASAS, whose reduced photometric observations will soon be publicly available. As a result, we are able to utilize both photometric and color information of the 11,076 ASAS eclipsing binaries. Combining the ASAS eclipsing binaries with ROSAT The soft X-ray components of stellar spectra are primarily thought to originate from coronal activity. Stars on the main sequence that possess sizable outer convection zones exhibit increasing levels of coronal activity with increasing rotation rate. In particular, the ratio of X-ray to bolometric luminosity L X /L bol for G-F stars increases in proportion to the inverse of the Rossby number Ro = P/τ c squared (Schmitt et al. 1985), where P and τ c are the orbital period and convective turnover time of the primary, respectively. This relation holds as long as Ro ≥ 1. As is readily shown in Fig. 1, an overwhelming majority of the ASAS eclipsing binaries possess orbital periods shorter than ten days (which is purely a result of observational selection) and hence, are tidally locked. It follows that most of the individual stars in Fig. 1 are rapidly rotating. In order to study the relationship between coronal activity, rotation, and spectral type in the ASAS eclipsing binaries, a relatively deep wide-field X-ray survey is required. For these purposes, the best data set to date is given by the ROSAT All Sky Survey (RASS), which we briefly describe below. Combined ROSAT and ASAS EB catalogue Data for the RASS were taken primarily between 1990 and 1991 and holes in the survey were filled with pointed observations in 1997. The sky was scanned in 2 • strips along the ecliptic and therefore regions of low ecliptic latitude were more thoroughly surveyed. The RASS covers between 0.1-2.4 keV in photon energy and entries are divided into a bright and faint source catalogue, consisting of 18,811 and 105,924 entries, respectively. Details on the ROSAT satellite and the RASS can be found in Voges et al. (1999). We combine the ASAS eclipsing binary catalogue with the RASS Bright and Faint Source catalogues. The angular resolution of ASAS and RASS is ∼ 15 ′′ /pixel and ∼ 50 ′′ /pixel, respectively. The ROSAT angular resolution of = 50 ′′ determines whether or not a ROSAT source and a given ASAS eclipsing binary coincide with one another. In other words, as long as the angular separation between an ASAS eclipsing binary and an entry in RASS is less than 50 ′′ , we assume that they are the same object. The distribution of angular separations for the coincident RASS and ASAS eclipsing binary sources are given in Fig. 2 (311) to the bright source catalogue. The angular distribution of the ASAS-RASS coincident sources is plotted in the right panel of Fig. 2. For angular radius of 50 ′′ , the possibility that our matches are polluted by background X-ray sources may draw concern. We construct a false catalogue of eclipsing binaries by randomly shifting the right ascension and declination of each of the ASAS eclipsing binaries in between +5 and −5 degrees, similarly to the test performed by Geske et al. (2006). The resulting combination of the false eclipsing binary catalogue and RASS indicates the level of random coincidence. From this exercise, we find that only 1.4% of the matches between the real ASAS eclipsing binary catalogue and RASS are false at a separation of 50 ′′ . As a further check, we loosen the criteria for determining a match between the catalogues from an angular separation of 50" up to 90". For the latter separation, we find that the number of matches between both catalogues increases to 942, where 615 and 327 belong to the faint and bright source catalogue, respectively. Thus, the increase in number of coincidences is largely due to sources belonging to the faint source catalogue. Though we increase the number of matches by ∼ 10% by relaxing our criteria for coincidence, the number of false matches increases to ∼ 3.5%. Nevertheless, from here on, we restrict our study to more conservative sample, which corresponds to a maximum angular separation of 50 ′′ between RASS and ASAS eclipsing binary catalogues. Incidence of X-ray Activity The majority, if not all, of W UMa type variables are thought to be strong Xray emitters. Cruddace and Dupree (1984) observed 17 W UMa stars, of which 14 showed X-ray activity. In the sample of Stȩpień et al. (2001) 57 out of 100 W UMa were X-ray sources (26 out of 31 closer than 100 pc). In the work of Geske et al. (2006) 140 X-ray sources were identified among 1022 galactic contact binaries, which constitutes 14%. However, when the sample is distance limited the percentage is higher -about 60% objects within 200 pc, 77% within 150 pc and 94% within 125 pc have detectable X-ray emission. In our distance unlimited sample of 5,376 EC stars 379 exhibit X-ray activity, which constitutes 7%. We do not have distance estimates for objects which are not coronally active, so we cannot estimate the change of detection rate with distance. However, since the ASAS and NSVS projects are similar in the means of instrumentation and thus sky resolution and magnitude range, both detection rates should behave similarly with distance. We assume that roughly 80% of contact binaries closer that 150 pc display X-ray activity and in the following studies we will take a look both at the full sample of active binaries and at this closer subsample. Since we did not limit our study to contact binaries only, we can estimate the incidence of X-ray activity among semi-detached and detached binaries. The former group contains 9% active sources (266 out of 2,957) and the latter 7% (191 out of 2,758). These values are close to the one for EC group, which suggests that there is a similar number of active binaries of each type. Fluxes and Luminosities in the Optical and X-ray As previously mentioned, there exists a relatively tight relationship between the stellar activity and the Rossby number for main sequence stars with convective envelopes. The resulting picture is that of magnetic dynamo processes converting convective and rotational energy into magnetic energy. Since magnetic structures are relatively buoyant in comparison to their surroundings, they have a tendency to rise to the surface, away from the region of their birth. The magnetic energy is dissipated high up in the atmosphere, above the photosphere, where the effective resistivity becomes large at relatively low densities. It is not clear how precisely this picture applies to close binary systems. The combination of the ASAS eclipsing binary catalogue with RASS provides a sizable sample (836 sources) of binary stars that display some level of coronal activity. However, understanding of the relationship between the spectral type and rotation is less straightforward for binary systems than for single stars. Since a binary consists of two stellar components, it is not clear which of the individual components is responsible for the bulk of the stellar activity, though is it clear which component dominates the total stellar emission. For example, the V-band magnitude of any of the ASAS eclipsing binaries that we refer to is in reality, the A. A. V-band magnitude at the maximum of the light curve. Therefore, the magnitude and color of the ASAS eclipsing binaries plotted in this paper are indicative of both components (but mostly of the primary), though either object may be dominant in X-rays. For contact binaries the level of uncertainty is even greater. Neither of the components can truly be thought of as a main sequence stars from the perspective of a color-magnitude diagram. That is, a theoretical understanding of the relationship between surface temperature, effective gravity, and mass is not well understood for contact binaries. Bolometric Luminosity of Eclipsing Binaries We parametrize the bolometric luminosity L bol by the absolute magnitude in the V-band. L bol,⋆ = L bol,⊙ × 10 (M bol,⊙ −M bol,⋆ )/2.5 (1) M bol,⋆ = M V,⋆ + BC(2) Absolute visual magnitudes (M V ) are calculated from (V-I ) color; detached and semi detached systems were treated as single stars and for them we used a main sequence fit by Hawley et al. (1999), adopting solar metallicity. For contact binaries we applied a fit of Ruciński and Duerbeck (1997). With the help of Hipparcos parallax measurements, they derived a useful distance calibration for 40 nearby W UMa (contact) binaries. Since the authors claim, that the coefficients in the formula have large and non-Gaussian errors, we calculated distances to each star adopting two formulae -one utilizing V-band magnitude, and the other using I-band. We rejected 29 contact objects whose distance estimates differed by more than 20%. Bolometric corrections (BC ) were calculated from a polynomial fit to the data given by Flower (1996), from the BC − T eff relation, independent of luminosity classes. Effective temperatures were derived from the (V-I ) colour, based on calibration of Ramírez and Meléndez (2005). We place the X-ray active ASAS eclipsing binaries on a color-magnitude diagram in Fig. 3. In this and following figures "EC best" (filled circles) denotes a subsample of the EC binaries, which are unambiguously classified as EC. Stars, whose type is uncertain are represented with open circles. Such object are assigned multiple type in the process of the ASAS automated classification, e.g. EC/ESD or EC/BCEP/DSCT or EC/RRC (Pojmański, 2002). It happens usually when a light curve is of poor quality (eg. for faint stars), or when the amplitude is small (eg. low amplitude pulsating stars, single spotted rotating stars, etc.). Note that all the EC binaries lie above the solid line representing the main sequence color-magnitude relation of Hawley et al. (1999) for local stars. Nevertheless, most of the EC sample lie close to the fit, with the majority lying about one magnitude above the main sequence relation. This is not unusual, since some of the Fig. 3. The colour-magnitude diagram for EC, based on calibration of Ruciński and Duerbeck (1997). EC best is a subsample of all EC binaries, in which objects have a unique EC classification (see text). The line represents a MS relation for single stars derived by Hawley et al. (1999), which was adopted for ESD and ED variables. contact binaries are the systems that have already evolved off the main sequence. Also, some scatter is probably due to mass ratio differences -similar spread is observed on a color-magnitude diagram for Hipparcos systems (Fig. 1 of Ruciński and Duerbeck 1997). The objects lying very far above the MS line might simply not be contact binaries (which is most probable in case of open circles), thus they do not obey an adequate color-magnitude relation. For ESD and ED binaries in our sample, we assume that the luminosity of the system is dominated by the primary. Therefore we treat binaries which are either in marginal contact (ESDs) or detached (EDs) as single stars. This assumption may be dangerous when studying a relationship between X-ray emission and intrinsic physical parameters of the combined system e.g., the total bolometric luminosity, a size of the convection zone or orbital period. First, both stars may not be in close tidal contact and thus, the rotation period may differ from the orbital period. Furthermore, the primary star may not be a dominant source of the stellar activity. The distance distribution for the three classes of eclipsing binaries that display coronal activity is displayed in Fig. 4. Systems that are in close contact (EC) are observed to further distances than semidetached and detached systems (ESD and ED), although most of the long tail among ECs is occupied by systems of multiple type (open circles in Fig. 3). Since ED are on average more luminous than EC and they are generally visible to further distances, this may suggest that EC have a higher level of X-ray activity than ESD and ED. On the other hand this may simply be a result of the selection effect -we observe twice as many EC than ED or ESD. It is important to note that in the whole calculation process described in this section we did not account for interstellar extinction. This is justifiable in a close neighbourhood of the Sun, where we can assume that stars are unreddened, but be- comes a significant factor at larger distances and can introduce a high scatter when not taken into consideration. Due to this effect and the fact that the sample completeness deteriorates quickly with distance (as shown in Section 3.2), we choose a subsample of active binaries that lie within 150 pc from the Sun which consists of 127 EC, 214 ESD and 148 ED variables. In further study we will look both at the full sample and this distance limited subsample. X-ray Luminosity The RASS data provide the number of photons per energy bin per unit time. The soft energy bin S corresponds to photons of energy 0.1-0.4 keV, while the hard bin H corresponds to photons with energies of 0.5-2.0 keV. Huensch et al. (1996) define a hardness ratio HR HR = H − S H + S ,(3) which is an important parameter in describing the X-ray spectral energy distribution. In terms of the hardness ratio, the X-ray flux f X reads f X = CNT R × (5.30 HR + 8.7) × 10 −12 erg cm −2 s −1 . where CNTR is the source count rate. The spectral energy distribution from which the above relation originates is based upon ROSAT observations of a complete, volume-limited sample of late-type giants (Huensch et al. 1996). This relation had been previously applied for W UMa type systems, whose X-ray emitting regions are believed to have similar properties as those of late-type giants (Chen et al. 2006, Stȩpień et al. 2001). Finally, we calculate the X-ray luminosity L X = 4πd 2 f X(5) of the eclipsing binary in question. The distance d was obtained from a distance modulus in the V-band. The X-ray vs. bolometric luminosity for the combined sample is given in Fig. 5. Almost all of the X-ray bright ASAS eclipsing binaries are active at a level between L X /L bol = 10 −3 and L X /L bol = 10 −4 In this sample, and for our choice of bolometric calibration, EC binaries have higher L bol than ESD and ED binaries, as was expected following Fig. 3. The final catalogue We produce a catalog of all ASAS eclipsing binaries for which we detected X-ray emission. All data are put in a single file "ASAS.ROSAT.cat" which contains 807 entries. The file is available for download from this website: http://www.astrouw.edu.pl/asas/?page=rosat Each line in the catalogue contains all basic information about a star, such as the ASAS and RASS IDs (ie coordinates), magnitudes, orbital period and all the information from the ACVS that might be useful for the catalogue users. Since the RASS catalogues are very large and are easily accessible, we only include a few quantities from RASS that were used in the course of this study, that is the hardness ratio and count rate and their errors. In addition, we supply the catalogue with the values that were calculated in the course of this study, such as distance, L bol , L X and log(Ro). The detailed column descriptions as well as a few exemplary lines are presented in Table 1. A few exemplary lines of the ASAS catalogue of X-ray active binaries ASAS.ROSAT.cat. Each of the 807 lines contains star's ASAS ID and RASS ID, angular separation of both sources on the sky, orbital period of the binary, MJD 0 (HJD 0 -2450000), V and I magnitudes at maximum light, distance d, bolometric luminosity L bol , hardness ratio and its error, count rate and its error, X-ray luminosity L X and a logarithm of Rossby number log(P/τ c ). Last two columns contain other ID from the literature (mostly GCVS) and ACVS variability type (eg. EC, ESD, ED). Fig. 6. The dependence of activity of the ASAS eclipsing binaries on the Rossby number, for objects with orbital periods less than 10 days (753). Left panel contains data points for all binaries and the right one for objects closer than 150 pc (439). Relationship Between Coronal Activity, Rotation and Color Binaries are more complicated than single stars, both on physical and observational grounds, particularly within the context of stellar activity. If indeed there is a physical connection between stellar activity, convection and rotation as is strongly suggested in the case of single dwarfs, then a similar trend may exist for binaries as well. Nevertheless, it seems difficult to easily disentangle -from both an observational and theoretical perspective -the salient physical parameters of a given binary for the purpose of quantifying its activity. Reasons for this difficulty have been previously mentioned. In Fig. 6 the relationship between activity and the primary's Rossby number, Ro, is given for the coincident sources whose orbital periods are less than 10 days, which constitutes 753 objects. In order to calculate the Rossby number, the turnover time τ c for each star was extracted by the empirical formula provided by Stȩpień (2003). The turnover time is a function of (B-V) color, so we transform our (V-I ) colors into (B-V) using the calibration of Caldwell et al. (1993). The fit of Stȩpień is valid for stars with (B − V) > 0.4, so for 34 objects which had this colour value lower, we adopt log(τ c ) = 0. Note that relative to the turnover time, all of the binaries in Fig. 6 are rapid rotators. It follows that the binaries in our sample should be in the "saturated" state, in analogy with the observed phenomena in rapidly rotating single late-type dwarfs. That is, the coronal activity, quantified by L X /L bol should be constant with decreasing Rossby number Ro. The scatter of EC and ESD variables on the activity -Rossby number diagram is even larger than ECs, and we do not observe a clear increase of X-ray activity with decreasing rotational period for either group. We note however, that ESDs and EDs are on average more active than ECs. To some degree, Fig. 6 displays the saturation effect in that the level of activity is roughly constant with decreasing Rossby number. The same diagram for the limited sample (d < 150pc) does not reduce the scatter in ESD and ED groups, but the trend of increasing activity with decreasing Ro is now clearly visible, especially among contact binaries. There are claims (Stȩpień et al. 2001;Chen et al. 2006) that the so-called "super-saturation" phenomena -a reduction in coronal and chromospheric activity with decreasing Ro -is observed in extremely rapidly rotating isolated late type dwarfs and W UMa binaries. This subtle effect is not observed in Fig. 6, which is not surprising with the high scatter of points. In Fig. 7, we plot an activity-color diagram for all the X-ray bright ASAS eclipsing binaries and for the subsample. Here, it is clear that the level of coronal activity indeed changes not only with the rotation, but surface temperature as well. With decreasing color index (V-I ), the surface temperature of the primary increases, while the depth of the its convection zone decreases. So, binaries that are earlier type are less active in comparison to binaries that are later type. Of course, in this context "early" and "late" refer to surface temperature alone, rather than surface temperature and rotation as in the case of isolated stars. Both Figs. 6 and 7 show that at a fixed overturn time and (V-I ) colour, the EC binaries are on average less active in comparison to ESD and ED systems. This is better visible on diagrams for the subsample of objects closer than 150 pc. Summary We had produced a catalogue of the combined ASAS eclipsing binaries and RASS X-ray sources. All data had been put in a single file "ASAS.ROSAT.cat" which is available for download from the ASAS website (http://www.astrouw.edu.pl/asas/?page=rosat) in the form presented in Table 1 (see Section 5 for details). The combination of the eclipsing binary catalogue with the RASS produces a total of 836 coincident sources (∼ a 7% yield). Among these are 379 contact binary stars (EC), which is the largest sample of X-ray active contact binary stars assembled up to date. Semi-detached (ESD) and detached (ED) binaries were also taken into account, resulting in 266 and 191 active variables, respectively). We observe similar incidence of X-ray activity in all 3 groups of variables, which is around 7% in the distance unlimited sample. An overwhelming majority of the coincident sources possesses orbital periods shorter than 10 days, so we expect tidal locking between the primary and the companion. Therefore, the orbital period of the binary may be also thought of as the rotation period of the stellar components. We analyzed the dependence of X-ray activity on Rossby number and colour. In comparison to typical isolated stars of similar primary mass, the ASAS eclipsing binaries display a higher level of coronal activity (see Fig. 5) and the maximum value L X /L bol ∼ 3 × 10 −3 , for the coincident sources is similar to that found in rapidly rotating isolated late-type dwarfs. Both the activity -Rossby number (Fig. 6) and the activity -colour (Fig. 7) diagrams display a large scatter and they do not tighten any of the already known relations. The coronally-active binaries in our sample display the saturation effect, while there is no clear evidence of the so-called "super-saturation" effect. However, for a given (V-I ) colour ECs are rotating more rapidly than the other two classes. And at the same time, for a given colour the level of activity of ECs is lower in comparison to ESDs and EDs. This slight downturn in activity among the three classes may in fact be an indirect manifestation of the saturation effect in Fig. 7. Fig. 1 . 1Orbital period distribution of the ASAS eclipsing binaries that were matched with X-ray sources in the RASS catalogue. Fig. 4 . 4Histograms of estimated distances of contact (EC), semi-detached (ESD) and detached (ED) eclipsing binaries. The most left image contains two histograms -the darker one is based on 'EC best' subsample. Fig. 5 . 5Bolometric and X-ray luminosity function of 807 of the ASAS eclipsing binaries (left) and a distance limited subsample -d < 150pc (right). The overwhelming majority of these systems lie below L X /L bol = 10 −3 (dotted line). EC binaries have higher L bol than ESD and ED groups (this is better visible on a colour figure). Fig. 7 . 7Color-Activity diagram for the ASAS eclipsing binaries coincident with the RASS. Left panel contains data points for all 807 binaries and the right one for 489 objects closer than 150 pc. (left panel). The combination of both catalogues produces 836 matches. The majority ofFig. 2. Angular separation between matched pairs of the ASAS eclipsing binaries and the RASS Xray sources is presented on the left panel. The right panel shows a distribution of objects in combined catalogues in equatorial coordinates. Blue circles correspond to ROSAT faint source catalogue, while red squares to ROSAT bright source catalogue. 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[ "AREA-PRESERVING MEAN CURVATURE FLOW OF ROTATIONALLY SYMMETRIC HYPERSURFACES WITH FREE BOUNDARIES", "AREA-PRESERVING MEAN CURVATURE FLOW OF ROTATIONALLY SYMMETRIC HYPERSURFACES WITH FREE BOUNDARIES" ]	[ "Kunbo Wang " ]	[]	[]	In this paper, we consider the area-preserving mean curvature flow with free Neumann boundaries. We show that for a rotationally symmetric n-dimensional hypersurface in R n+1 between two parallel hyperplanes will converge to a cylinder with the same area under this flow. We use the geometric properties and the maximal principle to obtain gradient and curvature estimates, leading to long-time existence of the flow and convergence to a constant mean curvature surface.	10.1007/s11425-000-0000-0	null	59,383,472	1712.05918	b49e21519983847e2dacab329fc971f7491cd4e5	AREA-PRESERVING MEAN CURVATURE FLOW OF ROTATIONALLY SYMMETRIC HYPERSURFACES WITH FREE BOUNDARIES 16 Dec 2017 Kunbo Wang AREA-PRESERVING MEAN CURVATURE FLOW OF ROTATIONALLY SYMMETRIC HYPERSURFACES WITH FREE BOUNDARIES 16 Dec 2017 In this paper, we consider the area-preserving mean curvature flow with free Neumann boundaries. We show that for a rotationally symmetric n-dimensional hypersurface in R n+1 between two parallel hyperplanes will converge to a cylinder with the same area under this flow. We use the geometric properties and the maximal principle to obtain gradient and curvature estimates, leading to long-time existence of the flow and convergence to a constant mean curvature surface. Introduction and the main results A Hypersurface M t in Eucilidean space is said to be evolving by mean curvature flow if each point X(·) of the surface moves, in time and space, in the direction of its unit normal with speed equal to the mean curvature H at that point. That is (1.1) ∂X ∂t = −Hν(x, t), where ν(x, t) is the outer unit normal. It was first studied by Brakle in [6] from the viewpoint of geometric measure theory. In [12], G. Huisken showed that any compact, convex hypersurface without boundary converges asymptotically to a round sphere in a finite time interval. Mean curvature flow is also the steepest descent flow for the area functional, evolving to minimal surfaces. In [13], Huisken initiated the idea of considering the following volume-preserving mean curvature flow, (1.2) ∂ ∂t X(x, t) = (h(t) − H(x, t))ν(x, t), where h(t) =´M t Hdµt M t dµt is the average of the mean curvature on M t . Huisken proved if the initial hypersurface M n 0 (n ≥ 2) is uniformly convex, then the evolution equation (1.2) has a smoothly solution M t for all times 0 ≤ t < ∞ and M t converges to a round sphere enclose the same volume as M 0 in the C ∞ -topology as t → ∞. In [20], Pihan studied the following area-preserving mean curvature flow. (1.3) ∂ ∂t X(x, t) = (1 − h(t)H(x, t))ν(x, t). Here h(t) =´M t Hdµt M t H 2 dµt . Pihan showed that if the initial hypersurface is compact without boundary, (1.3) has a unique solution for a short time under the assumption h(0) > 0. For n = 1, Pihan also showed that an initially closed, convex curve in the plane converges exponentially to a circle with the same length as the initial curve. For n ≥ 2, Mccoy in [16] showed that if the initial n-dimensional hypersurface M 0 is strictly convex then the evolution equation (1.3) has a smooth solution M t for all time 0 ≤ t < ∞, and M t converge, as t → ∞, in the C ∞ -topology, to a sphere with the same surface area as M 0 . In [11], Huang and Lin use the idea of iteration of Li in [14] and Ye in [23], in cases of volume preserving mean curvature flow and Ricci flow, respectively. And obtained the same result, by assuming that the initial hypersurface M 0 satisfies h(0) > 0 and´M 0 \|A\| 2 − 1 n H 2 dµ ≤ ǫ. In this paper, we study the area preserving mean curvature flow with free boundaries, where a restriction on the angles of boundaries with fixed hypersurfaces in Euclidean space are imposed. In this setting, there are some interesting works (cf. [7], [19] and [22]). In these papers, the authors study the mean curvature flow with Neumann and Dirichlet free boundaries. Let Σ be a fixed hypersurface smoothly embedded in R n+1 . We say X(x, t) is evolved by mean curvature flow with free Neumann bounary condition on Σ, if ∂X ∂t = −Hν, ∀(x, t) ∈ M n × [0, T ), ν Mt , ν Σ • X (x, t) = 0, ∀(x, t) ∈ ∂M n × [0, T ), X(·, 0) = M 0 , X(x, t) ⊂ Σ, ∀(x, t) ∈ ∂M n × [0, T ). The volume-preserving mean curvature flow with free Neumann boundaries was first studied by Athanassenas in [2]. Let M 0 be a complete n-dimensional hypersurface with boundary ∂M 0 = ∅. Assume M 0 is smoothly embedded in the domain G = {x ∈ R n+1 : 0 < x n+1 < d, d > 0}, We denote by Σ i (i = 1, 2), the two parallel hyperplanes bounding the domain G, and assume ∂M 0 ⊂ Σ i (i = 1, 2). Then Athanassenas proved the following theorem in [2]. Theorem 1.1. Let V, d ∈ R be given two positive constants. M 0 ⊂ G is a smooth, rotationally symmetric, initial hypersurface which intersects Σ i (i = 1, 2) orthogonally at the boundaries which encloses the volume V . Then the free Neumann boundaries problem for equation (1.2) has a unique solution on [0, +∞), which converges to a cylinder C ⊂ G of volume V under assumption \|M 0 \| ≤ V d as t → ∞. Other works on this problem were investigated in [3], [4], [17] and [18]. In this paper, we consider the following problem for area-preserving mean curvature flow with free Neumann boundaries. Problem 1.1. ∂ ∂t X(x, t) = (1 − h(t)H(x, t))ν(x, t), ∀(x, t) ∈ M n × [0, T ), ν Mt , ν Σi • X (x, t) = 0, ∀(x, t) ∈ ∂M n × [0, T ), i = 1, 2, X(·, 0) = M 0 , X(x, t) ⊂ Σ i , ∀(x, t) ∈ ∂M n × [0, T ), i = (1, 2). We prove the following main theorem for Problem 1.1. Main Theorem. Let V, d ∈ R + be given two constants and M 0 ⊂ G to be a smooth, rotationally symmetric, mean convex initial hypersurface which intersects Σ i (i = 1, 2) orthogonally at the boundaries. Then the solution to Problem 1.1 exists for all times t > 0 and converges to a cylinder of the same area with M 0 under the assumption \|M 0 \| ≤ V d . Remark 1.1. We say M 0 is mean convex if the mean curvature is positive everywhere. The condition of mean convex will be used to prove the equation (1.3) is strictly parabolic. In [20], Pihan shows that the equation (1.3) is strictly parabolic for a short time if h(0) > 0. And as the case of volume-preserving mean curvature flow in [3], Problem 1.1 is a Neumann boundary problem for strictly parabolic equation, from which we obtain the short time existence. And also see [19] and [22] for details of general cases. This paper is organized as follows. In Section 2, we give some definitions and preliminaries. In Section 3, we show some basic properties of this flow. We prove that the property of mean convexity can be preserved under equation (1.3) and the surfaces do not pinch off under the condition of our main theorem. In Section 4, we use the property of mean convexity and maximal principle to show the curvature estimates. Gradient and curvature estimates lead to long time existence to a constant curvature surface. And we prove our main theorem in Section 5. The methods we use here are those introduced by Athanassenas in [2], Ecker and Huisken in [9]. We use the free Neumann boundary condition to convert the boundary estimates to interior estimates (see Lemma 3.4, Lemma 4.1 and Theorem 4.1). We put the condition of mean convexity here is to give an upper and lower bounds for h(t) and v(x, t), which is crucial for our curvature estimates. Preliminaries We adopt the similar notations of Huisken in [12]. Let M be an n-dimensional Riemannian manifold. Vectors on M are denoted by X = {X i }, covectors by Y = {Y i } and mixed tensors by T = {T ik j }. The induced metric and the second fundamental form on M are denoted by g = g ij and A = {h ij } respectively. The surface area element of M is given by µ = det(g ij ). For tensors T ijkl and U ijkl on M, we have the inner product (T ijkl , U ijkl ) = g iα g jβ g kγ g lδ T ijkl U αβγδ and the norm \|T ijkl \| 2 = (T ijkl , T ijkl ). The trace of the second fundamental form, H = g ij h ij , is the mean curvature of M , and \|A\| 2 = g ik g jl h ij h kl is the square of the norm of the second fundamental form on M . We also denotẽ C = tr(A 3 ) = g ij g kl g mn h ik h jm h ln . If X : M n ֒→ R n+1 smoothly embeds M n into R n+1 , then the induced metric g is given by g ij = ∂X ∂xi (x), ∂X ∂xj (x) and the second fundamental forms h ij = ∂X ∂xi (x), ∂ν ∂xi (x) , where ·, · is the ordinary scalar product of vectors in R n+1 . The matrix of the Weingarten map of M is h i j (x) = g ik (x)h kj (x) . The eigenvalues of this matrix are the principal curvatures of M . The induced connection on M is given via the Christoffel Symbols. Γ k ij = 1 2 g kl ( ∂g jl ∂x i + ∂g il ∂x j − ∂g ij ∂x l ). The covariant derivative, for a vector field v = v j ∂ ∂x j is given by ∇ i v j = ∂v j ∂x i + Γ j ik v k , ∇ i v j = ∂v j ∂x i − Γ j ik v k . The Laplacian of T is △T = g ij ∇ i ∇ j T. The Riemannian curvature tensor on M can be given through the Gauss Equations R ijkl = h ik h jl − h il h jk . We denote \|M t \| to be the surface area of M t . We assume M 0 is mean convex and rotationally symmetric about an axis which intersects Σ i orthogonally. We denote this axis by x n+1 and use the parametrization ρ(x n+1 ) : [0, d] → R for the generating curve of a surface of revolution, which is a radius function. Basic properties From now on, we write [0, T ) to indicate the maximal time interval for which the flow exists. First we verify that the surface area does indeed remain fixed under the area-preserving flow (1.3), while the enclose volume does not decrease. The rotationally symmetric property is preserved under the equation (1.3). This is clear from the evolution equation, since the mean curvature and the normal are symmetric. Lemma 3.1. The surface area of M t remains constant throughout the flow, that is d dtˆM t dµ t ≡ 0. Proof. We apply the first variation of area formula to the vector field ∂X ∂t , extended appropriately, and the divergence theorem, That is, if E t ⊂ R n+1 is the (n + 1)-dimensional set enclosed by M t and the two parallel planes Σ i , then d dtˆM t dµ t =ˆM t div Mt ( ∂X ∂t )dµ t = −ˆM t (1 − hH)Hdµ t ≡ 0.d dt V ol(E t ) ≥ 0 Proof. We first extend ∂X ∂t to a vector field on the whole of E t , then apply the first variation of area formula and divergence theorem, d dt V ol(E t ) =ˆE t div R n+1 ( ∂X ∂t )dV =ˆ∂ Et < ∂X ∂t ν > dµ t =ˆM t < ∂X ∂t ν > dµ t +ˆΣ i < ∂X ∂t ν > dµ t =ˆM t dµ t − (´M t Hdµ t ) 2 Mt H 2 dµ t ≥ 0. Here we have use the free Neumann boundary condition to obtain the integral on Σ i is 0. As in [16] (Section 4), we have the following evolution equations. Proof. Since we consider the hypersurface has free boundaries, we can not directly use the maximal principle. Suppose the first time H(x, t) = 0 is attained at an interior point of M t , then from Lemma 3.3, we have Lemma 3.3. We have (1) ∂ ∂t g ij = 2(1 − hH)h ij ; (2) ∂ ∂t g ij = −2(1 − hH)h ij ; (3) ∂ ∂t ν = h∇H; (4) ∂ ∂t h ij = h△h ij + (1 − 2hH)h m i h mj + h\|A\| 2 h ij ; (5) ∂ ∂t h i j = h(△h i j + \|A\| 2 h i j ) − h i m h m j ; (6) ∂ ∂t H = h△H − (1 − hH)\|A\| 2 ; (7) ∂ ∂t \|A\| 2 = h(△\|A\| 2 − 2\|∇A\| 2 + 2\|A\| 4 ) − 2C; (8) ( ∂ ∂t − h△)H 2 = −2h\|∇H\| 2 − 2(1 − hH)H\|A\| 2 .( ∂ ∂t − h△)H 2 = −2h\|∇H\| 2 − 2(1 − hH)H\|A\| 2 . From the maximal principle, we know H(x, t) = 0 can not be attained at an interior point of M t . If t 0 is the first time such that lim H(x, t 0 ) = 0. By a reflection of Σ 1 and Σ 2 , we can define two pieces of new hypersurfaces outside the boundary, which satisfies the free Neumann boundaries conditions. DenoteM t0 to be the new hypersurface andρ(x n+1 ) its radius function . Precisely, ρ(x n+1 ) =    ρ(2d − x n+1 ) , d ≤ x n+1 ≤ 2d ρ(x n+1 ) , 0 ≤ x n+1 < d ρ(−x n+1 ) , −d ≤ x n+1 < 0 i.e.H(x 1 , x 2 , · · · , x n , 0, t 0 ) = lim xn+1→0 H(x, t 0 ),H(x 1 , x 2 , · · · , x n , d, t 0 ) = lim xn+1→d H(x, t 0 ). Then at the boundary points, ∂ ∂tH (x, t 0 ) ≤ 0, △H(x, t 0 ) > 0. So the maximal principle can still be applied, which proves the lemma. Now we will show that the radius of the hypersurface M t has uniform lower and upper bounds for any time t ∈ [0, T ). The method follows from [2](Lemma 1). Lemma 3.5. Under the conditions of the Main Theorem, there exist constant r and R only depending on n,d,V and \|M 0 \|such that r ≤ ρ(x n+1 , t) ≤ R for any t ∈ [0, T ). Proof. Given an initial surface M 0 , we denote by C the cylinder with the same enclosed volume V as M 0 in G. Assume that there is some t 0 > 0 such that M t0 pinches off. We project M t0 onto the plane Σ 1 , using the natural projection π : R n+1 → R n . Then \|M t0 \| ≥ \|π(M t0 )\|. Any M t has to intersect the cylinder C at least once by the volume constrain, that the volume of M t is not decreasing. Therefore \|M 0 \| = \|M t0 \| > \|π(M t0 )\| > \|π(C)\| = ω n ρ n C = V d . Here ρ C is the radius of C, and ω n is the volume of unit ball in R n , then we obtain a contradiction. For the upper bound, we assume that ρ(x n+1 , t) max = R(t), then we have \|M 0 \| = \|M t \| > ω n · (R(t) − ρ C ) n , which implies R(t) < ρ C + ( \|M 0 \| ω n ) 1 n . Curvature estimates Letx = (x 1 , · · · , x n , 0), and ω =x \|x\| to denote the unit outer normal to the cylinder intersecting M t at the point X(P, t). As in [2] and [21], we define the height function of M t to be u = X, ω . And define v(x, t) = ω, ν −1 = (ρ) 2 + 1, whereρ is the derivative of ρ about x n+1 . From Lemma 3.5, we can obtain the height estimate r ≤ u(x, t) ≤ R. Now we show that v has an upper bound under the assumption T < ∞. The kernel ideal is that according to our parametrization, the points where v tends to infinity are not the singular points of the evolving surface, and the \|A\| tends to zero at these points. Proof. Since M t is rotationally symmetric we have H = κ 1 + (n − 1)κ 2 , where κ 1 and κ 2 denote the principle curvatures. If we parameterize M by its radius function ρ ∈ C ∞ ([0, d]), then clearly H = −ρ (1 +ρ 2 ) 3 2 + n − 1 ρ(1 +ρ 2 ) 1 2 . Suppose t 0 is the first time such that lim Next, we will show an estimate of h(t). First, we prove the following lemma. Lemma 4.2. Under the assumption of the main theorem, we have C 1 ≤´M t Hdµ t ≤ C 2 , for all time t ∈ [0, T ). Here C 1 and C 2 are positive constants only depending on n, d, r, R, V and \|M 0 \|. Proof. First we show that´M t Hdµ t ≥ C 1 for some constant C 1 . This is a direct consequence of the first variation formula and mean curvature is positive. Since n\|M 0 \| = n\|M t \| ≤ˆM t H < X, ν > dµ t ≤ˆM t H\|X\|dµ t ≤ d 2 + R 2ˆM t Hdµ t , the lower bound for´M t Hdµ t is obtained. Next we show there is an upper bound for´M t Hdµ t , that there is a constant C 2 such that´M t Hdµ t ≤ C 2 . We still parameterize M t by its radius function ρ(x n+1 , t). We denote ω n to be the volume of unit ball in R n , and its surface area is nω n , then we have H = −ρ (1 +ρ 2 ) 3 2 + n − 1 ρ(1 +ρ 2 ) 1 2 . \|M t \| = nω nˆd 0 ρ n−1 1 +ρ 2 dx n+1 . Mt Hdµ t = nω nˆd 0 (−ρ (1 +ρ 2 ) ρ n−1 + (n − 1)ρ n−2 )dx n+1 . On one hand, we have´d 0 (n − 1)ρ n−2 dµ t ≤ (n − 1)dR n−2 . On the other hand, by our boundary conditons, d 0 −ρ (1 +ρ 2 ) ρ n−1 dx n+1 =ˆd 0 −ρ n−1 d(arctanρ)dx n+1 = −ρ n−1 · (arctanρ\| d 0 ) + (n − 1)ˆd 0 ρ n−2 ·ρ · arctanρdx n+1 ≤ (n − 1) π 2 ·ˆd 0 ρ n−2 1 +ρ 2 dx n+1 ≤ (n − 1)π 2rˆd 0 ρ n−1 1 +ρ 2 dx n+1 = (n − 1)π 2nω n \|M t \| 1 r = (n − 1)π 2nω n \|M 0 \| 1 r . Thus the upper bound is obtained. Proof. Now we use the Cauchy-Schwarz inequality (´M t Hdµ) 2 (´M t H 2 dµ)(´M t dµ) ≤ 1. From which we have 0 <´M t Hdµt M t H 2 dµt ≤´M t dµt M t Hdµt = \|M0\| M t Hdµt ≤ C 3 . Now we show that \|A\| 2 is bounded for any finite time interval. v(x, t) =< ω, ν > −1 . Clearly we have ∂ ∂t v = −v 2 w, ∂ ∂t ν = −v 2 · h∇H, ω . From [2] , we have △v = \|A\| 2 v − v 2 < ω, ∇H > + 2\|∇v\| 2 v − n − 1 u 2 · v. Then we obtain ( ∂ ∂t − h△)v = −h\|A\| 2 v − 2h\|∇v\| 2 v + (n − 1)hv u 2 , and ( ∂ ∂t − h△)v 2 = 2v · (−h\|A\| 2 v − 2h\|∇v\| 2 v + (n − 1)hv u 2 ) − 2h\|∇v\| 2 = −6h\|∇v\| 2 − 2hv 2 \|A\| 2 + 2(n − 1)hv 2 u 2 . We considering \|A\| 2 v 2 as in [12] and divide the points in M t into three sets. S t = {P ∈ M t \|ρ ≥ 0}. I t = {P ∈ M t \|ρ < 0, κ 1 κ 2 < α}. J t = {P ∈ M t \|ρ < 0, κ 1 κ 2 ≥ α}. Here α is a positive constant large enough. We will show that for all points in S t and I t , \|A\| 2 v 2 has uniform upper bounds for any t ∈ [0, T ). We split our proof into three cases. Case(1). If P ∈ S t , from H = −ρ (1+ρ 2 ) 3 2 + n−1 ρ(1+ρ 2 ) 1 2 > 0, we havë ρ < n − 1 ρ [1 + (ρ) 2 ]. Then we have \|A\| 2 v 2 = (ρ) 2 [1 + (ρ) 2 ] 2 + n − 1 ρ 2 ≤ (n − 1) 2 ρ 2 + n − 1 ρ 2 ≤ C. From now on, we denote by C to any constant depending on n, V, d, r, R and M 0 . Case (2). If P ∈ I t , then κ1 κ2 < α. We have −ρ·ρ (ρ) 2 +1 < α, and −ρ (ρ) 2 +1 < α r . Thus \|A\| 2 v 2 = (ρ) 2 [1 + (ρ) 2 ] 2 + n − 1 ρ 2 ≤ C. Case (3). For points in J t , we use the technique of maximal principle. First we have ( ∂ ∂t − h△)\|A\| 2 v 2 = \|A\| 2 · (−6h\|∇v\| 2 − 2hv 2 \|A\| 2 + 2(n − 1)hv 2 u 2 ) + v 2 · (−2h\|∇A\| 2 + 2h\|A\| 4 − 2C) − 2h∇\|A\| 2 · ∇v 2 = \|A\| 2 · (−6h\|∇v\| 2 − 2hv 2 \|A\| 2 + 2(n − 1)hv 2 u 2 ) + v 2 · (−2h\|∇A\| 2 + 2h\|A\| 4 − 2C) + h(−∇\|A\| 2 ∇v 2 − 4v\|A\|∇\|A\|∇v) = \|A\| 2 · (−6h\|∇v\| 2 − 2hv 2 \|A\| 2 + 2(n − 1)hv 2 u 2 ) + v 2 · (−2h\|∇A\| 2 + 2h\|A\| 4 − 2C) + h[−v −2 ∇v 2 ∇(\|A\| 2 v 2 ) + v −2 \|∇v 2 \| 2 \|A\| 2 − 4v\|A\|∇\|A\|∇v] + 2(n − 1)h\|A\| 2 v 2 u 2 ≤ −6h\|A\| 2 \|∇v\| 2 − 2hv 2 \|∇\|A\|\| 2 − 2Cv 2 − hv −2 ∇v 2 ∇(\|A\| 2 v 2 ) + 4h\|∇v\| 2 \|A\| 2 − 4hv\|A\|∇\|A\|∇v + 2(n − 1)h\|A\| 2 v 2 u 2 ≤ −hv 2 ∇v 2 ∇(\|A\| 2 v 2 ) + 2(n − 1)h\|A\| 2 v 2 u 2 − 2Cv 2 . We have used \|∇\|A\|\| ≤ \|∇A\| and Cauchy-Schwarz inequality. Since 2(n − 1)h\|A\| 2 u 2C ≤ C 4 · κ 2 1 + (n − 1)κ 2 2 κ 3 1 + (n − 1)κ 3 2 = C 4 · 1 κ1 + (n − 1) 1 κ1 ( κ2 κ1 ) 2 1 + (n − 1)( κ2 κ1 ) 3 ≤ C 5 · 1 κ 1 . Then, if κ 1 > C 5 , we have 2(n−1)h\|A\| 2 v 2 u 2 − 2Cv 2 < 0. Thus \|A\| 2 v 2 can not attain a maximal value by the maximal principle. And if κ 1 ≤ C 5 , we have \|A\| 2 = k 2 1 + n − 1 ρ 2 [1 + (ρ) 2 ] ≤ C, and \|A\| 2 v 2 ≤ CM 2 T . Therefore, \|A\| 2 ≤ CM 2 T v 2 min = C T .∂ ∂t \|∇ m A\| 2 = h△\|∇ m A\| 2 − 2h\|∇ m+1 A\| 2 + i+j+k=m ∇ i A * ∇ j A * ∇ k A * ∇ m A + r+s=m ∇ r A * ∇ s A * ∇ m A. We assume when l ≤ m, we have \|∇ l A\| 2 ≤ C l (T ). Then for n = m + 1, we have ∂ ∂t \|∇ m+1 A\| 2 ≤ h△\|∇ m+1 A\| 2 + C(T ) 1 · (\|∇ m+1 A\| 2 + 1). We choose f = \|∇ m+1 A\| 2 + N \|∇ m A\| 2 , where N is a constant large enough. Then ∂ ∂t f ≤ h△\|∇ m+1 A\| 2 + C(T ) 1 · (\|∇ m+1 A\| 2 + 1) + N h△\|∇ m A\| 2 − 2hN \|∇ m+1 A\| 2 + C(T ) 2 ≤ h△f − C(T ) 3 \|∇ m+1 A\| 2 + C(T ) 4 = h△f − C(T ) 3 (f − N \|∇ m A\| 2 ) + C(T ) 5 ≤ h△f − C(T ) 3 f + C(T ) 6 . Thus f ≤ C T . Proof of the main theorem Since the upper bound we derived above is a constant depending on T , \|A\| 2 may be unbounded when t tends to infinity. We will show that this will not happen and the initial hypersurface converges to a constant mean curvature surface. Theorem 5.1. The mean curvature H of the evolving surfaces converge to a constant as t → ∞. Then by Cauchy-Schwarz inequality, H = C for some constant. Proof. Since d dt V ol(E t ) =´M t (1 − hH)dµ t . Then we haveˆ∞ 0 d dt V ol(E t )dt =ˆ∞ 0ˆMt (1 − hH)dµ t dt = V ol(E ∞) − V ol(E 0 ) ≤ C. AREA-PRESERVING MEAN CURVATURE FLOW OF ROTATIONALLY SYMMETRIC HYPERSURFACES WITH FREE BOUNDAR Proof. Rotationally symmetric hypersurfaces of constant mean curvature in R n+1 are plane, sphere, cylinder, catenoid, unduloid and nodoid, they are known as the Delaunay surfaces (see [8]). Our boundary conditions excludes the possibilities of plane, sphere, catenoid and nodoid. In [2] (see Section 1), Athanassenas use the condition \|M 0 \| ≤ V d to exclude the existence of unduloids in G. So our possibility can only be the cylinder. Thus the Main Theorem is proved. Lemma 3. 2 . 2The volume enclosed by M t does not decrease throughout the flow. Lemma 3 . 4 . 34The mean curvature is positive on M t , t ∈ [0, T ). Furthermore on the boundaries ∂M t , we have limxn+1→0 H(x, t) = a(t) > 0, lim xn+1→d H(x, t) = b(t) > 0. Lemma 4 . 1 . 41If T < ∞, then v(x, t) ≤ M T < +∞ for any t ∈ [0, T ], in the limitation sense when t = T , i.e. v(x, T ) = lim t→T v(x, t). Here M T is a constant depending on n, d, T, r, R, V and \|M 0 \|. then we have H = 0 at this point, which is a contradiction with H > 0 everywhere. If lim t→T,xn+1→s v(x, t) = +∞, then lim t→T,xn+1→s\|A\| 2 (x, t) = (ρ) 2 (xn+1,t) [(ρ) 2 (xn+1,t)+1] 3 + n−1 ρ 2 [(ρ) 2 (xn+1,t)+1] = 0 which implies X(x 1 , · · · , x n ,s, T ) is not a singular point. So the maximal principal method in Lemma 3.4 can still be applied and we must have H > 0 at this point, which is still a contradiction. The boundary points case can be proved in the same way as in Lemma 3.4. Now we show that v(x, t) max < +∞ for all t ∈ [0, T ], by the continuity of v, we have v ≤ M T for some constant depending on T . Corollary 4. 1 . 1Under the assumption of the main theorem, we have 0 < h(t) ≤ C 3 for all time t ∈ [0, T ). Here C 3 is a constant only depending on n, d, r, R, V and \|M 0 \|. Theorem 4. 1 . 1If the maximal time interval [0, T ) is finite, i.e. T < +∞, then we have \|A\| 2 (x, t) ≤ C T , where C T is a constant depending only on T ,n, d, r, R, V and \|M 0 \|. Proof. First we compute the evolution equation of Corollary Corollary 4. 4 . 4T = +∞. Hdµt ) 2 Mt H 2 dµ t . 4.2. Under the assumption of the above theorem, we have a lower bound for h(t), namely, h(t) ≥ m T . Here, m T is a constant depending on T ,n,V ,d, r, α, R and \|M 0 \|.Proof. 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[ "TFROM: A Two-sided Fairness-Aware Recommendation Model for Both Customers and Providers", "TFROM: A Two-sided Fairness-Aware Recommendation Model for Both Customers and Providers" ]	[ "Yao Wu ", "Jian Cao ", "Guandong Xu [email protected] ", "Yudong Tan [email protected] ", "Yao Wu ", "Jian Cao ", "Guandong Xu ", "Yudong Tan ", "\nShanghai Jiao Tong University Shanghai\nChina\n", "\nDepartment of Air Ticket Business Ctrip.com International Ltd Shanghai Shanghai\nUniversity of Technology Sydney Sydney\nNew South WalesAustralia, China\n" ]	[ "Shanghai Jiao Tong University Shanghai\nChina", "Department of Air Ticket Business Ctrip.com International Ltd Shanghai Shanghai\nUniversity of Technology Sydney Sydney\nNew South WalesAustralia, China" ]	[]	At present, most research on the fairness of recommender systems is conducted either from the perspective of customers or from the perspective of product (or service) providers. However, such a practice ignores the fact that when fairness is guaranteed to one side, the fairness and rights of the other side are likely to reduce. In this paper, we consider recommendation scenarios from the perspective of two sides (customers and providers). From the perspective of providers, we consider the fairness of the providers' exposure in recommender system. For customers, we consider the fairness of the reduced quality of recommendation results due to the introduction of fairness measures. We theoretically analyzed the relationship between recommendation quality, customers fairness, and provider fairness, and design a two-sided fairness-aware recommendation model (TFROM) for both customers and providers. Specifically, we design two versions of TFROM for offline and online recommendation. The effectiveness of the model is verified on three real-world data sets. The experimental results show that TFROM provides better two-sided fairness while still maintaining a higher level of personalization than the baseline algorithms.	10.1145/3404835.3462882	[ "https://arxiv.org/pdf/2104.09024v1.pdf" ]	233,296,725	2104.09024	d40fbf5b0d97d616afeb6ab1cb4f4cf870e5900e	TFROM: A Two-sided Fairness-Aware Recommendation Model for Both Customers and Providers 19 Apr 2021 Yao Wu Jian Cao Guandong Xu [email protected] Yudong Tan [email protected] Yao Wu Jian Cao Guandong Xu Yudong Tan Shanghai Jiao Tong University Shanghai China Department of Air Ticket Business Ctrip.com International Ltd Shanghai Shanghai University of Technology Sydney Sydney New South WalesAustralia, China TFROM: A Two-sided Fairness-Aware Recommendation Model for Both Customers and Providers 19 Apr 202110.1145/nnnnnnn.nnnnnnnACM Reference Format: 2021. TFROM: A Two-sided Fairness-Aware Recommendation Model for Both Customers and Providers. In Proceedings of ACM Conference (Conference'17). ACM, New York, NY, USA, 10 pages. https://doi.org/10.1145/nnnnnnn.nnnnnnnTwo-sided FairnessFairness-aware RecommendationCustomer FairnessProvider Fairness At present, most research on the fairness of recommender systems is conducted either from the perspective of customers or from the perspective of product (or service) providers. However, such a practice ignores the fact that when fairness is guaranteed to one side, the fairness and rights of the other side are likely to reduce. In this paper, we consider recommendation scenarios from the perspective of two sides (customers and providers). From the perspective of providers, we consider the fairness of the providers' exposure in recommender system. For customers, we consider the fairness of the reduced quality of recommendation results due to the introduction of fairness measures. We theoretically analyzed the relationship between recommendation quality, customers fairness, and provider fairness, and design a two-sided fairness-aware recommendation model (TFROM) for both customers and providers. Specifically, we design two versions of TFROM for offline and online recommendation. The effectiveness of the model is verified on three real-world data sets. The experimental results show that TFROM provides better two-sided fairness while still maintaining a higher level of personalization than the baseline algorithms. INTRODUCTION At present, most recommender systems aim to maximize the customers' utility by learning their behavior and recommending items that best match their preferences. Research on customer behaviors [33] has proven that recommendations can indeed influence their decisions and result in a good customer experience. However, recommender systems can also bring unfavorable consequences, such as they may narrow the customers' vision [1], or superior items will receive increased attention so as to become dominant [27], while inferior items will be relegated to a lower position, which becomes an extremely vicious circle. As a possible unfavorable consequence, the unfairness in recommender systems in different aspects, such as racial/gender stereotypes [22], social polarization [12], position bias [27], has been a well-studied research topic. Problem Statement. Despite the different mechanisms which have been implemented to ensure the fairness of recommendations, these studies only consider the utility of one type of stakeholder in business and try to eliminate unfairness among their members. This makes sense on a platform where one side dominates. For example, employers have an absolute say in job hunting, and liminating inequality among employees may not harm the interests of employers. However, in most situations, there are multiple stakeholder types on a platform. If the interests of one side are enhanced, the interests of the other side will be damaged. Research has found that when the fairness of customers' recommendation quality is guaranteed, the exposure of providers will be greatly unfair [29]. Since on most platforms, customers and product providers are the two most important stakeholder types, in this paper, we consider the problem of fair recommendation for these two types of stakeholders. The interests of providers in a recommender system are mainly reflected in the positions of their products on the customers' recommendation lists. Providers want their products to be ranked as high as possible since products in higher positions can attract more attention, which in turn brings more orders and higher revenues. But maintaining fairness among providers is necessary to maintain a healthy market environment, which is beneficial to the long-term development of the platform. Customers would like to receive recommendations from the platform that meet their personal requirements, which is the main goal of recommendation algorithm design. However, due to the introduction of the fairness measurement, customer satisfaction may be lowered because the recommendation lists are not generated by only considering their preferences. It is a challenging task to maintain the fairness of recommendation and also maximize the level of personalization at the same time. State-of-the-art and Limitations. At present, only a few studies consider two-sided fairness in recommender system, among which [29] is one representative. Although [29] also discusses fairness between customers and providers, it is limited in that one provider only corresponds to one item so the problem becomes assuring fairness both for customers and the exposure of items. It does not conform to the reality that a provider often offers multiple items, which is what we are trying to solve in this paper. It is also worth noting that one provider offering one item is a special case of our problem. In addition, [29] also ignores the impact of the positions of items in the recommendation list on item exposure, and regards all the items appearing in a list as gaining the same exposure rate. However, according to the research [20], the top ranked items receive more attention. Therefore, we introduce the metrics considering the influence of position on exposure. Many papers classify fairness in recommender systems into individual fairness [32] and group fairness [3]. Individual fairness emphasizes the similar treatment of similar individual users, while group fairness emphasizes that the benefits of the group or the probability of receiving services accord with the demographic structure. In fact, both approaches are reasonable, and their differences stem from their objectives. Since most of the literature only considers fairness for one type of stakeholder, it is natural for them to either consider individual fairness or group fairness. In this paper, we try to ensure recommendation fairness both for providers and consumers so that individual fairness among customers and group fairness among providers are both considered. Approach and Contribution. When two-sided fairness is considered, recommendation becomes a multi-objective problem. Since these objectives conflict with each other, it is impossible to optimize each objective at the same time, but it is possible to find a relatively better trade-off among multiple objectives through algorithms. We theoretically analyze the relationship between recommendation quality, customer fairness, and provider fairness, and find the direction of problem optimization. We design two algorithms for online and offline application scenarios. In the offline scenario, we generate recommendation lists for all customers. In the online scenario, customers' requests arrive randomly and recommendation lists are generated for each request. The contributions of this paper are as follows: • We propose and formulate the problem of two-sided fairness in recommender systems. By considering the influence of product position in a recommendation list, we design new metrics to measure the individual fairness of customers, group fairness of providers and the quality of the recommendation results. • Through theoretical analysis, we design TFROM (a Two-sided Fairness-aware RecOmmendation Model), which is implemented in two versions for online and offline scenarios. TFROM can be easily applied to various existing recommender systems. • The experiment results on three real-world datasets show that TFROM can provide better two-sided fairness and still maintain a higher recommendation quality than the comparison algorithms. The rest of the paper is organized as follows. Section 2 discusses the related work. Section 3 formalizes the two-sided fairness problem. Section 4 presents the TFROM. The experiment results are detailed in Section 5. We conclude the paper in Section 6. RELATED WORK With the increasing maturity of recommendation technology, many researchers have begun to focus on metrics other than recommendation accuracy to measure the performance of recommender systems [13,30], and fairness is one of the important metrics. According to stakeholders considered in the algorithm, research on fairness in recommender systems can be divided into the following three categories [9]: those that consider customer-side fairness [6,34], those that consider provider-side fairness [23,26,39] and those that consider two-sided fairness [11]. Research which considers consumer-side fairness usually aims at eliminating discrimination suffered by some customers in the recommendation process and enables different customers to have the same experience. For example, a re-ranking algorithm is proposed in [15] that mitigates the bias of protected attributes such as gender or age. In [8], information on protected sensitive attributes is removed in graph embedding by learning a series of adversarial filters. Provider-side fairness usually focuses on providing a fair channel for different providers to reach their customers. For example, in [5], an amortization algorithm is designed that allows recommended items to gain exposure commensurate with their quality. In [31], a fair taxi route recommendation system is proposed so that taxi drivers can have a fair chance of accessing passengers. While the vast majority of relevant work considers only unilateral stakeholder fairness, we consider fairness from both sides. When both stakeholder sides are considered, there will be more objectives in the algorithm, and the fairness for the two sides may be in conflict with each other, bringing new problems which need to be solved. [35] discusses two-sided fairness in a ride-hailing platform to ensure fairness in driver income and user waiting time. [36] discusses the relationship between user fairness, item fairness and diversity in intent-aware ranking. In [29], an algorithm is designed based on a greedy strategy to ensure providers receive fair exposure and customers receive fair recommendation quality, which is consistent with the goal of our research. However, in the setting of [29], each item is treated as a provider, which is a special case in our research problem. In our model, we assume each provider can provide one or more items, which is more in line with the application scenario in real life. There are also some taxonomies that classify the fairness of recommendations from other perspectives. In some studies, the fairness of recommendations is divided into individual fairness [7,32] and group fairness [3,4,14]. Group fairness is intended to eliminate the influence of specific attributes on the recommendation results for different groups so that disadvantaged groups are offered the same opportunities as the advantaged groups, whereas the goal of individual fairness is to enable similar users to be treated similarly. Approaches can also be classified from the perspective of the time that the mechanism works in the system [38], and the fairness mechanism is divided into pre-processing [10], in-processing [2,8,37] and post-processing [21,25] approaches. Our study considers both individual and group fairness. Provider-side fairness focuses on the fairness of the groups of items provided by each provider, while from the perspective of customers we focus on fairness between individual customers. We propose a post-processing approach that further processes the existing recommendation results to obtain results that ensure two-sided fairness. PROBLEM FORMULATION In this paper, we consider a recommendation system with two types of stakeholders. The provider is the party who provides the recommended items or services, and each provider can provide one or more items or services. The customer is the one who receives the recommended result. We assume that there is a recommendation algorithm in the system, which provides a predicted rating matrix and the original recommendation lists for all customers based on . Our algorithm uses the obtained preference matrix to provide customers with recommendation results that are both in line with their preferences and ensure two-sided fairness. Notations We use the following notations: • = { 1 , 2 , ..., } is a set of customers. • = { 1 , 2 , ..., } is a set of recommended items. • = { 1 , 2 , ..., } is a set of providers supplying items. • is the set of items provided by provider . • = 1 , 1 , 1 , 2 , ..., , is a relevant rating matrix produced by the original recommendation algorithm of the system. • = {l 1 , l 2 , ..., l } is a set of original recommendation lists based on . • = {l 1 , l 2 , ..., l } is a set of recommendation lists finally outputted to customers. Exposure of items and providers As previously mentioned, the provider's interest in the recommendation system is reflected in the exposure of its items on the recommendation list, and the exposure of an item depends on its position on the customers' recommendation lists. According to research on user behavior, only top-ranked items tend to attract more attention before the user makes a decision [19], and even an item in position 5 is largely ignored [20]. To address this phenomenon, we need to give lower weights to lower-ranked items, and this loss of exposure changes very quickly at the beginning as the rank drops, whereas for lower ranked items, as the attention has been reduced, the changes in exposure tend to stabilize. Therefore, the exposure of an item can be defined as: = ∈ 1 l ( ) log 2 ( , + 1)(1) where 1 l ( ) equals 1 when is in l , and 0 otherwise. The symbol , represents the position of item in . A provider's total exposure can be viewed as an aggregation of the exposures of items it provides as = ∈ . Fairness in Providers' Exposure There are different definitions of exposure fairness in relevant studies. In some definitions, better items should have higher exposure [5], that is, it is fair if the exposure of an item is proportional to its quality. Some consider a more universal measure of fairness that gives all items the same exposure. In this paper, we consider these two kinds of exposure fairness at the same time, and our algorithm can support the optimization of these two kinds of fairness. Since a provider's exposure is the aggregation of its items' exposures, the provider who offers more items intuitively gains higher exposure, then the fairness between providers can be essentially transformed into the fairness between groups of recommended items. In line with the aforementioned idea of the two kinds of exposure fairness, we have the following definitions: D 1 (U F ). A recommendation result has the property of uniform fair exposure for providers if each provider receives exposure proportional to the number of items it offers. 1 \| 1 \| = 2 \| 2 \| , ∀ 1 , 2 ∈ . (2) D 2 ( W F ). A recommendation result is quality weighted fair exposure for providers if each provider receives exposure proportional to the sum of quality scores of items it offers in the recommendation lists of all customers. 1 ∈ 1 ∈ , = 2 ∈ 2 ∈ , , ∀ 1 , 2 ∈ .(3) Fairness of exposure can then be measured in terms of the dispersion of data, such as the variance of exposure, and the lower the degree of dispersion of the data, the more fairness it indicates. Quality of Recommendation The introduction of a fairness index into recommendation results will reduce the quality of recommendation results since some items with low ratings will be allocated to higher positions in the list in order to ensure fair exposure for providers. In this paper, we define the quality of the recommendation results as the degree to which the recommendation list matches the original list of the recommender system. It is worth pointing out that although the original recommendation list may still not be absolutely in line with customers' preferences in practice, it can be assumed that the original recommendation list has already reflected their preferences as much as possible. We use two classic metrics in information retrieval, namely discounted cumulative gain (DCG) and normalized discounted cumulative gain (NDCG) to measure the quality of recommendation [5]. DCG sums up the relative scores of all items in the recommended list and gives a logarithmic discount based on their ranks which is consistent with the idea of item exposure. ,l = ,l[1] + =2 ,l[ ] log 2 ( + 1)(4) NDCG further normalizes DCG by dividing the DCG value of a customer's original recommendation list l (IDCG) which is the ideal situation in terms of recommendation quality for results completely being in line with customer preferences. By dividing by the DCG of the original recommendation list, the difference in customers' scoring habits can be eliminated. When this value is equal to 1, it indicates that the result is fully in line with the customer's preference, and the smaller the value, the greater the loss of recommendation quality. Measuring the Fairness of Recommendation for Customers As previously mentioned, when the fair exposure of providers is taken into consideration, the quality of recommendations will be reduced. We introduce the idea of individual fairness and want the recommendation quality reduction to be equally allocated to every customer. We provide the following definition: D 3 (F ). The recommendation is fair for customers if each customer receives recommendation results with the same NDCG value. 1 = 2 , ∀ 1 , 2 ∈ .(6) We can also use variance of NDCG values of customers' recommendation results to measure the overall customer-side fairness. Trade-off between Customer Benefits and Provider Benefits In this section, we discuss the relationship between customer recommendation quality, customer fairness, and provider exposure fairness, and clarify the direction of our algorithm optimization. As demonstrated experimentally in [29], if the recommendation system is completely subordinate to customer preferences, it can result in vastly unfair provider exposure. When the algorithm tries to modify the original recommendation list to improve the fairness of the providers' exposure, we get the following three theorems, which describe the relationships between the three objectives: THEOREM 1. There is no such an algorithm that does not reduce the quality of recommendation results compared with the original list l , unless the algorithm directly recommends the original recommendation list to customers. THEOREM 2. When the recommendation quality decreases from the best situation, the fairness of customer recommendation quality decreases or remains unchanged. THEOREM 3. When the fairness of provider exposure increases, the fairness of customer recommendation quality decreases or remains unchanged. As mentioned in Section 3.4, if the algorithm directly recommends the original recommendation list l to customers, the recommendation quality of customers is at the best level (NDCG is equal to 1), and the quality among customers is fair. Since the original recommendation list is the optimal case of the customers' recommendation quality, THEOREM 1 must be correct. When the recommendation quality decreases, because the distribution of customers recommendation quality is not the same, there will be differences in the degree of quality loss between customers, which increases the variance of customer recommendation quality, and the customer-side fairness decreases. But at the same time, there may be such a special case that the degradation of quality is evenly distributed to all customers, so that the quality of customer recommendations remains the same, and the customer-side fairness is maintained, which is THEOREM 2. Although this special case does not necessarily exist in all data, we can distribute the loss of quality to all customers as much as possible when optimizing provider exposure, so as to make the decrease in customer-side fairness as small as possible. Combining THEOREM 1 and THEOREM 2, we can get the relationship between the fairness of exposure and the fairness of recommendation quality, which is THEOREM 3. This shows that we can improve the fairness of exposure while still maintaining the fairness of recommendation quality. Based on these three theorems, we can get the optimization direction of the problem. Regarding the unfairness problem of the providers' exposure when customers' preferences are fully satisfied, we can sacrifice part of the recommendation quality to optimize the fairness of exposure. At the same time, we can make the quality loss distributed to all customer as evenly as possible to maintain the fairness of customer recommendation quality. The design of our algorithms is based on this idea. A TWO-SIDED FAIRNESS-AWARE RECOMMENDATION MODEL (TFROM) The two-sided fairness problem discussed in Section 3 can be reduced to a knapsack problem which has been proven to be a nondeterministic polynomial complete problem. We analogize the length of the recommendation list as the capacity of the knapsack, the items to be recommended as the items put in the knapsack, and fairness as the objective. When further taking the quality of recommendation lists into consideration, the problem becomes more complicated. So we choose heuristic strategies to solve the problem. In this section, we propose TFROM to solve the two-sided fairness problem. TFROM consists of two algorithms designed for two scenarios. One is an offline scenario in which the system makes recommendation to all customers once at the same time, such as via an advertising push. The other is an online scenario where customers' requests arrive randomly, and the system needs to respond to each request within a short period of time and provide the recommendation results, for example, for online purchases. TFROM for Offline Scenario Recommendations can be generated for all customers in an offline fashion, such as advertising via email. In this situation, we need to select items from items for customers respectively. The exposure brought by the position of each item in the recommendation list can be calculated. If the length of the recommendation list and the total number of customers are known, the total exposure provided by a recommendation can be calculated as follows: = × =1 1 2 ( + 1)(7) Furthermore, based on the previous definitions of exposure fairness, we can calculate how much exposure each provider should obtain to reach a fair state based on the number of items provided by this provider as follows: = × \| \| ∈ \| \| (Uniform Fairness) (8) = × ∈ ∈ , ∈ ∈ ∈ , (Quality Weighted Fairness) (9) If all providers receive exposure equal to in a recommendation, then absolute exposure fairness is achieved between providers. Because only a fixed exposure value can be provided by the position in the recommendation list, the total exposure received by each provider is virtually impossible to be equal to the ideal value. But we can use these values as a benchmark for the exposure of each provider, and make the actual exposure as close to these benchmarks as possible, so as to ensure the fairness of exposure between providers as much as possible. In order to ensure customer fairness, we propose the following approach. In order to distribute the recommendation quality reduction as evenly as possible, customers who have experienced a lower loss in quality in the recommendation process should suffer more losses than customers who have experienced a higher loss in quality at present; in order to improve the overall recommendation quality for customers, it is necessary to give priority to items which are more relevant to customers as much as possible. Combining the aforementioned ideas, TFROM-offline works as follows. The algorithm recommends a list of items from position 1 to . When making a recommendation for a certain position, the algorithm must wait until all customers are recommended an item in this position before items in the next position can be recommended. For the first position, an item is selected for each customer according to the results of a recommendation algorithm in an arbitrary order. For each of the remaining positions, the customers will be sorted from high to low according to recommendation quality scores in terms of the items selected for them, and the rest items will be selected for them in this order. For each selection, the algorithm finds the highest-ranked and un-recommended item from a customer's original recommendation list l . If the item's provider's previous exposure plus the item's exposure at this position does not exceed the provider's fair exposure baseline , the item is selected, otherwise the algorithm will look for the next item along the original recommendation list l until a suitable item is found. If all the items in the l list do not meet the conditions, then this position will be skipped and will be allocated after all positions have been tentatively filled with items. Every time an item is selected, the exposure of the provider and the recommendation quality obtained by the customers are updated. This process is repeated until items in position have been selected. After this, the positions that are skipped before are re-allocated from high positions to low positions. Each time, TFROMoffline recommends an unrecommended item with the lowest provider exposure in the l list to further reduce the difference in exposure between providers and ensure that all positions are filled. The pseudo-code of TFROM-offline is shown in Algorithm 1. if rank == 1 then end for 39: end for 40: return l 1 , l 2 ,..., l ; TFROM for Online Scenario In an online situation, customer requests arrive randomly and the algorithm needs to respond to each request in a timely manner. In this case, the fairness of a single round of recommendation loses its meaning, and our target is changed into long-term fairness. We transform the customer's recommendation quality and the provider's exposure into a cumulative value. In addition, due to the different number of times the recommendation services are provided for each customer, the recommendation quality for each customer should be further divided by the number of times the recommendations are provided, which is the average of the recommendation quality obtained by each customer in the service process. At the same time, because the number of customer requests constantly increases, the total exposure cannot be determined in advance as in the offline situation, but changes dynamically with the arrival of customer requests. The fair exposure of providers thus needs to be recalculated as the total exposure changes: = _ × =1 1 2 ( + 1) where _ is the number of customer requests. If the algorithm continues to follow the idea of TFROM-offline and reduces the exposure difference between providers by only adjusting this customer's recommendation list, the quality of recommendations for him may be greatly reduced. Therefore, when filling vacancies in the recommendation list, we give priority to recommendation quality and directly select the remaining items with the highest preference scores in the l list. At the same time, when the number of customer requests is very small, the calculated fair exposure baseline of each provider is smaller than the exposure that will be provided by the recommendation, which will result in no items being selected in the early stage. When filling vacancies, TFROM-online selects items in the order of preference scores, and in this case, it happens to be directly recommending the l list to customers, which is also a very reasonable strategy in the initial stage of system startup. This shows that TFROM-online is suitable for the inception phase, and can naturally transition to the regular stage without additional operations. The pseudo-code of TFROM-online is shown in Algorithm 2. Time complexity The time complexity of TFROM-offline is analyzed as follows. Before selecting items for a certain position, TFROM-offline first sorts the customers according to the recommendation quality scores obtained by the items selected for their lists. The complexity of sorting customers is ( log( )) when using the Quick Sort Algorithm or the Merge Sort Algorithm. Then TFROM-offline iterates through the original list of customers receiving recommendations to find suitable items that will not exceed the fairness exposure of providers, and in the worst case, the algorithm needs to traverse the list again in the second stage. So in the worst case, the complexity of selecting items for a certain position is (2 2 log( )). Over the whole process, the algorithm needs to select items for positions, and the worst case time complexity of TFROM-offline is ( 2 log( )). Since TFROM-online only deals with a single customer at a time, its worst case time complexity is (2 ). EXPERIMENTS 5.1 Datasets and Metrics We conducted experiments on three datasets -a flight dataset from an online travel company Ctrip 1 , a Google local dataset and an Amazon review dataset. These three data sets cover three very important aspects of a customer's daily life, i.e., travel, local living and online shopping. Our code and datasets are released on Zenodo 2 . Ctrip Flight Dataset. We select the ticket order data on a popular international flight route from Shanghai to Seoul from 2017 to 2020, and treat tickets from the same airline, of the same class and in the same departure time as the same item, and the airline to which the ticket belongs as the provider. The entire dataset contains data of 3,814 customers, 6,006 kinds of air tickets and 25,190 orders, and it provides basic information on customers, as well as air ticket price, air ticket class, airline company of the ticket, flight time and other ticket information. We adopt the state-ofthe-art collaborative filtering air ticket recommendation algorithm [16] to process the data, and obtain the original recommendation lists and customer-item preference matrix. Google Local dataset. This dataset was released in [28] and contains reviews about local businesses from Google Maps. We consider businesses located in California, filter out businesses and customers with less than 10 reviews, and obtain a dataset containing 3,335 users, 4,927 businesses, and 97,658 reviews. We mainly consider information such as ratings, comment time, business location, etc., and adopt the state-of-the-art location-based latent factorization algorithm [17] to process the data. The reviews in this dataset are for businesses and do not provide information on the reviewed items. Therefore, for this dataset, we regard each business as a provider, and each provider only provides one item, which is also a special case in our problem. Amazon Review dataset. This dataset contains a variety of product reviews from Amazon and we use the data released in [18]. Because of the sheer volume of data, we pre-filter customers and items with less than 10 comments, and only consider reviews of items in the "Clothing Shoes and Jewelry" category, which has the largest number of reviews. We use the state-of-the-art matrix factorization model [24] to obtain the preference matrix. Since the data set does not provide information on the providers of the items, we randomly aggregate 1-100 items to simulate providers with different scales. The processed dataset contains 1,851 users, 7,538 items, 161 providers and 24,658 reviews. Metrics. We measure the variance of the provider's exposure to evaluate provider-side Uniform Fairness, measure the variance of the ratio of provider exposure and relevance in Equation(9) to evaluate Quality Weighted Fairness. Since the magnitude of the numerator and denominator is quite different, we use the [0,1]-normalized value for calculation. For the customer-side metrics, we measure the variance of the customer's recommendation quality to evaluate customerside fairness, and measure the sum of the customers' recommendation quality to evaluate the overall quality of recommendation results. The smaller the variance, the fairer the recommendation results. The greater the sum of , the smaller the loss of the recommendation quality. Compared Approaches We compare our proposed approach with the following algorithms. Top-k. This algorithm directly recommends the top-k items in the original recommendation list l , which is also the case for maximizing recommendation quality. All random. This algorithm randomly selects items from the customer's original recommendation list l to recommend. Minimum exposure. This algorithm selects items from the least exposed provider each time for recommendation. This is an algorithm ensure that the providers' exposure is as fair as possible. 5.2.4 FairRec. This is a state-of-the-art algorithm that guarantees two-sided fairness based on a greedy strategy [29], which ensures Uniform Fairness for providers by setting the minimum exposure, and fairness for customers using a greedy strategy. 5.2.5 An ILP-based fair ranking mechanism. This is an algorithm based on integer linear programming(ILP) proposed by [5] to ensure the Quality Weighted Fairness of provider exposure. This algorithm takes the absolute value of the difference between the two cumulative values as the objective, and the quality of recommendation as the limiting condition. Experiment Results and Analysis Results for the offline situation. We conducted experiments for the offline situation on three data sets and evaluated the results of the algorithms at different values. Recommendation quality. As shown in Figure 1(a), 2(a) and 3(a), Top-k, FairRec and TFROM-offline-Uniform produce higher recommendation quality than All random and Minimum exposure methods. Of these, Top-k achieves the maximum value for recommendation quality as its results are completely in line with the customer's preferences. It is worth noting that the quality loss of TFROMoffline-Quality-Weighted is large when is small, because the algorithm selects items with low relevance when adjusting the exposure fairness. In this case, the cost, i.e., the recommendation quality loss, is large, which results in excessive loss. This situation will be alleviated as increases. All random and Minimum exposure algorithm cause a large loss in the recommendation quality due to the lack of special treatment for the recommendation quality. In summary, TFROM-offline is capable of maintaining an acceptable loss of recommendation quality (less than 10% in most cases), and in practice, this loss of quality is spread evenly across multiple items so the customer experience will not be changed much and the customers may not even be aware of it. Customer-side Fairness. The associated results are shown in Figures 1(b), 2(b) and 3(b). It can be seen that all the algorithms provide good customer fairness for all datasets, and the results of TFROMoffline are also at a high level in comparison algorithms. In principle, the all random algorithm guarantees fairness for customers, because it carries out the same operation for all customers. However, it only makes a recommendation to each customer once in offline situations, which leads to unsatisfactory results. The Minimum exposure algorithm does not operate on customer fairness, so the main reason for the good effect is that there is a large loss in Provider-side Fairness. As can be seen from Figures 1(c), 2(c) and 3(c), TFROM-offline-Uniform, the minimum exposure algorithm and all random algorithm stably provide fair exposure results on all three datasets as grows when considering Uniform Fairness. From the results, it can also be seen that if customer preferences are completely respected, the inequality of providers will increase with the increase of and will even increase exponentially. It is worth noting that FairRec's results are not good. Although FairRec is designed to ensure fair exposure at the level of individual items, it is not as good as it would have been if multiple items had been aggregated to a provider. The results of Quality Weighted Fairness are shown in 1(d), 2(d) and 3(d). It can be seen that TFROM-offline-Quality-Weighted can provide better and more stable fairness on the three datasets. Although the All random algorithm can provide better fairness on the Google dataset in some cases, the performance on the other two datasets is not satisfactory. It is worth noting that the results of the ILP-based method seem to indicate it has limited optimization capabilities based on the results of Top-k algorithm, which may be caused by insufficient solution set space due to pre-filtering. Results for the online situation. We generate a random sequence of customer requests to simulate the online scenario, and evaluate the changes of the aforementioned metrics during the recommendation process. The length of the sequence is set to 10 times the number of customers, so that each customer can receive multiple recommendations. At the same time, we also test the performance of TFROM-online with two kinds of provider-side fairness. Since FairRec is not suitable for online scenarios, we do not compare it in this experiment. Recommendation quality. It can be seen from Figures 4(a), 5(a) and 6(a) that with the continuous arrival of customer requests, the recommendation quality of the aforementioned algorithms grows basically linearly. Top-k still achieves the maximum value for recommendation quality, whereas the Minimum exposure algorithm achieves the worst result by completely ignoring recommendation quality. TFROM-online has a small amount of loss on the three data sets. Of these, the loss of TFROM-online-Uniform is larger because the goal of uniform fairness forces the algorithm to select items with lower relevance in the recommendation list of a single customer. This loss is accumulated during the recommendation process, but in fact, for a single recommendation, the loss of quality is completely acceptable to a customer. Customer-side Fairness. The results are shown in Figures 4(b), 5(b) and 6(b). It is worth noting that customer fairness decreases firstly and then increases with the number of requests (variance and mean deviation increases first and then decreases). This is because at the beginning, most customers have not received recommendation, and the customer's recommendation quality changes from zero to a value greater than zero. This change is dramatic, resulting in the rapid improvement of the deviation of recommendation quality. After all the customers have received at least one recommendation, the subsequent recommendation will reduce the differences between customers and the system tends to be stable. Compared with other algorithms, TFROM-online can reach a good level of customer fairness in the end which shows that TFROMonline can maintain the fairness of customer recommendation quality in the long-term recommendation process. Provider-side Fairness. As shown in Figures 4(c), 5(c) and 6(c), if no action is taken on exposure unfairness as in the Top-k algorithm, the degree of unfairness will continue to increase along with the recommendation process and will reach an unacceptable level. At the same time, the other algorithms can provide very good exposure fairness, which is consistent with the results of offline scenarios. As for Quality Weighted Fairness, the experiment results are basically the same as the offline scenarios. Compared with the comparison algorithms, TFROM-online-Quality-Weighted can provide recommendation results more consistently and more fairly. CONCLUSIONS In this paper, we consider the issue of fairness in a recommendation system from two sides, i.e., customers and providers. The objective of our study is to ensure fairness for both sides while maintaining a high level of personalization in the recommendation results. We model the providers that provide multiple items and ensure fairness among them at the group level and consider two kinds of fairness definitions, while from the customer perspective, we ensure fairness between individual customers. Aiming at both offline and online scenarios, we design post-processing heuristic algorithms to ensure two-sided fairness, which enables our method to be easily applied to various existing recommendation systems in various scenarios, and helps them to improve the fairness of the system. Experiments on three real-world datasets show that our algorithms provide better two-sided fairness than the comparison algorithms while losing only a little recommendation quality. ..., : Accumulated Exposure of providers up to last recommendation;: Average recommendation quality of customers up to last recommendation; _ : The number of times that the recommendation services customer has received up to last time recommendation; Output: l : Recommendation results for the coming customer ;1: l ← [ 2 https://zenodo.org/record/4527725#.YCMxKegzZPY of the ratio of exposure and relevance Figure 1 : 1Experiment of the ratio of exposure and relevance Figure 2 : 2Experiment of the ratio of exposure and relevance Figure 3 : 3Experiment Results on Amazon Dataset in the Offline Scenario recommendation quality, and as the overall level of recommendation quality is very low, which reduces the difference in customer recommendation quality. 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[ "⋆ Supported by the Austrian Science Fund (FWF) under grant S11409-N23", "⋆ Supported by the Austrian Science Fund (FWF) under grant S11409-N23" ]	[ "Florian Lonsing \nResearch Division of Knowledge Based Systems Institute of Logic and Computation\nWienTUAustria\n", "Uwe Egly \nResearch Division of Knowledge Based Systems Institute of Logic and Computation\nWienTUAustria\n" ]	[ "Research Division of Knowledge Based Systems Institute of Logic and Computation\nWienTUAustria", "Research Division of Knowledge Based Systems Institute of Logic and Computation\nWienTUAustria" ]	[ "LNCS" ]	The QRAT (quantified resolution asymmetric tautology) proof system simulates virtually all inference rules applied in state of the art quantified Boolean formula (QBF) reasoning tools. It consists of rules to rewrite a QBF by adding and deleting clauses and universal literals that have a certain redundancy property. To check for this redundancy property in QRAT, propositional unit propagation (UP) is applied to the quantifier free, i.e., propositional part of the QBF. We generalize the redundancy property in the QRAT system by QBF specific UP (QUP). QUP extends UP by the universal reduction operation to eliminate universal literals from clauses. We apply QUP to an abstraction of the QBF where certain universal quantifiers are converted into existential ones. This way, we obtain a generalization of QRAT we call QRAT + . The redundancy property in QRAT + based on QUP is more powerful than the one in QRAT based on UP. We report on proof theoretical improvements and experimental results to illustrate the benefits of QRAT + for QBF preprocessing.	10.1007/978-3-319-94205-6_12	[ "https://arxiv.org/pdf/1804.02908v2.pdf" ]	4,709,346	1804.02908	abb8a3afcd559159b863b4c1d58f52997fcce910	⋆ Supported by the Austrian Science Fund (FWF) under grant S11409-N23 SpringerCopyright Springer2018. 2018 Florian Lonsing Research Division of Knowledge Based Systems Institute of Logic and Computation WienTUAustria Uwe Egly Research Division of Knowledge Based Systems Institute of Logic and Computation WienTUAustria ⋆ Supported by the Austrian Science Fund (FWF) under grant S11409-N23 LNCS Springer2018. 2018arXiv:1804.02908v2 [cs.LO] QRAT + : Generalizing QRAT by a More Powerful QBF Redundancy Property ⋆ The QRAT (quantified resolution asymmetric tautology) proof system simulates virtually all inference rules applied in state of the art quantified Boolean formula (QBF) reasoning tools. It consists of rules to rewrite a QBF by adding and deleting clauses and universal literals that have a certain redundancy property. To check for this redundancy property in QRAT, propositional unit propagation (UP) is applied to the quantifier free, i.e., propositional part of the QBF. We generalize the redundancy property in the QRAT system by QBF specific UP (QUP). QUP extends UP by the universal reduction operation to eliminate universal literals from clauses. We apply QUP to an abstraction of the QBF where certain universal quantifiers are converted into existential ones. This way, we obtain a generalization of QRAT we call QRAT + . The redundancy property in QRAT + based on QUP is more powerful than the one in QRAT based on UP. We report on proof theoretical improvements and experimental results to illustrate the benefits of QRAT + for QBF preprocessing. Introduction In practical applications of propositional logic satisfiability (SAT), it is necessary to establish correctness guarantees on the results produced by SAT solvers by proof checking [7]. The DRAT (deletion resolution asymmetric tautology) [22] approach has become state of the art to generate and check propositional proofs. The logic of quantified Boolean formulas (QBF) extends propositional logic by existential and universal quantification of the propositional variables. Despite the PSPACE-completeness of QBF satisfiability checking, QBF technology is relevant in practice due to the potential succinctness of QBF encodings [4]. DRAT has been lifted to QBF to obtain the QRAT (quantified RAT) proof system [8,10]. QRAT allows to represent and check (un)satisfiability proofs of QBFs and compute Skolem function certificates of satisfiable QBFs. The QRAT system simulates virtually all inference rules applied in state of the art QBF reasoning tools, such as Q-resolution [15] including its variant long-distance Qresolution [13,24], and expansion of universal variables [3]. A QRAT proof of a QBF in prenex CNF consists of a sequence of inference steps that rewrite the QBF by adding and deleting clauses and universal literals that have the QRAT redundancy property. Informally, checking whether a clause C has QRAT amounts to checking whether all possible resolvents of C on a literal l ∈ C (under certain restrictions) are propositionally implied by the quantifierfree CNF part of the QBF. The principle of redundancy checking by inspecting resolvents originates from the RAT property in propositional logic [12] and was generalized to first-order logic in terms of implication modulo resolution [14]. Instead of a complete (and thus computationally hard) propositional implication check on a resolvent, the QRAT system relies on an incomplete check by propositional unit propagation (UP). Thereby, it is checked whether UP can derive the empty clause from the CNF augmented by the negated resolvent. Hence redundancy checking in QRAT is unaware of the quantifier structure, which is entirely ignored in UP. We generalize redundancy checking in QRAT by making it aware of the quantifier structure of a QBF. To this end, we check the redundancy of resolvents based on QBF specific UP (QUP). It extends UP by the universal reduction (UR) operation [15] and is a polynomial-time procedure like UP. UR is central in resolution based QBF calculi [1,15] as it shortens individual clauses by eliminating universal literals depending on the quantifier structure. We apply QUP to abstractions of the QBF where certain universal quantifiers are converted into existential ones. The purpose of abstractions is that if a resolvent is found redundant by QUP on the abstraction, then it is also redundant in the original QBF. Our contributions are as follows: (1) by applying QUP and QBF abstractions instead of UP, we obtain a generalization of the QRAT system which we call QRAT + . In contrast to QRAT, redundancy checking in QRAT + is aware of the quantifier structure of a QBF. We show that (2) the redundancy property in QRAT + based on QUP is more powerful than the one in QRAT based on UP. QRAT + can detect redundancies which QRAT cannot. As a formal foundation, we introduce (3) a theory of QBF abstractions used in QRAT + . Redundancy elimination by QRAT + or QRAT can lead to (4) exponentially shorter proofs in certain resolution based QBF calculi, which we point out by a concrete example. Note that here we do not study the power of QRAT or QRAT + as proof systems themselves, but the impact of redundancy elimination. Finally, we report on experimental results (5) to illustrate the benefits of redundancy elimination by QRAT + and QRAT for QBF preprocessing. Our implementation of QRAT + and QRAT for preprocessing is the first one reported in the literature. Preliminaries We consider QBFs φ := Π.ψ in prenex conjunctive normal form (PCNF) with a quantifier prefix Π := Q 1 B 1 . . . Q n B n and a quantifier free CNF ψ not containing tautological clauses. The prefix consists of quantifier blocks Q i B i , where B i are blocks (i.e., sets) of propositional variables and Q i ∈ {∀, ∃} are quantifiers. We have B i ∩ B j = ∅, Q i = Q i+1 and Q n = ∃. The CNF ψ is defined precisely over the variables vars(φ) = vars(ψ) := B 1 ∪ . . . ∪ B n in Π so that all variables are quantified, i.e., φ is closed. The quantifier Q(Π, l) of literal l is Q i if the variable var(l) of l appears in B i . The set of variables in a clause C is vars( C) := {x \| l ∈ C, var(l) = x}. A literal l is existential if Q(Π, l) = ∃ and universal if Q(Π, l) = ∀. If Q(Π, l) = Q i and Q(Π, k) = Q j , then l ≤ Π k iff i ≤ j. We extend the ordering ≤ Π to an arbitrary but fixed ordering on the variables in every block B i . An assignment τ : vars(φ) → {⊤, ⊥} maps the variables of a QBF φ to truth constants ⊤ (true) or ⊥ (false). Assignment τ is complete if it assigns every variable in φ, otherwise τ is partial. By τ (φ) we denote φ under τ , where each occurrence of variable x in φ is replaced by τ (x) and x is removed from the prefix of φ, followed by propositional simplifications on τ (φ). We consider τ as a set of literals such that, for some variable x, x ∈ τ if τ (x) = ⊤ andx ∈ τ if τ (x) = ⊥. An assignment tree [10] T of a QBF φ is a complete binary tree of depth \|vars(φ)\| + 1 where the internal (non-leaf) nodes of each level are associated with a variable of φ. An internal node is universal (existential) if it is associated with a universal (existential) variable. The order of variables along every path in T respects the extended order ≤ Π of the prefix Π of φ. An internal node associated with variable x has two outgoing edges pointing to its children: one labelled withx and another one labelled with x, denoting the assignment of x to false and true, respectively. Each path τ in T from the root to an internal node (leaf) represents a partial (complete) assignment. A leaf at the end of τ is labelled by τ (φ), i.e., the value of φ under τ . An internal node associated with an existential (universal) variable is labelled with ⊤ iff one (both) of its children is (are) labelled with ⊤. The QBF φ is satisfiable (unsatisfiable) iff the root of T is labelled with ⊤ (⊥). Given a QBF φ and its assignment tree T , a subtree T ′ of T is a pre-model [10] of φ if (1) the root of T is the root of T ′ , (2) for every universal node in T ′ both children are in T ′ , and (3) for every existential node in T ′ exactly one of its children is in T ′ . A pre-model T ′ of φ is a model [10] of φ, denoted by T ′ \|= t φ, if each node in T ′ is labelled with ⊤. A QBF φ is satisfiable iff it has a model. Given a QBF φ and one of its models T ′ , T ′′ is a rooted subtree of T ′ (T ′′ ⊆ T ′ ) if T ′′ has the same root as T ′ and the leaves of T ′′ are a subset of the leaves of T ′ . We consider CNFs ψ defined over a set B of variables without an explicit quantifier prefix. A model of a CNF ψ is a model τ of the QBF ∃B.ψ which consists only of the single path τ . We write τ \|= ψ if τ is a model of ψ. For CNFs ψ and ψ ′ , ψ ′ is implied by ψ (ψ \|= ψ ′ ) if, for all τ , it holds that if τ \|= ψ then τ \|= ψ ′ . Two CNFs ψ and ψ ′ are equivalent (ψ ≡ ψ ′ ), iff ψ \|= ψ ′ and ψ ′ \|= ψ. We define notation to explicitly refer to QBF models. For QBFs φ and φ ′ , φ ′ is implied by φ (φ \|= t φ ′ ) if, for all T , it holds that if T \|= t φ then T \|= t φ ′ . QBFs φ and φ ′ are equivalent (φ ≡ t φ ′ ) iff φ \|= t φ ′ and φ ′ \|= t φ, and satisfiability equivalent (φ ≡ sat φ ′ ) iff φ is satisfiable whenever φ ′ is satisfiable. Satisfiability equivalence of CNFs is defined analogously and denoted by the same symbol '≡ sat '. The Original QRAT Proof System Before we generalize QRAT, we recapitulate the original proof system [10] and emphasize that redundancy checking in QRAT is unaware of quantifier structures. Definition 1 ( [10]). The outer clause of clause C on literal l ∈ C with respect to prefix Π is the clause OC(Π, C, l) := {k \| k ∈ C, k ≤ Π l, k = l}. The outer clause OC(Π, C, l) ⊂ C of C on l ∈ C contains only literals that are smaller than or equal to l in the variable ordering of prefix Π, excluding l. Definition 3 ([10] ). Clause C has property QIOR (quantified implied outer resolvent) on literal l ∈ C with respect to QBF Π.ψ iff ψ \|= OR(Π, C, D, l) for all D ∈ ψ withl ∈ D. Property QIOR relies on checking whether every possible outer resolvent OR of some clause C on a literal is redundant by checking if OR is propositionally implied by the quantifier-free CNF ψ of the given QBF Π.ψ. If C has QIOR on literal l ∈ C then, depending on whether l is existential or universal and side conditions, either C is redundant and can be removed from QBF Π.ψ or l is redundant and can be removed from C, respectively, resulting in a satisfiabilityequivalent QBF. Theorem 1 ([10]). Given a QBF φ := Π.ψ and a clause C ∈ ψ with QIOR on an existential literal l ∈ C with respect to QBF φ ′ := Π.ψ ′ where ψ ′ := ψ \ {C}. Then φ ≡ sat φ ′ .(ψ ∪ {C}) where C has QIOR on a universal literal l ∈ C with respect to φ 0 . Let φ ′ := Π.(ψ ∪ {C ′ }) with C ′ := C \ {l}. Then φ ≡ sat φ ′ . Note that in Theorems 1 and 2 clause C is actually removed from the QBF for the check whether C has QIOR on a literal. Checking propositional implication (\|=) as in Definition 3 is co-NP hard and hence intractable. Therefore, in practice a polynomial-time incomplete implication check based on propositional unit propagation (UP) is applied. The use of UP is central in the QRAT proof system. Definition 4 (propositional unit propagation, UP). For a CNF ψ and clause C, let ψ ∧ C ⊢ 1 ∅ denote the fact that propositional unit propagation (UP) applied to ψ ∧ C produces the empty clause, where C is the conjunction of the negation of all the literals in C. If ψ ∧ C ⊢ 1 ∅ then we write ψ ⊢ 1 C to denote that C can be derived from ψ by UP (since ψ \|= C). Definition 5 ( [10]). Clause C has property AT (asymmetric tautology) with respect to a CNF ψ iff ψ ⊢ 1 C. AT is a propositional clause redundancy property that is used in the QRAT proof system to check whether outer resolvents are redundant, thereby replacing propositional implication (\|=) in Definition 3 by unit propagation (⊢ 1 ) as follows. Definition 6 ( [10]). Clause C has property QRAT (quantified resolution asymmetric tautology) on literal l ∈ C with respect to QBF Π.ψ iff, for all D ∈ ψ withl ∈ D, the outer resolvent OR(Π, C, D, l) has AT with respect to CNF ψ. Example 2. Consider φ := ∃x 1 ∀u∃x 2 .(C ∧ D) with C := (x 1 ∨ u ∨ x 2 ) and D := (x 1 ∨ū ∨x 2 ) from Example 1. C does not have AT with respect to CNF D, but C has QRAT on x 2 with respect to QBF ∃x 1 ∀u∃x 2 .(D) since OR(Π, C, D, x 2 ) = (x 1 ∨ u ∨x 1 ∨ū) has AT with respect to CNF D. QRAT is a restriction of QIOR, i.e., a clause that has QRAT also has QIOR but not necessarily vice versa. Therefore, the soundness of removing redundant clauses and literals based on QRAT follows right from Theorems 1 and 2. Based on the QRAT redundancy property, the QRAT proof system [10] consists of rewrite rules to eliminate redundant clauses, denoted by QRATE, to add redundant clauses, denoted by QRATA, and to eliminate redundant universal literals, denoted by QRATU. In a QRAT satisfaction proof (refutation), a QBF is reduced to the empty formula (respectively, to a formula containing the empty clause) by applying the rewrite rules. The QRAT proof systems has an additional rule to eliminate universal literals by extended universal reduction (EUR). We do not present EUR because it is not affected by our generalization of QRAT, which we define in the following. Observe that QIOR and AT (and hence also QRAT) are based on propositional implication (\|=) and unit propagation (⊢ 1 ), i.e., the quantifier structure of the given QBF is not exploited. QRAT + : A More Powerful QBF Redundancy Property We make redundancy checking of outer resolvents in QRAT aware of the quantifier structure of a QBF. To this end, we generalize QIOR and AT by replacing propositional implication (\|=) and unit propagation (⊢ 1 ) by QBF implication (\|= t ) and QBF unit propagation, respectively. Thereby, we obtain a more general and more powerful notion of the QRAT redundancy property, which we call QRAT + . First, in Proposition 2 we point out a property of QIOR (Definition 3) which is due to the following result from related work [20]: if we attach a quantifier prefix Π to equivalent CNFs ψ and ψ ′ , then the resulting QBFs are equivalent. Proposition 1 ( [20]). Given CNFs ψ and ψ ′ such that vars(ψ) = vars(ψ ′ ) and a quantifier prefix Π defined precisely over vars(ψ). If ψ ≡ ψ ′ then Π.ψ ≡ t Π.ψ ′ . Proposition 2. If clause C has QIOR on literal l ∈ C with respect to QBF Π.ψ, then Π.ψ ≡ t Π.(ψ ∧ OR(Π, C, D, l)) for all D ∈ ψ withl ∈ D. Proof. Since C has QIOR on literal l ∈ C with respect to QBF Π.ψ, by Definition 3 we have ψ \|= OR(Π, C, D, l) for all D ∈ ψ withl ∈ D, and further also ψ ≡ ψ ∧ OR(Π, C, D, l). Then Π.ψ ≡ t Π.(ψ ∧ OR(Π, C, D, l)) by Proposition 1. ⊓ ⊔ By Proposition 2 any outer resolvent OR of some clause C that has QIOR with respect to some QBF Π.ψ is redundant in the sense that it can be added to the QBF Π.ψ in an equivalence preserving way (≡ t ), i.e., OR is implied by the QBF Π.ψ (\|= t ). This is the central characteristic of our generalization QRAT + of QRAT. We develop a redundancy property used in QRAT + which allows to, e.g., remove a clause C from a QBF Π.ψ in a satisfiability preserving way (like in QRAT, cf. Theorem 1.) if all respective outer resolvents of C are implied by the QBF Π.(ψ \ {C}). Since checking QBF implication is intractable just like checking propositional implication in QIOR, in practice we apply a polynomialtime incomplete QBF implication check based on QBF unit propagation. In the following, we develop a theoretical framework of abstractions of QBFs that underlies our generalization QRAT + of QRAT. Abstractions are crucial for the soundness of checking QBF implication by QBF unit propagation. Definition 7 (nesting levels, prefix/QBF abstraction). Let φ := Π.ψ be a QBF with prefix Π : = Q 1 B 1 . . . Q i B i Q i+1 B i+1 . . . Q n B n . For a clause C, levels(Π, C) := {i \| ∃l ∈ C, Q(Π, l) = Q i } is the set of nesting levels in C. 1 The abstraction of Π with respect to i with 0 ≤ i ≤ n produces the abstracted prefix Abs(Π, i) := Π for i = 0 and otherwise Abs(Π, i) : = ∃(B 1 ∪ . . . ∪ B i )Q i+1 B i+1 . . . Q n B n . The abstraction of φ with respect to i with 0 ≤ i ≤ n produces the abstracted QBF Abs(φ, i) := Abs(Π, i).ψ with prefix Abs(Π, i). Example 3. Given the QBF φ := Π.ψ with prefix Π := ∀B 1 ∃B 2 ∀B 3 ∃B 4 . We have Abs(φ, 0) = φ, Abs(φ, 1) = Abs(φ, 2) = ∃(B 1 ∪ B 2 )∀B 3 ∃B 4 .ψ, Abs(φ, 3) = Abs(φ, 4) = ∃(B 1 ∪ B 2 ∪ B 3 ∪ B 4 ).ψ. In an abstracted QBF Abs(φ, i) universal variables from blocks smaller than or equal to B i are converted into existential ones. If the original QBF φ has a model T , then all nodes in T associated to universal variables must be labelled with ⊤, in particular the universal variables that are existential in Abs(φ, i). Hence, for all models T of φ, every model T A of Abs(φ, i) is a subtree of T . Proposition 3. Given a QBF φ := Π.ψ with prefix Π := Q 1 B 1 . . . Q i B i . . . Q n B n and Abs(φ, i) for some arbitrary i with 0 ≤ i ≤ n. For all T and T A we have that if T \|= t φ and T A ⊆ T is a pre-model of Abs(φ, i), then T A \|= t Abs(φ, i). Proof. By induction on i. The base case i := 0 is trivial. As induction hypothesis (IH), assume that the claim holds for some i with 0 ≤ i < n, i.e., for all T and T A we have that if T \|= t φ and T A ⊆ T is a pre- model of Abs(φ, i), then T A \|= t Abs(φ, i). Consider Abs(φ, j) for j = i + 1, which is an abstraction of Abs(φ, i). We have to show that, for all T and T B we have that if T \|= t φ and T B ⊆ T is a pre-model of Abs(φ, j), then T B \|= t Abs(φ, j). We distinguish cases by the type of Q j in the abstracted prefix Abs(Π, i) = ∃(B 1 ∪ . . . ∪ B i )Q j B j . . . Q n B n of Abs(φ, i). If Q j = ∃ then Abs(Π, i) = Abs(Π, j) = ∃(B 1 ∪ . . . B i ∪ B j ) . . . Q n B n . Since Abs(φ, i) = Abs(φ, j) , the claim holds for Abs(φ, j) by IH. If Q j = ∀ then, towards a contradiction, assume that, for some T and T B , T \|= t φ and T B ⊆ T is a pre-model of Abs(φ, j), but T B \|= t Abs(φ, j) . Then the root of T B is labelled with ⊥, and in particular the nodes of all the variables which are existential in B j with respect to Abs(Π, j) are also labelled with ⊥. These existential variables appear along a single branch τ ′ in T B , i.e., τ ′ is a partial assignment of the variables in B j . Since T B ⊆ T A and Q j = ∀ in Abs(Π, i), the root of T A is labelled with ⊥ since there is the branch τ ′ containing the variables in B j whose nodes are labelled with ⊥ in T A . Hence T A \|= t Abs(φ, i), which is a contradiction to IH. Therefore, we conclude that T B \|= t Abs(φ, j). ⊓ ⊔ If an abstraction Abs(φ, i) is unsatisfiable then also the original QBF φ is unsatisfiable due to Proposition 3. We generalize Proposition 1 from CNFs to QBFs and their abstractions. Note that the full abstraction Abs(φ, i) for i := n of a QBF φ is a CNF, i.e., it does not contain any universal variables. Lemma 1. Let φ := Π.ψ and φ ′ := Π.ψ ′ be QBFs with the same prefix Π := Q 1 B 1 . . . Q i B i . . . Q n B n . Then for all i, if Abs(φ, i) ≡ t Abs(φ ′ , i) then φ ≡ t φ ′ . Proof. By induction on i := 0 up to i := n. The base case i := 0 is trivial. As induction hypothesis (IH), assume that the claim holds for some i with 0 ≤ i < n, i.e., if Abs(φ, i) ≡ t Abs(φ ′ , i) then φ ≡ t φ ′ . Let j = i + 1 and consider Abs(φ, j) and Abs(φ ′ , j), which are abstractions of Abs(φ, i) and Abs(φ ′ , i). We have Abs(Π, i) = ∃(B 1 ∪ . . . ∪ B i )Q j B j . . . Q n B n and Abs(Π, j) = ∃(B 1 ∪ . . . ∪ B j ) . . . Q n B n . We show that if Abs(φ, j) ≡ t Abs(φ ′ , j) then Abs(φ, i) ≡ t Abs(φ ′ , i), and hence also φ ≡ t φ ′ by IH. Assume that Abs(φ, j) ≡ t Abs(φ ′ , j). We distinguish cases by the type of Q j in Abs(Π, i). If Q j = ∃ then Abs(Π, i) = Abs(Π, j) = ∃(B 1 ∪ . . . B i ∪ B j ) . . . Q n B n , and hence Abs(φ, i) ≡ t Abs(φ ′ , i). If Q j = ∀, then towards a contradiction, assume that Abs(φ, j) ≡ t Abs(φ ′ , j) but Abs(φ, i) ≡ t Abs(φ ′ , i). Then there exists T such that T \|= t Abs(φ, i) but T \|= t Abs(φ ′ , i). Since T \|= t Abs(φ ′ , i) there exists a pre-model T A ⊆ T of Abs(φ ′ , j) such that the root of T A is labelled with ⊥, and in particular the nodes of all the variables which are existential in B j with respect to Abs(Π, j) (and universal with respect to Abs(Π, i)) are also labelled with ⊥. These existential variables appear along a single branch τ ′ in T A , i.e., τ ′ is a partial assignment of the variables in B j . Therefore we have T A \|= t Abs(φ ′ , j). Since T \|= t Abs(φ, i) and T A ⊆ T , we have T A \|= t Abs(φ, j) by Proposition 3, which contradicts the assumption that Abs(φ, j) ≡ t Abs(φ ′ , j). ⊓ ⊔ The converse of Lemma 1 does not hold. From the equivalence of two QBFs φ and φ ′ we cannot conclude that the abstractions Abs(φ, i) and Abs(φ ′ , i) are equivalent. In our generalization QRAT + of the QRAT system we check whether an outer resolvent of some clause C is implied (\|= t ) by an abstraction of the given QBF. If so then by Lemma 1 the outer resolvent is also implied by the original QBF. Below we prove that this condition is sufficient for the soundness of redundancy removal in QRAT + . To check QBF implication in an incomplete way and in polynomial time, in practice we apply QBF unit propagation, which is an extension of propositional unit propagation, to abstractions of the given QBF. Definition 8 (universal reduction, UR [15]). Given a QBF φ := Π.ψ and a non-tautological clause C, universal reduction (UR) of C produces the clause UR(Π, C) := C \ {l ∈ C \| Q(Π, l) = ∀, ∀l ′ ∈ C, Q(Π, l ′ ) = ∃ : var(l ′ ) ≤ Π var(l)}. Definition 9 (QBF unit propagation, QUP). QBF unit propagation (QUP) extends UP (Definition 4) by applications of UR. For a QBF φ := Π.ψ and a clause C, let Π.(ψ ∧ C) ⊢ 1∀ ∅ denote the fact that QUP applied to Π.(ψ ∧ C) produces the empty clause, where C is the conjunction of the negation of all the literals in C. If Π.(ψ ∧ C) ⊢ 1∀ ∅ and additionally Π.ψ \|= t Π.(ψ ∧ C) then we write φ ⊢ 1∀ C to denote that C can be derived from φ by QUP. In contrast to UP (Definition 4), deriving the empty clause by QUP by propagating C on a QBF φ is not sufficient to conclude that C is implied by φ. Example 4. Given the QBF Π.ψ with prefix Π := ∀u∃x and CNF ψ := (u ∨ x) ∧ (ū ∨ x) and the clause C := (x). We have Π.((u ∨x) ∧ (ū ∨ x) ∧ (x)) ⊢ 1∀ ∅ since propagating C = (x) produces (ū), which is reduced to ∅ by UR. However, Π.ψ \|= t Π.(ψ ∧ C) since Π.ψ is satisfiable whereas Π.(ψ ∧ C) is unsatisfiable. Note that Abs(Π.((u ∨x) ∧ (ū ∨ x) ∧x), 2) 1∀ ∅. To correctly apply QUP for checking whether some clause C (e.g., an outer resolvent) is implied by a QBF φ := Π.ψ and thus avoid the problem illustrated in Example 4, we carry out QUP on a suitable abstraction of φ with respect to C. Let i = max(levels(Π, C)) be the maximum nesting level of variables that appear in C. We show that if QUP derives the empty clause from the abstraction Abs(φ, i) augmented by the negated clause C, i.e., Abs(Π.(ψ∧C), i) ⊢ 1∀ ∅, then we can safely conclude that C is implied by the original QBF, i.e., Π.ψ \|= t Π.(ψ∧C). This approach extends failed literal detection for QBF preprocessing [16]. Lemma 2. Let Π.ψ be a QBF with prefix Π := Q 1 B 1 . . . Q n B n and C a clause such that vars(C) ⊆ B 1 . If Π.(ψ ∧ C) ⊢ 1∀ ∅ then Π.ψ ≡ t Π.(ψ ∧ C). Proof. By contradiction, assume T \|= t Π.ψ but T \|= t Π.(ψ ∧ C). Then there is a path τ ⊆ T such that τ (C) = ⊥. Since vars(C) ⊆ B 1 and Π.(ψ ∧ C) ⊢ 1∀ ∅, the QBF Π.(ψ∧C) is unsatisfiable and in particular T \|= t Π.(ψ∧C). Since τ (C) = ⊥, we have τ (C) = ⊤ and hence T \|= t Π.(ψ ∧ C), which is a contradiction. ⊓ ⊔ Lemma 3. Let Π.ψ be a QBF, C a clause, and i = max(levels(Π, C)). If Abs(Π.(ψ ∧ C), i) ⊢ 1∀ ∅ then Abs(Π.ψ, i) ≡ t Abs(Π.(ψ ∧ C), i). Proof. The claim follows from Lemma 2 since all variables that appear in C are existentially quantified in Abs(Π.(ψ ∧ C), i) in the leftmost quantifier block. ⊓ ⊔ Lemma 4. Let Π.ψ be a QBF, C a clause, and i = max(levels(Π, C)). If Abs(Π.(ψ ∧ C), i) ⊢ 1∀ ∅ then Π.ψ ≡ t Π.(ψ ∧ C). Proof. By Lemma 3 and Lemma 1. ⊓ ⊔ Lemma 4 provides us with the necessary theoretical foundation to lift AT (Definition 5) from UP, which is applied to CNFs, to QUP, which is applied to suitable abstractions of QBFs. The abstractions are constructed depending on the maximum nesting level of variables in the clause we want to check. Definition 10 (QAT). Let φ be a QBF, C a clause, and i = max(levels(Π, C)) Clause C has property QAT (quantified asymmetric tautology) with respect to φ iff Abs(φ, i) ⊢ 1∀ C. As an immediate consequence from the definition of QUP (Definition 9) and Lemma 3, we can conclude that a clause C has QAT with respect to a QBF Π.ψ if QUP derives the empty clause from the suitable abstraction of Π.ψ with respect to C (i.e., Abs(Π.(ψ ∧ C), i) ⊢ 1∀ ∅). Further, if C has QAT then we have Π.ψ ≡ t Π.(ψ ∧ C) by Lemma 4, i.e., C is implied by the given QBF Π.ψ. Example 5. Given the QBF φ := Π.ψ with Π := ∀u 1 ∃x 3 ∀u 2 ∃x 4 and ψ := (u 1 ∨x 3 ) ∧ (u 1 ∨x 3 ∨ x 4 ) ∧ (ū 2 ∨x 4 ). Clause (u 1 ∨x 3 ) has QAT with respect to Abs(φ, 2) with max(levels(C)) = 2 since ∀u 2 is still universal in the abstraction. By QUP clause (u 1 ∨x 3 ∨ x 4 ) becomes unit and clause (ū 2 ∨x 4 ) becomes empty by UR. However, clause (u 1 ∨x 3 ) does not have AT since ∀u 2 is treated as an existential variable in UP, hence clause (ū 2 ∨x 4 ) does not become empty by UR. In contrast to AT, QAT is aware of quantifier structures in QBFs as shown in Example 5. We now generalize QRAT to QRAT + by replacing AT by QAT. Similarly, we generalize QIOR to QIOR + by replacing propositional implication (\|=) and equivalence (Proposition 1), by QBF implication and equivalence (Lemma 4). Definition 11 (QRAT + ). Clause C has property QRAT + on literal l ∈ C with respect to QBF Π.ψ iff, for all D ∈ ψ withl ∈ D, the outer resolvent OR(Π, C, D, l) has QAT with respect to QBF Π.ψ. Definition 12 (QIOR + ). Clause C has property QIOR + on literal l ∈ C with respect to QBF Π.ψ iff Π.ψ ≡ t Π.(ψ ∧ OR(Π, C, D, l)) for all D ∈ ψ withl ∈ D. If a clause has QRAT then it also has QRAT + . Moreover, due to Proposition 2, if a clause has QIOR then it also has QIOR + . Hence QRAT + and QIOR + indeed are generalizations of QRAT and QIOR, which are strict, as we argue below. The soundness of removing redundant clauses and universal literals based on QIOR + (and on QRAT + ) can be proved by the same arguments as original QRAT, which we outline in the following. We refer to the appendix for full proofs. Definition 13 (prefix/suffix assignment [10]). For a QBF φ := Π.ψ and a complete assignment τ in the assignment tree of φ, the partial prefix and suffix assignments of τ with respect to variable x, denoted by τ x and τ x , respectively, are defined as τ x := {y → τ (y) \| y ≤ Π x, y = x} and τ x := {y → τ (y) \| y ≤ Π x}. For a variable x from block B i of a QBF, Definition 13 allows us to split a complete assignment τ into three parts τ x lτ x , where the prefix assignment τ x assigns variables (excluding x) from blocks smaller than or equal to B i , l is a literal of x, and the suffix assignment τ x assigns variables from blocks larger than B i . Prefix and suffix assignments are important for proving the soundness of satisfiability-preserving redundancy removal by QIOR + (and QIOR). Soundness is proved by showing that certain paths in a model of a QBF can safely be modified based on prefix and suffix assignments, as stated in the following. Lemma 5 (cf. Lemma 6 in [10]). Given a clause C with QIOR + with respect to QBF φ := Π.ψ on literal l ∈ C with var(l) = x. Let T be a model of φ and τ ⊆ T be a path in T . If τ (C \{l}) = ⊥ then τ x (D) = ⊤ for all D ∈ ψ withl ∈ D. Proof (sketch, see appendix). Let D ∈ ψ be a clause withl ∈ D and R := OR(Π, C, D, l) = (C \ {l}) ∪ OC(Π, D,l). By Definition 12, we have Π.ψ ≡ t Π.(ψ ∧ OR(Π, C, D, l)) for all D ∈ ψ withl ∈ D. The rest of the proof considers a path τ in T and works in the same way as the proof of Lemma 6 in [10]. ⊓ ⊔ Theorem 3. Given a QBF φ := Π.ψ and a clause C ∈ ψ with QIOR + on an existential literal l ∈ C with respect to QBF φ ′ := Π.ψ ′ where ψ ′ := ψ \ {C}. Then φ ≡ sat φ ′ . Proof (sketch, see appendix). The proof relies on Lemma 5 and works in the same way as the proof of Theorem 7 in [10]. A model T of φ is obtained from a model T ′ of φ ′ by flipping the assignment of variable x = var(l) on a path τ in T ′ to satisfy clause C. All D ∈ ψ withl ∈ D are satisfied by such modified τ . ⊓ ⊔ Proof (sketch, see appendix). The proof relies on Lemma 5 and works in the same way as the proof of Theorem 8 in [10]. A model T ′ of φ ′ is obtained from a model T of φ by modifying the subtree under the node associated to variable x = var(l). Suffix assignments of some paths τ in T are used to construct modified paths in T ′ under which clause C ′ is satisfied. All D ∈ ψ withl ∈ D are still satisfied after such modifications. ⊓ ⊔ Analogously to the QRAT proof system that is based on the QRAT redundancy property (Definition 6), we obtain the QRAT + proof system based on property QRAT + (Definition 11). The system consists of rewrite rules QRATE + , QRATA + , and QRATU + to eliminate or add redundant clauses, and to eliminate redundant universal literals. On a conceptual level, these rules in QRAT + are similar to their respective counterparts in the QRAT system. The extended universal reduction rule EUR is the same in the QRAT and QRAT + systems. In contrast to QRAT, QRAT + is aware of quantifier structures of QBFs because it relies on the QBF specific property QAT and QUP instead of on propositional AT and UP. The QRAT + system has the same desirable properties as the original QRAT system. QRAT + simulates virtually all inference rules applied in QBF reasoning tools and it is based on redundancy property QRAT + that can be checked in polynomial time by QUP. Further, QRAT + allows to represent proofs in the same proof format as QRAT. However, proof checking, i.e., checking whether a clause listed in the proof has QRAT + on a literal, must be adapted to the use of QBF abstractions and QUP. Consequently, the available QRAT proof checker QRATtrim [10] cannot be used out of the box to check QRAT + proofs. Notably, Skolem functions can be extracted from QRAT + proofs of satisfiable QBFs in the same way as in QRAT (consequence of Theorem 3, cf. Corollaries 26 and 27 in [10]). Hence like QRAT, QRAT + can be integrated in complete QBF workflows that include preprocessing, solving, and Skolem function extraction [5]. Exemplifying the Power of QRAT + In the following, we point out that the QRAT + system is more powerful than QRAT in terms of redundancy detection. In particular, we show that the rules QRATE + and QRATU + in the QRAT + system can eliminate certain redundancies that their counterparts QRATE and QRATU cannot eliminate. Definition 14. For n ≥ 1, let Φ C (n) := Π C (n).ψ C (n) be a class of QBFs with prefix Π C (n) and CNF ψ C (n) defined as follows. Π C (n) := ∃B 1 ∀B 2 ∃B 3 ∀B 4 ∃B 5 : B 1 := {x 4i+1 , x 4i+2 \| 0 ≤ i < n} B 2 := {u 2i+1 \| 0 ≤ i < n} B 3 := {x 4i+3 \| 0 ≤ i < n} B 4 := {u 2i+2 \| 0 ≤ i < n} B 5 := {x 4i+4 \| 0 ≤ i < n} ψ C (n) := n−1 i:=0 C(i) with C(i) := 6 j:=0 C i,j : C i,0 := (x 4i+1 ∨ u 2i+1 ∨ ¬x 4i+3 ) C i,1 := (x 4i+2 ∨ ¬u 2i+1 ∨ x 4i+3 ) C i,2 := (¬x 4i+1 ∨ ¬u 2i+1 ∨ ¬x 4i+3 ) C i,3 := (¬x 4i+2 ∨ u 2i+1 ∨ x 4i+3 ) C i,4 := (u 2i+1 ∨ ¬x 4i+3 ∨ x 4i+4 ) C i,5 := (¬u 2i+2 ∨ ¬x 4i+4 ) C i,6 := (¬x 4i+1 ∨ u 2i+2 ∨ ¬x 4i+4 ) Example 6. For n := 1, we have Φ C (n) with prefix Π C (n) := ∃x 1 , x 2 ∀u 1 ∃x 3 ∀u 2 ∃x 4 and CNF ψ C (n) := C(0) with C(0) := 6 j:=0 C 0,j as follows. C 0,0 := (x 1 ∨ u 1 ∨ ¬x 3 ) C 0,1 := (x 2 ∨ ¬u 1 ∨ x 3 ) C 0,2 := (¬x 1 ∨ ¬u 1 ∨ ¬x 3 ) C 0,3 := (¬x 2 ∨ u 1 ∨ x 3 ) C 0,4 := (u 1 ∨ ¬x 3 ∨ x 4 ) C 0,5 := (¬u 2 ∨ ¬x 4 ) C 0,6 := (¬x 1 ∨ u 2 ∨ ¬x 4 ) Proposition 4. For n ≥ 1, QRATE + can eliminate all clauses in Φ C (n) whereas QRATE cannot eliminate any clause in Φ C (n). Proof (sketch). For i and k with i = k, the sets of variables in C(i) and C(k) are disjoint. Thus it suffices to prove the claim for an arbitrary C(i). Clause C i,0 has QRAT + on literal x 4i+1 and can be removed. The relevant outer resolvents are OR 0,2 = OR(Π C (n), C i,0 , C i,2 , x 4i+1 ) and OR 0,6 = OR(Π C (n), C i,0 , C i,6 , x 4i+1 ), and we have OR 0,2 = OR 0,6 = (u 2i+1 ∨ ¬x 4i+3 ). Since max(levels(OR 0,2 )) = max(levels(OR 0,6 )) = 3, we apply QUP to the abstraction Abs(Φ C (n), 3). Note that variable u 2i+2 from block B 4 still is universal in the prefix of Abs(Φ C (n), 3). Propagating OR 0,2 and OR 0,6 , respectively, in either case makes C i,4 unit, finally C i,5 becomes empty under the derived assignment x 4i+4 since UR reduces the literal ¬u 2i+2 . After removing C i,0 , clauses C i,2 and C i,6 trivially have QRAT + on ¬x 4i+1 . Then C i,1 has QRAT + on x 4i+3 . Finally, the remaining clauses trivially have QRAT + . In contrast to that, QRATE cannot eliminate any clause in Φ C (n). Clause C i,5 does not become empty by UP since all variables are existential. The claim can be proved by case analysis of all possible outer resolvents. ⊓ ⊔ Definition 15. For n ≥ 1, let Φ L (n) := Π L (n).ψ L (n) be a class of QBFs with prefix Π L (n) and CNF ψ L (n) defined as follows. Π L (n) := ∀B 1 ∃B 2 ∀B 3 ∃B 4 : B 1 := {u 3i+1 , u 3i+2 \| 0 ≤ i < n} B 2 := {x 3i+1 , x 3i+2 \| 0 ≤ i < n} B 3 := {u 3i+3 \| 0 ≤ i < n} B 4 := {x 3i+3 \| 0 ≤ i < n} ψ L (n) := n−1 i:=0 C(i) with C(i) := 7 j:=0 C i,j : C i,0 := (¬u 3i+2 ∨ ¬x 3i+1 ∨ ¬x 3i+2 ) C i,1 := (¬u 3i+1 ∨ ¬x 3i+1 ∨ x 3i+2 ) C i,2 := (u 3i+1 ∨ x 3i+1 ∨ ¬x 3i+2 ) C i,3 := (u 3i+2 ∨ x 3i+1 ∨ x 3i+2 ) C i,4 := (¬x 3i+1 ∨ ¬x 3i+2 ∨ x 3i+3 ) C i,5 := (u 3i+3 ∨ ¬x 3i+3 ) C i,6 := (¬x 3i+1 ∨ x 3i+2 ∨ ¬x 3i+3 ) C i,7 := (¬u 3i+3 ∨ x 3i+3 ) Proposition 5. For n ≥ 1, QRATU + can eliminate the entire quantifier block ∀B 1 in Φ L (n) whereas QRATU cannot eliminate any universal literals in Φ L (n). Proof (sketch, see appendix). Formulas Φ L (n) are constructed based on a similar principle as Φ C (n) in Definition 14. E.g., clauses C i,0 and C i,1 have QRAT + but not QRAT on literals ¬u 3i+2 and ¬u 3i+1 . During QUP, clauses C i,5 and C i,7 become empty only due to UR, which is not possible when using UP. ⊓ ⊔ 6 Proof Theoretical Impact of QRAT and QRAT + As argued in the context of interference-based proof systems [6], certain proof steps may become applicable in a proof system only after redundant parts of the formula have been eliminated. We show that redundancy elimination by QRAT + or QRAT can lead to exponentially shorter proofs in the resolution based LQU +resolution [1] QBF calculus. Note that we do not compare the power of QRAT or QRAT + as proof systems themselves, but the impact of redundancy elimination on other proof systems. The following result relies only on QRATU, i.e., it does not require the more powerful redundancy property QRATU + in QRAT + . LQU + -resolution is a calculus that generalizes traditional Q-resolution [15]. It allows to generate resolvents on both existential and universal variables and admits tautological resolvents of a certain kind. LQU + -resolution is among the strongest resolution calculi currently known [1,2], yet the following class of QBFs provides an exponential lower bound on the size of LQU + -resolution proofs. Definition 16 ([2] ). For n > 1, let Φ Q (n) := Π Q (n).ψ Q (n) be the QUParity QBFs with Π Q (n) := ∃x 1 , . . . , x n ∀z 1 , z 2 ∃t 2 , . . . , t n and ψ Q (n) := C 0 ∧ C 1 ∧ n i:=2 C(i) where C 0 := (z 1 ∨ z 2 ∨ t n ), C 1 := (z 1 ∨z 2 ∨t n ), and C(i) := 7 j:=0 C i,j : C 2,0 := (x 1 ∨x 2 ∨ z 1 ∨ z 2 ∨t 2 ) C 2,1 := (x 1 ∨ x 2 ∨ z 1 ∨ z 2 ∨t 2 ) C 2,2 := (x 1 ∨ x 2 ∨ z 1 ∨ z 2 ∨ t 2 ) C 2,3 := (x 1 ∨x 2 ∨ z 1 ∨ z 2 ∨ t 2 ) C 2,4 := (x 1 ∨x 2 ∨z 1 ∨z 2 ∨t 2 ) C 2,5 := (x 1 ∨ x 2 ∨z 1 ∨z 2 ∨t 2 ) C 2,6 := (x 1 ∨ x 2 ∨z 1 ∨z 2 ∨ t 2 ) C 2,7 := (x 1 ∨x 2 ∨z 1 ∨z 2 ∨ t 2 ) C i,0 := (t i−1 ∨x i ∨ z 1 ∨ z 2 ∨t i ) C i,1 := (t i−1 ∨ x i ∨ z 1 ∨ z 2 ∨t i ) C i,2 := (t i−1 ∨ x i ∨ z 1 ∨ z 2 ∨ t i ) C i,3 := (t i−1 ∨x i ∨ z 1 ∨ z 2 ∨ t i ) C i,4 := (t i−1 ∨x i ∨z 1 ∨z 2 ∨t i ) C i,5 := (t i−1 ∨ x i ∨z 1 ∨z 2 ∨t i ) C i,6 := (t i−1 ∨ x i ∨z 1 ∨z 2 ∨ t i ) C i,7 := (t i−1 ∨x i ∨z 1 ∨z 2 ∨ t i ) Any refutation of Φ Q (n) in LQU + -resolution is exponential in n [2]. The QUParity formulas are a modification of the related LQParity formulas [2]. An LQParity formula is obtained from a QUParity formula Φ Q (n) by replacing ∀z 1 , z 2 in prefix Π Q (n) by ∀z and by replacing every occurrence of the literal pairs z 1 ∨z 2 andz 1 ∨z 2 in the clauses in ψ Q (n) by the literal z andz, respectively. Proposition 6. QRATU can eliminate either variable z 1 or z 2 from a QUParity formula Φ Q (n) to obtain a related LQParity formula in polynomial time. Proof. We eliminate z 2 (z 1 can be eliminated alternatively) in a polynomial number of QRATU steps. Every clause C with z 2 ∈ C has QRAT on z 2 since {z 1 ,z 1 } ⊆ OR for all outer resolvents OR. UP immediately detects a conflict when propagating OR. After eliminating all literals z 2 , the clauses containingz 2 trivially have QRAT onz 2 , which can be eliminated. Finally, z 1 including all of its occurrences is renamed to z. ⊓ ⊔ In the proof above the universal literals can be eliminated by QRATU in any order. Hence in this case the non-confluence [9,14] of rewrite rules in the QRAT and QRAT + systems is not an issue. LQU + -resolution has polynomial proofs for LQParity formulas [2]. Hence the combination of QRATU and LQU +resolution results in a calculus that is more powerful than LQU + -resolution. A related result [13] was obtained for the combination of QRATU and the weaker QU-resolution calculus [21]. Experiments We implemented QRAT + redundancy removal in a tool called QRATPre + for QBF preprocessing. 2 It applies rules QRATE + and QRATU + to remove redundant clauses and universal literals. We did not implement clause addition (QRATA + ) or extended universal reduction (EUR). QRATPre + is the first implementation of QRAT + and QRAT for QBF preprocessing. The preprocessors HQSpre [23] and Bloqqer [10] (which generates partial QRAT proofs to trace preprocessing steps) do not apply QRAT to eliminate redundancies. The following experiments were run on a cluster of Intel Xeon CPUs (E5-2650v4, 2.20 GHz) running Ubuntu 16.04.1. We used the benchmarks from the PCNF track of the QBFEVAL'17 competition. In terms of scheduling the non-confluent (cf. [9,14]) rewrite rules QRATE + and QRATU + , we have not yet optimized QRATPre + . Moreover, in general large numbers of clauses in formulas may cause run time overhead. In this respect, our implementation leaves room for improvements. We illustrate the impact of QBF preprocessing by QRAT + and QRAT on the performance of QBF solving. To this end, we applied QRATPre + in addition to the state of the art QBF preprocessors Bloqqer and HQSpre. In the experiments, first we preprocessed the benchmarks using Bloqqer and HQSpre, respectively, with a generous limit of two hours wall clock time. We considered 39 and 42 formulas where Bloqqer and HQSpre timed out, respectively, in their original form. Then we applied QRATPre + to the preprocessed formulas with a soft wall clock time limit of 600 seconds. When QRATPre + reaches the limit, it prints the formula with redundancies removed that have been detected so far. These preprocessed formulas are then solved. Table 1 shows the performance of our solver DepQBF [17] in addition to the top-performing solvers 3 RAReQS [11], CAQE [19], and Qute [18] from QBFEVAL'17, using limits of 7 GB and 1800 seconds wall clock time. The solvers implement different solving paradigms such as expansion or resolution-based QCDCL. The results clearly indicate the benefits of preprocessing by QRATPre + . The number of solved instances increases. Qute is an exception to this trend. We conjecture that QRATPre + blurs the formula structure in addition to Bloqqer and HQSpre, which may be harmful to Qute. We emphasize that we hardly observed a difference in the effectiveness of redundancy removal by QRAT + and QRAT on the considered benchmarks. The benefits of QRATPre + shown in Table 1 are due to redundancy removal by QRAT already, and not by QRAT + . However, on additional 672 instances from class Gent-Rowley (encodings of the Connect Four game) available from QBFLIB, QRATE + on average removed 54% more clauses than QRATE. We attribute this effect to larger numbers of quantifier blocks in the Gent-Rowley instances (median 73, average 79) compared to QBFEVAL'17 (median 3, average 27). The advantage of QBF abstractions in the QRAT + system is more pronounced on instances with many quantifier blocks. Conclusion We presented QRAT + , a generalization of the QRAT proof system, that is based on a more powerful QBF redundancy property. The key difference between the two systems is the use of QBF specific unit propagation in contrast to propositional unit propagation. Due to this, redundancy checking in QRAT + is aware of quantifier structures in QBFs, as opposed to QRAT. Propagation in QRAT + potentially benefits from the presence of universal variables in the underlying formula. This is exploited by the use of abstractions of QBFs, for which we developed a theoretical framework, and from which the soundness of QRAT + follows. By concrete classes of QBFs we demonstrated that QRAT + is more powerful than QRAT in terms of redundancy detection. Additionally, we reported on proof theoretical improvements of a certain resolution based QBF calculus made by QRAT (or QRAT + ) redundancy removal. A first experimental evaluation illustrated the potential of redundancy elimination by QRAT + . As future work, we plan to implement a workflow for checking QRAT + proofs and extracting Skolem functions similar to QRAT proofs [10]. In our QRAT + preprocessor QRATPre + we currently do not apply a specific strategy to handle the non-confluence of rewrite rules. We want to further analyze the effects of non-confluence as it may have an impact on the amount of redundancy detected. In our tool QRATPre + we considered only redundancy removal. However, to get closer to the the full power of the QRAT + system, it may be beneficial to also add redundant clauses or universal literals to a formula. A Appendix A.1 Example: Formula Φ L (1) The following example shows formula Φ L (1) from the class Φ L (n), which illustrates that QRATU + is more powerful than QRATU. Example 7 (related to Definition 15 on page 12). For n := 1, we have Φ L (n) with prefix Π L (n) := ∀u 1 , u 2 ∃x 1 , x 2 ∀u 3 ∃x 3 and CNF ψ L (n) := C(0) with C(0) := 7 j:=0 C 0,j as follows. C 0,0 := (¬u 2 ∨ ¬x 1 ∨ ¬x 2 ) C 0,1 := (¬u 1 ∨ ¬x 1 ∨ x 2 ) C 0,2 := (u 1 ∨ x 1 ∨ ¬x 2 ) C 0,3 := (u 2 ∨ x 1 ∨ x 2 ) C 0,4 := (¬x 1 ∨ ¬x 2 ∨ x 3 ) C 0,5 := (u 3 ∨ ¬x 3 ) C 0,6 := (¬x 1 ∨ x 2 ∨ ¬x 3 ) C 0,7 := (¬u 3 ∨ x 3 ) A.2 Proofs Proof of Lemma 5 on page 10: Lemma 5 Given a clause C with QIOR + with respect to QBF φ := Π.ψ on literal l ∈ C with var(l) = x. Let T be a model of φ and τ ⊆ T be a path in T . If τ (C \ {l}) = ⊥ then τ x (D) = ⊤ for all D ∈ ψ withl ∈ D. Proof (similar to proof of Lemma 6 in [10]). Let D ∈ ψ be a clause withl ∈ D and consider R := OR(Π, C, D, l) = (C \ {l}) ∪ OC(Π, D,l). By Definition 12, we have Π.ψ ≡ t Π.(ψ∧OR(Π, C, D, l)) for all D ∈ ψ withl ∈ D. Let T be a model of Π.ψ and τ ⊆ T a path in T . Since T \|= t Π.ψ and T \|= t Π.(ψ ∧ OR(Π, C, D, l)), we have τ (ψ) = ⊤ and τ (R) = ⊤. Assuming that τ (C \ {l}) = ⊥, we have τ (OC(Π, D,l)) = ⊤ since τ (R) = ⊤. The clause OC(Π, D,l) does not contain l and it contains only literals of variables from blocks smaller than or equal to the block containing x. Hence we have τ x (OC(Π, D,l)) = ⊤ for the prefix assignment τ x , and further τ x (D) = ⊤ since OC(Π, D,l) ⊆ D. ⊓ ⊔ Proof of Theorem 3 on page 10: Theorem 3 Given a QBF φ := Π.ψ and a clause C ∈ ψ with QIOR + on an existential literal l ∈ C with respect to QBF φ ′ := Π ′ .ψ ′ where ψ ′ := ψ \ {C} and Π ′ is the same as Π with variables and respective quantifiers removed that no longer appear in ψ ′ . Then φ ≡ sat φ ′ . Proof (similar to proof of Theorem 7 in [10]). We can adapt the prefix Π ′ of φ ′ to be the same as the prefix of φ in a satisfiability-preserving way. If φ is satisfiable then φ ′ is also satisfiable since every model of φ is also a model of φ ′ . Let T ′ be a model of φ ′ and T P := T ′ a pre-model of φ. Consider paths τ ⊆ T P in T P for which we have τ (ψ ′ ) = ⊤ but τ (C) = ⊥, where τ = τ xl τ x for var(l) = x and l ∈ C. Since τ (C) = ⊥ also τ (C \ {l}) = ⊥, and due to Lemma 5 we have τ x (D) = ⊤ for all D ∈ ψ withl ∈ D. We construct a premodel T of φ from T P by modifying all such paths τ ⊆ T P by flipping the assignment of x to obtain τ ′ := τ x lτ x such that τ ′ ⊆ T . (If we process multiple redundant clauses C, then cyclic modifications by assignment flipping cannot occur if we do the modifications in reverse chronological ordering as the clauses were detected redundant. This is the same principle of reconstructing solutions when using blocked clause elimination, for example.) Now τ ′ (C) = ⊤ and also τ ′ (D) = ⊤ since τ x (D) = ⊤, and τ and τ ′ have the same prefix assignment τ x . Hence T \|= t φ and thus φ is satisfiable. Proof (similar to proof of Theorem 8 in [10]). If φ ′ is satisfiable then φ is also satisfiable since every model of φ ′ is also a model of φ. Let T be a model of φ and T P := T be a pre-model of φ ′ . Consider paths τ ⊆ T P in T P for which we have τ (ψ) = ⊤ and τ (C) = ⊤ but τ (C ′ ) = ⊥. Since C ′ = C \ {l}, we have τ = τ x lτ x for var(l) = x and l ∈ C. Since l is universal, for every such τ there exists a path τ ′ ⊆ T P with τ ′ = τ xl ρ x , with τ x and ρ x being different suffix assignments of τ and τ ′ , respectively. We have τ ′ (ψ) = ⊤ and τ ′ (C) = ⊤ since τ ′ ⊆ T because T P = T , and also τ ′ (C ′ ) = ⊤ because l ∈ C butl ∈ τ ′ . Hence C ′ is satisfied by τ ′ due to some assignment k ∈ ρ x . Due to τ (C ′ ) = τ (C \ {l}) = ⊥ and Lemma 5 we have τ x (D) = ⊤ for all D ∈ ψ withl ∈ D and hence also τ ′ (D) = ⊤ since τ and τ ′ have the same prefix assignment τ x . We construct a pre-model T ′ of φ ′ from T P by modifying all paths τ = τ x lτ x for which τ (C ′ ) = ⊥ to be τ ′′ := τ x lρ x , where ρ x is the suffix assignment of path τ ′ = τ xl ρ x that corresponds to the other branchl of the universal literal l. These modifications in fact are a replacement of the subtree under τ x l. (As noted in the proof of Theorem 3 above, cyclic modifications cannot occur if we process multiple redundant clauses C, provided that we do the modifications in reverse chronological ordering as the clauses were detected redundant.) We have τ ′′ (C ′ ) = ⊤ due to its suffix assignment ρ x , and also τ ′′ (ψ) = ⊤. Therefore, T ′ \|= t φ ′ and hence φ ′ is satisfiable. ⊓ ⊔ Extended Proof Sketch of Proposition 5 on page 12: Proposition 5 For n ≥ 1, QRATU + can eliminate the entire quantifier block ∀B 1 in Φ L (n) whereas QRATU cannot eliminate any universal literals in Φ L (n). Proof (sketch). Formulas Φ L (n) are constructed based on a similar principle as Φ C (n) in Definition 14. For i and k with i = k, the sets of variables in C(i) and C(k) are disjoint. Thus it suffices to prove the claim for an arbitrary C(i). Clause C i,0 has QRAT + on literal ¬u 3i+2 . The relevant outer resolvent is OR 0,3 = OR(Π L (n), C i,0 , C i,3 , ¬u 3i+2 ) = (¬x 3i+1 ∨ ¬x 3i+2 ). We have max(levels(OR 0,3 )) = 2, and variable u 3i+3 is universal in Abs(Φ L (n), 2). Propagating OR 0,3 makes C i,4 unit, finally C i,5 becomes empty under the derived assignment x 3i+3 and since UR reduces the literal u 3i+3 . After removing literal ¬u 3i+2 from C i,0 , clause C i,3 trivially has QRAT + on u 3i+2 . The literals of variable u 3i+1 in C i,1 and C i,2 can be eliminated in a similar way, where clause C i,7 becomes empty by UR in QUP. In contrast to QRATU + , QRATU cannot eliminate any universal literals in Φ L (n). Clauses C i,5 and C i,7 in Φ L (n) do not become empty. All variables are existential since UP is applied. The claim can be proved by case analysis of all possible outer resolvents. ⊓ ⊔ Definition 2 ( 2[10]). Let C be a clause with l ∈ C and D be a clause withl ∈ D occurring in QBF Π.ψ. The outer resolvent of C with D on l with respect to Π is the clause OR(Π, C, D, l) := (C \ {l}) ∪ OC(Π, D,l). Example 1 . 1Given φ := ∃x 1 ∀u∃x 2 .(C ∧ D) with C := (x 1 ∨ u ∨ x 2 ) and D := (x 1 ∨ū∨x 2 ), we have OR(Π, C, D, x 1 ) = (u∨x 2 ), OR(Π, C, D, u) = (x 1 ∨x 1 ∨x 2 ), OR(Π, C, D, x 2 ) = (x 1 ∨ u ∨x 1 ∨ū), and OR(Π, D, C,ū) = (x 1 ∨x 1 ∨x 2 ). Computing outer resolvents is asymmetric since OR(Π, C, D, u) = OR(Π, D, C,ū). Theorem 2 ( 2[10]). Given a QBF φ 0 := Π.ψ and φ := Π. Theorem 4 . 4Given a QBF φ 0 := Π.ψ and φ := Π.(ψ ∪ {C}) where C has QIOR + on a universal literal l ∈ C with respect to φ 0 . Let φ ′ := Π.(ψ ∪ {C ′ }) with C ′ := C \ {l}. Then φ ≡ sat φ ′ . Given a QBF φ 0 := Π.ψ and φ := Π.(ψ ∪ {C}) where C has QIOR + on a universal literal l ∈ C with respect to φ 0 . Let φ ′ := Π.(ψ ∪ {C ′ }) with C ′ := C \ {l}. Then φ ≡ sat φ ′ . Table 1 . 1Solved instances (S ), solved unsatisfiable (⊥) and satisfiable ones (⊤), and total wall clock time in kiloseconds (K) including time outs on instances from QBFE-VAL'17. Different combinations of preprocessing by Bloqqer, HQSpre, and our tool QRATPre + . Prepr. by HQSpre and QRATPre + .(a) Original instances (no prepr.). Solver S ⊥ ⊤ Time CAQE 170 128 42 656K RAReQS 167 133 34 660K DepQBF 152 108 44 690K Qute 130 91 39 720K (b) Prepr. by QRATPre + only. Solver S ⊥ ⊤ Time CAQE 209 141 68 594K RAReQS 203 152 51 599K DepQBF 157 109 48 689K Qute 131 98 33 724K (c) Prepr. by Bloqqer only. Solver S ⊥ ⊤ Time RAReQS 256 180 76 508K CAQE 251 168 83 522K DepQBF 187 121 66 630K Qute 154 109 45 682K (d) Prepr. by Bloqqer and QRATPre + . Solver S ⊥ ⊤ Time RAReQS 262 178 84 492K CAQE 255 172 83 507K DepQBF 193 127 66 622K Qute 148 107 41 688K (e) Prepr. by HQSpre only. Solver S ⊥ ⊤ Time CAQE 306 197 109 415K RAReQS 294 194 100 429K DepQBF 260 171 89 494K Qute 255 171 84 497K (f) Solver S ⊥ ⊤ Time CAQE 314 200 114 407K RAReQS 300 195 105 418K DepQBF 262 177 85 488K Qute 250 169 81 500K In general, clauses C are always (implicitly) interpreted under a quantifier prefix Π. Source code of QRATPre + : https://github.com/lonsing/qratpreplus We excluded the top-performing solver AIGSolve due to observed assertion failures. V Balabanov, M Widl, J R Jiang, QBF Resolution Systems and Their Proof Complexities. In: SAT. LNCS. Springer8561Balabanov, V., Widl, M., Jiang, J.R.: QBF Resolution Systems and Their Proof Complexities. In: SAT. LNCS, vol. 8561, pp. 154-169. Springer (2014) Proof Complexity of Resolution-based QBF Calculi. O Beyersdorff, L Chew, M Janota, STACS. LIPIcs. 30Beyersdorff, O., Chew, L., Janota, M.: Proof Complexity of Resolution-based QBF Calculi. In: STACS. LIPIcs, vol. 30, pp. 76-89. 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EasyChair (2017) The science of brute force. M Heule, O Kullmann, Commun. ACM. 608Heule, M., Kullmann, O.: The science of brute force. Commun. ACM 60(8), 70-79 (2017) A Unified Proof System for QBF Preprocessing. M Heule, M Seidl, A Biere, IJCAR. LNCS. Springer8562Heule, M., Seidl, M., Biere, A.: A Unified Proof System for QBF Preprocessing. In: IJCAR. LNCS, vol. 8562, pp. 91-106. Springer (2014) Blocked Literals Are Universal. M Heule, M Seidl, A Biere, NASA Formal Methods. LNCS. Springer9058Heule, M., Seidl, M., Biere, A.: Blocked Literals Are Universal. In: NASA Formal Methods. LNCS, vol. 9058, pp. 436-442. Springer (2015) Solution Validation and Extraction for QBF Preprocessing. M Heule, M Seidl, A Biere, J. Autom. Reasoning. 581Heule, M., Seidl, M., Biere, A.: Solution Validation and Extraction for QBF Pre- processing. J. Autom. Reasoning 58(1), 97-125 (2017) Solving QBF with Counterexample Guided Refinement. M Janota, W Klieber, J Marques-Silva, E Clarke, Artif. Intell. 234Janota, M., Klieber, W., Marques-Silva, J., Clarke, E.: Solving QBF with Coun- terexample Guided Refinement. Artif. Intell. 234, 1-25 (2016) Inprocessing Rules. M Järvisalo, M Heule, A Biere, IJCAR. LNCS. Springer7364Järvisalo, M., Heule, M., Biere, A.: Inprocessing Rules. In: IJCAR. LNCS, vol. 7364, pp. 355-370. Springer (2012) A Little Blocked Literal Goes a Long Way. B Kiesl, M Heule, M Seidl, SAT. LNCS. Springer10491Kiesl, B., Heule, M., Seidl, M.: A Little Blocked Literal Goes a Long Way. In: SAT. LNCS, vol. 10491, pp. 281-297. Springer (2017) A Unifying Principle for Clause Elimination in First-Order Logic. B Kiesl, M Suda, CADE. LNCS. 10395SpringerKiesl, B., Suda, M.: A Unifying Principle for Clause Elimination in First-Order Logic. In: CADE. LNCS, vol. 10395, pp. 274-290. Springer (2017) Resolution for Quantified Boolean Formulas. Kleine Büning, H Karpinski, M Flögel, A , Inf. Comput. 1171Kleine Büning, H., Karpinski, M., Flögel, A.: Resolution for Quantified Boolean Formulas. Inf. Comput. 117(1), 12-18 (1995) Failed Literal Detection for QBF. F Lonsing, A Biere, SAT. LNCS. Springer6695Lonsing, F., Biere, A.: Failed Literal Detection for QBF. In: SAT. LNCS, vol. 6695, pp. 259-272. Springer (2011) DepQBF 6.0: A Search-Based QBF Solver Beyond Traditional QCDCL. F Lonsing, U Egly, CADE. LNCSSpringer10395Lonsing, F., Egly, U.: DepQBF 6.0: A Search-Based QBF Solver Beyond Tradi- tional QCDCL. In: CADE. LNCS, vol. 10395, pp. 371-384. Springer (2017) Dependency Learning for QBF. T Peitl, F Slivovsky, S Szeider, SAT. LNCS. Springer10491Peitl, T., Slivovsky, F., Szeider, S.: Dependency Learning for QBF. In: SAT. LNCS, vol. 10491, pp. 298-313. Springer (2017) CAQE: A Certifying QBF Solver. M N Rabe, L Tentrup, IEEERabe, M.N., Tentrup, L.: CAQE: A Certifying QBF Solver. In: FMCAD. pp. 136- 143. IEEE (2015) Preprocessing QBF. H Samulowitz, J Davies, F Bacchus, CP. LNCS. 4204SpringerSamulowitz, H., Davies, J., Bacchus, F.: Preprocessing QBF. In: CP. LNCS, vol. 4204, pp. 514-529. Springer (2006) Contributions to the Theory of Practical Quantified Boolean Formula Solving. A Van Gelder, CP. LNCS. 7514SpringerVan Gelder, A.: Contributions to the Theory of Practical Quantified Boolean For- mula Solving. In: CP. LNCS, vol. 7514, pp. 647-663. Springer (2012) DRAT-trim: Efficient Checking and Trimming Using Expressive Clausal Proofs. N Wetzler, M Heule, W A J Hunt, SAT. LNCS. Springer8561Wetzler, N., Heule, M., Hunt, W.A.J.: DRAT-trim: Efficient Checking and Trim- ming Using Expressive Clausal Proofs. In: SAT. LNCS, vol. 8561, pp. 422-429. Springer (2014) HQSpre -An Effective Preprocessor for QBF and DQBF. R Wimmer, S Reimer, P Marin, B Becker, TACAS. LNCS. Springer10205Wimmer, R., Reimer, S., Marin, P., Becker, B.: HQSpre -An Effective Preprocessor for QBF and DQBF. In: TACAS. LNCS, vol. 10205, pp. 373-390. 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[ "An Information-Spectrum Approach to Multiterminal Rate-Distortion Theory An Information-Spectrum Approach to Multiterminal Rate-Distortion Theory", "An Information-Spectrum Approach to Multiterminal Rate-Distortion Theory An Information-Spectrum Approach to Multiterminal Rate-Distortion Theory" ]	[ "Shengtian Yang [email protected] \nDepartment of Information Science & Electronic Engineering\nZhejiang University\n310027HangzhouChina\n", "Peiliang Qiu \nDepartment of Information Science & Electronic Engineering\nZhejiang University\n310027HangzhouChina\n", "S Yang \nDepartment of Information Science & Electronic Engineering\nZhejiang University\n310027HangzhouChina\n", "P Qiu \nDepartment of Information Science & Electronic Engineering\nZhejiang University\n310027HangzhouChina\n" ]	[ "Department of Information Science & Electronic Engineering\nZhejiang University\n310027HangzhouChina", "Department of Information Science & Electronic Engineering\nZhejiang University\n310027HangzhouChina", "Department of Information Science & Electronic Engineering\nZhejiang University\n310027HangzhouChina", "Department of Information Science & Electronic Engineering\nZhejiang University\n310027HangzhouChina" ]	[ "TO IEEE TRANSACTIONS ON INFORMATION THEORY FOR PEER REVIEW" ]	An information-spectrum approach is applied to solve the multiterminal source coding problem for correlated general sources, where sources may be nonstationary and/or nonergodic, and the distortion measure is arbitrary and may be nonadditive. A general formula for the rate-distortion region of the multiterminal source coding problem with the maximum distortion criterion under fixed-length coding is shown in this correspondence.	null	[ "https://arxiv.org/pdf/cs/0605006v1.pdf" ]	58,254	cs/0605006	ed47a61220c47018c496833114de8062dac596da	An Information-Spectrum Approach to Multiterminal Rate-Distortion Theory An Information-Spectrum Approach to Multiterminal Rate-Distortion Theory APRIL 28. 2006 Shengtian Yang [email protected] Department of Information Science & Electronic Engineering Zhejiang University 310027HangzhouChina Peiliang Qiu Department of Information Science & Electronic Engineering Zhejiang University 310027HangzhouChina S Yang Department of Information Science & Electronic Engineering Zhejiang University 310027HangzhouChina P Qiu Department of Information Science & Electronic Engineering Zhejiang University 310027HangzhouChina An Information-Spectrum Approach to Multiterminal Rate-Distortion Theory An Information-Spectrum Approach to Multiterminal Rate-Distortion Theory TO IEEE TRANSACTIONS ON INFORMATION THEORY FOR PEER REVIEW 1APRIL 28. 2006arXiv:cs/0605006v1 [cs.IT] SUBMITTEDIndex Terms Correlated general sourcesinformation spectrummultiterminal source codingrate-distortion regionside information An information-spectrum approach is applied to solve the multiterminal source coding problem for correlated general sources, where sources may be nonstationary and/or nonergodic, and the distortion measure is arbitrary and may be nonadditive. A general formula for the rate-distortion region of the multiterminal source coding problem with the maximum distortion criterion under fixed-length coding is shown in this correspondence. I. INTRODUCTION In this correspondence, we study the classic problem in multiterminal rate-distortion theory, i.e., multiterminal source coding problem. In this problem, M (M ≥ 2) correlated general sources have to be compressed separately from each other in a lossy fashion, i.e., with respect to a fidelity criterion, and then decoded by the common decoder which has access to a side information source that is correlated with the sources to be compressed. This situation is illustrated in Fig. 1, and it is also called distributed source coding. The well-known Slepian-Wolf coding problem and the Wyner-Ziv coding problem can be regarded as two special cases of this situation. These two special cases were solved in 1970's for stationary memoryless sources [1], [2], and later extended to the case of general sources [3], [4]. However, for this general problem, no conclusive results are available to date. Even for the special case that the sources are memoryless and stationary and the distortion measure is additive, only inner and outer bounds are derived in [5], [6], etc. In this correspondence, we adopt an information-spectrum approach to solve this open problem for general sources under maximum distortion criterions. We obtain the rate-distortion region for correlated general sources, which is the main contribution of this correspondence. The rest of this correspondence is organized as follows. In Section II, we first briefly introduce required notations and definitions in information-spectrum methods [7], and then formally state the multiterminal source coding problem. In Section III, the main theorem concerning the rate-distortion region is presented and discussed. All the proofs are finally given in Section IV. II. NOTATIONS AND DEFINITIONS A general source X with alphabet X is characterize by an infinite sequence {X n = (X (n) 1 , X (n) 2 , · · · , X (n) n )} ∞ n=1 of n-dimensional random variables X n taking values in the n-th Cartesian product X n , and in this correspondence, all the alphabets are assumed to be finite. Specifically, for M (M ≥ 2) correlated general sources, each general source X m (1 ≤ m ≤ M ) with alphabet X m is an infinite sequence denoted by {X n m = (X (n) m,1 , X (n) m,2 , · · · , X (n) m,n )} ∞ n=1 , and the whole group of correlated general sources is denoted by (X m ) m∈IM , where I M denotes the set {1, 2, · · · , M }. Analogously, any part of (X m ) m∈IM is denoted by (X m ) m∈A , where A ⊆ I M . In most situations, A is also assumed to be an ordered set, hence (X m ) m∈A is virtually a vector. Similar notations apply to any related quantities or functions, e.g., (X n m ) m∈A and (f (m) (X n m )) m∈A . For a sequence of real random variables {Z n } ∞ n=1 , the limit superior in probability p-lim sup n→∞ Z n and limit inferior in probability p-lim inf n→∞ Z n are defined by respectively. Then for general sources X, Y ,X andŶ , the spectral sup-entropy rate H(X), the spectral conditional supentropy rate H(X\|Y ), the spectral sup-mutual information rate I(X; Y ), the spectral inf-mutual information rate I(X; Y ) and the spectral inf-divergence rate of X conditioned on Y with respect toX conditioned onŶ are defined respectively by H(X) △ = p-lim sup n→∞ 1 n ln 1 P X n (X n ) ,(1)H(X\|Y ) △ = p-lim sup n→∞ 1 n ln 1 P X n \|Y n (X n \|Y n ) ,(2)I(X; Y ) △ = p-lim sup n→∞ 1 n ln P X n Y n (X n , Y n ) P X n (X n )P Y n (Y n ) ,(3)I(X; Y ) △ = p-lim inf n→∞ 1 n ln P X n Y n (X n , Y n ) P X n (X n )P Y n (Y n ) ,(4)D(X\|Y X \|Ŷ ) △ = p-lim inf n→∞ 1 n ln P X n \|Y n (X n \|Y n ) PX n \|Ŷ n (X n \|Y n ) .(5) In this correspondence, all the logarithms are calculated to the natural base e. Also note that in these definitions, the general source can be replaced by a sequence of random variables with arbitrary alphabets, and we denote such a sequence by Z = {Z n } ∞ n=1 with alphabets Z = {Z n } ∞ n=1 , where Z n takes values in Z n . We also call it a general source. If a general source is a process, that is, it satisfies the consistency condition, the notation {X n } ∞ n=1 is then replaced by the usual notation {X i } ∞ i=1 for a process and X n = (X 1 , X 2 , · · · , X n ). If the processes (X, Y ) are stationary and ergodic, the quantities H(X), H(X\|Y ), I(X; Y ) and I(X; Y ) are then given by H(X) = lim n→∞ 1 n H(X n ),H(X\|Y ) = lim n→∞ 1 n H(X n \|Y n ), I(X; Y ) = I(X; Y ) = lim n→∞ 1 n I(X n ; Y n ), where H(X n ) △ = E ln 1 P X n (X n ) ,(6)H(X n \|Y n ) △ = E ln 1 P X n \|Y n (X n \|Y n ) ,(7)I(X n ; Y n ) △ = H(X n ) − H(X n \|Y n ).(8) Moreover, if the stationary ergodic sources (X, Y ) are memoryless, we denote them by (X, Y ) for convenience, and H(X) = H(X), H(X\|Y ) = H(X\|Y ), I(X; Y ) = I(X; Y ) = I(X; Y ) due to the memoryless property. The multiterminal source coding system can be stated as follows. Given correlated general sources (X m ) m∈IM and the side information source S, the n-length source outputs (X n m ) m∈IM are separately encoded into a group of fixed-length codewords (φ (m) n (X n m )) m∈IM . Then the common decoder observes these codewords (φ (m) n (X n m )) m∈IM and the side information S n to reproduce the estimates (Y n m ) m∈IM = (ψ (m) n (S n , (φ (m) n (X n m )) m∈IM )) m∈IM of X n m . Accordingly the reproduced sequences form a group of general sources, namely, (Y m ) m∈IM with reproduction alphabets (Y m ) m∈IM . Here, the encoder (φ (m) n ) m∈IM is defined by φ (m) n : X n m → I L (m) n △ = {1, 2, · · · , L (m) n } and we denote the sequence {φ (m) n } ∞ n=1 and {φ n = (φ (m) n ) m∈IM } ∞ n=1 by φ (m) and φ respectively. The rate of each encoder φ (m) n is defined by R(φ (m) n ) △ = ln \|φ (m) n (X n m )\| n , where \|A\| denotes the cardinality of the set A. The decoder (ψ (m) n ) m∈IM is defined by ψ (m) n : S n × m ′ ∈IM I L (m ′ ) n → Y n m . and we denote the sequence {ψ n ) m∈IM } ∞ n=1 by ψ (m) and ψ respectively. Next, let us define a general distortion measure. A general distortion measure (d (k) ) k∈IK is a group of sequences d (k) = {d (k) n } ∞ n=1 of (measurable) functions d (k) n defined by d (k) n : m∈IM X n m × m∈IM Y n m → [0, +∞), where K is a positive constant integer and I K △ = {1, 2, · · · , K}. Then the fixed-length coding problem under the maximum distortion criterion is formulated as follows. A rate-distortion tuple ((R m ) m∈IM , (D k ) k∈IK ) is f m-achievable if and only if there exists a sequence (φ, ψ) of fixed-length codes such that (lim sup n→∞ R(φ (m) n )) m∈IM ≤ (R m ) m∈IM , (p-lim sup n→∞ d (k) n ((X n m ) m∈IM , ψ n (S n , (φ (m) n (X n m )) m∈IM ))) k∈IK ≤ (D k ) k∈IK . Here, for any (x m ) m∈IM , (y m ) m∈IM ∈ R M , we say that (x m ) m∈IM ≤ (y m ) m∈IM if and only if x m ≤ y m for all m ∈ I M . Accordingly, the f m-rate-distortion region and f m-distortion-rate region are defined respectively by R f m ((D k ) k∈IK \|(X m ) m∈IM , S) △ = {(R m ) m∈IM \|((R m ) m∈IM , (D k ) k∈IK ) is f m-achievable} and D f m ((R m ) m∈IM \|(X m ) m∈IM , S) △ = {(D k ) k∈IK \|((R m ) m∈IM , (D k ) k∈IK ) is f m-achievable}. The fixed-length coding problem under the average distortion criterion can be defined analogously. A rate-distortion tuple ((R m ) m∈IM , (D k ) k∈IK ) is f a-achievable if and only if there exists a sequence (φ, ψ) of fixed-length codes such that (lim sup n→∞ R(φ (m) n )) m∈IM ≤ (R m ) m∈IM , (lim sup n→∞ E[d (k) n ((X n m ) m∈IM , ψ n (S n , (φ (m) n (X n m )) m∈IM ))]) k∈IK ≤ (D k ) k∈IK . It can be easily shown that f m-achievability always implies f a-achievability for bounded distortion measures, and that f aachievability implies f m-achievability for those bounded distortion measures satisfying (25) defined in Section III when the sources are stationary and memoryless. III. MAIN RESULTS In this section, we investigate the sufficient and necessary condition on the f m-achievability of rate-distortion tuples, thus determining the f m-rate-distortion region and f m-distortion-rate region of multiterminal source coding for correlated general sources. The main result is stated in the following theorem and the proof is presented in Section IV. Theorem 1: For correlated general sources (X m ) m∈IM , a side information source S and distortion measures (d (k) ) k∈IK , the rate-distortion tuple ((R m ) m∈IM , (D k ) k∈IK ) is f m-achievable if and only if there exist general sources (Z (m) ) m∈IM with alphabet m∈IM Z (m) = { m∈IM Z (m) n } ∞ n=1 and a sequence h = {h n } ∞ n=1 of functions h n = (h (m) n ) m∈IM defined by h (m) n : S n × m ′ ∈IM Z (m ′ ) n → Y n m such that P (X n m )m∈I M S n (Z (m) n )m∈I M = P (X n m )m∈I M S n m∈IM P Z (m) n \|X n m , ∀n ≥ 1 (9) (p-lim sup n→∞ d (k) n ((X n m ) m∈IM , h n (S n , (Z (m) n ) m∈IM ))) k∈IK ≤ (D k ) k∈IK ,(10) and m∈A R m ≥ m∈A I(X m ; Z (m) ) − I \|A\| ((Z (m) ) m∈A ) − I((Z (m) ) m∈A ; S, (Z (m) ) m∈IM \A )(11) for any nonempty set A ⊆ I M , where I \|A\| ((Z (m) ) m∈A ) △ = p-lim inf n→∞ 1 n ln P (Z (m) n )m∈A ((Z (m) n ) m∈A ) m∈A P Z (m) n (Z (m) n ) .(12) Remark 1: In Theorem 1, we introduce a new quantity I \|A\| ((Z (m) ) m∈A ), which may be regarded as a generalized version of the spectral inf-mutual information rate. Also note that I \|A\| ((Z (m) ) m∈A ) = 0 when \|A\| = 1. Remark 2: The condition (9) may be loosened to the following one α n P (X n m )m∈I M S n m∈IM P Z (m) n \|X n m ≤ P (X n m )m∈I M S n (Z (m) n )m∈I M ≤ β n P (X n m )m∈I M S n m∈IM P Z (m) n \|X n m(13) with lim n→∞ α n = lim n→∞ β n = 1 (see Lemma 2 in Section IV). By Theorem 1, the f m-rate-distortion region and f m-distortion-rate region are determined as follows. Corollary 1: The f m-rate-distortion region for a constant distortion tuple (D k ) k∈K is R f m ((D k ) k∈IK \|(X m ) m∈IM , S) = (Z (m) )m∈I M R (Z (m) )m∈I M ((D k ) k∈IK ),(14) where R (Z (m) )m∈I M ((D k ) k∈IK ) △ = (R m ) m∈IM m∈A R m ≥ m∈A I(X m ; Z (m) ) − I \|A\| ((Z (m) ) m∈A ) − I((Z (m) ) m∈A ; S, (Z (m) ) m∈IM \A ), for any nonempty set A ⊆ I M , and (Z (m) ) m∈IM denotes all general sources that satisfy the conditions (9) and (10) with some sequence h of functions h n . The f m-distortion-rate region for a constant rate tuple (R m ) m∈IM is D f m ((R m ) m∈IM \|(X m ) m∈IM , S) = (Z (m) )m∈I M D (Z (m) )m∈I M ((R m ) m∈IM ),(15) where D (Z (m) )m∈I M ((R m ) m∈IM ) △ = (D k ) k∈IK (D k ) k∈IK ≥ (p-lim sup n→∞ d (k) n ((X n m ) m∈IM , h n (S n , (Z (m) n ) m∈IM ))) k∈IK for some sequence h of functions h n , and (Z (m) ) m∈IM denotes all general sources that satisfy the conditions (9) and (11). To have an insight into Theorem 1, we consider some special cases. First consider the case of one terminal with side information at the decoder, we then get the rate-distortion function of the Wyner-Ziv problem for general sources. h n : S n × Z n → Y n , such that P X n S n Zn = P X n S n P Zn\|X n , ∀n ≥ 1 (16) (p-lim sup n→∞ d (k) n (X n , h n (S n , Z n ))) k∈IK ≤ (D k ) k∈IK ,(17)R ≥ I(X; Z) − I(Z; S).(18) and hence the infimum of the achievable rate for a constant distortion tuple (D k ) k∈IK is given by inf{I(X; Z) − I(Z; S)},(19) where inf is over all Z and {h n } ∞ n=1 satisfying properties (16) and (17). Second, let us consider the case of two terminals without side information at the decoder. Then Theorem 1 is reduced to the following form. Corollary 3: For correlated general sources (X 1 , X 2 ) and distortion measures (d (k) ) k∈IK , the rate-distortion tuple (R 1 , R 2 , (D k ) k∈IK ) is f m-achievable if and only if there exist general sources (Z (1) , Z (2) ) with alphabet {Z (1) n × Z (2) n } ∞ n=1 and a sequence h = {h n } ∞ n=1 of functions h n = (h (1) n , h(2)n ) defined by h n : Z (1) n × Z (2) n → Y n 1 × Y n such that P X n 1 X n 2 Z (1) n Z (2) n = P X n 1 X n 2 P Z (1) n \|X n 1 P Z (2) n \|X n 2 ,(20)(p-lim sup n→∞ d (k) n (X n 1 , X n 2 , h n (Z (1) n , Z (2) n ))) k∈IK ≤ (D k ) k∈IK ,(21) and R 1 ≥ I(X 1 ; Z (1) ) − I(Z (1) ; Z (2) ),(22)R 2 ≥ I(X 2 ; Z (2) ) − I(Z (1) ; Z (2) ),(23)R 1 + R 2 ≥ I(X 1 ; Z (1) ) + I(X 2 ; Z (2) ) − I(Z (1) ; Z (2) ).(24) If for each k ∈ I K , the distortion measure d (k) n is additive, that is, it is defined by d (k) n (x 1 , x 2 , y 1 , y 2 ) = 1 n n i=1 d (k) (x 1,i , x 2,i , y 1,i , y 2,i ),(25) for all x 1 ∈ X n 1 , x 2 ∈ X n 2 , y 1 ∈ Y n 1 and y 2 ∈ Y n 2 , where d (k) is a nonnegative (measurable) function called memoryless distortion measure, then for stationary memoryless sources (X 1 , X 2 ), the sufficient condition given by [5] follows from Corollary 3. Corollary 4 ([5, Theorem 6.1]): For correlated stationary memoryless sources (X 1 , X 2 ) and memoryless distortion measure (d (k) ) k∈IK , the rate-distortion tuple (R 1 , R 2 , (D k ) k∈IK ) is f m-achievable (or f a-achievable) if there exist a pair of random variables (Z 1 , Z 2 ) with alphabet Z 1 × Z 2 and a pair of functions h (1) : Z 1 × Z 2 → Y 1 and h (2) : Z 1 × Z 2 → Y 2 such that P X1X2Z1Z2 = P X1X2 P Z1\|X1 P Z2\|X2 ,(26)(E[d (k) (X 1 , X 2 , h (1) (Z 1 , Z 2 ), h (2) (Z 1 , Z 2 ))]) k∈IK ≤ (D k ) k∈IK (27) R 1 ≥ I(X 1 ; Z 1 \|Z 2 ), R 2 ≥ I(X 2 ; Z 2 \|Z 1 ),(28)R 1 + R 2 ≥ I(X 1 , X 2 ; Z 1 , Z 2 ).(29) The proof is easy and hence omitted here. Note that the memoryless distortion measures are bounded since the alphabets are finite, and the sources (X 1 , X 2 , Z 1 , Z 2 ) are jointly stationary and memoryless, which implies the validity of the law of large number or the ergodic theorem. Though we have determined the whole f m-rate-distortion or f m-distortion-rate region for correlated general sources under the maximum distortion criterion, this does not mean that the multiterminal source coding problem has been solved. First, the f m-rate-distortion or f m-distortion-rate region in Corollary 1 are obviously incomputable in general. Second, even for the correlated memoryless sources, the single letter f m-rate-distortion (or f a-rate-distortion) region is still unknown. It seems that the usual treatment for memoryless sources or channels in information-spectrum methods can not be easily applied to this problem, and one of the difficulties is how to reduce the general function h n = (h (m) n ) m∈IM to a memoryless form if we do not use the method in [5]. But in any case, Theorem 1 does provide a very general sufficient condition. and it also gives some hints on the characterization of single letter rate-distortion region. Under the same settings of Corollary 4, we believe that for any f m-achievable rate-distortion tuple, there exits a pair of random variables (Z 1 , Z 2 ) which can be regarded as a mixture of infinite general sources satisfying the condition (9), and most of the general sources (with probability 1) achieve the same rate-distortion tuple. Finally, let us turn our attention to the multiterminal source coding problem for the mixed sources as an application example of Theorem 1. For simplicity, we will consider the settings of Corollary 3, and we assume that the source pair is a mixed source pair. Let (X 1α , X 2α ) and (X 1β , X 2β ) be two correlated general sources, and let us define a mixed source pair (X 1 , X 2 ) by P X n 1 X n 2 (x 1 , x 2 ) = αP X n 1α X n 2α (x 1 , x 2 ) + βP X n 1β X n 2β (x 1 , x 2 ) for x 1 ∈ X n 1 , x 2 ∈ X n 2 , where α > 0, β > 0 are constants such that α + β = 1. Then for this mixed source pair, we have the following corollary derived from Corollary 3. Corollary 5: For given correlated mixed sources (X 1 , X 2 ) with component sources (X 1α , X 2α ) and (X 1β , X 2β ) and distortion measures (d (k) ) k∈IK , the rate-distortion tuple (R 1 , R 2 , (D k ) k∈IK ) is f m-achievable if and only if there exist general sources (Z (1) , Z (2) ) with alphabet {Z (1) n × Z (2) n } ∞ n=1 and a sequence h = {h n } ∞ n=1 of functions h n = (h (1) n , h(2) n ) defined by h n : Z (1) n × Z (2) n → Y n 1 × Y n 2 such that P X n 1 X n 2 Z (1) n Z (2) n = P X n 1 X n 2 P Z (1) n \|X n 1 P Z (2) n \|X n 2 ,(30)(max{p-lim sup n→∞ d (k) n (X n 1α , X n 2α , h n (Z (1) n , Z (2) n )), p-lim sup n→∞ d (k) n (X n 1β , X n 2β , h n (Z (1) n , Z (2) n ))}) k∈IK ≤ (D k ) k∈IK ,(31) and R 1 ≥ max{I(X 1α ; Z (1) ), I(X 1β ; Z (1) )} − I(Z (1) ; Z (2) ),(32)R 2 ≥ max{I(X 2α ; Z (2) ), I(X 2β ; Z (2) )} − I(Z (1) ; Z (2) ),(33) R 1 + R 2 ≥ max{I(X 1α ; Z (1) ), I(X 1β ; Z (1) )} + max{I(X 2α ; Z (2) ), I(X 2β ; Z (2) )} − I(Z (1) ; Z (2) ). The proof is similar to the proof of Corollary 1 in [4] and hence omitted here. Lemma 3 and 4 in [4] are applied repeatedly in the proof. IV. PROOFS OF THEOREMS To prove Theorem 1, we first need to establish a series of lemmas. Lemma 1: Let X = {X n } ∞ n=1 ,X = {X n } ∞ n=1 and Y = {Y n } ∞ n=1 ,Ŷ = {Ŷ n } ∞ n=1 be arbitrary general sources with alphabets X = {X n } ∞ n=1 and Y = {Y n } ∞ n=1 respectively, then we have D(X\|Y X \|Ŷ ) ≥ 0.(35) Proof: For any γ > 0, Pr 1 n ln P Xn\|Yn (X n \|Y n ) PX n \|Ŷn (X n \|Y n ) < −γ = x∈Xn,Pr{ 1 n ln P Xn\|Yn (X n \|Y n ) PX n \|Ŷn (X n \|Y n ) < −γ} = 0 for any γ > 0, and this concludes (35) by the definition (5). Lemma 2 (An enhanced version of Lemma 1 in [4]): Let U = {U n } ∞ n=1 , V = {V n } ∞ n=1 and W = {W n } ∞ n=1 be arbitrary general sources with alphabets U = {U n }, V = {V n } and W = {W n } respectively, and they satisfy P UnVnWn ≥ c n P UnVn P Wn\|Vn with lim n→∞ c n = 1. Now let {B n } ∞ n=1 be a sequence of arbitrary (measurable) sets in U n × W n such that lim inf n→∞ Pr{(U n , W n ) ∈ B n } = ǫ.(37) Then, for any γ > 0, there exits a sequence f = {f n } ∞ n=1 of functions f n : V n → W n such that \|f n (V n )\| ≤ e n(I(V ;W )+γ) , lim inf n→∞ Pr{(U n , f n (V n )) ∈ B n } ≥ ǫ. The proof of Theorem 1 relies heavily on Lemma 1 in [4], but here we choose to present an enhanced version of Lemma 1 in [4], and its proof is greatly simplified by applying Theorem 5.5.1 in [7]. Proof: Let us define a distortion measure d n (v, w) = u∈Un P Un\|VnWn (u\|v, w)1{(u, w) ∈ B n } for all v ∈ V n , w ∈ W n . Clearly, d n is bounded, and we have lim sup n→∞ E[d n (V n , W n )] = lim sup n→∞ v∈Vn,w∈Wn P VnWn (v, w)d n (v, w) = lim sup n→∞ Pr{(U n , W n ) ∈ B n } = 1 − ǫ by the condition (37). Then from the proof of the direct part of Theorem 5.5.1 in [7], it follows that for any γ > 0, there exists a sequence f = {f n } ∞ n=1 of functions f n : V n → W n such that \|f n (V n )\| ≤ e n(I(V ;W )+γ) , lim sup n→∞ E[d n (V n , f (V n ))] ≤ 1 − ǫ. Finally, we have Pr{(U n , f n (V n )) ∈ B n } = 1 − Pr{(U n , f n (V n )) ∈ B n } = 1 − v∈Vn P Vn (v) u∈Un P Un\|Vn (u\|v)1{(u, f n (v)) ∈ B n } (a) ≥ 1 − c −1 n v∈Vn P Vn (v) u∈Un P Un\|VnWn (u\|v, f n (v))1{(u, f n (v)) ∈ B n } = 1 − c −1 n E[d n (V n , f (V n ))], where (a) follows from (36), and hence lim inf n→∞ Pr{(U n , f n (V n )) ∈ B n } ≥ 1 − lim sup n→∞ E[d n (V n , f (V n ))] c n ≥ 1 − lim sup n→∞ E[d n (V n , f (V n ))] lim n→∞ c n ≥ ǫ. The proof is completed. lim n→∞ Pr{((X n m ) m∈IM , S n , (f (m) n (X n m )) m∈IJ , (Z (m) n ) m∈IM \IJ ) ∈ B n } = 1,(39) and I \|A\| ((f (m) (X m )) m∈A∩IJ , (Z (m) ) m∈A\IJ ) ≥ I \|A\| ((Z (m) ) m∈A ) − J M γ 2 ,(40)I((f (m) (X m )) m∈A∩IJ , (Z (m) ) m∈A\IJ ; S, (f (m) (X m )) m∈(IM \A)∩IJ , (Z (m) ) m∈(IM \A)\IJ ) ≥ I((Z (m) ) m∈A ; S, (Z (m) ) m∈IM \A ) − J M γ 2(41) for any nonempty set A ⊆ I M . Clearly, the above conditions hold trivially when J = 0, and the lemma corresponds to the case J = M . So to prove the lemma, we only need to show that the above conditions also hold for the case of J + 1. Next, we define C n = ((x m ) m∈IM , s, (z (m) ) m∈IM \IJ ) ∈ m∈IM X n m × S n × m∈IM \IJ Z (m) n ((x m ) m∈IM , s, (f (m) n (x m )) m∈IJ , (z (m) ) m∈IM \IJ ) ∈ B n C ′ n = C n ∩ T (1) n ∩ T (2) n , C ′′ n = {(u, w) ∈ U n × W n \|u = ((x m ) m∈IM , s, (z (m) ) m∈IM \IJ+1 ), w = z (J+1) , ((x m ) m∈IM , s, (z (m) ) m∈IM \IJ ) ∈ C ′ n } where T (1) n = ((x m ) m∈IM , s, (z (m) ) m∈IM \IJ ) ∈ m∈IM X n m × S n × m∈IM \IJ Z (m) n 1 n ln P (f (m) n (X n m ))m∈A∩I J (Z (m) n ) m∈A\I J ((f (m) n (x m )) m∈A∩IJ , (z (m) ) m∈A\IJ ) m∈A∩IJ P f (m) n (X n m ) (f (m) n (x m )) m∈A\IJ P Z (m) n (z (m) ) ≥ I \|A\| ((f (m) (X m )) m∈A∩IJ , (Z (m) ) m∈A\IJ ) − 1 M γ 2 , for all A containing J + 1 ,(42)T (2) n = ((x m ) m∈IM , s, (z (m) ) m∈IM \IJ ) ∈ m∈IM X n m × S n × m∈IM \IJ Z (m) n 1 n ln P S n (f (m) n (X n m ))m∈I J (Z (m) n ) m∈I M \I J (s, (f (m) n (x m )) m∈IJ , (z (m) ) m∈IM \IJ ) P (f (m) n (X n m ))m∈A∩I J (Z (m) n ) m∈A\I J ((f (m) n (x m )) m∈A∩IJ , (z (m) ) m∈A\IJ ) P S n (f (m) n (X n m )) m∈(I M \A)∩I J (Z (m) n ) m∈(I M \A)\I J (s, (f (m) n (x m )) m∈(IM \A)∩IJ , (z (m) ) m∈(IM \A)\IJ ) ≥ I((f (m) (X m )) m∈A∩IJ , (Z (m) ) m∈A\IJ ; S, (f (m) (X m )) m∈(IM \A)∩IJ , (Z (m) ) m∈(IM \A)\IJ ) − 1 M γ 2 , for all nonempty A ⊆ I M . Clearly, by the definition (4) and (12), we have lim n→∞ Pr{(U n , W n ) ∈ C ′′ n } = lim n→∞ Pr{((X n m ) m∈IM , S n , (Z (m) n ) m∈IM \IJ ) ∈ C ′ n } ≥ 1 − lim n→∞ Pr{((X n m ) m∈IM , S n , (f (m) n (X n m )) m∈IJ , (Z (m) n ) m∈IM \IJ ) ∈ B n } − lim n→∞ Pr{((X n m ) m∈IM , S n , (Z (m) n ) m∈IM \IJ ) ∈ T (1) n ∩ T (2) n } = 1, and hence lim n→∞ Pr{(U n , W n ) ∈ C ′′ n } = 1. Then from Lemma 2, it follows that there exists a sequence f (J+1) = {f (J+1) n } ∞ n=1 of functions f (J+1) n : X n J+1 → Z (J+1) n such that \|f (J+1) n (X n J+1 )\| ≤ e n(I(XJ+1;Z (J+1) )+γ1) ,(44)lim n→∞ Pr{((X n m ) m∈IM , S n , f (J+1) n (X n m ), (Z (m) n ) m∈IM \IJ+1 ) ∈ C ′ n } = 1.(45) Hence, we have lim n→∞ Pr{((X n m ) m∈IM , S n , (f (m) n (X n m )) m∈IJ+1 , (Z (m) n ) m∈IM \IJ+1 ) ∈ B n } = lim n→∞ Pr{((X n m ) m∈IM , S n , f (J+1) n (X n m ), (Z (m) n ) m∈IM \IJ+1 ) ∈ C n } ≥ lim n→∞ Pr{((X n m ) m∈IM , S n , f (J+1) n (X n m ), (Z (m) n ) m∈IM \IJ+1 ) ∈ C ′ n } = 1. Furthermore, Let A be any nonempty subset in I M . If J + 1 ∈ A, we have I \|A\| ((f (m) (X m )) m∈A∩IJ+1 , (Z (m) ) m∈A\IJ+1 ) = I \|A\| ((f (m) (X m )) m∈A∩IJ , (Z (m) ) m∈A\IJ ) ≥ I \|A\| ((Z (m) ) m∈A ) − J M γ 2 > I \|A\| ((Z (m) ) m∈A ) − J + 1 M γ 2 . Otherwise, I \|A\| ((f (m) (X m )) m∈A∩IJ+1 , (Z (m) ) m∈A\IJ+1 ) = p-lim inf n→∞ 1 n ln P (f (m) n (X n m ))m∈A∩I J+1 (Z (m) n ) m∈A\I J+1 ((f (m) n (X n m )) m∈A∩IJ+1 , (Z (m) n ) m∈A\IJ+1 ) m∈A∩IJ+1 P f (m) n (X n m ) (f (m) n (X n m )) m∈A\IJ+1 P Z (m) n (Z (m) n ) (a) ≥ p-lim inf n→∞ 1 n ln P (f (m) n (X n m ))m∈A∩I J (Z (m) n ) m∈A\I J ((f (m) n (X n m )) m∈A∩IJ+1 , (Z (m) n ) m∈A\IJ+1 ) P Z (J+1) n (f (J+1) n (X n J+1 )) m∈A∩IJ P f (m) n (X n m ) (f (m) n (X n m )) m∈A\IJ+1 P Z (m) n (Z (m) n ) + p-lim inf n→∞ 1 n ln P (f (m) n (X n m ))m∈A∩I J (Z (m) n ) m∈A\I J+1 \|f (J+1) n (X n J+1 ) ((f (m) n (X n m )) m∈A∩IJ , (Z (m) n ) m∈A\IJ+1 \|f (J+1) n (X n J+1 )) P (f (m) n (X n m ))m∈A∩I J (Z (m) n ) m∈A\I J+1 \|Z (J+1) n ((f (m) n (X n m )) m∈A∩IJ , (Z (m) n ) m∈A\IJ+1 \|f (J+1) n (X n J+1 )) (b) ≥ I \|A\| ((f (m) (X m )) m∈A∩IJ , (Z (m) ) m∈A\IJ ) − 1 M γ 2 + D((f (m) n (X n m )) m∈A∩IJ (Z (m) n ) m∈A\IJ+1 \|f (J+1) n (X n J+1 ) (f (m) n (X n m )) m∈A∩IJ (Z (m) n ) m∈A\IJ+1 \|Z (J+1) n ) (c) ≥ I \|A\| ((Z (m) ) m∈A ) − J + 1 M γ 2 , where (a) follows from the property of the limit inferior in probability, (b) follows from (42) Analogously, if J + 1 ∈ A, we have I((f (m) (X m )) m∈A∩IJ+1 , (Z (m) ) m∈A\IJ+1 ; S, (f (m) (X m )) m∈(IM \A)∩IJ+1 , (Z (m) ) m∈(IM \A)\IJ+1 ) = I((f (m) (X m )) m∈A∩IJ , (Z (m) ) m∈A\IJ ; S, (f (m) (X m )) m∈(IM \A)∩IJ+1 , (Z (m) ) m∈(IM \A)\IJ+1 ) = p-lim inf n→∞ 1 n ln P S n (f (m) n (X n m ))m∈I J+1 (Z (m) n ) m∈I M \I J+1 (S n , (f (m) n (X n m )) m∈IJ+1 , (Z (m) n ) m∈IM \IJ+1 ) P (f (m) n (X n m ))m∈A∩I J (Z (m) n ) m∈A\I J ((f (m) n (X n m )) m∈A∩IJ , (Z (m) n ) m∈A\IJ ) P S n (f (m) n (X n m )) m∈(I M \A)∩I J+1 (Z (m) n ) m∈(I M \A)\I J+1 (S n , (f (m) n (X n m )) m∈(IM \A)∩IJ+1 , (Z (m) n ) m∈(IM \A)\IJ+1 ) (a) ≥ p-lim inf n→∞ 1 n ln P (f (m) n (X n m ))m∈A∩I J (Z (m) n ) m∈A\I J \|S n (f (m) n (X n m )) m∈(I M \A)∩I J (Z (m) n ) m∈(I M \A)\I J ((f (m) n (X n m )) m∈A∩IJ , (Z (m) n ) m∈A\IJ \|S n , (f (m) n (X n m )) m∈(IM \A)∩IJ+1 , (Z (m) n ) m∈(IM \A)\IJ+1 ) P (f (m) n (X n m ))m∈A∩I J (Z (m) n ) m∈A\I J ((f (m) n (X n m )) m∈A∩IJ , (Z (m) n ) m∈A\IJ ) + p-lim inf n→∞ 1 n ln P (f (m) n (X n m ))m∈A∩I J (Z (m) n ) m∈A\I J \|S n (fP (f (m) n (X n m ))m∈A∩I J (Z (m) n ) m∈A\I J \|S n (f (m) n (X n m )) m∈(I M \A)∩I J (Z (m) n ) m∈(I M \A)\I J ((f (m) n (X n m )) m∈A∩IJ , (Z (m) n ) m∈A\IJ \| S n , (f (m) n (X n m )) m∈(IM \A)∩IJ+1 , (Z (m) n ) m∈(IM \A)\IJ+1 ) (b) ≥ I((f (m) (X m )) m∈A∩IJ , (Z (m) ) m∈A\IJ ; S, (f (m) (X m )) m∈(IM \A)∩IJ , (Z (m) ) m∈(IM \A)\IJ ) − 1 M γ 2 + D((f (m) (X m )) m∈A∩IJ , (Z (m) ) m∈A\IJ \|S, (f (m) (X m )) m∈(IM \A)∩IJ+1 , (Z (m) ) m∈(IM \A)\IJ+1 (f (m) (X m )) m∈A∩IJ , (Z (m) ) m∈A\IJ \|S, (f (m) (X m )) m∈(IM \A)∩IJ , (Z (m) ) m∈(IM \A)\IJ ) (c) ≥ I((Z (m) ) m∈A ; S, (Z (m) ) m∈IM \A ) − J + 1 M γ 2 , where (a) follows from the property of the limit inferior in probability, (b) follows from (43) and (45), and (c) from Lemma 1 and (41). In the same way, it can also be shown that I((f (m) (X m )) m∈A∩IJ+1 , (Z (m) ) m∈A\IJ+1 ; S, (f (m) (X m )) m∈(IM \A)∩IJ+1 , (Z (m) ) m∈(IM \A)\IJ+1 ) ≥ I((Z (m) ) m∈A ; S, (Z (m) ) m∈IM \A ) − J + 1 M γ 2 for the case of J + 1 ∈ A. Therefore, the conditions (38), (39), (40) and (41) hold for J + 1. The lemma is hence proved by simply repeating the above argument M times. Now, we start to prove Theorem 1. Proof of Theorem 1: 1) Direct Part: Let γ 1 , γ 2 , γ 3 and γ 4 be arbitrary positive real numbers. We define T (1) n by T (1) n △ = ((x m ) m∈IM , s, (z (m) ) m∈IM ) ∈ m∈IM X n m × S n × m∈IM Z (m) n (d (k) n ((x m ) m∈IM , h n (s, (z (m) ) m∈IM ))) k∈IK ≤ (D k + γ 1 ) k∈IK . Clearly, by the condition (10), we have lim n→∞ Pr{((X n m ) m∈IM , S n , (Z (m) n ) m∈IM ) ∈ T (1) n } = 1. Then by Lemma 3, we obtain a group of sequences ( f (m) = {f (m) n } ∞ n=1 ) m∈IM of functions f (m) n : X n m → Z (m) n such that \|f (m) n (X n m )\| ≤ e ⌈n(I(Xm;Z (m) )+γ2)⌉ , lim n→∞ Pr{((X n m ) m∈IM , S n , (f (m) n (X n m )) m∈IM ) ∈ T (1) n } = 1,(46) and I \|A\| ((f (m) (X m )) m∈A ) ≥ I \|A\| ((Z (m) ) m∈A ) − γ 3 ,(47)I((f (m) (X m )) m∈A ; S, (f (m) (X m )) m∈IM \A ) ≥ I((Z (m) ) m∈A ; S, (Z (m) ) m∈IM \A ) − γ 3(48) for any nonempty set A ⊆ I M , where f (m) (X m ) △ = {f (m) n (X n m )} ∞ n=1 . Next, we specify the encoding and decoding procedures. We will present a pair of random encoding and decoding maps according to a given rate (R m ) m∈IM satisfying Φ (m) n (x m ) △ = G (m) n (f (m) n (x m )) for all x m ∈ X n m . Decoding Ψ n : S n × m∈IM I L (m) n (x m )) m∈IM ) = h n (s, (z (m) ) m∈IM ), where T (2) n △ = T (2,0) n ∩ B⊆IM ,B =∅ T (2,A) n ,(51)T (2,0) n △ = (s, (z (m) ) m∈IM ) ∈ S n × m∈IM Z (m) n 1 n ln 1 P f (m) n (X n m ) (z (m) ) ≤ I(X m ; Z (m) ) + 2γ 2 , ∀m ∈ I M , T (2,B) n △ = (s, (z (m) ) m∈IM ) ∈ S n × m∈IM Z (m) n 1 n ln P (f (m) n (X n m ))m∈B ((z (m) ) m∈B ) m∈B P f (m) n (X n m ) (z (m) ) ≥ I \|B\| ((f (m) (X m )) m∈B ) − γ 4 , 1 n ln P S n (f (m) n (X n m ))m∈I M (s, (z (m) ) m∈IM ) P (f (m) n (X n m ))m∈B ((z (m) ) m∈B )P S n (f (m) n (X n m )) m∈I M \B (s, (z (m) ) m∈IM \B ) ≥ I((f (m) (X m )) m∈B ; S, (f (m) (X m )) m∈IM \B ) − γ 4 . Otherwise, Ψ n (s, (Φ (m) n (x m )) m∈IM ) is defined to be an arbitrary fixed element in m∈IM Y n m . If the source outputs (x m ) m∈IM of the m terminals together with the side information s satisfy the following conditions: (1) ((x m ) m∈IM , s, (f (m) n (x m )) m∈IM ) ∈ T (1) n ∩ ( m∈IM X n m × T (2) n ) (2) There is no (z (m) ) m∈IM ∈ m∈IM Z (m) n such that (z (m) ) m∈IM = (f (m) n (x m )) m∈IM , (G (m) n (z (m) )) m∈IM = (Φ (m) n (x m )) m∈IM , (s, (z (m) ) m∈IM ) ∈ T (2) n , then (f (m) n (x m )) m∈IM is the unique element in m∈IM Z (m) n such that (G (m) n (f (m) n (x m ))) m∈IM = (Φ (m) n (x m )) m∈IM and (s, (f (m) n (x m )) m∈IM ) ∈ T (2) n . Furthermore, we have d (k) n ((x m ) m∈IM , Ψ n (s, (Φ (m) n (x m )) m∈IM )) = d (k) n ((x m ) m∈IM , h n (s, (f (m) n (x m )) m∈IM )) ≤ D k + γ 1 , ∀k ∈ I K . due to ((x m ) m∈IM , s, (f (m) n (x m )) m∈IM ) ∈ T (1) n . Hence, every pair ((x m ) m∈IM , s) ∈ T 1 (Φ n , Ψ n ) ∩ T 2 (Φ n , Ψ n ) satisfies (d (k) n ((x m ) m∈IM , Ψ n (s, (Φ (m) n (x m )) m∈IM ))) k∈IK ≤ (D k + γ 1 ) k∈IK , where T i (Φ n , Ψ n ) is defined for the code (Φ n , Ψ n ) by T i (Φ n , Ψ n ) △ = ((x m ) m∈IM , s) ∈ m∈IM X n m × S n ((x m ) m∈IM , s) satisfies condition i for i = 1, 2. Next, we estimate the probability of error. Let us define the probability of error P (n) e (Φ n , Ψ n ) for the code (Φ n , Ψ n ) by P (n) e (Φ n , Ψ n ) = Pr{∃k ∈ I K , s. t. d (k) n ((X n m ) m∈IM , Ψ n (S n , (Φ (m) n (X n m )) m∈IM )) > D k + γ 1 }. Then, the P (n) e (Φ n , Ψ n ) can be bounded as follows: P (n) e (Φ n , Ψ n ) ≤ 1 − Pr{((X n m ) m∈IM , S n ) ∈ T 1 (Φ n , Ψ n ) ∩ T 2 (Φ n , Ψ n )} = Pr{((X n m ) m∈IM , S n ) ∈ T 1 (Φ n , Ψ n ) ∩ T 2 (Φ n , Ψ n )} ≤ Pr{((X n m ) m∈IM , S n ) ∈ T 1 (Φ n , Ψ n )} + Pr{((X n m ) m∈IM , S n ) ∈ T 2 (Φ n , Ψ n )}.(52) Clearly, the first term in (52) can be bounded above by Pr{((X n m ) m∈IM , S n , (f (m) n (X n m )) m∈IM ) ∈ T (1) n } + Pr{(S n , (f (m) n (X n m )) m∈IM ) ∈ T (2) n } and hence converges to zero by (46) and (51). As for the second term in (52), the event ((X n m ) m∈IM , S n ) ∈ T 2 (Φ n , Ψ n ) can be rewritten as ((X n m ) m∈IM , S n ) ∈ B⊆IM ,B =∅ E n (Φ n , Ψ n , B), where E n (Φ n , Ψ n , B) = {((x m ) m∈IM , s) ∈ m∈IM X n m × S n \|∃(z (m) ) m∈B ∈ m∈B Z (m) n , s. t. (z (m) ) m∈B = (f (m) n (x m )) m∈B , (G (m) n (z (m) )) m∈B = (Φ (m) n (x m )) m∈B , (s, (z (m) ) m∈B , (f (m) n (x m )) m∈IM \B ) ∈ T (2) n }. Then we have Pr{((X n m ) m∈IM , S n ) ∈ T 2 (Φ n , Ψ n )} ≤ B⊆IM ,B =∅ Pr{((X n m ) m∈IM , S n ) ∈ E n (Φ n , Ψ n , B)}(53) In order to estimate Pr{((X n m ) m∈IM , S n ) ∈ T 2 (Φ n , Ψ n )}, let us estimate each term in the summation (53). For each nonempty set B ⊆ I M , we have Finally, let us use the diagonal method to complete the proof. By repeating the above argument with replacing γ 1 by a sequence {γ 1 (i)} ∞ i=1 which satisfies γ 1 (1) ≥ γ 1 (2) ≥ · · · > 0 and γ 1 (i) → 0 as i → ∞, we can conclude that there exists a sequence {(φ n , ψ n )} ∞ n=1 of codes which satisfies (lim sup n→∞ R(φ (m) n )) m∈IM ≤ (R m ) m∈IM , (p-lim sup n→∞ d (k) n ((X n m ) m∈IM , ψ n (S n , (φ (m) n (X n m )) m∈IM ))) k∈IK ≤ (D k ) k∈IK . Pr{((X n m ) m∈IM , S n ) ∈ E n (Φ n , Ψ n , B)} = gn P Gn (g n ) ((xm)m∈I M ,s)∈ m∈I M X n m ×S n P (X n m )m∈I M S n ((x m ) m∈IM , s)1{∃(z (m) ) m∈B ∈ m∈B Z (m) n , s. t. (z (m) ) m∈B = (f (m) n (x m )) m∈B , (g (m) n (z (m) )) m∈B = (g (m) n (f (m) n (x m ))) m∈B , (s, (z (m) ) m∈B , (f (m) n (x m )) m∈IM \B ) ∈ T (2) n } ≤ gn P Gn (g n ) ((xm)m∈I M ,s)∈ m∈I M X n m ×S n P (X n m )m∈I M S n ((x m ) m∈IM , s) (z (m) )m∈B ∈ m∈B Z (m) n ,(z (m) )m∈B =(f (m) n (xm))m∈B , (s,(z (m) )m∈B ,(f (m) n (xm)) m∈I M \B )∈T (2) n 1{(g (m) n (z (m) )) m∈B = (g (m) n (f (m) n (x m ))) m∈B } = ((xm)m∈I M ,s)∈ m∈I M X n m ×S n P (X n m )m∈I M S n ((x m ) m∈IM , s) (z (m) )m∈B ∈ m∈B Z (m) n ,(z (m) )m∈B =(f (m) n (xm))m∈B , (s,(z (m) )m∈B ,(f (m) n (xm)) m∈I M \B )∈T (2) n Pr{(G (m) n (z (m) )) m∈B = (G (m) n (f (m) n (x m ))) m∈B } (a) = ((xm)m∈I M ,s)∈ m∈I M X n m ×S n P (X n m )m∈I M S n ((x m ) m∈IM , s) (z (m) )m∈B ∈ m∈B Z (m)P S n (f (m) n (X n m )) m∈I M \B (s, (z (m) ) m∈IM \B ) (z (m) )m∈B ∈ m∈B Z (m) n , (s,(z (m) )m∈I M )∈T (2) n 1 m∈B L (m) n = (s,(z (m) )m∈I M )∈T (2) n P S n (f (m) n (X n m )) m∈I M \B (s, (z (m) ) m∈IM \B ) m∈B L (m) n (b) ≤ (s,(z (m) )m∈I M )∈T (2,0) n ∩T (2,B) n P S n (f (m) n (X n m )) m∈I M \B (s, (z (m) ) m∈IM \B ) m∈B L (m) n (c) ≤ (z (m) )m∈B ∈ m∈B Z (m) n 1 1 n ln P (f (m) n (X n m ))m∈B ((z (m) ) m∈B ) m∈B P f (m) n (X n m ) (z (m) ) ≥ I \|B\| ((f (m) (X m )) m∈B ) − γ 4 1 1 n ln 1 P f (m) n (X n m ) (z (m) ) ≤ I(X m ; Z (m) ) + 2γ 2 , ∀m ∈ B (s,(z (m) ) m∈I M \B )∈S n × m∈I M \B Z (m) n P S n (f (m) n (X n m )) m∈I M \B \|(f (m) n (X n m ))m∈B (s, (z (m) ) m∈IM \B \|(z (m) ) m∈B ) e n(I((f (m) (Xm))m∈B ;S,(f (m) (Xm)) m∈I M \B )−γ4) m∈B L (m) n ≤ (z (m) )m∈B ∈ m∈B Z (m) n 1 P (f (m) n (X n m ))m∈B ((z (m) n ) m∈B ) ≥ e n(I \|B\| ((f (m) (Xm))m∈B )− m∈B I(Xm;Z (m) )−2\|B\|γ2−γ4) Therefore, the proof of the direct part is established. 2) Converse Part: Since the rate pair ((R m ) m∈IM , (D k ) k∈IK is achievable, there exits a sequence {(φ n , ψ n )} ∞ n=1 such that (lim sup n→∞ R(φ (m) n )) m∈IM ≤ (R m ) m∈IM , (p-lim sup n→∞ d (k) n ((X n m ) m∈IM , ψ n (S n , (φ (m) n (X n m )) m∈IM ))) k∈IK ≤ (D k ) k∈IK . Let us define the general sources (Z (m) ) m∈IM by Z Therefore, the converse part is established. Fig. 1 . 1Separate compression of M correlated general sources with side information at the decoder Corollary 2 ([ 4 , 24Theorem 1]): For a general source X, a side information source S and distortion measures (d (k) ) k∈IK , the rate-distortion tuple (R, (D k ) k∈IK ) is f m-achievable if and only if there exist general sources Z with alphabet {Z n } ∞ n=1 and a sequence h = {h n } ∞ n=1 of functions defined by Lemma 3 : 3Let (X m ) m∈IM , S and (Z (m) ) m∈IM be arbitrary general sources satisfying (9). Now let {B n } ∞ n=1 be a sequence of arbitrary (measurable) sets B n in m∈IM X n m × S n × m∈IM Z(m) n such that lim n→∞ Pr{((X n m ) m∈IM , S n , (Z (m) n ) m∈IM ) ∈ B n } = 1. m∈IM , S n , (f (m) n (X n m )) m∈IM ) ∈ B n } = 1, and I \|A\| ((f (m) (X m )) m∈A ) ≥ I \|A\| ((Z (m) ) m∈A ) − γ 2 , I((f (m) (X m )) m∈A ; S, (f (m) (X m )) m∈IM \A ) ≥ I((Z (m) ) m∈A ; S, (Z (m) ) m∈IM \A ) − γ 2for any nonempty set A ⊆ I M , where f (m) (X m ) Proof: Supposing that for a given J (0 ≤ J < M ), we have \|f (m) n (X n m )\| ≤ e ⌈n(I(Xm;Z (m) )+γ1)⌉ , ∀m ∈ I J (38) P Now, let U n = ((X n m ) m∈IM , S n , (Z (m) n ) m∈IM \IJ+1 ), V n = X n J+1 , W n = Z UnVnWn = P UnVn P Wn\|Vn . ) m∈(I M \A)\I J+1 ( (f (m) n (X n m )) m∈A∩IJ , (Z (m) n ) m∈A\IJ \|S n , (f (m) n (X n m )) m∈(IM \A)∩IJ+1 , (Z (m) n ) m∈(IM \A)\IJ+1 ) n) m ; Z (m) ) − I \|A\| ((Z (m) ) m∈A ) − I((Z (m) ) m∈A ; S, (Z (m) ) m∈IM \A ) (49) for any nonempty set A ⊆ I M . Generation of random bins: For each m ∈ I M , assign each z (m) ∈ Z (m) n to one of the indices in I L (z (m) ) denote the index to which z (m) corresponds. m∈IM . For each m ∈ I M , the encoder Φ (m) n with respect to the m-th terminal is defined by n (x m )) m∈IM ), if there is one and only one (z (m) ) m∈IM ∈ m∈IM Z (m) n satisfying (G (m) n (z (m) )) m∈IM = (Φ (m) n (x m )) m∈IMand (s, (z (m) ) m∈IM ) ∈ T (2) n , ( ,(z (m) )m∈B ,(f (m) n (xm)) m∈I M \B )∈T(2) (xm)m∈I M ,s)∈ m∈I M X n m ×S n P (X n m )m∈I M S n ((x m ) m∈IM , s) (z (m) )m∈B ∈ m∈B Z (m) n , (s,(z (m) )m∈B ,(f (m) n (xm)) m∈I M \B )∈T (2) (m) ) m∈I M \B )∈S n × m∈I M \B Z (m) n ≤ e n(I((f (m) (Xm))m∈B ;S,(f (m) (Xm)) m∈I M \B )−γ4) m∈B L (m) n ≤ e −n(I \|B\| ((f (m) (Xm))m∈B )+I((f (m) (Xm))m∈B ;S,(f (m) (Xm)) m∈I M \B )− m∈B I(Xm;Z (m) e −n[ m∈B Rm−( m∈B I(Xm;Z (m) )−I \|B\| ((Z (m) )m∈B )−I((Z (m) )m∈B ;S,(Z (m) ) m∈I M \B ))+\|B\|(γ1−2γ2)−2(γ3+γ4)] where (a) follows from the property of the random bins, (b) and (c) from the definition (51), and (d) from (47), (48) and (50).Since γ 2 , γ 3 and γ 4 are arbitrary, let us defineγ 2 = γ 3 = γ m∈IM , S n ) ∈ E n (Φ n , Ψ n , B)} = 0for all nonempty set B ⊆ I M , and hence we havelim sup n→∞ P (n) e (Φ n , Ψ n ) ≤ lim sup n→∞ Pr{((X n m ) m∈IM , S n ) ∈ T 1 (Φ n , Ψ n )} + Pr{((X n m ) m∈IM , S n ) ∈ T 2 (Φ n , Ψ n )} ≤ lim sup n→∞ Pr{((X n m ) m∈IM , S n ) ∈ T 1 (Φ n , Ψ n )} + lim sup n→∞ Pr{((X n m ) m∈IM , S n ) ∈ T 2 (Φ n , Ψ n m∈IM , S n ) ∈ E n (Φ n , Ψ n , B)} = 0,which implies that there exists at least one sequence {(φ n , ψ n )} ∞ n=1 of codes satisfying p-lim sup n→∞ d (k) n ((X n m ) m∈IM , ψ n (S n , (φ (m) n (X n m )) m∈IM )) ≤ D k + γ 1 , ∀k ∈ I K and lim sup n→∞ R(φ (m) n ) = R m + γ 1 , ∀m ∈ I M . n all n = 1, 2, · · ·. Besides, let us define h n = ψ n , then we have(p-lim sup n→∞ d (k) n ((X n m ) m∈IM , h n (S n , (Z (m) n ) m∈IM ))) k∈IK = (p-lim sup n→∞ d (k) n ((X n m ) m∈IM , ψ n (S n , (φ (m) n (X n m )) m∈IM ))) k∈IK ≤ (D k ) k∈IK .Furthermore, for any γ > 0 and sufficiently large n, X m ; Z (m) ) − I \|A\| ((Z (m) ) m∈A ) − I((Z (m) ) m∈A ; S, (Z (m) ) m∈IM \A ) − 2M γ where (a) follows from [7, Lemma 2.6.2], (b) from the fact that P Z \|X n m ) ≤ 1 and \|A\| ≤ M , and (c) from the nonnegativity of I \|A\| ((Z (m) ) m∈A ) and I((Z (m) ) m∈A ; S, (Z (m) ) m∈IM \A ). Since γ is arbitrary, we have m∈A R m ≥ m∈A I(X m ; Z (m) ) − I \|A\| ((Z (m) ) m∈A ) − I((Z (m) ) m∈A ; S, (Z (m) ) m∈IM \A ). y∈YnP XnYn (x, y)1 P Xn\|Yn (x\|y) PX n \|Ŷn (x\|y) < e −nγ < x∈Xn,y∈Yn e −nγ P Yn (y)PX n \|Ŷn (x\|y) = e −nγ y∈Yn P Yn (y) x∈Xn PX n \|Ŷn (x\|y) = e −nγ , hence lim n→∞ Noiseless coding of correlated information sources. D Slepian, J K Wolf, IEEE Trans. Inform. Theory. 194D. Slepian and J. K. Wolf, "Noiseless coding of correlated information sources," IEEE Trans. Inform. Theory, vol. 19, no. 4, pp. 471-480, July 1973. The rate-distortion function for source coding with side information at the decoder. A D Wyner, J Ziv, IEEE Trans. Inform. Theory. 221A. D. Wyner and J. Ziv, "The rate-distortion function for source coding with side information at the decoder," IEEE Trans. Inform. Theory, vol. 22, no. 1, pp. 1-10, Jan. 1976. Coding theorems on correlated general sources. S Miyake, F Kanaya, IEICE Trans. Fundamentals. 9S. Miyake and F. Kanaya, "Coding theorems on correlated general sources," IEICE Trans. Fundamentals, vol. E78-A, no. 9, pp. 1063-1070, Sept. 1995. An information-spectrum approach to rate-distortion function with side information. K Iwata, J Muramatsu, IEICE Trans. Fundamentals. 6K. Iwata and J. Muramatsu, "An information-spectrum approach to rate-distortion function with side information," IEICE Trans. Fundamentals, vol. E85-A, no. 6, pp. 1387-1395, June 2002. Multiterminal source coding. T Berger, The Information Theory Approach to Communications. New YorkSpringer-VerlagT. Berger, "Multiterminal source coding," in The Information Theory Approach to Communications. New York: Springer-Verlag, July 1977, pp. 171-231. The Wyner-Ziv problem with multiple sources. M Gastpar, IEEE Trans. Inform. Theory. 5011M. Gastpar, "The Wyner-Ziv problem with multiple sources," IEEE Trans. Inform. Theory, vol. 50, no. 11, pp. 2762-2768, Nov. 2004. Information-Spectrum Methods in Information Theory. T S Han, SpringerBerlinT. S. Han, Information-Spectrum Methods in Information Theory. Berlin: Springer, 2003.	[]
[ "COLLECTIVE COGNITIVE AUTHORITY: EXPERTISE LOCATION VIA SOCIAL LABELING", "COLLECTIVE COGNITIVE AUTHORITY: EXPERTISE LOCATION VIA SOCIAL LABELING" ]	[ "Terrell G Russell [email protected] \nUniversity of North Carolina at Chapel Hill\n100 Manning Hall Chapel HillNC\n" ]	[ "University of North Carolina at Chapel Hill\n100 Manning Hall Chapel HillNC" ]	[]	The problem of knowing who knows what is multifaceted. Knowledge and expertise lie on a spectrum and one's expertise in one topic area may have little bearing on one's knowledge in a disparate topic area. In addition, we continue to learn new things over time. Each of us see but a sliver of our acquaintances' and co-workers' areas of expertise. By making explicit and visible many individual perceptions of cognitive authority, this work shows that a group can know what its members know about in a relatively efficient and inexpensive manner.	null	[ "https://arxiv.org/pdf/1204.3353v1.pdf" ]	9,461,757	1204.3353	2ec2f995595d8556b58ee2bd11172a458cd30e27	COLLECTIVE COGNITIVE AUTHORITY: EXPERTISE LOCATION VIA SOCIAL LABELING 2012 Terrell G Russell [email protected] University of North Carolina at Chapel Hill 100 Manning Hall Chapel HillNC COLLECTIVE COGNITIVE AUTHORITY: EXPERTISE LOCATION VIA SOCIAL LABELING 2012 The problem of knowing who knows what is multifaceted. Knowledge and expertise lie on a spectrum and one's expertise in one topic area may have little bearing on one's knowledge in a disparate topic area. In addition, we continue to learn new things over time. Each of us see but a sliver of our acquaintances' and co-workers' areas of expertise. By making explicit and visible many individual perceptions of cognitive authority, this work shows that a group can know what its members know about in a relatively efficient and inexpensive manner. INTRODUCTION Cognitive authority is the foil to administrative authority (Wilson 1983). Administrative authority is that which one has through rank or position. Cognitive authority is that which is granted to you by others because of what they think you know about. Cognitive authority is a subjective measurement and should be respected as such. There are no right answers to questions of cognitive authority, although, when taken collectively, an assessment of it can be seen as a barometer of one's standing among peers. Making a collective assessment visible, bringing the tacit individual knowledge into the realm of the explicit, and performing a sanity check on that assessment is the thrust of this paper. This work shows that a group's evaluations of an individual's areas of expertise can be gathered and potentially serve as useful loose credentials; loose credentials that may be useful when more expensive or heavyweight reputation cues may not be viable. BACKGROUND We satisfice; we satisfy with what is sufficient (Simon 1957). We use what information we have to make decisions that we deem to be good enough at the time. We often seek out more information before making a decision but we have, what Simon called, "bounded rationality." We have imperfect information, limited attention and money, limited processing power and limited time, but we still need to make decisions. Choo's Decision Behavior Model shows us that contextualized decision making happens within organizations based on cognitive limits, information quality and availability, and the values of the organization (Choo 1996). These inputs are handled with bounded rationality and within the confines of performance concerns, and whether the decision is good enough, among other simplifications. This decision making behavior is both rationally expected and observed. Even knowing we will never have perfect information when working in these limited environments, we can arguably make better decisions if we can improve or increase the amount of information on hand when making decisions. Having more good information reduces uncertainty about the environment surrounding a decision, but it does not necessarily reduce equivocality. To reduce equivocality, or ambiguity, of the information we have on hand, we need sensemaking and a perspective that comes from "retrospective interpretations" of earlier data and decisions (Choo 1996). We need to have seen this before and know what it means. What we need to make good decisions, in addition to good information, is called expertise. There is a vast amount of latent, untapped information in the environment around us. Some of it is in the built world, some of it is in the natural world (too big, too small, hidden in non-visible wavelengths, etc.), and some of it is in the heads of those around us. Cross noted that 85% of managers immediately mentioned specific people when asked "to describe sources of information important to successful completion of their project" (Cross & Sproull 2004). They went on to write: As one manager said, "I mean the whole game is just being the person that can get the client what they need with [the Firm's] resources behind you. This almost always seems to mean knowing who knows what and figuring out a way to bring them to your client's issue" (R6). Very few of the named people were simply organizationally designated "experts"; most were described as partners in information relationships. If we are informed by the right people before making decisions, and they help us decide what we are looking for (Belkin, Oddy & Brooks 1982), then we may improve our knowledge and understanding of a situation or problem at the time when we need to decide. Knowing from whom we should get our information, when we are not sure of what we need, is a hard problem. arXiv:1204.3353v1 [cs.SI] 16 Apr 2012 Figure 1. Study design. 10 groups, their Self and Group lists about each participant, the three types of similarity ratings. There was also a survey and a set of interviews which captured context and sentiment. Expertise location, for this reason, has been a focus of the knowledge management field for many years. Knowledge management has also focused on the process of organizational learning and dissemination of that learning within the organization. In many cases, this has been done through the tracking of created documents and other knowledge artifacts (Martin 2008). An additional approach should consist of uncovering that which has not yet been recorded -that information which is in the heads of a group's membership. We should be equipped to hold up a mirror to help reflect an organization's insights and expertise back on itself. We need to help uncover the dark corners where we are not sure about the expertise in the room. With a regimen of self-reflection, iterated over time, I hope this problem can be made less hard. I think we can discover whom to ask for the relatively low cost of a little sustained individual effort and some focused record-keeping in the distributed network. STUDY DESIGN Because cognitive authority is a subjective measurement, there is no objective way to measure its "precision", no yard stick by which to measure its correctness. However, if, with repeated exposure and more familiarity, a group's assessment and the assessment of the individual they are assessing become more similar over time, then I argue that this is a signal of the collective assessment's relative validity. This validity is what makes the group's opinion matter. This validity is what gives a group's opinion its weight. This study had 10 different groups of individuals, mostly coworkers, use free text keywords to label each others' areas of expertise ( Figure 1). The participating groups consisted of members from a family retail business, a dentist's office, two distributed software development groups, a museum education staff, a writer's network, a legal non-profit, a global engineering firm, an academic faculty group, and an academic administrative office. Results were shared back into the group and made visible, and the process was repeated for up to five total rounds. The resulting product was an aggregated, weighted list of words associated with each person's areas of expertise. Inspired by the distribution of keywords affiliated with individual URLs at delicious.com, each individual's weighted list can be viewed as a specific fingerprint in the multidimensional space created by all possible keywords. While individuals were labeling each other during each round, they were also labeling their own areas of expertise. With each passing round, ala the Delphi model (Dalkey & Helmer 1963, Rohrbaugh 1979, Stewart 1987, Luo & Wildemuth 2009), the participants were presented with what their group had said about them and had the opportunity to update (by adding, removing, or abstaining) their list of keywords about their own areas of expertise. The original Delphi study was run in the 1950s and 1960s by the RAND corporation to help the US Government determine the nuclear capabilities of the Soviet Union (Helmer & Rescher 1959, Dalkey & Helmer 1963. They were studying the unknown military futures market by asking a variety of experts to answer a battery of questions. The answers were collated and then distributed back to the experts for additional rounds of answering the same questions -but critically, with the collective opinions of the other experts to aid their synthesis. Rowe and Wright write that, "in particular, the structure of the technique is intended to allow access to the positive attributes of interacting groups (knowledge from a variety of sources, creative synthesis, etc.), while pre-empting their negative aspects (attributable to social, personal and political conflicts, etc.)" (Rowe & Wright 1999). Over the following four decades, the Delphi method has been refined and used in many other areas besides military futures, including social science predictions (Linstone & Turoff 1975, Rowe, Wright & McColl 2005, Hsu & Sandford 2007. Most research has suggested that with proper preparation and consideration for expert subjects, questionnaires, and evaluation, a Delphi study can run from three to five rounds, with four being the most common number of iterations (Hsu & Sandford 2007). Some prior Delphi studies have used post-task surveys to sample participants' reactions -from satisfaction (Van De Van & Delbecq 1974) to confidence (Scheibe, Skutsch & Schofer 1975, Boje & Murnighan 1982 to difficulty and enjoyableness (Rohrbaugh 1979). I employed some of the same types of questions here, especially considering the subjects were being asked to formalize their informal knowledge about one another. A traditional Delphi study involves 1) an objective facilitator who gives "controlled feedback" in the aggregate, 2) a collection of independent experts in a domain (anonymous, to each other), and 3) a series of evalua-tions (iterations) designed to have the collective opinion of the experts predict the future in that particular domain (Rowe & Wright 1999). In this study, the co-workers are the experts, their labels are handled by the software and not attributed to any individual labeler, and there are 5 rounds of reflection and labeling. Sample results from the labeling exercise demonstrating the Self and Group lists. Data collection for this study was done primarily through a custom web interface that guided the participants through three stages per round, for up to five rounds. The three stages included reflection on the existing labels in the system (see an example in Figure 2), a chance to add, update, or delete labels about their own areas of expertise (Self), and a chance to do the same for each of their peers (Group). Regarding the inherent "fuzziness" of this type of data, Latour, Woolgar, and Nelson suggest to us that where there is a lack of contention, a social fact will be defined (Latour & Woolgar 1986, Nelson 1993. Social tagging phenomena have demonstrated a stabilization of tagging behavior over time (Russell 2006, Golder & Huberman 2005. Together, these suggest the first hypothesis: Hypothesis 1. As the social fact of what a person knows is collectively molded by the group, a consensus will appear and converge. The similarity of a group's opinion and an individual's opinion will increase over time. That is, a similarity rating comparing two groups of words will increase from round to round of labeling. Furthermore, the warranting principle suggests that we give more credence to information provided by others, rather than information within the control of a particular other (Walther & Parks 2002, Walther, Heide, Hamel & Shulman 2009. Online or offline, information that is known to be easily manipulated is less trusted. Additionally, Delphi-style studies increase the confidence levels of the participants (Rowe et al. 2005). This suggests the second hypothesis: Hypothesis 2. Group members will have confidence in this system and exhibit increased trust in one another. RESULTS Similarity ratings were generated in three distinct ways: Human coded (gold standard), Amazon's Mechanical Turk (for scale), and an Algorithmic (automated) solution based on a bag-of-words assumption. These three methods, along with some survey results, are described here. Trained Human Similarity (HumanSim) The HumanSim ratings were designed to serve as a sanity check for this type of data collection and analysis. Six trained raters coded a sample of the entire dataset and made 2348 comparisons of sets of words and rated their similarity on a 1-7 Likert scale with 1 meaning very dissimilar and 7 meaning nearly identical. They compared both weighted and unweighted lists and evaluated both random pairings (a Self list from one person with a Group evaluation of a different person) and lists "belonging" to a study participant (where the Self and Group lists were about that participant). Each comparison was evaluated in an average of 15 seconds. . HumanSim: Trained humans determined that random pairings of Self and Group were dissimilar and that the Self and Group lists for the same individual were similar. When the labels were weighted, the similarity ratings were more tightly clustered at the high end of the scale. A two-way analysis of variance (see Figure 4) illustrates the significance of both independent variables. There were no significant interaction effects. The main effect between the random design and study design was significant with a p-value of 0.000. The main effect for weightedness was significant at the 0.01 level with a pvalue of 0.0013. Trained humans determined that labels attributed to one's areas of expertise by one's peers are similar to the labels given by someone about their own areas of expertise. The next two methods of generating ratings are an attempt to do the same thing, but cheaper and more quickly. Untrained Mechanical Turk Similarity (TurkSim) The second of the three methods of calculating similarity of the study data was via Amazon's Mechanical Turk. This method of analysis evaluated a series of 8773 Human Intelligence Tasks (HITs) for a total cost of $219.33. Similar to the HumanSim comparisons, the Turkers were asked to rate the similarity of two sets of words on a 1-7 Likert scale. A rating of 7 meant they felt the two lists were extremely similar and conveyed the same information. The Turkers rated data from each member of each group from each of the rounds. When looking at the similarity ratings across the entire study (see Figure 5), it is clear the agreement that the lists convey similar information, but there is not a dramatic rise in that agreement over time. The range of similarity scores tightened around the mean as the rounds progressed. This does show consensus-building, but not an increase in the raw similarity of the data being evaluated. Untrained Mechanical Turk workers determined that labels attributed to one's areas of expertise by one's peers are similar to the labels given by someone about their own areas of expertise. However, that similarity did not measurably increase over time. The ranges collapsed more quickly (Round 2 vs. Round 5) when all the words were shown, rather than only the words that appeared in the lists multiple times ("common" words). Algorithmic Similarity (AlgSim) The third method for evaluating similarity of the study data was automated and tested here to see whether it produced reproducible results, similar to that of our Hu-manSim gold standard and TurkSim methods. Figure 6. Similarity Rating Flow: For the AlgSim ratings, both the Self list and the Group list were "sense disambiguated" by using a package of the WordNet project which determines which version of each word is being used (i.e. bank: money vs river). Then each list is compared as a group against the other and rated by the algorithm in Equation 1. The ratings in this section were computationally generated based on an algorithm (Equation 1) originally defined to calculate the similarity of sentences in the English language (Mihalcea, Corley & Strapparava 2006). It uses a naive bag-of-words approach and ignores context such as word order. This naivety is a positive for this analysis since there is no implicit word order in the sets of labels being evaluated. However, the labels are related in that they are part of the same set and this algorithm allows a straightforward approach to generating a value for that relationship. The resulting similarity scores were in the range [0..1] and cannot be directly compared to the human-generated Likert scale scores of 1-7 from the prior two sections. The original lists of labels were sense disambiguated using the WordNet database (Pedersen, Patwardhan & Michelizzi 2004, Pedersen & Kolhatkar 2009 and then compared against one another. AlgSim(A, B) = 1 2     w∈{A} (maxSim(w, B) * idf (w)) w∈{A} idf (w) + w∈{B} (maxSim(w, A) * idf (w)) w∈{B} idf (w)     (1) The algorithm took each word in set A and found the most similar word in set B (represented by maxSim(w, B)) and then multiplied by the information content of that word (represented by idf (w)). This summation was normalized across the information content of the entire list ( w∈{A} idf (w)). After each list was compared one to the other, the similarity values were averaged for the final AlgSim value for two lists A and B. Using this approach, the same analysis that was performed by the Mechanical Turk workers could be duplicated much more efficiently. The results are presented here. Figure 7. AlgSim: The automatic algorithm detected a greater similarity in the later rounds than in the earlier rounds. It produced a tighter range of scores when it had more information to use (when using all the words rather than just the words used more than once in each list). Using this method the scores appear low on an absolute scale, but should be interpreted relatively. There was a clear rise in similarity score from Round 1 to Round 5 (see Figure 7) for both evaluation techniques (using all the words and only the words which occurred more than once). for the case where all words were evaluated showed a statistically significant main effect with an alpha of 0.01 and p-value of 0.0015. However, looking at round by round post-hoc ANOVA analysis when all the words are used (Table 2), the only significant differences appear to occur after the original feedback loop between Rounds 1 and 2 (p-values less than 0.05) when the participants are initially faced with their group members' feedback and labels. No other single round or span of rounds show an effect. This same analysis did not produce significant effects when only the words appearing more than once were used. Survey The survey accompanying this study consisted of 54 questions and was completed by 56 of the 64 participants across all 10 groups. The survey results listed in Table 4 represent the 11 original questions as well as the aggregated results from the seven included scales (see Table 3). The scores are on a 1-7 Likert scale representing agreement: 1=Extremely Disagree, 4=Neutral, 7=Extremely Agree. All but one of the original items scored with mild to strong agreement. The highest ratings of agreement were received by the statements regarding comfort and familiarity of the group members with one another's areas of expertise. Additionally, nearly all participants rated this to be an interesting exercise. Slightly lower ratings were received by the items regarding the results of the exercise. The participants believed the system gave them somewhat good and new information that they found useful. They also thought the system did not necessarily gather all the important areas of their expertise and that they would not necessarily use the information to help them make decisions moving forward. Being averages, the aggregate scales are relatively mild and all fit between 3 and 5, straddling the Neutral rating. However, they showed similar results to the original items. At the top of the list, the participants believed this exercise provided good data quality and was easy to use and clear to understand. The participants rated the items regarding the results of the exercise and its fit within their organization slightly higher than neutral. -I would find this system useful in my job. -Using this system enables me to accomplish tasks more quickly. -Using this system increases my productivity. Effort Expectancy (Venkatesh et al. 2003) -My interaction with this system would be clear and understandable. -It would be easy for me to become skillful at using this system. -I would find this system easy to use. -Learning to operate this system would be easy for me. Facilitating Conditions (Venkatesh et al. 2003) -I have the resources necessary to use this system. -I have the knowledge necessary to use this system. -This system is not compatible with other systems I use. (reverse coded) Anxiety (Venkatesh et al. 2003) -I feel apprehensive about using this system. -It scares me to think that I could lose a lot of information using this system by hitting the wrong key. -I hesitate to use this system for fear of making mistakes I cannot correct. -This system is somewhat intimidating to me. Data Quality (Wang & Strong 1996) -This system produced data in conformance with the actual or true values. -This system produced data that is applicable and relevant to my job. -This system produced data that is intelligible and clear. -This system produced data that is easily accessible. DISCUSSION H1 The data supports the view that this method provides a baseline for concluding that a group's opinion about a person's areas of expertise can give good information. A consensus appeared, was agreed to by the individual being labeled, and somewhat converged over time as the language and norms of the group were negotiated in a shared space. This finding comes with the caveat that the participants knew one another well enough or had enough experience with one another to feel the data being provided was of good enough quality. When conducted outside of wellknown groups, this finding may not hold as both participant identity and the promise of future interactions are not as strong. H2 This hypothesis was found to be partially supported. Participants did have confidence in the system to collect and then report the type of information they were expecting it to report. They thought the data would be quality data and they trusted it for what it was. However, they did not report that the trust in the data carried over to increased trust in the other participants. The study design forced the group members to already be acquainted with one another and have existing working relationships. This means that the participants began the study with a fairly high degree of trust. This study provided no support for the idea that participants' trust levels increased because of the exercise. It would be interesting to ask a specific set of questions about colleague trust of a set of group members who were just beginning to work together or of group members who knew each other in a less formal environment than their salaried jobs. CONCLUSION Throughout this work, there was a sense that a kind of signaling was happening among members of the group, but implicitly through the data. The participants did not report they brought the ongoing results of this study into their face-to-face conversations. In a social space, judgment from others is ever-present and constant. By bringing that sense of evaluation out in the open and working together, a new collective artifact was produced that changed over time. This artifact held importance in the minds of the participants as they were looking squarely at the group's judgment of their skillsets and areas of expertise. Their value to the group was laid bare in some sense, made explicit. The participants reported the first round was the hardest -tough on the psyche -but also exciting, and potentially rewarding. They also reported ongoing concern about being pigeon-holed for what they have already done or already shared. Perhaps this type of public judgment encourages exploration and continued learning on the behalf of the individual. Perhaps it discourages sharing in the longer term. Overall, this research has provided insight into how familiar groups of individuals in the workplace can understand what their colleagues think of their areas of expertise. This work has shown that, with simple keywords, group members can convey the salient areas of expertise of their colleagues to a degree that is deemed "similar" and of "high quality" by both third parties and those being evaluated. Identity formation and negotiation is alive and well, and this research fits within the frames drawn by (Goffman 1959) and (Tajfel & Turner 1986) and furthered by (boyd 2002, boyd 2008). We perform and we understand ourselves in part by understanding the reflections that come back to us from others (Marchionini 2009). In a fast-moving networked workplace, this ability to gain insight into the knowledge of others with a simple trustable lookup may prove valuable. Tapping into the collective understanding and distilled opinion of those around us could be a useful tool or sanity check against both direct and indirect individual claims of expertise. Equally, it could serve as a weapon against misplaced modesty, allowing us to collectively reward those who deserve to be given credit when credit is due. What remains an open question is whether this type of collective opinion mapping works in an environment beyond the walls of the relatively small, trusted workplace, where people know one another (stable identity) and have many incentives to behave and only say positive, professional things about one another ("the shadow of the future" (Axelrod 1984)). I hope to begin answering this larger question soon with work based on the open internet. We now live in an ever-shrinking world of always-on connectivity and powerful communication devices. Since these devices are two-way, they provide a voice (and a distribution platform) to millions who, prior, have never had a voice. This is a remarkable achievement and serves as a testament to the incredible advance of technology and our collective striving for equality with regards to opinions and freedom of speech. However, with monumental increases in the number of voices and opinions being shared, we demand a requisite increase in the power of tools to help us filter all this newfound information. We need good knobs to help us determine where to direct our always-limited and increasingly precious amount of attention. The freedom to listen to anyone has to be balanced with the practicality of not being able to listen to everyone. We need tools that help us serve both of these needs, albeit not at the same time. The tools need to be flexible enough to let us listen to whomever, whenever and wherever we want, and to reserve the right to change our minds at a later time. Finding good sources of information is hard. Knowing whom to listen to when the subject matter is beyond one's personal experience is a daunting and important problem, but one that can be reduced to an engineering problem with the right approach. Figure 2 . 2Figure 2. Sample results from the labeling exercise demonstrating the Self and Group lists. Figure 3 . 3Similarity Rating Flow: For both the Human-Sim and TurkSim (next section) ratings, a Self list and a Group list were compared against one another and rated on a 1-7 scale with a high rating of 7. Figure 4 4Figure 4. HumanSim: Trained humans determined that random pairings of Self and Group were dissimilar and that the Self and Group lists for the same individual were similar. When the labels were weighted, the similarity ratings were more tightly clustered at the high end of the scale. Figure 5 . 5TurkSim: Untrained Mechanical Turk workers evaluated the two sets of labels about study participants' areas of expertise to be similar (average rating of 5) with the range tightening noticeably as the rounds continued. Table 1 . 1AlgSim: Repeated measures ANOVA by RoundDf Sum Sq Mean Sq F value Pr(>F) Round 1 0.32 0.32 10.22 0.0015 Residuals 286 8.85 0.03 The repeated measures analysis of variance (Table 1) of the AlgSim scores across time (represented by Round) Table 2 . 2AlgSim: Post-hoc ANOVA p-values by RoundRound 1 Round 2 Round 3 Round 4 Round 2 0.0249* - - - Round 3 0.0058* 0.6428 - - Round 4 0.0016 0.3906 0.6783 - Round 5 0.0014 0.3480 0.6062 0.9072 Table 3 . 3Survey: Items from Selected ScalesResult Demonstrability (Moore & Benbasat 1991) -I would have no difficulty telling others about the results of using this system. -I believe I could communicate to others the consequences of using this system. -The results of using this system are apparent to me. -I would have difficulty explaining why using this system may or may not be beneficial. (reverse coded) Relative Advantage (Moore & Benbasat 1991) -Using this system would enable me to accomplish tasks more quickly. -Using this system would improve the quality of work I do. -Using this system would make it easier to do my job. -Using this system would enhance my effectiveness on the job. -Using this system would give me greater control over my work. Performance Expectancy (Venkatesh, Morris, Davis & Davis 2003) Table 4 . 4Survey Scales and RatingsOriginal Items Average Rating I am comfortable with my group's tags about my areas of expertise. 5.439 I am happy with my group's tags about my areas of expertise. 5.351 I am familiar with my group members' ar- eas of expertise. 5.333 This was an interesting exercise. 5.196 My group members are familiar with my areas of expertise. 5.175 My group did not list important areas of my expertise. 4.764 I am confident that this system gives me new information. 4.696 This was a useful exercise. 4.679 I am confident that this system gives me good information. 4.643 I am willing to incorporate output from this system into my decision making. 4.607 I would be more comfortable with my group's tags if the tags were not anony- mous. 3.298 Scale Average Rating Data Quality 4.709 Effort Expectancy 4.670 Result Demonstrability 4.299 Facilitating Conditions 4.250 Performance Expectancy 3.836 Relative Advantage 3.742 Anxiety (reverse coded) 3.036 The Evolution of Cooperation, Basic Books. 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[ "Modified Cross-Correlation for Efficient White-Beam Inelastic Neutron Scattering Spectroscopy", "Modified Cross-Correlation for Efficient White-Beam Inelastic Neutron Scattering Spectroscopy" ]	[ "K Tomiyasu \nDepartment of Physics\nTohoku University\n980-8578AobaSendaiJapan\n", "M Matsuura \nInstitute for Materials Research\nTohoku University\n980-8577AobaSendaiJapan\n", "H Kimura \nInstitute for Multidisciplinary Research for Advanced Materials\nTohoku University\n980-8577AobaSendaiJapan\n", "K Iwasa \nDepartment of Physics\nTohoku University\n980-8578AobaSendaiJapan\n", "K Ohoyama \nInstitute for Materials Research\nTohoku University\n980-8577AobaSendaiJapan\n", "T Yokoo \nNeutron Science Laboratory\nHigh Energy Accelerator Organization\n305-0801OhoTsukubaJapan\n", "S Itoh \nNeutron Science Laboratory\nHigh Energy Accelerator Organization\n305-0801OhoTsukubaJapan\n", "E Kudoh \nDepartment of Information and Communication Engineering\nTohoku Institute of Technology\n982-8577TaihakuSendaiJapan\n", "T Sato \nWPI Advanced Institute for Materials Research\nTohoku University\n980-8577AobaSendaiJapan\n", "M Fujita \nInstitute for Materials Research\nTohoku University\n980-8577AobaSendaiJapan\n" ]	[ "Department of Physics\nTohoku University\n980-8578AobaSendaiJapan", "Institute for Materials Research\nTohoku University\n980-8577AobaSendaiJapan", "Institute for Multidisciplinary Research for Advanced Materials\nTohoku University\n980-8577AobaSendaiJapan", "Department of Physics\nTohoku University\n980-8578AobaSendaiJapan", "Institute for Materials Research\nTohoku University\n980-8577AobaSendaiJapan", "Neutron Science Laboratory\nHigh Energy Accelerator Organization\n305-0801OhoTsukubaJapan", "Neutron Science Laboratory\nHigh Energy Accelerator Organization\n305-0801OhoTsukubaJapan", "Department of Information and Communication Engineering\nTohoku Institute of Technology\n982-8577TaihakuSendaiJapan", "WPI Advanced Institute for Materials Research\nTohoku University\n980-8577AobaSendaiJapan", "Institute for Materials Research\nTohoku University\n980-8577AobaSendaiJapan" ]	[]	We describe a method of white-beam inelastic neutron scattering for improved measurement efficiency. The method consists of matrix inversion and selective extraction. The former is to resolve each incident energy component from the white-beam data, and the latter eliminates contamination by elastic components, which produce strong backgrounds that otherwise obfuscate the inelastic scattering components. In this method, the optimal experimental condition to obtain high efficiency will strongly depend on the specific aim of the individual experiments.PACS numbers:	10.1016/j.nima.2012.03.001	[ "https://arxiv.org/pdf/1111.4281v3.pdf" ]	119,300,615	1111.4281	68529d9769a6e13a8f663fa08aefd684a692b504	Modified Cross-Correlation for Efficient White-Beam Inelastic Neutron Scattering Spectroscopy 22 Mar 2012 K Tomiyasu Department of Physics Tohoku University 980-8578AobaSendaiJapan M Matsuura Institute for Materials Research Tohoku University 980-8577AobaSendaiJapan H Kimura Institute for Multidisciplinary Research for Advanced Materials Tohoku University 980-8577AobaSendaiJapan K Iwasa Department of Physics Tohoku University 980-8578AobaSendaiJapan K Ohoyama Institute for Materials Research Tohoku University 980-8577AobaSendaiJapan T Yokoo Neutron Science Laboratory High Energy Accelerator Organization 305-0801OhoTsukubaJapan S Itoh Neutron Science Laboratory High Energy Accelerator Organization 305-0801OhoTsukubaJapan E Kudoh Department of Information and Communication Engineering Tohoku Institute of Technology 982-8577TaihakuSendaiJapan T Sato WPI Advanced Institute for Materials Research Tohoku University 980-8577AobaSendaiJapan M Fujita Institute for Materials Research Tohoku University 980-8577AobaSendaiJapan Modified Cross-Correlation for Efficient White-Beam Inelastic Neutron Scattering Spectroscopy 22 Mar 2012(Dated: March 23, 2012)PACS numbers: We describe a method of white-beam inelastic neutron scattering for improved measurement efficiency. The method consists of matrix inversion and selective extraction. The former is to resolve each incident energy component from the white-beam data, and the latter eliminates contamination by elastic components, which produce strong backgrounds that otherwise obfuscate the inelastic scattering components. In this method, the optimal experimental condition to obtain high efficiency will strongly depend on the specific aim of the individual experiments.PACS numbers: I. INTRODUCTION Inelastic neutron scattering has come to be recognized as indispensable in modern materials science, because a material's spin and lattice dynamics provides unique information about a system's Hamiltonian. A complete description of these excitations in momentum (Q) and energy (E) space is needed to fully reconstruct the interactions that govern a material's behavior on the atomic scale. However, the technique normally requires a large volume of sample, often on the order of several cubic centimeters 1 . This is a very significant limitation in research to develop new materials with novel functions. In neutron diffraction experiments, however, the development of time-of-flight (TOF) technique allowed the use of a white beam for increase in measurement efficiency compared with the conventional method using a monochromatic beam. Each wavelength (energy) component can be resolved by TOF, as shown in Fig. 1(a), and finally merged into a single diffraction \|Q\| pattern for a powder sample (time focusing) or a Q map for a single-crystal sample. Unfortunately, the same cannot be applied to inelastic scattering, because the different E i (incident energy) and E f (final energy) components are entangled at the same TOF for the same pixel on the detector, as shown in Fig. 1(b). Thus, it has been considered that either E i or E f must be monochromatized or must be analyzed, either of which incurs a large loss in neutron intensity. Another remarkable method, called cross-correlation, was developed over four decades ago as an extension to the white-beam diffraction 2 . The method basically involves extracting the elastic components and removing the inelastic components 3 . As shown in Fig. 2, a special mechanical chopper modulates a white incident pulsed beam with a pseudorandom open/close sequence, and Ntimes cyclic phase shifts of the modulation generate a set of N data with different E i contrast. Then, on the basis of the contrast, the data for each E i can be mathematically resolved. The mathematical formalization is given below. Here, for convenience, parameters and functions are renamed and redefined from those in the original papers 2,3 . The intensity detected at a specific TOF at a specific pixel of the detector, I obs (p) (p = 1, . . . , N ), is described by I obs (p) = N j=1 F (j + p)I(j) + B(1) where p is the phase shift of the sequence (phase of sequence chopper), F (k) is the k-th element in the sequence F consisting of only 0 (close) and 1 (open), F (k + N ) is defined to be equal to F (k) for k = 1, . . . , N , j is an index for E i , I(j) is the intensity coming from the j-th E i component in a white incident pulsed beam for j = 1, . . . , N , and B is the background. The pseudorandom sequence F is restricted by N = 2 n − 1 (n : integer), (2a) F ′ (k) = 2F (k) − 1, (2b) N k=1 F ′ (k) = 1, (2c) N k=1 F ′ (k)F ′ (k + k ′ ) = (N + 1)δ 0,k ′ − 1. (2d) This type of sequence F is currently called a maximum length sequence, which is generated by a simple recurrence formula and is widely applied in the field of digital communications 4 . Combining the above equations, one can resolve each E i component, I(j) = 2 N + 1 N p=1 F ′ (j + p)I obs (p) − 2 N + 1 B.(3) It should be noted, however, that the method cannot be directly applied to inelastic scattering because the elastic components and their large statistical errors obfuscate the very weak inelastic components. This is probably why the method has not been realized thus far in an actual instrument dedicated to inelastic scattering. In fact, for the new CORELLI instrument under construction at the Spallation Neutron Source (SNS) in Oak Ridge, Tennessee, the method will be used mainly to study diffusive elastic scattering such as in frustrated systems and ionic conductors 5 . This paper presents a modification to this method for a more practical white-beam inelastic neutron scattering setup. The proposed method has two novel aspects: the introduction of an inverse matrix representation and a proposed method for selective extraction. The former affords a different solution to Eq. (1) to resolve each E i component in the white-beam data. The latter eliminates contamination by elastic components, which otherwise produce strong backgrounds. Finally, we present estimates of some instrumental specifications for the TOF polarized neutron spectrometer, POLANO, being constructed at J-PARC. II. INVERSE MATRIX REPRESENTATION We formalize an alternative solution to Eq. (1) without the conditions Eqs. (2a)-(2d). Here, the measurement of I obs (p) (p = 1, . . . , N ) is the same as in the original method except for the kind of sequence. Ignoring B for simplicity, Eq. (1) can be represented by I obs =F I,(4) where I obs is the vector [I obs (p)] (p = 1, . . . , N );F is the matrix [F p=1 , F p=2 , . . . , F p=N ]; F p is the sequence vec- tor [F (j + p)] (j = 1, . . . , N ); and I is the vector [I(j)]. Hence, one can resolve I by I =F −1 I obs .(5) Consider, for example, the sequence (0, 1, 1, 0, 1) at p = 1:      I obs (1) I obs (2) I obs (3) I obs (4) I obs (5)      =      0 1 1 0 1 1 0 1 1 0 0 1 0 1 1 1 0 1 0 1 1 1 0 1 0           I(1) I(2) I(3) I(4) I(5)      .(6) Hence, one can obtain      I(1) I(2) I(3) I(4) I(5)      = 1 3      −1 −1 −1 2 2 2 −1 −1 −1 2 2 2 −1 −1 −1 −1 2 2 −1 −1 −1 −1 2 2 −1           I obs (1) I obs (2) I obs (3) I obs (4) I obs (5)      .(7) Thus, almost all types of sequences can be used as long as F −1 exists. Taking into account B again, one can also identify a sequence to minimize \|F −1 (B, B, B, B, B)\|, for example, by trial and error with many numerical trials. It should be noted that this general matrix formalization is not considered superior to the maximum length sequence. However, the general matrix formalization does afford an advantage when the conditions of Eqs. (2a)−(2d) are not satisfied on actual instrumentation, for example, because of insufficient switching speed between 0 and 1 for high E i range or high resolution. In this paper, we use the general matrix formalization only because selective extraction, proposed in the next section, also does not fulfill the conditions. III. SELECTIVE EXTRACTION We explain the proposed selective extraction method using the above example in Eq. (6). First, one needs to prepare another chopper with the inverted sequencefrom open/close to close/open, that is, from 1/0 to 0/1. The inverted chopper gives another data set,      J obs (1) J obs (2) J obs (3) J obs (4) J obs (5)      =      1 0 0 1 0 0 1 0 0 1 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1           I(1) I(2) I(3) I(4) I(5)      ,(8) where [J obs (p)] is the raw data obtained for the phase p. Then, we consider the case where I(3) is the elastic component for the targeted TOF and pixel of the detector. Our purpose is to remove I(3). Thus, by selectively extracting only the arrays in which the third column is 0 (sequence chopper closed) from Eqs. (6) and (8), one can reconstruct a good quality data set:      J obs (1) J obs (2) I obs (3) J obs (4) I obs (5)      =      1 0 0 1 0 0 1 0 0 1 0 1 0 1 1 0 1 0 1 0 1 1 0 1 0           I(1) I(2) I(3) I(4) I(5)      .(9) Hence,      J obs (1) J obs (2) I obs (3) J obs (4) I obs (5)      =      1 0 1 0 0 1 0 1 0 1 1 1 0 1 1 0 1 1 1 0         I(1) I(2) I(4) I(5)    .(10) This equation can be solved by dropping one array and using the inverse matrix, or by the least squares method. Further, we can reincorporate the background term B:      J obs (1) J obs (2) I obs (3) J obs (4) I obs (5)      =      1 0 1 0 1 0 1 0 1 1 0 1 1 1 1 0 1 1 0 1 1 1 1 0 1           I(1) I(2) I(4) I(5) B      .(11) Hence,      I(1) I(2) I(4) I(5) B      =      0 0 0 −1 1 −1 0 0 0 1 0 −1 1 0 0 0 0 1 −1 0 1 1 −1 1 −1           J obs (1) J obs (2) I obs (3) J obs (4) I obs (5)      .(12) We emphasize that the selective extraction method is applicable not only when the third channel is elastic but also when an arbitrary channel is elastic. One can remove the elastic components at all TOFs and pixels with only the two data sets. IV. STATISTICAL EFFICIENCY Price and Sköld reported that the cross-correlation method is not always better than the conventional monochromatic method in terms of statistical efficiency 6 (5) in Eq. (7), and {∆I(1)} 2 = (−1) 2 · J obs (4) + 1 2 · I obs (5) in Eq. (12), where ∆I(1) denotes a statistical error of I(1). In addition, since N measurements with cyclic rotation of sequence are needed to resolve [I obs (p)] to [I(j)] with only one-measurement statistics, the use of white beam does not overall multiply the measurement efficiency so much. In the use of maximum length sequence, a statistical advantage can be obtained only for special j(s) (E i channel(s)), in which the signal of interest is more than twice the average counts per channel 6 . Hence, the cross-correlation methods would be suitable only in the cases of phonon resonance, magnon resonance, or elastic scattering. The above situation is essentially the same as that in our modified cross-correlation method with selective extraction. Here, we roughly estimate the statistical efficiency based on the assumption that [I(j)] consists of only inelastic scattering counts with similar magnitude and without huge elastic components. First, in the cross-correlation methods without selective extraction including the original one and the aforementioned generalmatrix one, the error of I (nsl) (j) is described by {∆I (nsl) (j)} 2 = N p=1 (Â −1 jp ) 2 I (nsl) obs (p) ∼ a 2 j I (nsl) obs ∼ N open a 2 j I (nsl) ,(13) where the superscript (nsl) denotes the non-use of selective extraction,Â −1 corresponds to the inverse matrix, a 2 j = N p=1 (Â −1 jp ) 2 , I(nsl)η (nsl) j ≡ I (nsl) (j) ∆I (nsl) (j) ∼ c j N open a j · I (nsl) ∼ √ 2c j √ N a j · I (nsl) ,(14) where I (nsl) (j) ≡ c j I (nsl) . Next, in the modified crosscorrelation method, the error of I (sl) (j) is described by {∆I (sl) (j)} 2 = N p=1 (B −1 j ′ p ) 2 {K (sl) obs (p)} ∼ N p=1 (B −1 j ′ p ) 2 {I (nsl) obs (p)/2} ∼ 1 2 b 2 j ′ I (nsl) obs ∼ 1 2 N open b 2 j ′ I (nsl) ,(15) where the superscript (sl) denotes the use of selective extraction, K 2 j ′ = N p=1 (B −1 j ′ p ) 2 , and j ′ denotes the row number in B −1 with which I (sl) (j) can be obtained (e.g., j ′ = 1 for j = 1 and j ′ = 3 for j = 4 in Eq. (12)). Hence, the statistical efficiency η (sl) j is estimated by η (sl) j ≡ I (sl) (j) ∆I (sl) (j) = I (nsl) (j)/2 ∆I (sl) (j) ∼ c j √ 2 N open b j ′ · I (nsl) ∼ c j √ N b j ′ · I (nsl) ,(16) where I (sl) (j) = I (nsl) (j)/2 since selective extraction involves the use of two data sets again. Then, in the conventional monochromatic experiments, the efficiency η (mono) j required to obtain the same data set of all E i (j = 1, . . . , N ) over the same total measurement time using monochromatic beam is estimated by η (mono) j = c j I (nsl) c j I (nsl) = √ c j · I (nsl) ,(17) which corresponds to the measurements described bŷ A =Ê (identity matrix) in the general matrix formalization. Thus, the ratios of the statistical efficiencies are estimated by η (nsl) j η (mono) j ∼ 2c j N a 2 j , (18a) η (sl) j η (mono) j ∼ c j N b 2 j ′ . (18b) Using example sequences summarized in the Appendix, we numerically calculated the values of a j and b j ′ and the criteria of c j to retrieve the I(j) of interest more efficiently than the use of conventional monochromatic beam. For the original cross-correlation method with maximum length sequence without selective extraction, the results are a 2 j = 0.44, 0.23, 0.12, 0.062, 0.031, and 0.016 for all j at N = 7, 15, 31, 63, 127, and 255, respectively. Hence, (η ) > 1 (j = N : I(j)) gives c j > 6.2, 5.8, 6.7, 9.4, 12, and 15. The latter criteria are harder than the former ones in the absence of huge elastic components, as expected. In addition, since c j increases as N increases in the latter, it would be better to set N less than about 60. In this way, both the original and modified crosscorrelation methods can efficiently give only the I(j) components with relatively large c j among [I(j)]. Therefore, it is important to tune the experimental conditions, such as the ranges of E i , TOF, and pixel used, such that the components of interest become the strongest in intensity among [I(j)]. In this sense, we would like to emphasize that the modified method with selective extraction can remove not only an elastic channel but also an arbitrary one channel of no interest with relatively strong intensity among [I(j)]; for example, spurious neutrons scattered on unexpected paths and spurious neutrons coming from the previous flame. In practice, (1) for elastic scattering, the original method without selective extraction could be safely applied, as has been reported in the past 2,3,5,6 , since elastic scattering is normally the strongest among [I(j)]. However, the modified method might improve the data as an insurance to remove a spurious channel. (2) For quasielastic scattering and low-energy inelastic scattering, which will have the next strongest intensity and will necessarily overlap with the strongest elastic scattering in TOF, the modified method will be effective to remove the elastic scattering. (3) For higher energy modes with relatively weak intensity, one must remove not only an elastic channel but also quasielastic and low-energy inelastic channels, which contaminate the higher energy data as a source of statistical errors. Therefore, it would be better to narrow the E i range from white to quasimonochromatic, which will allow us to avoid all the elastic, quasielastic, and low energy inelastic scatterings by TOF, in conjunction with the modified method to remove an arbitrary channel of no interest again. In any case, to efficiently utilize the cross-correlation methods, users should recognize that the optimal experimental condition will highly depend on the specific aim of the individual experiments. V. SPECIFICATIONS FOR PRACTICAL IMPLEMENTATION We are constructing a TOF polarized neutron spectrometer called POLANO at a decoupled moderator at J-PARC. Because polarization devices such as Heusler crystals and spin filters lose a large proportion of neutrons, we need a method to gain an increase of over a factor of 10 in measurement efficiency. Our method of white-beam inelastic neutron scattering is one candidate, though it is applicable to both unpolarized and polarized inelastic neutron scattering. In this section, taking this spectrometer as an example, we present some specifications for the sequence chopper. For this system, we set the distance between moderator and sample L 1 as 17.0 m, the distance between sample and detector L 2 as 2.0 m, and the distance between sequence chopper and sample L 3 as 2.0 m. The time width values at the decoupled moderator at J-PARC, ∆t m , were used for each E i (Table I). For each E i , the energy resolution ∆E/E i was kept ≤ 0.04. Then, using the analytical formula for energy resolution 7 , we evaluated the opening time per E i channel required at the sequence chopper, ∆t sc1 . Also, on the assumption that a sequence chopper is alternately closed without generating a sequence as an example, as shown in Fig. 3, we estimated the condition of ∆t sc2 necessary to avoid contamination by the elastic tails from neighboring E i channels. The condition can be estimated by ∆t d ≤ ∆t ch , where ∆t d is the time width of the elastic component at the pixel on the detector with E i , and ∆t ch is the time difference between two TOFs of neighboring E i channels at the pixel, as defined in Fig. 3. The two parameters are described by other parameters, ∆t d = L 1 + L 2 − ∆tm ∆tm+∆tsc2 (L 1 − L 3 ) ∆tm ∆tm+∆tsc2 (L 1 − L 3 ) · ∆t m ,(19a) ∆t ch = L 1 + L 2 L 1 − L 3 · (2∆t sc2 ) .(19b) As summarized in Table I, there exists a solution for ∆t sc . For example, over a wide E i range of 10-80 meV, ∆t sc = 9 µsec simultaneously satisfies the constraint of ∆E/E i ≤ 0.04 and the avoidance of the tails from neighboring E i channels. This opening time can be realized by a set of counter-rotating disk choppers with the following parameters: 700 mm-φ, 20 mm/channel, and 350 Hz 8 . In addition, for the sequence chopper, note that an arbitrary sequence can be generated by printing a sequence clockwise and then counterclockwise on counterrotating disk choppers. The sequence chopper generates 0.5 · (π · 700 (mm))/20 (mm) = 55 E i channels. VI. SUMMARY We developed a modified cross-correlation method for an increase in measurement efficiency of inelastic neutron scattering. First, a different solution using an inverse matrix representation was formalized to resolve each E i component in the white-beam data. Second, a method of selective extraction was proposed to avoid contamination by elastic components. Third, taking spectrometer POLANO at J-PARC as an example, practical specifications for the sequence chopper were estimated. Experimental situations giving high efficiency, however, would be quite limited and complex. The example sequences used for the calculations in Sec. 4 are summarized. For the original cross-correlation method, (1, 0, 0, 1, 1, 1, 0) at N = 7, (1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0) at N = 15, (0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0) at N = 31, (1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0) at N = 63, (1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, FIG. 1 : 1(Color online) Plot of TOF against position without sequence chopper. Solid arrows indicate the most probable neutrons. Dotted lines sectionalize each Ei channel. (a) Diffraction. (b) Inelastic scattering. FIG. 2 : 2(Color online) Example of TOF-position diagrams with sequence chopper (N = 5) at sequence phases p = 1 (a) and p = 2 (b). By cyclically shifting the phase, a raw data set of I obs (1), I obs (2), I obs(3), I obs (4), and I obs (5) is obtained at a specific TOF and at a specific pixel on the detector. After the measurements, each Ei component of I(1), I(2), I(3), I(4), and I(5) can be resolved mathematically. . The reason is that white-beam data counts [I obs (p)] and [J obs (p)] are inevitably large and are accompanied by large statistical errors [∆I obs (p)] = I obs (p) and [∆J obs (p)] = J obs (p) , which are propagated to [I(j)] with further enhancement by adding and subtracting [I obs (p)] and [J obs (p)]. For example, {∆I(1)} 2 = (−1/3) 2 · I obs (1) + (−1/3) 2 · I obs (2) + (−1/3) 2 · I obs (3) + (2/3) 2 ·I obs (4)+(2/3) 2 ·I obs open is the number of opening channels (∼ N/2), and I (nsl) = N j=1 I (nsl) (j)/N . Hence, the statistical effi the use of two data sets: one from the original chopper and the other from the inverted chopper,B −1 corresponds to the final inverse matrix like in Eq. (12), b Price's criterion of c j > 2 for all the N ; the I(j) of interest should be more than twice the average counts per channel 6 . For the modified crosscorrelation method, b 2 j ′ = 0.88, 0.39, 0.22, 0.15, 0.096, and 0.059 (j ′ = N : I(j)) and b 2 N = 0.28, 0.31, 0.63, 0.78, 0.20, and 0.30 (j ′ = N : constant background). FIG. 3 : 3(Color online) Diagram showing TOF and position with sequence chopper in the estimated specifications for POLANO. Solid arrows indicate the most probable neutrons, and dotted arrows indicate most inaccurate neutrons. The sequence chopper is alternately closed using a set of counterrotating disk choppers. The alternating closing parts do not generate the sequence. TABLE I : IResults of numerical estimation of system specifications for POLANO. The values ∆tsc1 and ∆tsc2 are obtained so as to satisfy ∆E/Ei ≤ 0.04 and avoid contamination by the elastic tails from neighboring Ei channels, respectively. All the time widths are defined as full widths at half maximum.Ei (meV) ∆tm (µsec) ∆tsc1 (µsec) ∆tsc2 (µsec)5.0 50 ≤ 31 ≥ 11 10 32 ≤ 22 ≥ 7 20 20 ≤ 16 ≥ 4 40 13 ≤ 11 ≥ 3 80 9 ≤ 9 ≥ 2 AcknowledgmentsWe thank Dr S. Rosenkranz for the helpful discussion and Dr H. Hiraka, Professor Y. Noda, and Professor K. Yamada for their encouragement. This study was financially supported by Grants-in-Aid for Young Scientists (B) (22740209), Priority Areas (22014001), and Scientific Researches (S) (21224008) and (A) (22244039) from the MEXT of Japan. The study was also supported by the Neutron Scattering Program Advisory Committee of IMSS, KEK (2009S09). ) at N = 255. For the modified cross-correlation method, the original chopper sequences are. at N = 31N = 127. 127N = 63. 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1) at N = 255, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0) at N = 127, (1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0) at N = 255. For the modified cross-correlation method, the original chopper sequences are (1, 1, 1, 1, 0, 1, 1) at N = 7, (1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1) at N = 15, (1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1) at N = 31, (1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1) at N = 63, (0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0) at at N = 127, (0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1) at N = 255. . K Tomiyasu, M Fujita, A I Kolesnikov, R I Bewley, M J Bull, S M Bennington, Appl. Phys. Lett. 9492502and references thereinK. Tomiyasu, M. Fujita, A. I. Kolesnikov, R. I. Bewley, M. J. Bull, and S. M. Bennington, Appl. Phys. Lett. 94, 092502 (2009), and references therein. . K Sköld, Nucl. Instrum. Methods. 63114K. Sköld, Nucl. Instrum. Methods 63, 114 (1968). . E G , P Pellionisz, Nucl. Instrum. Methods. 92125E. g., P. Pellionisz, Nucl. Instrum. Methods 92, 125 (1971). S W Golomb, Shift Register Sequences. San FranciscoHolden-Day, IncS. W. Golomb, Shift Register Sequences (Holden-Day, Inc., San Francisco, 1967). . S Rosenkranz, R Osborn, Pramana J. Phys. 71705S. Rosenkranz and R. Osborn, Pramana J. Phys. 71, 705 (2008). . D L Price, K Sköld, Nucl. Inst. and Methods. 82208D. L. Price and K. Sköld, Nucl. Inst. and Methods 82, 208 (1970). . K Tomiyasu, S Itoh, Physica B. 1110and references thereinK. Tomiyasu and S. Itoh, Physica B 385-386, 1110 (2006), and references therein. /{(π ·700(mm/cycle))(2·350(cycles/sec))}. /{(π ·700(mm/cycle))(2·350(cycles/sec))}.	[]
[ "Observation of the doubly radiative decay η ′ → γγπ 0", "Observation of the doubly radiative decay η ′ → γγπ 0", "Observation of the doubly radiative decay η ′ → γγπ 0", "Observation of the doubly radiative decay η ′ → γγπ 0" ]	[ "50a Destefanis ", "F. De 50c ", "Mori 50a ", "Y ; R 50c ", "Farinelli 21A", "L 21b ", "Fava 50B", "F 50c ", "G Feldbauer 22 ", "C Q. ; S Felici 20A", "Sosio 50A", "S 50c ", "Spataro 50A", "G X 50c ", "\nCOMSATS Institute of Information Technology\nDefence Road, Off Raiwind Road54000Lahore, LahorePakistan\n", "50a Destefanis ", "F. De 50c ", "Mori 50a ", "Y ; R 50c ", "Farinelli 21A", "L 21b ", "Fava 50B", "F 50c ", "G Feldbauer 22 ", "C Q. ; S Felici 20A", "Sosio 50A", "S 50c ", "Spataro 50A", "G X 50c ", "\nCOMSATS Institute of Information Technology\nDefence Road, Off Raiwind Road54000Lahore, LahorePakistan\n" ]	[ "COMSATS Institute of Information Technology\nDefence Road, Off Raiwind Road54000Lahore, LahorePakistan", "COMSATS Institute of Information Technology\nDefence Road, Off Raiwind Road54000Lahore, LahorePakistan" ]	[ "S. Pacetti 20B , Y. Pan", "S. Pacetti 20B , Y. Pan" ]	Based on a sample of 1.31 billion J/ψ events collected with the BESIII detector, we report the study of the doubly radiative decay η ′ → γγπ 0 for the first time, where the η ′ meson is produced via the J/ψ → γη ′ decay. The branching fraction of η ′ → γγπ 0 inclusive decay is measured to be B(η ′ → γγπ 0 ) Incl. = (3.20 ± 0.07(stat) ± 0.23(sys)) × 10 −3 , while the branching fractions of the dominant process η ′ → γω and the nonresonant component are determined to be B(η ′ → γω) × B(ω → γπ 0 ) = 3 (23.7 ± 1.4(stat) ± 1.8(sys)) × 10 −4 and B(η ′ → γγπ 0 )NR = (6.16 ± 0.64(stat) ± 0.67(sys)) × 10 −4 , respectively. In addition, the M 2 γγ -dependent partial widths of the inclusive decay are also presented.	10.1103/physrevd.96.012005	[ "https://arxiv.org/pdf/1612.05721v5.pdf" ]	119,406,066	1612.05721	805fb0198aa8f1149579583c577cdf63cd4180c7	Observation of the doubly radiative decay η ′ → γγπ 0 3 Aug 2017 50a Destefanis F. De 50c Mori 50a Y ; R 50c Farinelli 21A L 21b Fava 50B F 50c G Feldbauer 22 C Q. ; S Felici 20A Sosio 50A S 50c Spataro 50A G X 50c COMSATS Institute of Information Technology Defence Road, Off Raiwind Road54000Lahore, LahorePakistan Observation of the doubly radiative decay η ′ → γγπ 0 S. Pacetti 20B , Y. Pan X. Y. Song403 Aug 2017(BESIII Collaboration) 2numbers: 1340Gp1340Hq1320Jf1440Be Based on a sample of 1.31 billion J/ψ events collected with the BESIII detector, we report the study of the doubly radiative decay η ′ → γγπ 0 for the first time, where the η ′ meson is produced via the J/ψ → γη ′ decay. The branching fraction of η ′ → γγπ 0 inclusive decay is measured to be B(η ′ → γγπ 0 ) Incl. = (3.20 ± 0.07(stat) ± 0.23(sys)) × 10 −3 , while the branching fractions of the dominant process η ′ → γω and the nonresonant component are determined to be B(η ′ → γω) × B(ω → γπ 0 ) = 3 (23.7 ± 1.4(stat) ± 1.8(sys)) × 10 −4 and B(η ′ → γγπ 0 )NR = (6.16 ± 0.64(stat) ± 0.67(sys)) × 10 −4 , respectively. In addition, the M 2 γγ -dependent partial widths of the inclusive decay are also presented. Based on a sample of 1.31 billion J/ψ events collected with the BESIII detector, we report the study of the doubly radiative decay η ′ → γγπ 0 for the first time, where the η ′ meson is produced via the J/ψ → γη ′ decay. The branching fraction of η ′ → γγπ 0 inclusive decay is measured to be B(η ′ → γγπ 0 ) Incl. = (3.20 ± 0.07(stat) ± 0.23(sys)) × 10 −3 , while the branching fractions of the dominant process η ′ → γω and the nonresonant component are determined to be B(η ′ → γω) × B(ω → γπ 0 ) = (23.7 ± 1.4(stat) ± 1.8(sys)) × 10 −4 and B(η ′ → γγπ 0 )NR = (6.16 ± 0.64(stat) ± 0.67(sys)) × 10 −4 , respectively. In addition, the M 2 γγ -dependent partial widths of the inclusive decay are also presented. I. INTRODUCTION The η ′ meson provides a unique stage for understanding the distinct symmetry-breaking mechanisms present in low-energy quantum chromodynamics (QCD) [1][2][3][4][5] and its decays play an important role in exploring the effective theory of QCD at low energy [6]. Recently, the doubly radiative decay η ′ → γγπ 0 was studied in the frameworks of the linear σ model (LσM) and the vector meson dominance (VMD) model [7,8]. It has been demonstrated that the contributions from the VMD are dominant. Experimentally, only an upper limit of the nonresonant branching fraction of B(η ′ → γγπ 0 ) NR < 8 × 10 −4 at the 90% confidence level has been determined by the GAMS-2000 experiment [9]. In this article, we report the first measurement of the branching fraction of the inclusive η ′ → γγπ 0 decay and the determination of the M 2 γγ dependent partial widths, where M γγ is the invariant mass of the two radiative photons. The inclusive decay is defined as the η ′ decay into the final state γγπ 0 including all possible intermediate contributions from the ρ and ω mesons below the η ′ mass threshold and the nonresonant contribution from the excited vector meson above the η ′ mass threshold. Since the contribution from mesons above the η ′ threshold actually derives from the low-mass tail and looks like a contact term, we call this contribution 'nonresonant'. The branching fraction for the nonresonant η ′ → γγπ 0 decay is obtained from a fit to the γπ 0 invariant mass distribution by excluding the coherent contributions from the ρ and ω intermediate states. The measurement of the M 2 γγ dependent partial widths will provide direct inputs to the theoretical calculations on the transition form factors of η ′ → γγπ 0 and improve the theoretical understanding of the η ′ decay mechanisms. II. EXPERIMENTAL DETAILS The source of η ′ mesons is the radiative J/ψ → γη ′ decay in a sample of 1.31 × 10 9 J/ψ events [10,11] collected by the BESIII detector. Details on the features and capabilities of the BESIII detector can be found in Ref. [12]. The response of the BESIII detector is modeled with a Monte Carlo (MC) simulation based on geant4 [13]. The program evtgen [14] is used to generate a J/ψ → γη ′ MC sample with an angular distribution of 1+cos 2 θ γ , where θ γ is the angle of the radiative photon relative to the positron beam direction in the J/ψ rest frame. The decays η ′ → γω(ρ), ω(ρ) → γπ 0 are generated using the helicity amplitude formalism. For the nonresonant η ′ → γγπ 0 decay, the VMD model [7,8] is used to generate the MC sample with ρ(1450) or ω(1650) exchange. Inclusive J/ψ decays are generated with kkmc [15] generator; the known J/ψ decay modes are generated by evtgen [14] with branching fractions setting at Particle Data Group (PDG) world average values [16]; the remaining unknown decays are generated with lundcharm [17]. III. EVENT SELECTION AND BACKGROUND ESTIMATION Electromagnetic showers are reconstructed from clusters of energy deposits in the electromagnetic calorimeter (EMC). The energy deposited in nearby time-of-light (TOF) counters is included to improve the reconstruction efficiency and energy resolution. The photon candidate showers must have a minimum energy of 25 MeV in the barrel region (\| cos θ\| < 0.80) or 50 MeV in the end cap region (0.86 < \| cos θ\| < 0.92). Showers in the region between the barrel and the end caps are poorly measured and excluded from the analysis. In this analysis, only the events without charged particles are subjected to further analysis. The average event vertex of each run is assumed as the origin for the selected candidates. To select J/ψ → γη ′ , η ′ → γγπ 0 (π 0 → γγ) signal events, only the events with exactly five photon candidates are selected. To improve resolution and reduce background, a five-constraint kinematic (5C) fit imposing energymomentum conservation and a π 0 mass constraint is performed to the γγγπ 0 hypothesis, where the π 0 candidate is reconstructed with a pair of photons. For events with more than one π 0 candidate, the combination with the smallest χ 2 5c is selected. Only events with χ 2 5c < 30 are retained. The χ 2 5C distribution is shown in Fig. 1 with events in the η ′ signal region of \|M γγπ 0 − M η ′ \| < 25 MeV (M η ′ is the η ′ nominal mass from PDG [16]). In order to suppress the multi-π 0 backgrounds and remove the miscombined π 0 candidates, an event is vetoed if any two of five selected photons (except for the combination for the π 0 candidate) satisfies \|M γγ − M π 0 \| < 18 MeV/c 2 , where M π 0 is the π 0 nominal mass. After the application of the above requirements, the most energetic photon is taken as the primary photon from the J/ψ decay, and the remaining two photons and the π 0 are used to reconstruct the η ′ candidates. Figure 2 shows the γγπ 0 invariant mass spectrum. Detailed MC studies indicate that no peaking background remains after all the selection criteria. The sources of backgrounds are divided into two classes. Background events of class I are from J/ψ → γη ′ with η ′ decaying into final states other than the signal final states. These background events accumulate near the lower side of the η ′ signal region and are mainly from η ′ → π 0 π 0 η (η → γγ), η ′ → 3π 0 and η ′ → γγ, as shown as the (green) dotted curve in Fig. 2. Background events in class II are mainly from J/ψ decays to final states without η ′ , such as J/ψ → γπ 0 π 0 and J/ψ → ωη (ω → γπ 0 , η → γγ) decays, which contribute a smooth distribution under the η ′ signal region as displayed as the (pink) dotdashed curve in Fig. 2. IV. SIGNAL YIELDS AND BRANCHING FRACTIONS A fit to the γγπ 0 invariant mass distribution is performed to determine the inclusive η ′ → γγπ 0 signal yield. The probability density function (PDF) for the signal component is represented by the signal MC shape, which is obtained from the signal MC sample generated with an incoherent mixture of ρ, ω and the nonresonant components according to the fractions obtained in this analysis. Both the shape and the yield for the class I background are fixed to the MC simulations and their expected intensities. The shape for the class II background is described by a third-order Chebychev polynomial, and the corresponding yield and PDF parameters are left free in the fit to data. The fit range is 0.70−1.10 GeV/c 2 . Figure 2 shows the results of the fit. The fit quality assessed with the binned distribution is χ 2 /n.d.f = 108/95 = 1.14. The signal yield and the MC-determined signal efficiency for the inclusive η ′ decay are summarized in Table I. In this analysis, the partial widths can be obtained by studying the efficiency-corrected signal yields for each given M 2 γγ bin i for the inclusive η ′ → γγπ 0 decay. The resolution in M 2 γγ is found to be about 5×10 2 (MeV/c 2 ) 2 from the MC simulation, which is much smaller than 1.0 × 10 4 (MeV/c 2 ) 2 , a statistically reasonable bin width, and hence no unfolding is necessary. The η ′ signal yield in each M 2 γγ bin is obtained by performing bin-by-bin fits to the γγπ 0 invariant mass distributions using the fit procedure described above. Thus the background-subtracted, efficiency-corrected signal yield can be used to obtain the partial width for each given M 2 γγ interval, where the PDG value is used for the total width of the η ′ meson [16]. The results for dΓ(η ′ → γγπ 0 )/dM 2 γγ in each M 2 γγ interval are listed in Table II and depicted in Fig. 3, where the contributions from each component obtained from the MC simulations are normalized with the yields by fitting to M γπ 0 as displayed in Fig. 4. Assuming that the inclusive decay η ′ → γγπ 0 can be attributed to the vector mesons ρ and ω and the nonresonant contribution, we apply a fit to the γπ 0 invariant mass to determine the branching fraction for the nonresonant η ′ → γγπ 0 decay using the η ′ signal events with \|M γγπ 0 − m η ′ \| < 25 MeV/c 2 . In the fit, the ρω interference is considered, but possible interference between the ω (ρ) and the nonresonant process is neglected. To validate our fit, we also determine the product branching fraction for the decay chain η ′ → γω, ω → γπ 0 . Figure 4 shows the M γπ 0 distribution. Since the doubly radiative photons are indistinguishable, two entries are filled into the histogram for each event. For the PDF of the coherent ω and ρ produced in η ′ → γγπ 0 , we use [ε(M γπ 0 ) × E 3 γ η ′ × E 3 γ ω(ρ) × \|BW ω (M γπ 0 ) + αe iθ BW ρ (M γπ 0 )\| 2 ×B 2 η ′ ×B 2 ω(ρ) ]⊗G(0, σ), where ε(M γπ 0 ) is the detection efficiency determined by the MC simulations; E γ η ′ (ω/ρ) is the energy of the transition photon in I: Observed η ′ signal yields (N η ′ ) and detection efficiencies (ǫ) for inclusive η ′ → γγπ 0 , η ′ → γω(ω → γπ 0 ), and the nonresonant η ′ → γγπ 0 decays. The measured branching fractions c in this work, comparison of values from the PDG [16] and theoretical predictions are listed. The first errors are statistical and the second ones are systematic. -a The product branching fraction B(η ′ → γω) · B(ω → γπ 0 ). b The product branching fraction B(η ′ → γω) · B(ω → γπ 0 ) from PDG [16]. c The product branching fraction B(η ′ → γρ 0 ) · B(ρ 0 → γπ 0 ) is determined to be (1.92 ± 0.16(stat)) × 10 −4 using the fitted yield in Fig. 4, which is in agreement with the PDG value of (1.75 ± 0.23) × 10 −4 [16]. the rest frame of η ′ (ω/ρ); BW ω (M γπ 0 ) is a relativistic Breit-Wigner (BW) function, and BW ρ (M γπ 0 ) is a relativistic BW function with mass-dependent width [18]. The masses and widths of the ρ and ω meson are fixed to their PDG values [16]. B 2 η ′ (ω/ρ) is the Blatt-Weisskopf centrifugal barrier factor for the η ′ (ω/ρ) decay vertex with radius R = 0.75 fm [19,20], and B 2 η ′ (ω/ρ) is used to damp the divergent tail due to the factor E 3 γ η ′ (ω/ρ) . The Gaussian function G(0, σ) is used to parameterize the detector resolution. The combinatorial background is produced by the combination of the π 0 and the photon from the η ′ meson, and its PDF is described with a fixed shape from the MC simulation. The ratio of yields between the combinatorial backgrounds and the coherent sum of ρ-ω signals is fixed from the MC simulations. The shape of the nonresonant signal η ′ → γγπ 0 is determined from the MC simulation, and its yield is determined in the fit. The background from the class I as discussed above is fixed to the shape and yield of the MC simulation. Finally, the shape from the class II background is obtained from the η ′ mass sidebands (738−788 and 1008−1058 MeV/c 2 ), and its normalization is fixed in the fit. The M γπ 0 mass range used in the fit is 0.20−0.92 GeV/c 2 . In the fit, the interference phase θ between the ρ-and ω-components is allowed. Due to the low statistics of the ρ meson contribution, we fix the ratio α of ρ and ω intensities to the value for the ratio of B(η ′ → γρ) · B(ρ → γπ 0 ) and B(η ′ → γω) · B(ω → γπ 0 ) from the PDG [16]. Figure 4 shows the results. The yields for the vector mesons ρ, ω and their interference are determined to be (183 ± 15), (2340 ± 141), and (174 ± 92), respectively. The signal yields and efficiencies as well as the corresponding branching fractions for the η ′ → γω(ω → γπ 0 ) and nonresonant decays are summarized in Table I. η ′ → γγπ 0 (Inclusive) η ′ → γω, ω → γπ 0 η ′ → γγπ 0 (Nonresonant) N η ′ 3435 ± V. SYSTEMATIC UNCERTAINTIES The systematic uncertainties on the branching fraction measurements are summarized in Table III. The uncertainty due to the photon reconstruction is determined to be 1% per photon as described in Ref. [21]. The uncertainties associated with the other selection criteria, kinematic fit with χ 2 5C < 30, the number of photons equal to 5 and π 0 veto (\|M γγ − M π 0 \| > 18 MeV/c 2 ) are studied with the control sample J/ψ → γη ′ , η ′ → γω, ω → γπ 0 decay, respectively. The systematic error in each of the applied selection criteria is numerically estimated from the ratio of the number of events with and without the corresponding requirement. The corresponding resulting efficiency differences between data and MC (2.7%, 0.5%, and 1.9% , respectively) are taken to be representative of the corresponding systematic uncertainties. In the fit for the inclusive η ′ decay, the signal shape is fixed to the MC simulation. The uncertainty due to 2 ) 2 (GeV/c γ γ 2 M 0 0.2 0.4 0.6 ) 2 ) 2 (keV/(GeV/c γ γ 2 )/dM γ γ 0 π → ' η ( Γ d 0 1 2 3 4 Total ω γ → ' η ρ γ → ' η 0 π γ γ → ' η Non-resonantω γ → ' η ρ γ → ' η interference ω - ρ 0 π γ γ → ' η Non-resonant Combinatorial BG Class II background FIG. 4: Distribution of the invariant mass M γπ 0 and fit results in the η ′ mass region. The points with error bars are data; the (black) dotted-curve is from the ω-contribution; the (red) long dashed-curve is from the ρ-contribution; the (blue) short dashed-curve is the contribution of ρ-ω interference; the (green) long dashed curve is the nonresonance; the (pink) histogram is from the class II background; the (black) short dot-dashed curve is the combinatorial backgrounds of η ′ → γω, γρ. The (blue) solid line shows the total fit function. the signal shape is considered by convolving a Gaussian function to account for the difference in the mass resolution between data and MC simulation. In the fit to the γπ 0 distribution, alternative fits with the mass resolution left free in the fit and the radius R in the barrier factor changed from 0.75 fm to 0.35 fm are performed, and the changes of the signal yields are taken as the uncertainty due to the signal shape. In the fit to the M γγπ 0 distribution, the signal shape is described with an incoherent sum of contributions from processes involving ρ and ω and nonresonant processes obtained from MC simulation, where the nonresonant process is modeled with the VMD model. A fit with an alternative signal model for the different components, i.e. a coherent sum for the ρ-, ω-components and a uniform angular distribution in phase space (PHSP) for the nonresonant process, is performed. The resultant changes in the branching fractions are taken as the uncertainty related to the signal model. An alternate fit to the M γπ 0 distribution is performed, where the PDF of the nonresonant decay is extracted from the PHSP MC sample. The changes in the measured branching fractions are considered to be the uncertainty arising from the signal model. In the fit to the M γπ 0 distribution, the uncertainty due to the fixed relative ρ intensity is evaluated by changing its expectation by one standard deviation. An alternative fit in which the ratio of yields between combinatorial backgrounds and the coherent sum of ρ − ω signals is changed by one standard deviation from the MC simulation is performed, and the change observed in the signal yield is assigned as the uncertainty. A series of fits using different fit ranges is performed and the maximum change of the branching fraction is taken as a systematic uncertainty. The uncertainty due to the class I background is estimated by varying the numbers of expected background events by one standard deviation according to the errors on the branching fraction values in PDG [16]. The uncertainty due to the class II background is evaluated by changing the order of the Chebychev polynomial from 3 to 4 for the fit to the η ′ inclusive decay, and varying the ranges of η ′ sidebands for the fit to the γπ 0 invariant mass distribution, respectively. The number of J/ψ events is N J/ψ = (1310.6 ± 10.5) × 10 6 [10,11], corresponding to an uncertainty of 0.8%. The branching fractions for the J/ψ → γη ′ and π 0 → γγ decays are taken from the PDG [16], and the corresponding uncertainties are taken as a systematic uncertainty. The total systematic errors are 7.1%, 7.7%, 10.8% for the inclusive decay, ω contribution and nonresonant decay, respectively, as summarized in Table III. VI. SUMMARY In summary, with a sample of 1.31×10 9 J/ψ events collected with the BESIII detector, the doubly radiative decay η ′ → γγπ 0 has been studied. The branching fraction of the inclusive decay is measured for the first time to be B(η ′ → γγπ 0 ) Incl. = (3.20±0.07(stat)±0.23(sys))×10 −3 . The M 2 γγ dependent partial decay widths are also determined. In addition, the branching fraction for the nonresonant decay is determined to be B(η ′ → γγπ 0 ) NR = (6.16 ± 0.64(stat) ± 0.67(sys)) × 10 −4 , which agrees with the upper limit measured by the GAMS-2000 experiment [9]. As a validation of the fit, the product branching fraction with the omega intermediate state involved is obtained to be B(η ′ → γω) · B(ω → γπ 0 ) = (2.37 ± 0.14(stat) ± 0.18(sys)) × 10 −3 , which is consistent with the PDG value [16]. These results are useful to test QCD calculations on the transition form factor, and provide valuable inputs to the theoretical understanding of the light meson decay mechanisms. FIG. 1 : 1Distribution of the χ 2 5C of the 5C kinematic fit for the inclusive η ′ decay. Dots with error bars are data; the heavy (black) solid-curve is the sum of signal and expected backgrounds from MC simulations; the light (red) solid-curves is signal components which are normalized to the fitted yields; the (green) dotted-curve is the class I background; and the (pink) dot-dashed-curve is the class II background. FIG. 2 : 2Results of the fit to M γγπ 0 for the selected inclusive η ′ → γγπ 0 signal events. The (black) dots with error bars are the data. FIG. 3 : 3Partial width (in keV) versus M 2 γγ for the inclusive η ′ → γγπ 0 decay. The error includes the statistic and systematic uncertainties. The (blue) histogram is the sum of an incoherent mixture of ρ-ω and the nonresonant components from MC simulations; the (back) dotted-curves is ωcontribution; the (red) dot-dashed-curve is the ρ-contribution; and the (green) dashed-curve is the nonresonant contribution. All the components are normalized using the yields obtained inFig. 4. PACS numbers: 13.40.Gp, 13.40.Hq, 13.20.Jf, 14.40.Be TABLE TABLE II : IIResults for dΓ(η ′ → γγπ 0 )/dM 2 γγ (in units of keV/(GeV/c 2 ) 2 ) for thirteen intervals of M 2 γγ . The first uncertainties are statistical and the second systematic. M 2 γγ ((GeV/c 2 ) 2 ) [0.0, 0.01] [0.01, 0.04] [0.04, 0.06] [0.06, 0.09] [0.09, 0.12] dΓ(η ′ → γγπ 0 )/M 2 γγ 3.17 ± 0.44 ± 0.24 2.57 ± 0.18 ± 0.19 2.60 ± 0.15 ± 0.18 1.87 ± 0.12 ± 0.14 1.76 ± 0.11 ± 0.13 M 2 γγ ((GeV/c 2 ) 2 ) [0.12, 0.16] [0.16, 0.20] [0.20, 0.25] [0.25, 0.28] [0.28, 0.31] dΓ(η ′ → γγπ 0 )/M 2 γγ 1.63 ± 0.10 ± 0.12 1.76 ± 0.09 ± 0.13 1.97 ± 0.10 ± 0.14 2.00 ± 0.17 ± 0.15 1.07 ± 0.20 ± 0.08 M 2 γγ ((GeV/c 2 ) 2 ) [0.31, 0.36] [0.36, 0.42] [0.42, 0.64] dΓ(η ′ → γγπ 0 )/M 2 γγ 0.34 ± 0.06 ± 0.03 0.12 ± 0.03 ± 0.01 0.06 ± 0.01 ± 0.01 TABLE III : IIISummary of relative systematic uncertainties (%) for the branching fraction measurements. Here η ′ Incl. , η ′ ω and η ′ NR represent the inclusive η ′ → γγπ 0 , η ′ → γω(ω → γπ 0 ) and nonresonant decays, respectively.Incl.η ′ η ′ ω η ′ NR Photon detection 5.0 5.0 5.0 5C kinematic fit 2.7 2.7 2.7 Number of photons 0.5 0.5 0.5 π 0 veto 1.9 1.9 1.9 Signal shape 0.5 1.5 2.3 Signal model 1.7 1.0 4.3 ρ relative intensity - 1.3 4.9 Combinatorial backgrounds - 1.3 0.8 Fit range 0.8 1.6 2.1 Class I background 0.1 0.2 0.6 Class II background 0.3 1.8 4.2 Cited branching fractions 3.1 3.1 3.1 Number of J/ψ events 0.8 0.8 0.8 Total systematic error 7.1 7.7 10.8 AcknowledgmentsThe BESIII Collaboration thanks the staff of BEPCII and the IHEP computing center for their strong sup-port. This work is supported in part by National People's Republic of China. 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[ "Resummation Effects in Vector-Boson and Higgs Associated Production", "Resummation Effects in Vector-Boson and Higgs Associated Production" ]	[ "S Dawson \nDepartment of Physics\nBrookhaven National Laboratory\nUpton11973NYUSA\n", "T Han \nPittsburgh Particle Physics Astrophysics and Cosmology Center (PITT PACC\nDepartment of Physics and Astronomy\nUniversity of Pittsburgh\n15260PittsburghPAUSA\n", "W K Lai \nPittsburgh Particle Physics Astrophysics and Cosmology Center (PITT PACC\nDepartment of Physics and Astronomy\nUniversity of Pittsburgh\n15260PittsburghPAUSA\n", "A K Leibovich \nPittsburgh Particle Physics Astrophysics and Cosmology Center (PITT PACC\nDepartment of Physics and Astronomy\nUniversity of Pittsburgh\n15260PittsburghPAUSA\n", "I Lewis \nDepartment of Physics\nBrookhaven National Laboratory\nUpton11973NYUSA\n" ]	[ "Department of Physics\nBrookhaven National Laboratory\nUpton11973NYUSA", "Pittsburgh Particle Physics Astrophysics and Cosmology Center (PITT PACC\nDepartment of Physics and Astronomy\nUniversity of Pittsburgh\n15260PittsburghPAUSA", "Pittsburgh Particle Physics Astrophysics and Cosmology Center (PITT PACC\nDepartment of Physics and Astronomy\nUniversity of Pittsburgh\n15260PittsburghPAUSA", "Pittsburgh Particle Physics Astrophysics and Cosmology Center (PITT PACC\nDepartment of Physics and Astronomy\nUniversity of Pittsburgh\n15260PittsburghPAUSA", "Department of Physics\nBrookhaven National Laboratory\nUpton11973NYUSA" ]	[]	Fixed-order QCD radiative corrections to the vector-boson and Higgs associated production channels, pp → V H (V = W ± , Z), at hadron colliders are well understood. We combine higher order perturbative QCD calculations with soft-gluon resummation of both threshold logarithms and logarithms which are important at low transverse momentum of the V H pair. We study the effects of both types of logarithms on the scale dependence of the total cross section and on various kinematic distributions. The next-to-next-to-next-to-leading logarithmic (NNNLL) resummed total cross sections at the LHC are almost identical to the fixed-order perturbative next-to-next-toleading order (NNLO) rates, indicating the excellent convergence of the perturbative QCD series.Resummation of the V H transverse momentum (p T ) spectrum provides reliable results for small values of p T and suggests that implementing a jet-veto will significantly decrease the cross sections.	10.1103/physrevd.86.074007	[ "https://arxiv.org/pdf/1207.4207v2.pdf" ]	53,420,927	1207.4207	65bbc73e1dd1cc1bc0d724262d30fd2f27ab5138	Resummation Effects in Vector-Boson and Higgs Associated Production 23 Jul 2012 S Dawson Department of Physics Brookhaven National Laboratory Upton11973NYUSA T Han Pittsburgh Particle Physics Astrophysics and Cosmology Center (PITT PACC Department of Physics and Astronomy University of Pittsburgh 15260PittsburghPAUSA W K Lai Pittsburgh Particle Physics Astrophysics and Cosmology Center (PITT PACC Department of Physics and Astronomy University of Pittsburgh 15260PittsburghPAUSA A K Leibovich Pittsburgh Particle Physics Astrophysics and Cosmology Center (PITT PACC Department of Physics and Astronomy University of Pittsburgh 15260PittsburghPAUSA I Lewis Department of Physics Brookhaven National Laboratory Upton11973NYUSA Resummation Effects in Vector-Boson and Higgs Associated Production 23 Jul 2012(Dated: May 22, 2014)arXiv:1207.4207v2 [hep-ph] Fixed-order QCD radiative corrections to the vector-boson and Higgs associated production channels, pp → V H (V = W ± , Z), at hadron colliders are well understood. We combine higher order perturbative QCD calculations with soft-gluon resummation of both threshold logarithms and logarithms which are important at low transverse momentum of the V H pair. We study the effects of both types of logarithms on the scale dependence of the total cross section and on various kinematic distributions. The next-to-next-to-next-to-leading logarithmic (NNNLL) resummed total cross sections at the LHC are almost identical to the fixed-order perturbative next-to-next-toleading order (NNLO) rates, indicating the excellent convergence of the perturbative QCD series.Resummation of the V H transverse momentum (p T ) spectrum provides reliable results for small values of p T and suggests that implementing a jet-veto will significantly decrease the cross sections. I. INTRODUCTION The recent discovery of a Higgs-like particle [1,2] has brought our understanding of electroweak symmetry breaking to a deeper level. Now it is imperative to study the detailed properties of this particle in the hope of finding any hints for new physics beyond the Standard Model (SM). An important Higgs production mechanism at hadron colliders is the associated production of a Higgs boson and a vector boson, V H (V = W ± , Z) [3]. At the Tevatron, the process qq ′ → V H with the decay of the vector boson to leptons and of the Higgs to the bb and W + W − channels has provided important sensitivity to a light Higgs boson [4,5]. At the LHC, the production rate for associated V H production is small, but with ∼ 30 fb −1 a light Higgs in association with a W or Z can potentially be observed in the boosted regime via H → bb [6]. Reliable predictions are essential for the observation and study of the V V H couplings in this channel [7,8]. The rate for associated V H production is perturbatively known to next-to-next-to-leading order (NNLO), i.e. O(α 2 s ) [9,10]. At next-to-leading order (NLO), the QCD corrections are identical to those of the Drell-Yan process for an off-shell gauge boson, qq ′ → V * [11][12][13]. At NNLO, however, the ZH process receives a small additional contribution from the gg initial state, gg → ZH [9]. The NLO rates are available in the general purpose MCFM [14] program, while the total rate can be found to NNLO using the VH@NNLO code [9]. Infrared finite results in higher-order QCD processes occur due to a cancellation of virtual and real soft divergences. The fixed-order calculation is reliable providing all of the scales are of the same order of magnitude. When the invariant mass M V H of the final state particles W H or ZH approaches the center-of-mass energy of the colliding partons, there is less phase space available for real emission. While the infrared divergences will still cancel, large Sudakov logarithms will remain. These logarithms can spoil the convergence of the perturbative series and need to be resummed to all orders for reliable results in this threshold region [15]. Threshold corrections involve terms of the form α n s log 2n−1 (1−z) (1−z) , which are large when z = M 2 V H /ŝ ∼ 1, whereŝ is the partonic center-of-mass (c.m.) energy-squared [16][17][18][19][20]. Similarly, large logarithms of the form α n s log 2n−1 M We consider the process pp → V H + X and present results from both the threshold resummation and the transverse momentum resummation of large logarithms separately for LHC energies. Since the final state particles are color-singlets, both types of resummation can be straight-forwardly adopted from results in the literature for the Drell-Yan process [20][21][22][23][24]. (We do not discuss the joint resummation of the logarithms [25]). Section II contains a brief review of the resummation formalisms we apply. Details are relegated to several appendices. Section III presents results for the total cross section including the resummation of threshold logarithms and a discussion of the theoretical uncertainties, while Sections IV A and IV B contain some kinematic distributions resulting from the resummation of p T,V H and threshold logarithms, respectively. Finally, Section V discusses the relevance of our results to searches at the LHC. II. RESUMMATION FORMALISM In this section we briefly review the transverse momentum and threshold resummation formalism that we utilize in deriving our numerical results. A. Transverse-Momentum Resummation The discussion of the transverse momentum resummation follows that of Grazzini et al. [21]. The hard scattering process under consideration is Higgs boson production in association with a vector boson in hadronic collisions AB → V * + X → V H + X ,(1) where V = W ± , Z and X is the hadronic remnant of a collision. We apply the well known impact-parameter space (b-space) resummation [26,27] to the partonic cross section, dσ V H dM 2 V H dp 2 T,V H = dσ resum V H dM 2 V H dp 2 T,V H + dσ f inite V H dM 2 V H dp 2 T,V H ,(2) where p T,V H is the transverse momentum of the [21]: dσ f inite V H dM 2 V H dp 2 T,V H f.o = dσ V H dM 2 V H dp 2 T,V H f.o − dσ resum V H dM 2 V H dp 2 T,V H f.o ,(3) where the subscript f.o. refers to a fixed order expansion. In the low transverse momentum To correctly account for momentum conservation, transverse momentum resummation is performed in impact-parameter space: region, p T,V H ≪ M V H ,M 2 V H dσ resum V H dM 2 V H dp 2 T,V H = M 2 V Ĥ s ∞ 0 db b 2 J 0 (bp T,V H )W V H (b, M V H ,ŝ, µ r , µ f ) ,(4) where J 0 (x) is the 0 th order Bessel function and µ r , µ f are the renormalization/factorization scales. By performing a Mellin transformation 1 it is possible to factor the terms that are finite and logarithmically enhanced as p T,V H → 0: W V H N (b, M V H , µ r , µ f ) = H V H N M V H , α s (µ r ), M V H µ r , M V H µ f , M V H Q × exp G N α s (µ r ), L, M V H µ r , M V H Q ,(5) where L = ln Q 2 b 2 /b 2 0 with b 0 = 2 exp(−γ E ),log M 2 V H p 2 T,V H = log Q 2 p 2 T,V H + log M 2 V H Q 2 .(6) The scale Q, termed the resummation scale, is introduced to parameterize this arbitrariness and is the same as that in Eq. (5). To keep the separation between the finite and 1 The Mellin transformation of a function h(z) is defined as h N = 1 0 dzz N −1 h(z). logarithmically enhanced terms meaningful, the scale Q has to be chosen to be close to M V H . As mentioned in the previous paragraph, all of the logarithmically enhanced contributions are contained in G N . The divergent pieces can be reorganized such that G N is written as an expansion that is order-by-order smaller by α s [21]: G N α s , L, M V H µ r , M V H Q = Lg 1 N (α s L) + n=2 α s π n−2 g n N α s L, M V H µ r , M V H Q ,(7) where g n N = 0 for α s L = 0 and Lg 1 N contains the leading log (LL) terms α n s L n+1 , g 2 N contains the next-to-leading log (NLL) terms α n s L n , etc. Since the large logarithms are associated with collinear and soft divergences from real radiation, the functions g i N are only dependent on the initial state partons and are independent of the specific hard process under consideration. Explicit expressions for the LL and NLL terms needed for pp → V H + X are given in Appendix A. The resummed distribution is valid in the low p T,V H ≪ M V H region, while the perturbative expansion is valid in the high p T,V H ∼ M V H region. However, as Qb approaches zero the logarithm L grows uncontrollably. As a result, the resummed distribution makes an unacceptably large contribution to the high p T,V H region. This problem can be solved via the replacement [28] L →L = log Q 2 b 2 /b 2 0 + 1 , such thatL ≈ L for Qb ≫ 1 andL ≈ 0 for Qb ≪ 1. Hence, usingL, the resummed contribution maintains the correct dependence on the large logarithms at low p T,V H and does not make unwarranted contributions to the high p T,V H region. This replacement has the added benefit of reproducing the correct fixed order cross section once the transverse momentum is integrated [21]. The process-dependent function H is finite as p T,V H → 0. Hence, its Mellin transform H N does not contain any dependence on b and can be computed as an expansion in α s , H V H N M V H , α s , M V H µ r , M V H µ r , M V H Q = σ 0 (α s , M V H ) 1 + n=1 α s π n H V H(n) N M V H µ r , M V H µ f , M V H Q ,(8) where σ 0 is the Born-level partonic cross section for qq ′ → V H. At NLL accuracy, only the first hard coefficient H V H(1) N is needed. The value of this coefficient is given in Appendix A. B. Threshold resummation In the original approach to threshold resummation [16,18], the resummation is performed after taking the Mellin transformation of the hadronic cross section [29,30]. The Mellintransformed hadronic cross section can then be factored into the product of the partonic cross section and the parton luminosity. The threshold logarithms for V H production are of the form ln(1 − z), where z = M 2 V H /ŝ, and are contained in the partonic cross section. After resummation, an inverse-Mellin transformation is performed to obtain the physical cross section. This leads to a new divergence due to the presence of the Landau pole in α s . Prescriptions for how to perform the inverse-Mellin transformation have been developed to remove this problem. The resummation of threshold logarithms for Drell-Yan production has been extensively studied [31][32][33][34]. More recently, techniques using soft-collinear effective theory (SCET) [35][36][37][38] have been developed in which the resummation is performed in momentum space, obviating the need to go to Mellin space. This in turn removes the problem of the Landau pole. In this paper, we will generalize the SCET resummation results of [20] to the case of V H production. The leading singular terms at threshold in the hadronic differential cross section can be written as 1 τ σ 0 dσ dM 2 V H = 1 τ dz z C(z, M V H , µ f )L τ z , µ f ,(9) where τ = M 2 V H /s with s the hadronic c.m. energy-squared, L is the parton luminosity, L(y, µ f ) = 1 y dx x f q (x, µ f )f q ′ y x , µ f + (q ↔ q ′ ) ,(10) and σ 0 is the Born level partonic cross section for qq ′ → V H and is defined such that C(z, M V H , µ f ) = δ(1 − z) + O(α s ). In the threshold region, z ∼ 1, C(z, M V H , µ f ) can be factorized into a hard contribution and a soft contribution, C(z, M V H , µ f ) = H(M V H , µ f )S(M V H (1 − z), µ f ).(11) The L k Γ cusp γ V , γ φ C V ,s DY LO NLL 2n − 1 ≤ k ≤ 2n 2-loop 1-loop tree-level NLO NNLL 2n − 3 ≤ k ≤ 2n 3-loop 2-loop 1-loop NNLO NNNLL 2n − 5 ≤ k ≤ 2n 4-loop 3-loop 2-loop The final result is found from that for the Drell-Yan process [20] C(z, M V H , µ f ) = C V (−M 2 V H , µ h ) 2 U(M V H , µ h , µ s , µ f ) z −η (1 − z) 1−2η ×s DY ln M 2 V H (1 − z) 2 µ 2 s z + ∂ η , µ s e −2γ E η Γ(2η) ,(12) where η = 2a Γ (µ s , µ f ),dσ dM 2 V H matched = dσ dM 2 V H threshold resum − dσ dM 2 V H threshold f.o. + dσ dM 2 V H f.o. .(13) Here dσ dM 2 V H threshold resum is the result obtained using the threshold resummation formula of Eq. (12), dσ dM 2 V H f.o. is the fixed-order perturbative result and dσ dM 2 V H threshold f.o. is obtained from the fixed-order result by keeping only the leading threshold singularity in C. The order of the logarithmic approximation in the resummed result and the corresponding fixed-order results used in the matching of Eq. (13) are summarized in Table I. 2 III. SCALE DEPENDENCE OF THE CROSS SECTION In this section, we study the scale dependence of the total cross section for V H production at the LHC, beginning with the sensitivity of the resummed threshold distributions to the hard, soft, and factorization scales. Near the threshold, τ ≡ M 2 V H /s → 1, the threshold logarithms are enhanced, leading to potentially large scale violations. The naive choice for the soft scale is µ s ∼ M V H (1 − τ ). We follow the prescription of Ref. [20] to determine a sensible range of parameters for the soft scale. A low value of µ s is found empirically from the scale where the one-loop correction tos DY is minimal, µ (I) s = M V H (1 − τ ) 2 √ 1 + 100τ .(14) Alternatively, an upper scale for the soft variation can be chosen as the value where the one-loop correction tos DY drops below 10%, µ (II) s = M V H (1 − τ ) 0.9 + 12τ .(15) Empirically, the forms of µ (I,II) s M V H are insensitive to M V H . Here and henceforth, we adopt the Higgs mass value M H = 125 GeV.(16) We investigate the numerical effects of the scale variation by plotting the differential cross section of the threshold resummation of Eq. (12) and varying the soft, hard, and factorization scales. It is customary to measure the size of QCD corrections by a K-factor (K) typically defined as the ratio of a higher order cross section to the lowest order cross section: dσ dM 2 V H ≡ K dσ dM 2 V H LO ,(17) where dσ dM 2 V H is a distribution defined at higher order in QCD. To study the scale variation arising from threshold resummation, we investigate the K- factor of Eq. (17) defined with dσ dM 2 V H ≡ dσ dM 2 V H threshold−resum .(18) To isolate the effects of the scale variation due to threshold resummation from effects of the scale variation due to parton distribution functions (PDFs) and running α s , the K-factor is evaluated by using the NNLO MSTW20008 [40] PDF set and the 3-loop value of α s for all orders of the threshold resummed cross section and the LO cross section. Figure 1 shows the scale variation of this choice of K-factor as a function of τ at NLL between the dotted curves, NNLL between the dashed curves, and NNNLL between the solid curves for ZH production at M ZH = 1 TeV. The soft scale variation in pp → ZH, with µ h and µ f held constant, is shown in Fig. 1(a). The variation in the NLL result is significant, but the Ref. [7] also includes the NLO electroweak effects [41], assuming complete factorization of the QCD and electroweak corrections. In the G µ renormalization scheme, these corrections reduce the total Higgs and vector boson associated rates by about O(5%). IV. KINEMATIC DISTRIBUTIONS A. Transverse-Momentum Distributions We now give numerical results for the resummed transverse-momentum distributions. The distributions are computed at NLL-NLO accuracy with NLO MSTW2008 68% confidence level PDFs [40] and the 2−loop evolution of α s using the formulae of Appendix A. The numerical results were found by modifying the program HqT2. background typically has more hard jets than the V H signal, a jet veto may be applied to suppress this background. We note that vetoing jets with a minimum transverse momentum can be approximated by placing an upper limit on the V H transverse momentum and, as can be seen in Figs. 3(a) and 3(b), the perturbative calculation is unreliable in this regime. Hence, to fully account for the effects of a jet veto the soft-gluon resummation is needed. There has been much recent work on the systematic resummation of the large logarithms associated with jet vetoes [44][45][46][47][48]. To approximate the effect on the total cross section of a veto on jets with transverse momentum larger than p T,V H we define σ(p T,V H ) = p T,V H 0 dq T,V H dσ dq T,V H ,(19) where dσ/dq T,V H is the matched transverse-momentum distribution at NLL-NLO in Eqs. (2) and (3). The decrease of the K-factor at higher τ values is due to the PDF effect. To see this, we artificially adopt the NLO MSTW2008 68% confidence level PDFs and 2-loop α s for the NLO fixed-order result, the leading threshold singularity of the NLO-fixed-order result and the threshold-resummed result at NNLL, as well as the LO denominator, and show the Kfactors of these results with √ s = 14 TeV for pp → ZH + X and pp → W H + X in Fig. 6(a) and 6(b) respectively. This is to isolate the effects of PDFs from a dynamical origin. The choice of scales is the same as in Fig. 5. We note that the monotonic increase of the Kfactor distributions in Fig. 6 is drastically different from that in Fig. 5. This demonstrates the importance of a consistent choice of PDFs as in Fig. 5. To examine the convergence of the perturbative series, we plot the K-factors for the resummed results at NLL, NNLL and NNNLL with √ s = 14 TeV for pp → ZH + X in Fig. 7, using NNLO MSTW2008 68% confidence level PDFs and 3-loop α s for all the resummed results as well as the LO denominator. We see from Fig. 7 that the difference between NNLL and NNNLL is tiny (< 1%), confirming the excellent convergence of the perturbative series at this order especially after leaving out the PDF effect. V. CONCLUSIONS Given the exciting discovery of a Higgs-like particle at the LHC [1,2], it becomes imperative to determine its properties. Thus its production rate at the LHC must be calculated as accurately as possible. Since the gauge boson-Higgs associated production is one of the channels that unambiguously probes the V V H coupling with V = W ± or Z, it is of particular interest. We combined the long-known fixed-order perturbative QCD calculations for In this appendix, we list the functions needed for the p T,V H resummation of Section II A [50,51]. All formulae in this appendix can be found in Ref. [21], but we include them for the convenience of the reader. First, the coefficients of the QCD beta function are normalized according to the expansion d ln α s (µ 2 ) d ln µ 2 = β(α s (µ 2 )) = − ∞ n=0 β n α s (µ 2 ) 4π n+1 .(20) At LL only the function g 1 N is needed and the Born level contribution arises only from qq ′ scattering [21], g 1 N (α s L) = 4A 1 q β 0 λ + ln(1 − λ) λ λ ≡ β 0 4π α s (µ r )L, β 0 = 33 − 2n f 3 , A 1 q = 4 3 = C F ,(21) and L = ln Q 2 b 2 b 2 0 . At NLL the functions g 2 N and H V H(1) N are needed [21], g 2 N (α s L, M V H µ r , M V H Q ) = 4B 1 q,N β 0 ln(1 − λ) − 16A 2 q β 2 0 λ 1 − λ + ln(1 − λ) + 4A 1 q β 0 λ 1 − λ + ln(1 − λ) ln Q 2 µ 2 r + 4A 1 q β 1 β 3 0 1 2 ln 2 (1 − λ) + ln(1 − λ) 1 − λ + λ 1 − λ β 1 = 2 153 − 19n f 3 A 2 q = C F 2 67 6 − π 2 2 − 5 9 n f B 1 q,N = − 3 2 C F + 2γ 1 qq,N + A 1 q ln M 2 V H Q 2 .(22) and n f is the number of light flavors. The anomalous dimensions γ ab,N are the Mellin transforms of the DGLAP splitting functions, P ab [52]: γ ab,N = ∞ n=1 α s π n γ n ab,N ≡ 1 0 dzz N −1 P ab (z)(23) The process dependence arises through H Q 2 µ 2 f ,(25) where, A V H = C F −8 + 2π 2 3 .(26) where a γ is the anomalous exponent of γ defined by a γ (ν, µ) = − αs(µ) αs(ν) dα γ(α) β(α) , (28) and S is the Sudakov exponent S(ν, µ) = − αs(µ) αs(ν) dα Γ cusp (α) β(α) α αs(ν) dα ′ β(α ′ ) .(29) The cusp anomalous dimension is known to three-loops [55,56]. The coefficients are Γ 0 = 4C F , Γ 1 = 4C F 67 9 − π 2 3 C A − 20 9 T F n f , Γ 2 = 4C F C 2 A 245 6 − 134π 2 27 + 11π 4 45 + 22 3 ζ 3 + C A T F n f − 418 27 + 40π 2 27 − 56 3 ζ 3 +C F T F n f − 55 3 + 16ζ 3 − 16 27 T 2 F n 2 f .(33) The four-loop coefficient Γ 3 has not yet been calculated, so we use the Padé approximate Γ 3 = Γ 2 2 /Γ 1 . The anomalous dimension γ V can be obtained from the partial three-loop on-shell quark form factor [57]. The coefficients are γ V 0 = −6C F , γ V 1 = C 2 F (−3 + 4π 2 − 48ζ 3 ) + C F C A − 961 27 − 11π 2 3 + 52ζ 3 + C F T F n f 260 27 + 4π 2 3 , γ V 2 = C 3 F −29 − 6π 2 − 16π 4 5 − 136ζ 3 + 32π 2 3 ζ 3 + 480ζ 5 +C 2 F C A − 151 2 + 410π 2 9 + 494π 4 135 − 1688 3 ζ 3 − 16π 2 3 ζ 3 − 240ζ 5 +C F C 2 A − 139345 1458 −7163π The final anomalous dimension, γ φ , is known from the NNLO calculation of the Altarelli-Parisi splitting function [56]. The coefficients are γ φ 0 = 3C F , γ φ 1 = C 2 F 3 2 − 2π 2 + 24ζ 3 + C F C A 17 6 + 22π 2 9 − 12ζ 3 − C F T F n f 2 3 + 8π 2 9 , γ φ 2 = C 3 F 29 2 + 3π 2 + 8π 4 5 + 68ζ 3 − 16π 2 3 ζ 3 − 240ζ 5 +C 2 F C A 151 4 −205π The other functions needed are the Wilson coefficient C V and the soft functions DY . The Wilson coefficient C V has the expansion, C V (−M 2 −iǫ, µ) = 1+ C F α s 4π −L 2 + 3L − 8 + π 2 6 +C F α s 4π 2 (C F H F +C A H A +T F n f H f ),(36) where L = ln(M 2 /µ 2 ) − iπ, and H F = L 4 2 − 3L 3 + 25 2 − π 2 6 L 2 + − 45 2 −3π This agrees with the corresponding expression in [58]. The soft function to two-loops is s DY (ℓ, µ) = 1 + C F α s 4π 2ℓ 2 + π 2 3 + C F α s 4π 2 (C F W F + C A W A + T F n f W f ),(38) where W F = 2ℓ 4 + 2π 2 occur when the V H system is produced with small transverse momentum p T,V H [21, 22]. The techniques for resumming both types of logarithms to all orders are well known and the fixed order perturbative and resummed calculations can be consistently matched at intermediate values of the kinematic variables. hard function H(M V H , µ f ) and soft function S(M V H (1 − z), µ f ), evaluated at µ f , are obtained by renormalization group running from the hard scale µ h ∼ M V H and soft scale µ s ∼ M V H(1 − τ ), respectively, to sum the threshold logarithms to all orders in α s . and C V ands DY are the perturbatively calculable Wilson coefficient and soft Wilson loop coefficient, respectively. Eq. (10) with C given by Eq. (12) is defined only for η > 0. For η < 0, an analytic continuation is required. The analytic expressions for a Γ , C V ,s DY and U which are necessary for our numerical calculations are given in Appendix B. Eq. (12) is only valid in the threshold region z ∼ 1. To obtain a formula valid for all values of z, we match the threshold-resummed result with the fixed-order result, s NNLL and NNNLL curves have little dependence on the soft scale, justifying the ad hoc choices of µ (I,II) s . The K-factor grows rapidly as τ increases, as expected. The sensitivity to the hard scale is shown inFig. 1(b), with fixed µ s and µ f . The hard scale is set by the invariant mass of the V H pair, and again we find that at NNLL and NNNLL, there is little dependence on µ h , showing excellent convergence of the perturbation series. Finally, we show the factorization scale dependence inFig. 1(c). The factorization scale dependence is small even at NLL.We have also considered the scale dependence of the matched result for the total cross section. Analytic expressions for the LO and NLO fixed order results are found in Refs.[9,[11][12][13], and we use the computer code VH@NNLO for the fixed order NNLO results. The matched curves are found using the threshold resummation results of Eq. (13). InFig. 2, we use the MSTW2008 68% confidence level PDFs, and use LO PDFs for the LO and the NLL-LO matched curves, NLO PDFs for the NLO and NNLL-NLO matched curves, and NNLO PDFs for the NNLO and NNNLL-NNLO matched curves, and use 1, 2 and 3-loop evolution of α s respectively. We include the small contribution from the gg initial state in the ZH NNLO and NNNLL-NNLO matched curves. The results for ZH production at √ s = 8 TeV and √ s = 14 TeV are shown in Figs. 2(a) and 2(b), respectively. We have chosen the central scale to be µ 0 = M ZH . The top and bottom quark loops from the gg initial state contribute σ t,b loops gg = 0.06 pb at √ s = 14 TeV with µ f = µ 0 . This is the reason for the larger splitting between the NLO and NNLO curves than is seen in the W H results below. The fixed-order and matched curves have the renormalization/factorization scales set equal, µ r = µ f . The matched and resummed curves have the hard scale, µ h = 2M V H , and the soft scale, µ s ). The NNNLL-NNLO matched curve is almost identical to the NNLO fixed order curve, and the resummation has little effect at this order. On the other hand, the NNLL-NLO matched curve increases the fixed order NLO result (at µ f = µ 0 ) by about 7%. FIG. 1 :FIG. 2 : 12The (a) soft scale, (b) hard scale, and (c) factorization scale dependence of the threshold resummed cross section for pp → ZH at NLL between the dotted lines dotted, NNLL between the dashed lines, and NNNLL between the solid lines normalized to the LO result (the K-factor is defined in Eqs. (17) and (18)). The invariant mass M ZH is fixed at 1 TeV. in Figs. 2(c) and 2(d). These figures show the sum of W + H and W − H production. As in the ZH case, the NNLO and NNNLL-NNLO matched results for W H production are quite close and show little scale variation. The NNLL resummation increases the NLO fixed order result by ∼ 3%. The uncertainties in the ZH and W H cross sections from PDFs, renormalization and Scale dependence of the fixed order (dashed) and threshold resummed matched (solid) cross sections for (a,b) ZH and (c,d) W H production at (a,c) √ s = 8 TeV and (b,d) √ s = 14 TeV. The NNLO and NNNLL-NNLO matched ZH results include the contribution from the gg initial state.factorization scale dependence, and the determination of α s have been investigated by the LHC Higgs Cross Section Working Group for the NNLO total cross section[7]. They find a total uncertainty at √ s = 8 TeV of O(4%) for W H and O(5%) for ZH production for a 125 GeV Higgs boson. Our results show that including the resummation of threshold logarithms to NNNLL accuracy does not induce any further uncertainties. We note that 0 [21, 42, 43]. The factorization and renormalization scales are set to the central values of µ f = µ r = M V + M H . Also, the resummation scale is set equal to the invariant mass of the vector boson and Higgs pair, i.e., Q = M V H . Figures 3(a) and 3(b) show the transverse-momentum distribution for ZH and W H production, respectively, at √ s = 14 TeV. The matched transverse-momentum distribution defined by Eqs. (2) and (3) (solid), resummed (dot-dash), fixed order expansion of the resummed (dashed), and fixed-order perturbative (dotted) distributions are shown separately. As expected, the fixed-order expansion of the resummed and perturbative distributions are in good agreement. Hence, the finite piece, defined to be the difference between the perturbative distribution and fixed-order expansion of the resummed distribution as in Eq. (3), is negligible at low transverse momentum and the matched distribution is dominated by the resummed contribution. The transverse-momentum distribution is peaked around 5 GeV for both W H and ZH production. For comparison, in Figs. 3(c) and 3(d) we present the normalized matched transversemomentum distributions for ZH and W H production, respectively, at both √ s = 8 TeV (dashed) and √ s = 14 TeV (solid). The position of the peak of the transverse distribution is not significantly different between the two LHC energies. However, the distribution at √ s = 14 TeV has a longer tail than at √ s = 8 TeV. This can be understood by noting that higher transverse-momentum events correspond to higher partonic center-of-mass energies. Since events with higher partonic center-of-mass energies are more easily accessible at √ s = 14TeV than at √ s = 8 TeV, we would expect there to be larger fraction of high p T,V H events at √ s = 14 TeV than at √ s = 8 TeV. Hence, the transverse-momentum distribution has a FIG. 3 :FIG. 4 : 34Transverse-momentum distributions for (a,c) ZH, and (b,d) W H production at the LHC. In (a) and (b), the matched distribution is shown with a solid line, the resummed distribution with a dot-dash line, the fixed-order expansion of the resummed distribution with a dashed line and the fixed-order perturbative distribution with a dotted line at √ s = 14 TeV. In (c) and (d), the normalized matched transverse-momentum distributions are shown for both √ s = 8 TeV (dashed) and √ s = 14 TeV (solid) LHC. longer tail for √ s = 14 TeV. Finally, we comment how the transverse-momentum resummation can effect the analysis of kinematical cuts on the signal cross section, particularly in relation to jet vetoes. At hadron machines, the V H production with Higgs decaying to bb has large QCD backgrounds. To reduce the backgrounds and effectively trigger on the signal, one usually considers leptonic decays of the vector boson. However, if the vector boson decay contains missing energy, W → ℓν or Z → νν, semileptonic decays of tt can be a significant background. Since the tt Integrated matched transverse-momentum distributions normalized to the total cross section for both (a) ZH and (b) W H production. Results for √ s = 8 TeV and √ s = 14 TeV are shown with dashed and solid lines, respectively. Figure 4 4shows this cross section normalized to the total p T,V H resummed and matched cross section as a function of p T,V H for (a) ZH and (b) W H production for both √ s = 8 TeV (dashed) and √ s = 14 TeV (solid). As noted before in the discussion of Figs. 3(c) and 3(d), at √ s = 14 TeV, there is expected to be a larger fraction of high transverse-momentum jets than at √ s = 8 TeV. Hence, σ(p T,V H )/σ grows more slowly at √ s = 14 TeV than at √ s = 8 TeV. From the figures we see that the effects of a 20 (30) GeV p T,V H cut decreases the NLO cross section by ∼ 45% (∼ 33%) and ∼ 50% (∼ 37%) at section, we give numerical results for the invariant-mass distributions including threshold resummation and matching, using the analytic formulae of Appendix B. Since the distributions vary over many orders of magnitude, it is easier to see the effects in the K-factor, as defined in Eq. (17). Figures 5(a) and 5(b) show the K-factor versus τ at NNLL-NLO with √ s = 14 TeV for pp → ZH +X and pp → W H +X, respectively. The K-factor for the matched result of Eq. (13) is shown with solid lines, the threshold-resummed contribution with dot-dashed lines, the fixed-order perturbative contribution with dashed lines, and the contribution from the leading threshold singularity of the fixed-order perturbative piece with dotted lines. Here we use MSTW2008 68% confidence level PDFs [40]. The scales are chosen to be µ f = M ZH , µ h = 2M ZH and µ s = 1 2 (µ I s + µ II s ) as in Section III. For the NLO fixed-order result, the leading threshold singularity of the NLO fixed-order result and the threshold-resummed result at NNLL, the NLO PDFs and 2-loop α s are used, whereas for the LO fixed-order denominator of the K-factor, we use the LO PDFs and 1-loop α s . As expected, the leading singularity and fixed-order results (the two lower curves) are close to each other, since the leading singularity dominates in the fixed-order result. On the other hand, the resummation effect is significant at high τ , as seen by the large enhancement of the NNLL (the two upper curves) from the NLO result of ∼ 20% for both ZH and W H at τ = 0.3. FIG. 5 : 5K-factor distributions at √ s = 14 TeV for (a) ZH and (b) W H production. The NNLL-NLO matched result is shown with solid lines, the NNLL threshold resummed result with dotdashed lines, the leading threshold singularity of the NLO fixed-order result with dotted lines, and the NLO fixed-order result with dashed lines. VFIG. 6 : 6H production[9] with soft-gluon resummation of both threshold logarithms and logarithms which are important at low transverse momentum of the V H pair.After a brief overview of the resummation formalism, we carried out detailed numerical analyses at the LHC for √ s = 8 TeV and 14 TeV. The overall corrections from NNLO fixed K-factor distributions at √ s = 14 TeV for (a) ZH and (b) W H production. The NNLL-NLO matched result is shown with solid lines, the NNLL threshold resummed result with dotdashed lines, the leading threshold singularity of the NLO fixed-order result with dashed lines, and the NLO fixed-order result with dotted lines. The NLO PDFs and 2-loop α s are adopted for all the results as well as the LO denominator. V H system and as p T,V H → 0 and can be computed at fixed order in α sdσ resum V H dM 2 V H dp 2 T,V H contains the resummation of the log M 2 V H p 2 T,V H enhanced terms. Since all the logarithmically enhanced terms are factored into the resummed piece, the remaining contribution dσ f inite V H dM 2 V H dp 2 T,V H is finite the resummed distribution is dominant, while in the high transverse momentum region, p T,V H ∼ M V H , the perturbative expansion of the cross section dominates.Using Eq. (3), the two regions can be consistently matched in the intermediate p T,V H region, maintaining theoretical accuracy. H contains the finite hard scattering coefficients, and G contains the process independent logarithmically enhanced terms. Hence, all the terms that are divergent as p T,V H → 0 are exponentiated into the function G N , achieving the all-orders resummation. The split between the finite and logarithmically enhanced terms is somewhat arbitrary; that is, a finite shift in the invariant mass M V H can alter the separation: TABLE I : IApproximation schemes for threshold resummation given a fixed order matched to a logarithmic approximation as in Eq.(13).Fixed order Log. Accuracy∼ α n s The matched cross sections for W H production at √ s = 8 TeV and 14 TeV are shown µ h dependence of pp→ZH at M ZH =1 TeV M ZH < µ h < 2M ZH0.05 0.1 0.15 0.2 0.25 0.3 0.35 τ=M ZH 2 /s 1 1.5 2 2.5 3 dσ/dM ZH 2 K-factor NLL NNLL NNNLL µ s dependence of pp→ZH at M ZH =1 TeV µ s (I) < µ s < µ s (II) µ f =M ZH , µ h =2M ZH (a) 0.05 0.1 0.15 0.2 0.25 0.3 0.35 τ=M ZH 2 /s 1 1.5 2 2.5 3 dσ/dM ZH 2 K Factor NLL NNLL NNNLL µ f =M ZH , µ s =(µ s (I) +µ s (II) )/2 (b) 0.05 0.1 0.15 0.2 0.25 0.3 0.35 τ=M ZH 2 /s 1 1.5 2 2.5 3 dσ/dM ZH 2 K-factor NLL NNLL NNNLL pp→ZH+X, √s=14 TeV, M H =125 GeV FIG. 7: pp → ZH + X K factor distribution at √ s = 14 TeV for the threshold resummed piece at various orders of the logarithmic approximation, using the same PDFs for all curves.order calculations are sizable, increaing the LO rate by a factor as large as about 30%[7].After implementing threshold resummation, the dependence of the cross section and various kinematic distributions on the soft and hard scales, as well as on the factorization scale is very weak, indicating the reliability of the calculations. The NNLL threshold resummed total cross section increases the fixed-order NLO result by about 7%, while the NNNLL resummed result has little impact on the NNLO fixed order rate, demonstrating the excellent convergence of the perturbation series. The transverse-momentum spectrum of the V H system is calculated via soft and collinear gluon resummation. The distribution is peaked near 5 GeV and the spectrum is slightly harder at the center-of-mass energy of 14 TeV than at 8 TeV. Using the matched transversemomentum distribution, we have also calculated the effect on the NLO cross section of placing an upper bound on the p T of the V H system. Since such an upper bound on the transverse momentum of the V H system limits the amount of transverse momentum a jet may carry in V H + X events, we expect the upper bound on the p T,V H of the V H system to approximate a jet veto. As a final remark, our calculations can be easily extended to other electroweak pair production processes with the same color structures which arise via qq ′ annihilation at leading order, such as the EW gauge boson pairs and the Higgs pair production H 0 A 0 , H 0 H ± , A 0 H ± and H + H − [49]. A: p T V Resummation0.05 0.1 0.15 0.2 0.25 τ=M ZH 2 /s 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2 dσ/dM ZH 2 K-factor NLL NNLL NNNLL Appendix +C F C A T F n f −2 243 − 83π 4 45 + 7052 9 ζ 3 − 88π 2 9 ζ 3 − 272ζ 5 +C 2 F T F n f 5906 27 − 52π 2 9 − 56π 4 27 + 1024 9 ζ 3 34636 729 + 5188π 2 243 + 44π 4 45 − 3856 27 ζ 3 +C F T 2 F n 2 f 19336 729 − 80π 2 27 − 64 27 ζ 3 . +C F C A T F n f 40 −2 9 − 247π 4 135 + 844 3 ζ 3 + 8π 2 3 ζ 3 + 120ζ 5 +C F C 2 A − 1657 36 + 2248π 2 81 − π 4 18 − 1552 9 ζ 3 + 40ζ 5 +C 2 F T F n f −46 + 20π 2 9 + 116π 4 135 − 272 3 ζ 3 1336π 2 81 + 2π 4 45 + 400 9 ζ 3 +C F T 2 F n 2 f − 68 9 + 160π 2 81 − 64 9 ζ 3 . 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[ "Multivariate approximation in downward closed polynomial spaces ", "Multivariate approximation in downward closed polynomial spaces " ]	[ "Albert Cohen [email protected] \nUMR 7598\nSorbonne Universités\nUPMC Univ Paris 06\nCNRS\nLaboratoire Jacques-Louis Lions\n4, place Jussieu75005ParisFrance\n", "Giovanni Migliorati [email protected] \nSorbonne Universités\nUPMC Univ\nParis 06\n\nUMR 7598\nLaboratoire Jacques-Louis Lions\nCNRS\n4, place Jussieu75005ParisFrance\n" ]	[ "UMR 7598\nSorbonne Universités\nUPMC Univ Paris 06\nCNRS\nLaboratoire Jacques-Louis Lions\n4, place Jussieu75005ParisFrance", "Sorbonne Universités\nUPMC Univ\nParis 06", "UMR 7598\nLaboratoire Jacques-Louis Lions\nCNRS\n4, place Jussieu75005ParisFrance" ]	[]	The task of approximating a function of d variables from its evaluations at a given number of points is ubiquitous in numerical analysis and engineering applications. When d is large, this task is challenged by the so-called curse of dimensionality. As a typical example, standard polynomial spaces, such as those of total degree type, are often uneffective to reach a prescribed accuracy unless a prohibitive number of evaluations is invested. In recent years it has been shown that, for certain relevant applications, there are substantial advantages in using certain sparse polynomial spaces having anisotropic features with respect to the different variables. These applications include in particular the numerical approximation of high-dimensional parametric and stochastic partial differential equations. We start by surveying several results in this direction, with an emphasis on the numerical algorithms that are available for the construction of the approximation, in particular through interpolation or discrete least-squares fitting. All such algorithms rely on the assumption that the set of multi-indices associated with the polynomial space is downward closed. In the present paper we introduce some tools for the study of approximation in multivariate spaces under this assumption, and use them in the derivation of error bounds, sometimes independent of the dimension d, and in the development of adaptive strategies.	10.1007/978-3-319-72456-0_12	[ "https://arxiv.org/pdf/1612.06690v1.pdf" ]	119,321,413	1612.06690	25debec0ed5b9985d8e95a02502b54d80413b804	Multivariate approximation in downward closed polynomial spaces * 20 Dec 2016 December 15, 2016 Albert Cohen [email protected] UMR 7598 Sorbonne Universités UPMC Univ Paris 06 CNRS Laboratoire Jacques-Louis Lions 4, place Jussieu75005ParisFrance Giovanni Migliorati [email protected] Sorbonne Universités UPMC Univ Paris 06 UMR 7598 Laboratoire Jacques-Louis Lions CNRS 4, place Jussieu75005ParisFrance Multivariate approximation in downward closed polynomial spaces * 20 Dec 2016 December 15, 2016arXiv:1612.06690v1 [math.NA] * This research is supported by Institut Universitaire de France and the ERC AdV project BREAD. 1AMS classification numbers: 41A6541A2541A1041A0565N1265N15 Keywords: Multivariate approximationerror estimatesconvergence ratesin- terpolationleast squaresdownward closed setparametric PDEsPDEs with stochas- tic dataadaptive approximation The task of approximating a function of d variables from its evaluations at a given number of points is ubiquitous in numerical analysis and engineering applications. When d is large, this task is challenged by the so-called curse of dimensionality. As a typical example, standard polynomial spaces, such as those of total degree type, are often uneffective to reach a prescribed accuracy unless a prohibitive number of evaluations is invested. In recent years it has been shown that, for certain relevant applications, there are substantial advantages in using certain sparse polynomial spaces having anisotropic features with respect to the different variables. These applications include in particular the numerical approximation of high-dimensional parametric and stochastic partial differential equations. We start by surveying several results in this direction, with an emphasis on the numerical algorithms that are available for the construction of the approximation, in particular through interpolation or discrete least-squares fitting. All such algorithms rely on the assumption that the set of multi-indices associated with the polynomial space is downward closed. In the present paper we introduce some tools for the study of approximation in multivariate spaces under this assumption, and use them in the derivation of error bounds, sometimes independent of the dimension d, and in the development of adaptive strategies. Introduction The mathematical modeling of complex physical phenomena often demands for functions that depend on a large number of variables. One typical instance occurs when a quantity of interest u is given as the solution to an equation written in general form as P(u, y) = 0, (1.1) where y = (y j ) j=1,...,d ∈ R d is a vector that concatenates various physical parameters which have an influence on u. Supposing that we are able to solve the above problem, either exactly or approximately by numerical methods, for any y in some domain of interest U ⊂ R d we thus have access to the parameter-to-solution map y → u(y). (1. 2) The quantity u(y) may be of various forms, namely: 1. a real number, that is, u(y) ∈ R; 2. a function in some Banach space, for example when (1.1) is a partial differential equation (PDE); 3. a vector of eventually large dimension, in particular when (1.1) is a PDE whose solution is numerically approximated using some numerical method with fixed discretization parameters. As a guiding example which will be further discussed in this paper, consider the elliptic diffusion equation Here, a and ψ j are given functions in L ∞ (D), and the y j range in finite intervals that, up to renormalization, can all be assumed to be [−1, 1]. In this example y = (y j ) j≥1 is countably infinite-dimensional, that is, d = ∞. The standard weak formulation of (1.3) in H 1 0 (D), D a∇u∇v = D f v, v ∈ H 1 0 (D), is ensured to be well-posed for all such a under the so-called uniform ellipticity assumption j≥1 \|ψ j \| ≤ a − r, a.e. on D, (1.5) for some r > 0. In this case, the map y → u(y) acts from U = [−1, 1] N to H 1 0 (D). However, if we consider the discretization of (1.3) in some finite element space V h ⊂ H 1 0 (D), where h refers to the corresponding mesh size, using for instance the Galerkin method, then the resulting map y → u h (y), acts from U = [−1, 1] N to V h . Likewise, if we consider a quantity of interest such as the flux q(u) = Σ a∇u · σ over a given interface Σ ⊂ D with σ being the outward pointing normal vector, then the resulting map y → q(y) = q(u(y)), acts from U = [−1, 1] N to R. In all three cases, the above maps act from U to some finite-or infinite-dimensional Banach space V , which is either H 1 0 , V h or R. In the previous instances, the functional dependence between the input parameters y and the output u(y) is described in clear mathematical terms by equation (1.1). In other practical instances, the output u(y) can be the outcome of a complex physical experiment or numerical simulation with input parameter y. However the dependence on y might not be given in such clear mathematical terms. In all the abovementioned cases, we assume that are we are able to query the map (1.2) at any given parameter value y ∈ U , eventually up to some uncertainty. Such uncertainty may arise due to: (i) measurement errors, when y → u(y) is obtained by a physical experiment, or (ii) computational errors, when y → u(y) is obtained by a numerical computation. The second type of errors may result from the spatial discretization when solving a PDE with a given method, and from the round-off errors when solving the associated discrete systems. One common way of modeling such errors is by assuming that we observe u at the selected points y up to a an additive noise η which may depend on y, that is, we evaluate y → u(y) + η(y), (1.6) where η satisfies a uniform bound η L ∞ (U,V ) := sup y∈U η(y) V ≤ ε,(1.7) for some ε > 0 representing the noise level. Queries of the exact u(y) or of the noisy u(y) + η(y) are often expensive since they require numerically solving a PDE, or setting up a physical experiment, or running a time-consuming simulation algorithm. A natural objective is therefore to approximate the map (1.2) from some fixed number m of such queries at points {y 1 , . . . , y m } ∈ U . Such approximations y → u(y) are sometimes called surrogate or reduced models. Let us note that approximation of the map (1.2) is sometimes a preliminary task for solving other eventually more complicated problems, such as: 1. Optimization and Control, i.e. find a y which minimizes a certain criterion depending on u(y). In many situations, the criterion takes the form of a convex functional of u(y), and the minimization is subject to feasibility constraints. See e.g. the monographs [3,30] and references therein for an overview of classical formulations and numerical methods for optimization problems. 2. Inverse Problems, i.e. find an estimate y from some data depending on the output u(y). Typically, we face an ill-posed problem, where the parameter-tosolution map does not admit a global and stable inverse. Nonetheless, developing efficient numerical methods for approximating the parameter-to-solution map, i.e. solving the so-called direct problem, is a first step towards the construction of numerical methods for solving the more complex inverse problem, see e.g. [33]. 3. Uncertainty Quantification, i.e. describe the stochastic properties of the solution u(y) in the case where the parameter y is modeled by a random variable distributed according to a given probability density. We may for instance be interested in computing the expectation or variance of the V -valued random variable u(y). Note that this task amounts in computing multivariate integrals over the domain U with respect to the given probability measure. This area also embraces, among others, optimization and inverse problems whenever affected by uncertainty in the data. We refer to e.g. [24] for the application of polynomial approximation to uncertainty quantification, and to [32] for the Bayesian approach to inverse problems. There exist many approaches for approximating an unknown function of one or several variables from its evaluations at given points. One of the most classical approach consists in picking the approximant in a given suitable n-dimensional space of elementary functions, such that n ≤ m. (1.8) Here, by "suitable" we mean that the space should have the ability to approximate the target function to some prescribed accuracy, taking for instance advantage of its smoothness properties. By "elementary" we mean that such functions should have simple explicit form which can be efficiently exploited in numerical computations. The simplest type of such functions are obviously polynomials in the variables y j . As a classical example, we may decide to use, for some given k ∈ N 0 , the total degree polynomial space of order k, namely Note that since u(y) is V -valued, this means that we actually use the V -valued polynomial space V k := V ⊗ P k =    y → \|ν\|≤k w ν y ν : w ν ∈ V    . Another classical example is the polynomial space of degree k in each variable, namely Q k := span y → y ν : ν ∞ = max j=1,...,d ν j ≤ k . A critical issue encountered by choosing such spaces is the fact that, for a fixed value of k, the dimension of P k grows with d like d k , and that of Q k grows like k d , that is, exponentially in d. Since capturing the fine structure of the map (1.2) typically requires a large polynomial degree k in some coordinates, we expect in view of (1.8) that the number of needed evaluations m becomes prohibitive as the number of variables becomes large. This state of affairs is a manifestation of the so-called curse of dimensionality. From an approximation theoretic or information-based complexity point of view, the curse of dimensionality is expressed by the fact that functions in standard smoothness classes such as C s (U ) cannot be approximated in L ∞ (U ) with better rate then n −s/d by any method using n degrees of freedom or n evaluations, see e.g. [18,31]. Therefore, in high dimension, one is enforced to give up on classical polynomial spaces of the above form, and instead consider more general spaces of the general form P Λ := span{y → y ν : ∈ Λ}, (1.9) where Λ is a subset of N d 0 with a given cardinality n := #(Λ). In the case of infinitely many variables d = ∞, we replace N d 0 by the set F := ℓ 0 (N, N 0 ) := {ν = (ν j ) j≥1 : #(supp(ν)) < ∞}, of finitely supported sequences of nonnegative integers. For V -valued functions, we thus use the space V Λ = V ⊗ P Λ := y → ν∈Λ w ν y ν : w ν ∈ V . Note that V Λ = P Λ in the particular case where V = R. The main objective when approximating the map (1.2) is to maintain a reasonable trade-off between accuracy measured in a given error norm and complexity measured by n, exploiting the different importance of each variable. Intuitively, large polynomial degrees should only be allocated to the most important variables. In this sense, if d is the dimension and k is the largest polynomial degree in any variable appearing in Λ, we view Λ as a very sparse subset of {0, . . . , k} d . As generally defined by (1.9), the space P Λ does not satisfy some natural properties of usual polynomial spaces such as closure under differentiation in any variable, or invariance by a change of basis when replacing the monomials y ν by other tensorized basis functions of the form φ ν (y) = j≥1 φ νj (y j ), where the univariate functions {φ 0 , . . . , φ k } form a basis of P k for any k ≥ 0, for example with the Legendre or Chebyshev polynomials. In order to fulfill these requirements, we ask that the set Λ has the following natural property. Definition 1.1 A set Λ ⊂ N d 0 or Λ ⊂ F is downward closed if and only if ν ∈ Λ and ν ≤ ν =⇒ ν ∈ Λ, where ν ≤ ν means that ν j ≤ ν j for all j. Downward closed sets are also called lower sets. We sometimes use the terminology of downward closed polynomial spaces for the corresponding P Λ . To our knowledge, such spaces have been first considered in [23] in the bivariate case d = 2 and referred to as polynômes pleins. Their study in general dimension d has been pursued in [25] and [16]. The objective of the present paper is to give a survey of recent advances on the use of downward closed polynomial spaces for high-dimensional approximation. The outline is the following. We review in Section 2 several polynomial approximation results obtained in [1,2] in which the use of well-chosen index sets allows one to break the curse of dimensionality for relevant classes of functions defined on U = [−1, 1] d , e.g. such as those occuring when solving the elliptic PDE (1.3) with parametric diffusion coefficient (1.4). Indeed, we obtain an algebraic convergence rate n −s , where s is independent of d in the sense that such a rate may even hold when d = ∞. Here, we consider the error between the map (1.2) and its approximant in either norms L ∞ (U, V ) = L ∞ (U, V, dµ) or L 2 (U, V ) = L 2 (U, V, dµ), where dµ is the uniform probability measure, dµ := j≥1 dy j 2 . We also consider the case of lognormal diffusion coefficients of the form a = exp(b), b = b(y) = j≥1 y j ψ j ,(1.10) where the y j are i.i.d. standard Gaussians random variables. In this case, we have U = R d and the error is measured in L 2 (U, V, dγ) where dγ := j≥1 g(y j )dy j , g(t) := 1 √ 2π e −t 2 /2 , (1.11) is the tensorized Gaussian probability measure. The above approximation results are established by using n-term truncations of polynomial expansions, such as Taylor, Legendre or Hermite, which do not necessarily result in downward closed index sets. In the present paper we provide a general approach to establish similar convergence rates with downward closed polynomial spaces. The coefficients in the polynomial expansions cannot be computed exactly from a finite number of point evaluations of (1.2). One first numerical procedure that builds a polynomial approximation from point evaluations is interpolation. In this case the number m of samples is exactly equal to the dimension n of the polynomial space. We discuss in Section 3 a general strategy to choose evaluation points and compute the interpolant in arbitrarily high dimension. One of its useful features is that the evaluations and interpolants are updated in a sequential manner as the polynomial space is enriched. We study the stability of this process and its ability to achieve the same convergence rates in L ∞ established in Section 2. A second numerical procedure for building a polynomial approximation is the least-squares method, which applies to the overdetermined case m > n. To keep the presentation concise, we confine to results obtained in the analysis of this method only for the case of evaluations at random points. In Section 4 we discuss standard least squares, both in the noisy and noiseless cases, and in particular explain under which circumstances the method is stable and compares favorably with the best approximation error in L 2 . Afterwards we discuss the more general method of weighted least squares, which allows one to optimize the relation between the dimension of the polynomial space n and the number of evaluation points m that warrants stability and optimal accuracy. The success of interpolation and least squares is critically tied to the choice of proper downward closed sets (Λ n ) n≥1 with #(Λ n ) = n. The set Λ * n that gives the best polynomial approximation among all possible downward closed sets of cardinality n is often not accessible. In practice we need to rely on some a-priori analysis to select "suboptimal yet good" sets. An alternative strategy is to select the sequence (Λ n ) n≥1 in an adaptive manner, that is, make use of the computation of the approximation for Λ n−1 in order to choose Λ n . We discuss in Section 5 several adaptive and nonadaptive strategies which make use of the downward closed structure of such sets. While our paper is presented in the framework of polynomial approximation, the concept of downward closed set may serve to define multivariate approximation procedures in other nonpolynomial frameworks. At the end of the paper we give some remarks on this possible extension, including, as a particular example, approximation by sparse piecewise polynomial spaces using hierarchical bases, such as sparse grid spaces. Let us finally mention that another class of frequently used methods in highdimensional approximation is based on Reproducing Kernel Hilbert Space (RKHS) or equivalently Gaussian process regression, also known as kriging. In such methods, for a given Mercer kernel K(·, ·) the approximant is typically searched by minimizing the associated RKHS norm among all functions agreeing with the data at the evaluation points, or equivalently by computing the expectation of a Gaussian process with covariance function K conditional to the observed data. Albeit natural competitors, these methods do not fall in the category discussed in the present paper, in the sense that the space where the approximation is picked varies with the evaluation points. It is not clear under which circumstances they may also break the curse of dimensionality. where for each ν = (ν j ) j≥1 ∈ F the function φ ν : U → R has the tensor product form φ ν (y) = j≥1 φ νj (y j ), and u ν ∈ V . Here we assume that (φ k ) k≥0 is a sequence of univariate polynomials such that φ 0 ≡ 1 and the degree of φ k is equal to k. This implies that {φ 0 , . . . , φ k } is a basis of P k and that the above product only involves a finite number of factors, even in the case where d = ∞. Thus, we obtain polynomial approximations of (1.2) by fixing some sets Λ n ⊂ F with #(Λ n ) = n and defining u Λn := ν∈Λn u ν φ ν . (2.2) Before discussing specific examples, let us make some general remarks on the truncation of countable expansions with V -valued coefficients, not necessarily of tensor product or polynomial type. Definition 2.1 The series (2.1) is said to converge conditionally with limit u in a given norm · if and only if there exists an exhaustion (Λ n ) n≥1 of F (which means that for any ν ∈ F there exists n 0 such that ν ∈ Λ n for all n ≥ n 0 ), with the convergence property lim n→+∞ u − u Λn = 0. (2.3) The series (2.1) is said to converge unconditionally towards u in the same norm, if and only if (2.3) holds for every exhaustion (Λ n ) n≥1 of F . As already mentioned in the introduction, we confine our attention to the error norms L ∞ (U, V ) or L 2 (U, V ) with respect to the uniform probability measure dµ. We are interested in establishing unconditional convergence, as well as estimates of the error between u and its truncated expansion, for both types of norm. In the case of the L 2 norm, unconditional convergence can be established when (φ ν ) ν∈F is an orthonormal basis of L 2 (U ). In this case we know from standard Hilbert space theory that if (1.2) belongs to L 2 (U, V ) then the inner products u ν := U u(y)φ ν (y) dµ, ν ∈ F , are elements of V , and the series (2.1) converges unconditionally towards u in L 2 (U, V ). In addition, the error is given by u − u Λn L 2 (U,V ) =   ν / ∈Λn u ν 2 V   1/2 , (2.4) for any exhaustion (Λ n ) n≥1 . Let us observe that, since dµ is a probability measure, the L ∞ (U, V ) norm controls the L 2 (U, V ) norm, and thus the above holds whenever the map u is uniformly bounded over U . For the L ∞ norms, consider an expansion (2.1) where the functions φ ν : U → R are normalized such that φ ν L ∞ (U) = 1, for all ν ∈ F . Then ( u ν V ) ν∈F ∈ ℓ 1 (F ), and it is easily checked that, whenever the expansion (2.1) converges conditionally to a function u in L ∞ (U, V ), it also converges unconditionally to u in L ∞ (U, V ). In addition, for any exhaustion (Λ n ) n≥1 , we have the error estimate u − u Λn L ∞ (U,V ) ≤ ν / ∈Λn u ν V . (2.5) The above estimate is simply obtained by triangle inequality, and therefore generally it is not as sharp as (2.4). One particular situation is when (φ ν ) ν∈F is an orthogonal basis of L 2 (U ) normalized in L ∞ . Then, if u ∈ L 2 (U, V ) and the u ν := 1 φ ν 2 L 2 (U,V ) U u(y)φ ν (y) dµ, ν ∈ F , satisfy ( u ν V ) ν∈F ∈ ℓ 1 (F ), we find on the one hand that (2.1) converges unconditionally to a limit in L ∞ (U, V ) and in turn in L 2 (U, V ). On the other hand, we know that it converges toward u ∈ L 2 (U, V ). Therefore, its limit in L ∞ (U, V ) is also u. A crucial issue is the choice of the sets Λ n that we decide to use when defining the n-term truncation (2.2). Ideally, we would like to use the set Λ n which minimizes the truncation error in some given norm · among all sets of cardinality n. In the case of the L 2 error, if (φ ν ) ν∈F is an orthonormal basis of L 2 (U ), the estimate (2.4) shows that the optimal Λ n is the set of indices corresponding to the n largest u ν V . This set is not necessarily unique, in which case any realization of Λ n is optimal. In the case of the L ∞ error, there is generally no simple description of the optimal Λ n . However, when the φ ν are normalized in L ∞ (U ), the right-hand side in the estimate (2.5) provides an upper bound for the truncation error. This bound is minimized by again taking for Λ n the set of indices corresponding to the n largest u ν V , with the error now bounded by the ℓ 1 tail of the sequence ( u ν V ) ν∈F , in contrast to the ℓ 2 tail which appears in (2.4). The properties of a given sequence (c ν ) ν∈F which ensure a certain rate of decay n −s of its ℓ q tail after one retains its n largest entries are well understood. Here, we use the following result, see [12], originally invoked by Stechkin in the particular case q = 2. This result says that the rate of decay is governed by the ℓ p summability of the sequence for values of p smaller than q. In view of (2.4) or (2.5), application of the above result shows that ℓ p summability of the sequence ( u ν V ) ν∈F implies a convergence rate n −s when retaining the terms corresponding to the n largest u ν V in (2.1). From (2.4), when (φ ν ) ν∈F is an orthonormal basis, we obtain s = 1 p − 1 2 if p < 2. From (2.5), when the φ ν are normalized in L ∞ (U ), we obtain s = 1 p − 1 if p < 1. In the present setting of polynomial approximation, we mainly consider four types of series corresponding to four different choices of the univariate functions φ k : • Taylor (or power) series of the form ν∈F t ν y ν , t ν := 1 ν! ∂ ν u(y = 0), ν! := j≥1 ν j !, (2.6) with the convention that 0! = 1. • Legendre series of the form ν∈F w ν L ν (y), L ν (y) = j≥1 L νj (y j ), w ν := U u(y)L ν (y) dµ, (2.7) where (L k ) k≥0 is the sequence of Legendre polynomials on [−1, 1] normalized with respect to the uniform measure 1 −1 \|L k (t)\| 2 dt 2 = 1, so that (L ν ) ν∈F is an orthonormal basis of L 2 (U, dµ). • Renormalized Legendre series of the form ν∈F w ν L ν (y), L ν (y) = j≥1 L νj (y j ), w ν :=   j≥1 (1 + 2ν j )   1/2 w ν , (2.8) where ( L k ) k≥0 is the sequence of Legendre polynomials on [−1, 1] with the stan- dard normalization L k L ∞ ([−1,1]) = L k (1) = 1, so that L k = (1 + 2k) −1/2 L k . • Hermite series of the form ν∈F h ν H ν (y), H ν (y) = j≥1 H νj (y j ), h ν := U u(y)H ν (y) dγ, (2.9) with (H k ) k≥0 being the sequence of Hermite polynomials normalized according to R \|H k (t)\| 2 g(t)dt = 1, and dγ given by (1.11). In this case U = R d and (H ν ) ν∈F is an orthonormal basis of L 2 (U, dγ). We may therefore estimate the L 2 error resulting from the truncation of the Legendre series (2.7) by application of (2.4), or the L ∞ error resulting from the truncation of the Taylor series (2.6) or renormalized Legendre series (2.8) by application of (2.5). According to Lemma 2.1, we derive convergence rates that depend on the value of p such that of the coefficient sequences ( t ν V ) ν∈F , ( w ν V ) ν∈F , ( w ν V ) ν∈F or ( h ν V ) ν∈F belong to ℓ p (F ). In a series of recent papers such summability results have been obtained for various types of parametric PDEs. We refer in particular to [13,1] for the elliptic PDE (1.3) with affine parameter dependence (1.4), to [21,2] for the lognormal dependence (1.10), and to [9] for more general PDEs and parameter dependence. One specific feature is that these conditions can be fulfilled in the infinite-dimensional framework. We thus obtain convergence rates that are immune to the curse of dimensionality, in the sense that they hold with d = ∞. Here, we mainly discuss the results established in [1,2] which have the specificity of taking into account the support properties of the functions ψ j . One problem with this approach is that the sets Λ n associated to the n largest values in these sequences are generally not downward closed. In the next sections, we revisit these results in order to establish similar convergence rates for approximation in downward closed polynomial spaces. Summability Results The summability results in [1,2] are based on certain weighted ℓ 2 estimates which can be established for the previously defined coefficient sequences under various relevant conditions for the elliptic PDE (1.3). We first report below these weighted estimates. The first one from [1] concerns the affine parametrization (1.4). Here, we have V = H 1 0 (D) and V ′ denotes its dual H −1 (D). Theorem 2.1 Assume that ρ = (ρ j ) j≥1 is a sequence of positive numbers such that j≥1 ρ j \|ψ j (x)\| ≤ a(x) − r, x ∈ D, (2.10) for some fixed number r > 0. Then, one has ν∈F (ρ ν t ν V ) 2 < ∞, ρ ν = j≥1 ρ νj j , (2.11) as well as ν∈F β(ν) −1 ρ ν w ν V 2 = ν∈F β(ν) −2 ρ ν w ν V 2 < ∞, (2.12) with β(ν) := j≥1 (1 + 2ν j ) 1/2 . The constants bounding these sums depend on r, f V ′ , a min and a L ∞ . A few words are in order concerning the proof of these estimates. The first estimate (2.11) is established by first proving that the uniform ellipticity assumption (1.5) implies the ℓ 2 summability of the Taylor sequence ( t ν V ) ν∈F . Since the assumption (2.10) means that (1.5) holds with the ψ j replaced by ρ j ψ j , this gives the ℓ 2 summability of the Taylor sequence for the renormalized map y → u(ρy), ρy = (ρ j y j ) j≥1 , which is equivalent to (2.11). The second estimate is established by first showing that j≥1 ρ j \|ψ j \| ≤ a − r implies finiteness of the weighted Sobolev-type norm ν∈F ρ 2ν ν! U ∂ ν u(y) 2 V j≥1 (1 − \|y j \|) 2νj dµ < ∞. Then, one uses the Rodrigues formula L k (t) = d dt k √ 2k+1 k! 2 k (t 2 − 1) k in each variable y j to bound the weighted ℓ 2 sum in (2.12) by this norm. Remark 2.1 As shown in [1], the above result remains valid for more general classes of orthogonal polynomials of Jacobi type, such as the Chebyshev polynomials which are associated with the univariate measure dt 2π √ 1−t 2 . The second weighted ℓ 2 estimate from [2] concerns the lognormal parametrization (1.10). Theorem 2.2 Let r ≥ 0 be an integer. Assume that there exists a positive sequence ρ = (ρ j ) j≥1 such that j≥1 exp(−ρ 2 j ) < ∞ and such that j≥1 ρ j \|ψ j (x)\| = K < C r := ln 2 √ r , x ∈ D. (2.13) Then, one has ν∈F ξ ν h ν 2 V < ∞, (2.14) where ξ ν := ν ℓ ∞ ≤r ν ν ρ 2 ν = j≥1 r l=0 ν j l ρ 2l j , ν ν := j≥1 ν j ν j , with the convention that k l = 0 when l > k. The constant bounding this sum depends on f V ′ , j≥1 exp(−ρ 2 j ) and on the difference C r − K. Similar to the weighted ℓ 2 estimate (2.12) for the Legendre coefficients, the proof of (2.14) follows by first establishing finiteness of a weighed Sobolev-type norm ν ℓ ∞ ≤r ρ 2ν ν! U ∂ ν u(y) 2 V dγ < ∞, under the assumption (2.13) in the above theorem. Then one uses the Rodrigues formula H k (t) = (−1) k √ k! g (k) (t) g(t) , with g given by (1.11), in each variable y j to bound the weighted ℓ 2 sum in (2.14) by this norm. In summary, the various estimates expressed in the above theorems all take the form ν∈F (ω ν c ν ) 2 < ∞, where c ν ∈ { t ν V , w ν V , w ν V , h ν V }, or equivalently ω ν ∈ {ρ ν , ρ ν β(ν) −1 , ρ ν β(ν) −2 , ξ 1/2 ν }. Then, one natural strategy for establishing ℓ p summabilty of the sequence (c ν ) ν∈F is to invoke Hölder's inequality, which gives, for all 0 < p < 2, ν∈F \|c ν \| p 1/p ≤ ν∈F (ω ν c ν ) 2 1/2 ν∈F \|κ ν \| q 1/q < ∞, 1 q := 1 p − 1 2 , where the sequence (κ ν ) ν∈F is defined by κ ν := ω −1 ν . Therefore ℓ p summability of (c ν ) ν∈F follows from ℓ q summability of (κ ν ) ν∈F with 0 < q < ∞ such that 1 q = 1 p − 1 2 . This ℓ q summability can be related to that of the univariate sequence b = (b j ) j≥1 , b j := ρ −1 j . Indeed, from the factorization ν∈F b qν = j≥1 n≥0 b nq j , one readily obtains the following elementary result, see [12] for more details. Lemma 2.2 For any 0 < q < ∞, one has b ∈ ℓ q (N) and b ℓ ∞ < 1 ⇐⇒ (b ν ) ν∈F ∈ ℓ q (F ). In the case ω ν = ρ ν , i.e. κ ν = b ν , this shows that the ℓ p summability of the Taylor coefficients ( t ν V ) ν∈F follows if the assumption (2.10) holds with b = (ρ −1 j ) j≥1 ∈ ℓ q (N) and ρ j > 1 for all j. By a similar factorization, it is also easily checked that for any algebraic factor of the form α(ν) := j≥1 (1 + c 1 ν j ) c2 with c 1 , c 2 ≥ 0, one has b ∈ ℓ q (N) and b ℓ ∞ < 1 ⇐⇒ (α(ν)b ν ) ν∈F ∈ ℓ q (F ). This allows us to reach a similar conclusion in the cases ω ν = β(ν) −1 ρ ν or ω ν = β(ν) −2 ρ ν , which correspond to the Legendre coefficients ( w ν V ) ν∈F and ( w ν V ) ν∈F , in view of (2.12). Likewise, in the case where ω ν = ξ 1/2 ν , using the factorization ν∈F κ q ν = j≥1 n≥0 r l=0 n l ρ 2l j −q/2 , it is shown in [2] that the sum on the left converges if b ∈ ℓ q , provided that r was chosen large enough such that q > 2 r . This shows that the ℓ p summability of the Hermite coefficients ( h ν V ) ν∈F follows if the assumption (2.13) holds with b = (ρ −1 j ) j≥1 ∈ ℓ q (N). Note that, since the sequence b can be renormalized, we may replace (2.13) by the condition sup x∈D j≥1 ρ j \|ψ j (x)\| < ∞, (2.15) without a specific bound. Approximation by Downward Closed Polynomials The above results, combined with Lemma 2.1, allow us to build polynomial approximations u Λn with provable convergence rates n −s in L ∞ or L 2 by n-term trunctation of the various polynomial expansions. However, we would like to obtain such convergence rates with sets Λ n that are in addition downward closed. Notice that if a sequence (κ ν ) ν∈F of nonnegative numbers is monotone nonincreasing, that is ν ≤ ν =⇒ κ ν ≤ κ ν , then the set Λ n corresponding to the n largest values of κ ν (up to a specific selection in case of equal values) is downward closed. More generally, there exists a sequence (Λ n ) n≥1 of downward closed realizations of such sets which is nested, i.e. Λ 1 ⊂ Λ 2 . . ., with Λ 1 = 0 F := (0, 0, . . .). For any sequence (κ ν ) ν∈F tending to 0, in the sense that #{ν : \|κ ν \| > δ} < ∞ for all δ > 0, we introduce its monotone majorant ( κ ν ) ν∈F defined by κ ν := max ν≥ν \|κ ν \|, that is the smallest monotone nonincreasing sequence that dominates (κ ν ) ν∈F . In order to study best n-term approximations using downward closed sets, we adapt the ℓ q spaces as follows. Definition 2.2 For 0 < q < ∞, we say that (κ ν ) ν∈F ∈ ℓ ∞ (F ) belongs to ℓ q m (F ) if and only if its monotone majorant ( κ ν ) ν∈F belongs to ℓ q (F ). We are now in position to state a general theorem that gives a condition for approximation using downward closed sets in terms of weighted ℓ 2 summability. Theorem 2.3 Let (c ν ) ν∈F and (ω ν ) ν∈F be positive sequences such that ν∈F (ω ν c ν ) 2 < ∞, and such that (κ ν ) ν∈F ∈ ℓ q m (F ) for some 0 < q < ∞ with κ ν = ω −1 ν . Then, for any 0 < r ≤ 2 such that 1 q > 1 r − 1 2 , there exists a nested sequence (Λ n ) n≥1 of downward closed sets such that #(Λ n ) = n and   ν / ∈Λn c r ν   1/r ≤ Cn −s , s := 1 q + 1 2 − 1 r > 0. (2.16) Proof 2.1 With ( κ ν ) ν∈F being the monotone majorant of (κ ν ) ν∈F , we observe that A 2 := ν∈F ( κ −1 ν c ν ) 2 ≤ ν∈F (κ −1 ν c ν ) 2 = ν∈F (ω ν c ν ) 2 < ∞. We pick a nested sequence (Λ n ) n≥1 of downward closed sets, such that Λ n consists of the indices corresponding to the n largest κ ν . Denoting by ( κ n ) n≥1 the decreasing rearrangement of ( κ ν ) ν∈F , we observe that n κ q n ≤ n j=1 κ q j ≤ B q , B := ( κ ν ) ν∈F ℓ q < ∞. With p such that 1 p = 1 r − 1 2 , we find that   ν / ∈Λn c r ν   1/r ≤   ν / ∈Λn ( κ −1 ν c ν ) 2   1/2   ν / ∈Λn κ p ν   1/p ≤ A   κ p−q n+1 ν / ∈Λn κ q ν   1/p ≤ AB(n + 1) 1/p−1/q , where we have used Hölder's inequality and the properties of ( κ n ) n≥1 . This gives (2.16) with C := AB. We now would like to apply the above result with c ν ∈ { t ν V , w ν V , w ν V , h ν V }, and the corresponding weight sequences ω ν ∈ {ρ ν , ρ ν β(ν) −1 , ρ ν β(ν) −2 , ξ 1/2 ν }, or equiv- alently κ ν ∈ {b ν , b ν β(ν), b ν β(ν) 2 , ξ −1/2 ν }. In the case of the Taylor series, where κ ν = b ν , we readily see that if b j < 1 for all j ≥ 1, then the sequence (κ ν ) ν∈F is monotone nonincreasing, and therefore Lemma 2.2 shows that b ∈ ℓ q implies (κ ν ) ν∈F ∈ ℓ q m (F ). By application of Theorem 2.3 with the value r = 1, this leads to the following result. Theorem 2.4 If (2.10) holds with (ρ −1 j ) j≥1 ∈ ℓ q (N) for some 0 < q < ∞ and ρ j > 1 for all j, then u − u Λn L ∞ (U,V ) ≤ Cn −s , s := 1 q − 1 2 , where u Λn is the truncated Taylor series and Λ n is any downward closed set corresponding to the n largest κ ν . In the case of the Legendre series, the weight κ ν = b ν β(ν) is not monotone nonincreasing due to the presence of the algebraic factor β(ν). However, the following result holds. Lemma 2.3 For any 0 < q < ∞ and for any algebraic factor of the form α(ν) := j≥1 (1 + c 1 ν j ) c2 with c 1 , c 2 ≥ 0, one has b ∈ ℓ q (N) and b ℓ ∞ < 1 ⇐⇒ (α(ν)b ν ) ν∈F ∈ ℓ q m (F ). Proof 2.2 The implication from right to left is a consequence of Lemma 2.2, and so we concentrate on the implication from left to right. For this it suffices to find a majorant κ ν of κ ν := α(ν)b ν which is monotone nonincreasing and such that ( κ ν ) ν∈F ∈ ℓ q (F ). We notice that for any τ > 1, there exists C = C(τ, c 1 , c 2 ) ≥ 1 such that (1 + c 1 n) c2 ≤ Cτ n , n ≥ 0. For some J ≥ 1 and τ to be fixed further, we may thus write κ ν ≤ κ ν := C J J j=1 (τ b j ) νj j>J (1 + c 1 ν j ) c2 b νj j . Since b ℓ ∞ < 1 we can take τ > 1 such that θ := τ b ℓ ∞ < 1. By factorization, we find that ν∈F κ q ν = C Jq   J j=1   n≥0 θ qn       j>J   n≥0 (1 + c 1 n) qc2 b nq j     . The first product is bounded by (1 − θ q ) −J . Each factor in the second product is a converging series which is bounded by 1 + cb q j for some c > 0 that depends on c 1 , c 2 and b ℓ ∞ . It follows that this second product converges. Therefore ( κ ν ) ν∈F belongs to ℓ q (F ). Finally, we show that κ ν is monotone nonincreasing provided that J is chosen large enough. It suffices to show that κ ν+ej ≤ κ ν for all ν ∈ F and for all j ≥ 1 where e j := (0, . . . , 0, 1, 0, . . . ), is the Kronecker sequence of index j. When j ≤ J this is obvious since κ ν+ej = τ b j κ ν ≤ θ κ ν ≤ κ ν . When j > J, we have κ ν+ej κ −1 ν = b j 1 + c 1 (ν j + 1) 1 + c 1 ν j c2 . Noticing that the sequence a n := 1+c1(n+1) 1+c1n c2 converges toward 1 and is therefore bounded, and that b j tends to 0 as j → +∞, we find that for J sufficiently large, the right-hand side in the above equation is bounded by 1 for all ν and j > J. From Lemma 2.3, by applying Theorem 2.3 with r = 1 or r = 2, we obtain the following result. Theorem 2.5 If (2.10) holds with (ρ −1 j ) j≥1 ∈ ℓ q (N) for some 0 < q < ∞ and ρ j > 1 for all j, then u − u Λn L 2 (U,V ) ≤ Cn −s , s := 1 q , where u Λn is the truncated Legendre series and Λ n is any downward closed set corresponding to the n largest κ ν where κ ν := b ν β(ν). If q < 2, we also have u − u Λn L ∞ (U,V ) ≤ Cn −s , s := 1 q − 1 2 , with Λ n any downward closed set corresponding to the n largest κ ν where κ ν : = b ν β(ν) 2 . Finally, in the case of the Hermite coefficients, which corresponds to the weight κ ν := j≥1 r l=0 ν j l b −2l j −1/2 , (2.17) we can establish a similar summability result. Lemma 2.4 For any 0 < q < ∞ and any integer r ≥ 1 such that q > 2 r , we have b ∈ ℓ q (N) =⇒ (κ ν ) ν∈F ∈ ℓ q (F ), where κ ν is given by (2.17). In addition, for any integer r ≥ 0, the sequence (κ ν ) ν∈F is monotone nonincreasing. Proof 2.3 For any ν ∈ F and any k ≥ 1 we have κ ν+e k = r l=0 ν k + 1 l b −2l k −1/2 j≥1 j =k r l=0 ν j l b −2l j −1/2 ≤ r l=0 ν k l b −2l k −1/2 j≥1 j =k r l=0 ν j l b −2l j −1/2 = κ ν , and therefore the sequence (κ ν ) ν∈F is monotone nonincreasing. Now we check that (κ ν ) ν∈F ∈ ℓ q (F ), using the factorization ν∈F κ q ν = j≥1 n≥0 r l=0 n l b −2l j −q/2 ≤ j≥1 n≥0 n r ∧ n −q/2 b q(r∧n) j . (2.18) where the inequality follows from the fact that the value l = n ∧ r := min{n, r} is contained in the sum. The jth factor F j in the rightmost product in (2.18) may be written as F j = 1 + b q j + · · · + b (r−1)q j + C r,q b rq j , where C r,q := n≥r n r −q/2 = (r!) q/2 n≥0 (n + 1) · · · (n + r) −q/2 < ∞, (2.19) since we have assumed that q > 2/r. This shows that each F j is finite. If b ∈ ℓ q (N), there exists an integer J ≥ 0 such that b j < 1 for all j > J. For such j, we can bound F j by 1 + (C r,q + r − 1)b q j , which shows that the product converges. From this lemma, and by application of Theorem 2.3 with the value r = 2, we obtain the following result for the Hermite series. Theorem 2.6 If (2.15) holds with (ρ −1 j ) j≥1 ∈ ℓ q (N) for some 0 < q < ∞, then u − u Λn L 2 (U,V ) ≤ Cn −s , s := 1 q , where u Λn is the truncated Hermite series and Λ n is a downward closed set corresponding to the n largest κ ν given by (2.17). In summary, we have established convergence rates for approximation by downward closed polynomial spaces of the solution map (1.2) associated to the elliptic PDE (1.3) with affine or lognormal parametrization. The conditions are stated in terms of the control on the L ∞ norm of j≥1 ρ j \|ψ j \|, where the ρ j have a certain growth measured by the ℓ q summability of the sequence b = (b j ) j≥1 = (ρ −1 j ) j≥1 . This is a way to quantify the decay of the size of the ψ j , also taking their support properties into account, and in turn to quantify the anisotropic dependence of u(y) on the various coordinates y j . Other similar results have been obtained with different PDE models, see in particular [12]. In the above results, the polynomial approximants are constructed by truncation of infinite series. The remainder of the paper addresses the construction of downward closed polynomial approximants from evaluations of the solution map at m points {y 1 , . . . , y m } ∈ U , and discusses the accuracy of these approximants. Interpolation Sparse Interpolation by Downward Closed Polynomials Interpolation is one of the most standard processes for constructing polynomial approximations based on pointwise evaluations. Given a downward closed set Λ ⊂ F of finite cardinality, and a set of points Γ ⊂ U, #(Γ) = #(Λ), we would like to build an interpolation operator I Λ , that is, I Λ u ∈ V Λ is uniquely characterized by I Λ u(y) = u(y), y ∈ Γ, for any V -valued function u defined on U . In the univariate case, it is well known that such an operator exists if and only if Γ is a set of pairwise distinct points, and that additional conditions are needed in the multivariate case. Moreover, since the set Λ may come from a nested sequence (Λ n ) n≥1 as discussed in Section 2, we are interested in having similar nestedness properties for the corresponding sequence (Γ n ) n≥1 , where #(Γ n ) = #(Λ n ) = n. Such a nestedness property allows us to recycle the n evaluations of u which have been used in the computation of I Λn u, and use only one additional evaluation for the next computation of I Λn+1 u. It turns out that such hierarchical interpolants can be constructed in a natural manner by making use of the downward closed structure of the index sets Λ n . This construction is detailed in [8] but its main principles can be traced from [23]. In order to describe it, we assume that the parameter domain is of either form U = [−1, 1] d or [−1, 1] N , with the convention that d = ∞ in the second case. However, it is easily check that the construction can be generalized in a straightforward manner to any domain with Cartesian product form U = × k≥1 J k , where the J k are finite or infinite intervals. We start from a sequence of pairwise distinct points T = (t k ) k≥0 ⊂ [−1, 1]. We denote by I k the univariate interpolation operator on the space V k := V ⊗ P k associated with the k-section {t 0 , . . . , t k } of this sequence, that is, I k u(t i ) = u(t i ), i = 0, . . . , k, for any V -valued function u defined on [−1, 1]. We express I k in the Newton form I k u = I 0 u + k l=1 ∆ l u, ∆ l := I l − I l−1 ,(3.1) and set I −1 = 0 so that we can also write I k u = k l=0 ∆ l u. Obviously the difference operator ∆ k annihilates the elements of V k−1 . In addition, since ∆ k u(t j ) = 0 for j = 0, . . . , k − 1, we have ∆ k u(t) = α k B k (t), where B k (t) := k−1 l=0 t − t l t k − t l . The coefficient α k ∈ V can be computed inductively, since it is given by α k = α k (u) := u(t k ) − I k−1 u(t k ), that is, the interpolation error at t k when using I k−1 . Setting B 0 (t) := 1, we observe that the system {B 0 , . . . , B k } is a basis for P k . It is sometimes called a hierarchical basis. In the multivariate setting, we tensorize the grid T , by defining y ν := (t νj ) j≥1 ∈ U, ν ∈ F . We first introduce the tensorized operator I ν := j≥1 I νj , recalling that the application of a tensorized operator ⊗ j≥1 A j to a multivariate function amounts in applying each univariate operator A j by freezing all variables except the jth one, and then applying A j to the unfrozen variable. It is readily seen that I ν is the interpolation operator on the tensor product polynomial space V ν = V ⊗ P ν , P ν := j≥1 P νj , associated to the grid of points Γ ν = × j≥1 {t 0 , . . . , t νj }. This polynomial space corresponds to the particular downward closed index set of rectangular shape Λ = R ν := { ν : ν ≤ ν}. Defining in a similar manner the tensorized difference operators ∆ ν := j≥1 ∆ νj , we observe that I ν = j≥1 I νj = j≥1 ( νj l=0 ∆ l ) = ν∈Rν ∆ ν . The following result from [8] shows that the above formula can be generalized to any downward closed set in order to define an interpolation operator. We recall its proof for sake of completeness. Theorem 3.1 Let Λ ⊂ F be a finite downward closed set, and define the grid Γ Λ := {y ν : ν ∈ Λ}. Then, the interpolation operator onto V Λ for this grid is defined by I Λ := ν∈Λ ∆ ν . (3.2) Proof 3.1 From the downward closed set property, V ν ⊂ V Λ for all ν ∈ Λ. Hence the image of I Λ is contained in V Λ . With I Λ defined by (3.2), we may write I Λ u = I ν u + ν∈Λ, ν ν ∆ ν u, for any ν ∈ Λ. Since y ν ∈ Γ ν , we know that I ν u(y ν ) = u(y ν ). On the other hand, if ν ν, this means that there exists a j ≥ 1 such that ν j > ν j . For this j we thus have ∆ ν u(y) = 0 for all y ∈ U with the jth coordinate equal to t νj by application of ∆ νj in the jth variable, so that ∆ ν u(y ν ) = 0. The interpolation property I Λ u(y ν ) = u(y ν ) thus holds, for all ν ∈ Λ. The decomposition (3.2) should be viewed as a generalization of the Newton form (3.1). In a similar way, its terms can be computed inductively: if Λ = Λ ∪ {ν} where Λ is a downward closed set, we have Therefore, if (Λ n ) n≥1 is any nested sequence of downward closed index sets, we can compute I Λn by n iterations of ∆ ν u = α ν B ν ,I Λi u = I Λi−1 u + α ν i B ν i , where ν i ∈ Λ i is such that Λ i = Λ i−1 ∪ {ν i }. Note that (B ν ) ν∈Λ is a basis of P Λ and that any f ∈ V Λ has the unique decom- position f = ν∈Λ α ν B ν , where the coefficients α ν = α ν (f ) ∈ V are defined by the above procedure. Also note that α ν (f ) does not depend on the choice of Λ but only on ν and f . Stability and Error Estimates The pointwise evaluations of the function u could be affected by errors, as modeled by (1 .6) and (1.7). The stability of the interpolation operator with respect to such perturbations is quantified by the Lebesgue constant, which is defined by L Λ := sup I Λ f L ∞ (U,V ) f L ∞ (U,V ) , where the supremum is taken over the set of all V -valued functions f defined everywhere and uniformly bounded over U . It is easily seen that this supremum is in fact independent of the space V , so that we may also write L Λ := sup I Λ f L ∞ (U) f L ∞ (U) , where the supremum is now taken over real-valued functions. Obviously, we have u − I Λ (u + η) L ∞ (U,V ) ≤ u − I Λ u L ∞ (U,V ) + L Λ ε, where ε is the noise level from (1.7). The Lebesgue constant also allows us to estimate the error of interpolation u − I Λ u L ∞ (U,V ) for the noiseless solution map in terms of the best approximation error in the L ∞ norm: for any u ∈ L ∞ (U, V ) and any u ∈ V Λ we have u − I Λ u L ∞ (U,V ) ≤ u − u L ∞ (U,V ) + I Λ u − I Λ u L ∞ (U,V ) , which by infimizing over u ∈ V Λ yields u − I Λ u L ∞ (U,V ) ≤ (1 + L Λ ) inf u∈VΛ u − u L ∞ (U,V ) . We have seen in Section 2 that for relevant classes of solution maps y → u(y), there exist sequences of downward closed sets (Λ n ) n≥1 with #(Λ n ) = n, such that inf u∈VΛ n u − u L ∞ (U,V ) ≤ Cn −s , n ≥ 1, for some s > 0. For such sets, we thus have u − I Λn u L ∞ (U,V ) ≤ C(1 + L Λn )n −s . (3.3) This motivates the study of the growth of L Λn as n → +∞. For this purpose, we introduce the univariate Lebesgue constants L k := sup I k f L ∞ ([−1,1]) f L ∞ ([−1,1]) . Note that L 0 = 1. We also define an analog quantity for the difference operator D k := sup ∆ k f L ∞ ([−1,1]) f L ∞ ([−1,1]) . In the particular case of the rectangular downward closed sets Λ = R ν , since I Λ = I ν = ⊗ j≥1 I νj , we have L Rν = j≥1 L νj . Therefore, if the sequence T = (t k ) k≥0 is such that L k ≤ (1 + k) θ , k ≥ 0,(3.4) for some θ ≥ 1, we find that L Rν ≤ j≥1 (1 + ν j ) θ = (#(R ν )) θ , for all ν ∈ F . For arbitrary downward closed sets Λ, the expression of I Λ shows that L Λ ≤ ν∈Λ j≥1 D νj . Therefore, if the sequence T = (t k ) k≥0 is such that D k ≤ (1 + k) θ , k ≥ 0,(3.5) we find that L Λ ≤ ν∈Λ j≥1 (1 + ν j ) θ = ν∈Λ (#(R ν )) θ ≤ ν∈Λ (#(Λ)) θ = (#(Λ)) θ+1 . The following result from [8] shows that this general estimate is also valid under the assumption (3.4) on the growth of L k . Theorem 3.2 If the sequence T = (t k ) k≥0 is such that (3.4) or (3.5) holds for some θ ≥ 1, then L Λ ≤ (#(Λ)) θ+1 , for all downward closed sets Λ. One noticeable feature of the above result is that the bound on L Λ only depends on #(Λ), independently of the number of variables, which can be infinite, as well as of the shape of Λ. We are therefore interested in univariate sequences T = (t k ) k≥0 such that L k and D k have moderate growth with k. For Chebyshev or Gauss-Lobatto points, given by C k := cos 2l + 1 2k + 2 π : l = 0, . . . , k and G k := cos l k π : l = 0, . . . , k , it is well known that the Lebesgue constant has logarithmic growth L k ∼ ln(k), thus slower than algebraic. However these points are not the k section of a single sequence T , and therefore they are not convenient for our purposes. Two examples of univariate sequences of interest are the following. • The Leja points: from an arbitrary t 0 ∈ [−1, 1] (usually taken to be 1 or 0), this sequence is recursively defined by t k := argmax k−1 l=0 \|t − t l \| : t ∈ [−1, 1] . Note that this choice yields hierarchical basis functions B k that are uniformly bounded by 1. Numerical computations of L k for the first 200 values of k indicates that the linear bound \|e − e l \| : \|e\| ≤ 1 . L k ≤ 1 + k,(3. These points have the property of accumulating in a regular manner on the unit circle according to the so-called Van der Corput enumeration [4]. It is proven in [5] that the linear bound (3.6) holds for the Lebesgue constant of the complex interpolation operator on the unit disc associated to these points. The sequence of real parts t k := ℜ(e k ), is defined after eliminating the possible repetitions corresponding to e k = e l for two different values of k = l. These points coincide with the Gauss-Lobatto points for values of k of the form 2 n + 1 for n ≥ 0. A quadratic bound D k ≤ (1 + k) 2 , is established in [6]. If we use such sequences, application of Theorem 3.2 gives bounds of the form L Λ ≤ (#(Λ)) 1+θ , for example with θ = 2 when using the ℜ-Leja points, or θ = 1 when using the Leja points provided that the conjectured bound (3.6) holds. Combining with (3.3), we obtain the convergence estimate u − I Λn u L ∞ (U,V ) ≤ Cn −(s−1−θ) , which reveals a serious deterioration of the convergence rate when using interpolation instead of truncated expansions. However, for the parametric PDE models discussed in Section 2, it is possible to show that this deterioration actually does not occur, based on the following lemma which relates the interpolation error to the summability of coefficient sequences in general expansions of u. Lemma 3.1 Assume that u admits an expansion of the type (2.1), where φ ν L ∞ (U) ≤ 1 which is unconditionally convergent towards u in L ∞ (U, V ). Assume in addition that y → u(y) is continuous from U equipped with the product topology toward V . If the univariate sequence T = (t k ) k≥0 is such that (3.4) or (3.5) holds for some θ ≥ 1, then, for any downward closed set Λ, u − I Λ u L ∞ (U,V ) ≤ 2 ν / ∈Λ π(ν) u ν V , π(ν) := j≥1 (1 + ν j ) θ+1 . (3.7) Proof 3.2 The unconditional convergence of (2.1) and the continuity of u with respect to the product topology allow us to say that the equality in (2.1) holds everywhere in U . We may thus write I Λ u = I Λ ν∈F u ν φ ν = ν∈F u ν I Λ φ ν = ν∈Λ u ν φ ν + ν / ∈Λ u ν I Λ φ ν , where we have used that I Λ φ ν = φ ν for every ν ∈ Λ since φ ν ∈ P Λ . For the second sum on the right-hand side, we observe that for each ν / ∈ Λ, I Λ φ ν = ν∈Λ ∆ ν φ ν = ν∈Λ∩Rν ∆ ν φ ν = I Λ∩Rν φ ν , since ∆ ν annihilates P ν whenever ν ≤ ν. Therefore u − I Λ u = ν ∈Λ u ν (I − I Λ∩Rν )φ ν , where I stands for the identity operator. This implies u − I Λ u L ∞ (U,V ) ≤ ν ∈Λ (1 + L Λ∩Rν ) u ν V ≤ 2 ν ∈Λ L Λ∩Rν u ν V . Since (3.4) or (3.5) holds, we obtain from Theorem 3.2 that L Λ∩Rν ≤ (#(Λ ∩ R ν )) θ+1 ≤ (#(R ν )) θ+1 = π(ν), which yields (3.7). We can apply the above lemma with the Taylor series (2.6) or the renormalized Legendre series (2.8). This leads us to analyze the ℓ 1 tail of the sequence (c ν ) ν∈F where c ν is either π(ν) t ν V or π(ν) w ν V . If (2.10) holds, we know from Theorem 2.1 that this sequence satisfies the bound ν∈F (ω ν c ν ) 2 < ∞, where ω ν is either π(ν) −1 ρ ν or π(ν) −1 β(ν) −2 ρ ν . Since π(ν) has algebraic growth similar to β(ν), application of Lemma 2.3 and of Theorem 2.3 with the value r = 1, leads to the following result. Theorem 3.3 If (2.10) holds with (ρ −1 j ) j≥1 ∈ ℓ q (N) for some 0 < q < ∞ and ρ j > 1 for all j, then u − I Λn u L ∞ (U,V ) ≤ Cn −s , s := 1 q − 1 2 , where Λ n is any downward closed set corresponding to the n largest κ ν where κ ν is either π(ν)b ν or π(ν)β(ν) 2 b ν . Discrete Least Squares Approximations Discrete Least Squares on V -valued Linear Spaces Least-squares fitting is an alternative approach to interpolation for building a polynomial approximation of u from V Λ . In this approach we are given m observations u 1 , . . . , u m of u at points y 1 , . . . , y m ∈ U ⊆ R d where m ≥ n = #(Λ). We first discuss the least-squares method in the more general setting of V -valued linear spaces, V n := V ⊗ Y n , where Y n is the space of real-valued functions defined everywhere on U such that dim(Y n ) = n. In the next section, we discuss more specifically the case where Y n = P Λ . Here we study the approximation error in the L 2 (U, V, dµ) norm for some given probability measure dµ, when the evaluation points y i are independent and drawn according to this probability measure. For notational simplicity we use the shorthand · := · L 2 (U,V,dµ) . The least-squares method selects the approximant of u in the space V n as u L := argmin u∈Vn 1 m m i=1 u(y i ) − u i 2 V . In the noiseless case where u i := u(y i ) for any i = 1, . . . , m, this also writes u L = argmin u∈VΛ u − u m , (4.1) where the discrete seminorm is defined by f m := 1 m m i=1 f (y i ) 2 V 1/2 . Note that f 2 m is an unbiased estimator of f 2 since we have E( f 2 m ) = f 2 . Let {φ 1 , . . . , φ n } denote an arbitrary L 2 (U, dµ) orthonormal basis of the space Y n . If we expand the solution to (4.1) as n j=1 c j φ j , with c j ∈ V , the V -valued vector c = (c 1 , . . . , c n ) t is the solution to the normal equations Gc = d, (4.2) where the matrix G has entries G j,k = 1 m m i=1 φ j (y i )φ k (y i ), and where the V -valued data vector d = (d 1 , . . . , d n ) t is given by d j := 1 m m i=1 u i φ j (y i ). This linear system always has at least one solution, which is unique when G is nonsingular. When G is singular, we may define u L as the unique minimal ℓ 2 (R n , V ) norm solution to (4.2). In the subsequent analysis, we sometimes work under the assumption of a known uniform bound u L ∞ (U,V ) ≤ τ. (4.3) We introduce the truncation operator z → T τ (z) := z, if z V ≤ τ, z z V , if z V > τ, and notice that it is a contraction: T τ (z) − T τ ( z) V ≤ z − z V for any z, z ∈ V . The truncated least-squares approximation is defined by u T := T τ • u L . Note that, in view of (4.3), we have u(y)− u T (y) V ≤ u(y)− u L (y) V for any y ∈ U and therefore u − u T ≤ u − u L . Note that the random matrix G concentrates toward its expectation which is the identity matrix I as m → ∞. In other words, the probability that G is ill-conditioned becomes very small as m increases. The truncation operator aims at avoiding instabilities which may occur when G is ill-conditioned. As an alternative proposed in [15], we may define for some given A > 1 the conditioned least-squares approximation by u C := u L , if cond(G) ≤ A, u C := 0, otherwise, where cond(G) := λ max (G)/λ min (G) is the usual condition number. The property that G − I 2 ≤ δ for some 0 < δ < 1 amounts to the norm equivalence (1 − δ) f 2 ≤ f 2 m ≤ (1 + δ) f 2 , f ∈ V n . It is well known that if m ≥ n is too much close to n, least-squares methods may become unstable and inaccurate for most sampling distributions. For example, if U = [−1, 1] and Y n = P n−1 is the space of algebraic polynomials of degree n − 1, then with m = n the estimator coincides with the Lagrange polynomial interpolation which can be highly unstable and inaccurate, in particular for equispaced points. Therefore, m should be sufficiently large compared to n for the probability that G is ill-conditioned to be small. This trade-off between m and n has been analyzed in [11], using the function y → k n (y) := n j=1 \|φ j (y)\| 2 , which is the diagonal of the integral kernel of the L 2 (U, dµ) projector on Y n . This function depends on dµ, but not on the chosen orthonormal basis. It is strictly positive in U under minimal assumptions on the orthonormal basis, for example if one element of the basis is the constant function over all U . Obviously, the function k n satisfies U k n dµ = n. We define K n := k n L ∞ (U) ≥ n. The following results for the least-squares method with noiseless evaluations were obtained in [11,27,7,15] for real-valued functions, however their proof extends in a straightforward manner to the present setting of V -valued functions. They are based on a probabilistic bound for the event G − I 2 > δ using the particular value δ = 1 2 , or equivalently the value A = 1+δ 1−δ = 3 as a bound on the condition number of G. then the following hold. (i) The matrix G satisfies the tail bound u satisfies (4.3), then the truncated least-squares estimator satisfies, in the noiseless case, Pr G − I 2 > 1 2 ≤ 2m −r . (ii) IfE( u − u T 2 ) ≤ (1 + ζ(m)) inf u∈Vn u − u 2 + 8τ 2 m −r , where ζ(m) := 4κ ln(m) → 0 as m → +∞, and κ is as in (4.4). (iii) The conditioned least-squares estimator satisfies, in the noiseless case, E( u − u C 2 ) ≤ (1 + ζ(m)) inf u∈Vn u − u 2 + 2 u 2 m −r , where ζ(m) is as in (ii). (iv) If u satisfies (4.3), then the estimator u E ∈ {u L , u T , u C } satisfies, in the noiseless case, u − u E ≤ (1 + √ 2) inf u∈Vn u − u L ∞ (U,V ) , (4.5) with probability larger than 1 − 2m −r . In the case of noisy evaluations modeled by (1.6)-(1.7), the observations are given by u i = u(y i ) + η(y i ). (4.6) The following result from [7] shows that (4.5) holds up to this additional perturbation. u − u E ≤ (1 + √ 2) inf u∈Vn u − u L ∞ (U,V ) + √ 2ε, with probability larger than 1 − 2n −r , where ε is the noise level in (1.7). Similar results, with more general assumptions on the type of noise, are proven in [11,28,15]. Downward Closed Polynomial Spaces and Weighted Least Squares Condition (4.4) shows that K n gives indications on the number m of observations required to ensure stability and accuracy of the least-squares approximation. In order to understand how demanding this condition is with respect to m, it is important to have sharp upper bounds for K n . Such bounds have been proven when the measure dµ on U = [−1, 1] d has the form dµ = C d j=1 (1 − y j ) θ1 (1 + y j ) θ2 dy j ,(4.7) where θ 1 , θ 2 > −1 are real shape parameters and C is a normalization constant such that U dµ = 1. Sometimes (4.7) is called the Jacobi measure, because the Jacobi polynomials are orthonormal in L 2 (U, dµ). Remarkable instances of the measure (4.7) are the uniform measure, when θ 1 = θ 2 = 0, and the Chebyshev measure, when θ 1 = θ 2 = − 1 2 . When Y n = P Λ is a multivariate polynomial space and Λ is a downward closed multi-index set with #(Λ) = n, it is proven in [7,26] that K n satisfies an upper bound which only depends on n and on the choice of the measure (4.7) through the values of θ 1 and θ 2 . Lemma 4.1 Let dµ be the measure defined in (4.7). Then it holds K n ≤ n ln 3 ln 2 , if θ 1 = θ 2 = − 1 2 , n 2 max{θ1,θ2}+2 , if θ 1 , θ 2 ∈ N 0 . (4.8) A remarkable property of both algebraic upper bounds in (4.8) is that the exponent of n is independent of the dimension d, and of the shape of the downward closed set Λ. Both upper bounds are sharp in the sense that equality holds for multi-index sets of rectangular type Λ = R ν corresponding to tensor product polynomial spaces. As an immediate consequence of Theorem 4.1 and Lemma 4.1, we have the next corollary. Other types of results on the accuracy of least squares have been recently established in [14], under conditions of the same type as (4.9). In some situations, for example when n is very large, the conditions (4.9) might require a prohibitive number of observations m. It is therefore a legitimate question to ask whether there exist alternative approaches with less demanding conditions than (4.9) between m and n. At best, we would like that m is of order only slightly larger than n, for example by a logarithmic factor. In addition, the above analysis does not apply to situations where the basis functions φ k are unbounded, such as when using Hermite polynomials in the expansion (2.9). It is thus desirable to ask for the development of approaches that also cover this case. These questions have an affirmative answer by considering weighted least-squares methods, as proposed in [19,22,15]. In the following, we survey some results from [15]. For the space V n = V ⊗ Y n , the weighted least-squares approximation is defined as u W := argmin u∈Vn 1 m m i=1 w i u(y i ) − u i 2 V , for some given choice of weights w i ≥ 0. This estimator is again computed by solving a linear system of normal equations now with the matrix G with entries G j,k = 1 m m i=1 w(y i )φ j (y i )φ k (y i ). Of particular interest to us are weights of the form w i = w(y i ), where w is some nonnegative function defined on U such that U w −1 dµ = 1. (4.10) We then denote by dσ the probability measure dσ := w −1 dµ,(4.11) and we draw the independent points y 1 , . . . , y m from dσ. The case w ≡ 1 and dσ = dµ corresponds to the previously discussed standard (unweighted) least-squares estimator u L . As previously done for u L , we associate to u W a truncated estimator u T and a conditioned estimator u C , by replacing u L with u W in the corresponding definitions. Let us introduce the function y → k n,w (y) := n j=1 w(y)\|φ j (y)\| 2 , where once again {φ 1 , . . . , φ n } is an arbitrary L 2 (U, dµ) orthonormal basis of the space Y n . Likewise, we define K n,w := k n,w L ∞ (U) . The following result, established in [15] for real-valued functions, extends Theorem 4.1 to this setting. Its proof in the V -valued setting is exactly similar. The choice (4.12) also gives K n,w = k n,w L ∞ (U) = n, and leads to the next result, as a consequence of the previous theorem. then the same conclusions of Theorem 4.1 hold true with u L replaced by u W , with w given by (4.12) and the weights taken as w i = w(y i ). The above theorem ensures stability and accuracy of the weighted least-squares approximation, under the minimal condition that m is linearly proportional to n, up to a logarithmic factor. Clearly this is an advantage of weighted least squares compared to standard least squares, since condition (4.4) is more demanding than (4.14) in terms of the number of observations m. However, this advantage comes with some drawbacks that we now briefly recall, see [15] for an extensive description. In general (4.11) and (4.13) are not product measures, even if dµ is one. Therefore, the first drawback of using weighted least squares concerns the efficient generation of independent samples from multivariate probability measures, whose computational cost could be prohibitively expensive, above all when the dimension d is large. In some specific settings, for example downward closed polynomial spaces Y n = P Λ with #(Λ) = n, and when dµ is a product measure, this drawback can be overcome. We refer to [15], where efficient sampling algorithms have been proposed and analyzed. For any m and any downward closed set Λ, these algorithms generate m independent samples with proven bounds on the required computational cost. The dependence on the dimension d and m of these bounds is linear. For the general measure (4.11) the efficient generation of the sample is a nontrivial task, and remains a drawback of such an approach. The second drawback concerns the use of weighted least squares in a hierarchical context, where we are given a nested sequence Λ 1 ⊂ . . . ⊂ Λ n of downward closed sets, instead of a single such set Λ. Since the measure (4.13) depends on n, the sets (Λ n ) n≥1 are associated to different measures (dσ n ) n≥1 . Hence, recycling samples from the previous iterations of the adaptive algorithm is not as straighforward as in the case of standard least squares. As a final remark, let us stress that the above results of Theorem 4.3 and Theorem 4.4 hold for general approximation spaces Y n other than polynomials. Adaptive Algorithms and Extensions Selection of Downward Closed Polynomial Spaces The interpolation and least-squares methods discussed in Section 3 and Section 4 allow us to construct polynomial approximations in V Λ = V ⊗ P Λ of the map (1.2) from its pointwise evaluations, for some given downward closed set Λ. For these methods, we have given several convergence results in terms of error estimates either in L ∞ (U, V ) or L 2 (U, V, dµ). In some cases, these estimates compare favorably with the error of best approximation min u∈VΛ u − u measured in such norms. A central issue which still needs to be addressed is the choice of the downward closed set Λ, so that this error of best approximation is well behaved, for a given map u. Ideally, for each given n, we would like to use the set Λ n = argmin Λ∈Dn min u∈VΛ u − u , where D n is the family of all downward closed sets Λ of cardinality n. However such sets Λ n are not explicitely given to us, and in addition the resulting sequence (Λ n ) n≥1 is generally not nested. Concrete selection strategies aim to produce "suboptimal yet good" nested sequences (Λ n ) n≥1 different from the above. Here, an important distinction should be made between nonadaptive and adaptive selection strategies. In nonadaptive strategies, the selection of Λ n is made in an a-priori manner, based on some available information on the given problem. The results from Section 2.3 show that, for relevant instances of solution maps associated to parametric PDEs, there exist nested sequences (Λ n ) n≥1 of downward closed sets such that #(Λ n ) = n and min u∈VΛ n u − u decreases with a given convergence rate n −s as n → ∞. In addition, these results provide constructive strategies for building the sets Λ n , since these sets are defined as the indices associated to the n largest κ ν := max ν≥ν κ ν like in Theorem 2.5, or directly to the n largest κ ν like in Theorem 2.4 and Theorem 2.6, and since the κ ν are explicitely given numbers. In the case where we build the polynomial approximation by interpolation, Theorem 3.3 shows that a good choice of Λ n is produced by taking κ ν to be either π(ν)b ν or π(ν)β(ν) 2 b ν where b = (ρ −1 j ) j≥1 is such that (2.10) holds. In the case where we build the polynomial approximation by least-squares methods, the various results from Section 4 show that under suitable assumptions, the error is nearly as good as that of best approximation in L 2 (U, V, dµ) with respect to the relevant probability measure. In the affine case, Theorem 2.5 shows that a good choice of Λ n is produced by taking κ ν to be b ν β(ν) where b = (ρ −1 j ) j≥1 is such that (2.10) holds. In the lognormal case Theorem 2.6 shows that a good choice of Λ n is produced by taking κ ν to be given by (2.17) where b = (ρ −1 j ) j≥1 is such that (2.15) holds. Let us briefly discuss the complexity of identifying the downward closed set Λ n associated to the n largest κ ν . For this purpose, we introduce for any downward closed set Λ its set of neighbors defined by N (Λ) := {ν ∈ F \ Λ such that Λ ∪ {ν} is downward closed}. We may in principle define Λ n = {ν 1 , . . . , ν n } by the following induction. • Take ν 1 = 0 F as the null multi-index. • Given Λ k = {ν 1 , . . . , ν k }, choose a ν k+1 maximizing κ ν over ν ∈ N (Λ k ). In the finite-dimensional case d < ∞, we observe that N (Λ k ) is contained in the union of N (Λ k−1 ) with the set consisting of the indices ν k + e j , j = 1, . . . , d, where e j is the Kroenecker sequence with 1 at position j. As a consequence, since the values of the κ ν have already been computed for ν ∈ N (Λ k−1 ), the step k of the induction requires at most d evaluations of κ ν , and therefore the overall computation of Λ n requires at most nd evaluations. In the infinite-dimensional case d = ∞, the above procedure cannot be practically implemented, since the set of neighbors has infinite cardinality. This difficulty can be circumvented by introducing a priority order among the variables, as done in the next definitions. Obviously, if (c ν ) ν∈F is anchored, one of the sets Λ n corresponding to its n largest values is anchored. It is also readily seen that all sequences ( κ ν ) ν∈F that are used in Theorems 2.4, 2.5, 2.6 or 3.3 for the construction of Λ n are anchored, provided that the sequence b = (ρ −1 j ) j≥1 is monotone nonincreasing. This is always the case up to a rearrangement of the variables. For any anchored set Λ, we introduce the set of its anchored neighbors defined by N (Λ) := {ν ∈ N (Λ) : ν j = 0 if j > j(Λ) + 1},(5.1) where j(Λ) := max{j : ν j > 0 for some ν ∈ Λ}. We may thus modify in the following way the above induction procedure. • Take ν 1 = 0 F as the null multi-index. • Given Λ k = {ν 1 , . . . , ν k }, choose a ν k+1 maximizing κ ν over ν ∈ N (Λ k ). This procedure is now feasible in infinite dimension. At each step k the number of active variables is limited by j(Λ k ) ≤ k − 1, and the total number of evaluations of κ ν needed to construct Λ n does not exceed 1 + 2 + · · · + (n − 1) ≤ n 2 /2. In adaptive strategies the sets Λ n are not a-priori selected, but instead they are built in a recursive way, based on earlier computations. For instance, one uses the previous set Λ n−1 and the computed polynomial approximation u Λn−1 to construct Λ n . If we impose that the sets Λ n are nested, this means that we should select an index ν n / ∈ Λ n−1 such that Λ n := Λ n−1 ∪ {ν n }. The choice of the new index ν n is further limited to N (Λ n−1 ) if we impose that the constructed sets Λ n are downward closed, or to N (Λ n−1 ) if we impose that these sets are anchored. Adaptive methods are known to sometimes perform significantly better than their nonadaptive counterpart. In the present context, this is due to the fact that the apriori choices of Λ n based on the sequences κ ν may fail to be optimal. In particular, the guaranteed rate n −s based on such choices could be pessimistic, and better rates could be obtained by other choices. However, convergence analysis of adaptive methods is usually more delicate. We next give examples of possible adaptive strategies in the interpolation and least-squares frameworks. Adaptive Selection for Interpolation We first consider polynomial approximations obtained by interpolation as discussed in Section 3. The hierarchical form I Λ u = ν∈Λ α ν B ν ,(5.2) may formally be viewed as a truncation of the expansion of u in the hierarchical basis ν∈F α ν B ν , which however may not always be converging, in contrast to the series discussed in Section 2. Nevertheless, we could in principle take the same view, and use for Λ n the set of indices corresponding to the n largest terms of (5.2) measured in some given metric L p (U, V, dµ). This amounts in choosing the indices of the n largest w ν α ν V , where the weight w ν is given by w ν := B ν L p (U,dµ) . This weight is easily computable when dµ is a tensor product measure, such as the uniform measure. In the case where p = ∞ and if we use the Leja sequence, we know that B ν L ∞ (U) = 1 and therefore this amounts to choosing the largest α ν V . This selection strategy is not practially feasible since we cannot afford this exhaustive search over F . However, it naturally suggests the following adaptive greedy algorithm, which has been proposed in [8]. • Initialize Λ 1 := {0 F } with the null multi-index. • Assuming that Λ n−1 has been selected and that the (α ν ) ν∈Λn−1 have been computed, compute the α ν for ν ∈ N (Λ n−1 ). • Set ν n := argmax{w ν α ν V : ν ∈ N (Λ n−1 )}. (5.3) • Define Λ n := Λ n−1 ∪ {ν n }. In the case where p = ∞ and if we use the Leja sequence, this strategy amounts in picking the index ν n that maximizes the interpolation error u(y ν ) − I Λn−1 u(y ν ) V among all ν in N (Λ n−1 ). By the same considerations as previously discussed for the a-priori selection of Λ n , we find that in the finite-dimensional case, the above greedy algorithm requires at most dn evaluation after n steps. When working with infinitely many variables (y j ) j≥1 , we replace the infinite set N (Λ n ) in the algorithm by the finite set of anchored neighbors N (Λ n ) defined by (5.1). Running n steps of the resulting greedy algorithm requires at most n 2 /2 evaluations. Remark 5.1 A very similar algorithm has been proposed in [20] in the different context of adaptive quadratures, that is, for approximating the integral of u over the domain U rather than u itself. In that case, the natural choice is to pick the new index ν n that maximizes \| U ∆ ν u dµ\| over N (Λ n ) or N (Λ n ). The main defect of the above greedy algorithm is that it may fail to converge, even if there exist sequences (Λ n ) n≥1 such that I Λn u converges toward u. Indeed, if ∆ ν u = 0 for a certain ν, then no index ν ≥ ν will ever be selected by the algorithm. As an example, if u is of the form u(y) = u 1 (y 1 )u 2 (y 2 ), where u 1 and u 2 are nonpolynomial smooth functions such that u 2 (t 0 ) = u 2 (t 1 ), then the algorithm could select sets Λ n with indices ν = (k, 0) for k = 0, . . . , n − 1, since the interpolation error at the point (t k , t 1 ) vanishes. One way to avoid this problem is to adopt a more conservative selection rule which ensures that all of F is explored, by alternatively using the rule (5.3), or picking the multi-index ν ∈ N (Λ n ) which has appeared at the earliest stage in the neighbors of the previous sets Λ k . This is summarized by the following algorithm. • Initialize Λ 1 := {0 F } with the null multi-index. • Assuming that Λ n−1 has been selected and that the (α ν ) ν∈Λn−1 have been computed, compute the α ν for ν ∈ N (Λ n−1 ). • If n is even, set ν n := argmax{w ν α ν V : ν ∈ N (Λ n−1 )}. (5.4) • If n is odd, set ν n := argmin{k(ν) : ν ∈ N (Λ n−1 )}, k(ν) := min{k : ν ∈ N (Λ k )}. • Define Λ n := Λ n−1 ∪ {ν n }. Even with such modifications, the convergence of the interpolation error produced by this algorithm is not generally guaranteed. Understanding which additional assumptions on u ensure convergence at some given rate, for a given univariate sequence T such as Leja points, is an open problem. Remark 5.2 Another variant to the above algorithms consists in choosing at the iteration k more than one new index at a time within N (Λ k−1 ) or N (Λ k−1 ). In this case, we have n k := #(Λ k ) ≥ k. For example we may choose the smallest subset of indices that retains a fixed portion of the quantity ν∈Λ k−1 w ν α ν V . This type of modification turns out to be particularly relevant in the least-squares setting discussed in the next section. Adaptive Selection for Least Squares In this section we describe adaptive selections in polynomial spaces, for the leastsquares methods that have been discussed in Section 4. We focus on adaptive selection algorithms based on the standard (unweighted) least-squares method. As a preliminary observation, it turns out that the most efficient available algorithms for adaptive selection of multi-indices might require the selection of more than one index at a time. Therefore, we adopt the notation that n k := #(Λ k ) ≥ k, where the index k denotes the iteration in the adaptive algorithm. As discussed in Section 4, stability and accuracy of the least-squares approximation is ensured under suitable conditions between the number of samples and the dimension of the approximation space, see e.g. condition (4.9). Hence, in the development of reliable iterative algorithms, such conditions need to be satisfied at each iteration. When dµ is the measure (4.7) with shape parameters θ 1 , θ 2 , condition (4.9) takes the form of m k ln m k ≥ κ n s k , (5.5) where m k denotes the number of samples at iteration k, and s = ln 3/ ln 2, if θ 1 = θ 2 = − 1 2 , 2 max{θ 1 , θ 2 } + 2, if θ 1 , θ 2 ∈ N 0 . Since n k increases with k, the minimal number of samples m k that satisfies (5.5) has to increase as well at each iteration. At this point, many different strategies can be envisaged for progressively increasing m k such that (5.5) remains satisfied at each iteration k. We may now present a first adaptive algorithm based on standard least squares. • Initialize Λ 1 := {0 F } with the null multi-index. • Assuming that Λ k−1 has been selected, compute the least-squares approximation u L = ν∈Λ k−1 ∪N (Λ k−1 ) c ν φ ν of u in V Λ k−1 ∪N (Λ k−1 ) , using a number of samples m k that satisfies condition (5.5) with n k = #(Λ k−1 ∪ N (Λ k−1 )). • Set ν k := argmax ν∈N (Λ k−1 ) \|c ν \| 2 . (5.6) • Define Λ k := Λ k−1 ∪ {ν k }. Similarly to the previously discussed interpolation algorithms, in the case of infinitely many variables (y j ) j≥1 the set N (Λ k ) is infinite and should be replaced by the finite set of anchored neighbors N (Λ k ) defined by (5.1). As for interpolation, we may define a more conservative version of this algorithm in order to ensure that all of F is explored. For example, when k is even, we define ν k according to (5.6), and when k is odd we pick for ν k the multi-index ν ∈ N (Λ k ) which has appeared at the earliest stage in the neighbors of the previous sets Λ k . The resulting algorithm is very similar to the one presented for interpolation, with obvious modifications due to the use of least squares. As announced at the beginning, it can be advantageous to select more than one index at a time from N (Λ k−1 ), at each iteration k of the adaptive algorithm. For describing the multiple selection of indices from N (Λ k−1 ), we introduce the so-called bulk chasing procedure. Given a finite set R ⊆ N (Λ k−1 ), a nonnegative function E : R → R and a parameter α ∈ (0, 1], we define the procedure bulk := bulk(R, E, α) that computes a set F ⊆ R of minimal positive cardinality such that ν∈F E(ν) ≥ α ν∈R E(ν). A possible choice for the function E is E(ν) = E L (ν) := \|c ν \| 2 , ν ∈ R, where c ν is given from an available least-squares estimator u L = ν∈Λ c ν φ ν , that has been already computed on any downward closed set R ⊂ Λ ⊆ Λ k−1 ∪ N (Λ k−1 ). Another choice for E is E(ν) = E M (ν) := φ ν , u − u L m k−1 , ν ∈ R, where u L is the truncation to Λ k−1 of a least-squares estimator u L = ν∈Λ c ν φ ν that has been already computed on any downward closed set Λ k−1 ⊂ Λ ⊆ Λ k−1 ∪ N (Λ k−1 ), using a number of samples m k−1 that satisfies condition (5.5) with n k = #(Λ). The discrete norm in E M (ν) uses the same m k−1 evaluations of u that have been used to compute the least-squares approximation u L on Λ. Both E L (ν) and E M (ν) should be viewed as estimators of the coefficient u, φ ν . The estimator E M (ν) is of Monte Carlo type and computationally cheap to calculate. Combined use of the two estimators leads to the next algorithm for greedy selection with bulk chasing, that has been proposed in [29]. • Initialize Λ 1 := {0 F } with the null multi-index, and choose α 1 , α 2 ∈ (0, 1]. • Assuming that Λ k−1 has been selected, set F 1 = bulk( N (Λ k−1 ), E M , α 1 ),(5.7) where E M uses the least-squares approximation u L = ν∈Λ c ν φ ν of u in V Λ that has been calculated at iteration k − 1 on a downward closed set Λ k−1 ⊂ Λ ⊆ Λ k−1 ∪ N (Λ k−1 ) using a number of samples m k−1 that satisfies (5.5) with n k = #(Λ). • Compute the least-squares approximation u L = ν∈Λ k−1 ∪F1 c ν φ ν (5.8) of u on V Λ k−1 ∪F1 using a number of samples m k that satisfies (5.5) with n k = #(Λ k−1 ∪ F 1 ). • Set F 2 = bulk(F 1 , E L , α 2 ), (5.9) where E L uses the least-squares approximation u L computed on Λ k−1 ∪ F 1 . • Define Λ k = Λ k−1 ∪ F 2 . The set N (Λ k−1 ) can be large, and might contain many indices that are associated to small coefficients. Discarding these indices is important in order to avoid unnecessary computational burden in the calculation of the least-squares approximation. The purpose of the bulk procedure (5.7) is to perform a preliminary selection of a set F 1 ⊆ N (Λ k−1 ) of indices, using the cheap estimator E M . At iteration k, E M in (5.7) uses the estimator computed in (5.8) at iteration k − 1 and truncated to Λ k−1 . Afterwards, at iteration k, the least-squares approximation in (5.8) is calculated on Λ k−1 ∪ F 1 , using a number of samples m k which satisfies condition (5.5), with n k = #(Λ k−1 ∪ F 1 ). The second bulk procedure (5.9) selects a set F 2 of indices from F 1 , using the more accurate estimator E L . The convergence rate of the adaptive algorithm depends on the values given to the parameters α 1 and α 2 . Finally we mention some open issues related to the development of adaptive algorithms using the weighted least-squares methods discussed in Section 4, instead of standard least squares. In principle the same algorithms described above can be used with the weighted least-squares estimator u W replacing the standard least-squares estimator u L , provided that, at each iteration k, the number of samples m k satisfies m k ln m k ≥ κ n k , and that the samples are drawn from the optimal measure, see Theorem 4.4. This ensures that at each iteration k of the adaptive algorithm, the weighted least-squares approximation remains stable and accurate. However, no guarantees on stability and accuracy are ensured if the above conditions are not met, for example when the samples from previous iterations are recycled. Approximation in Downward Closed Spaces: beyond Polynomials The concept of downward closed approximation spaces can be generalized beyond the polynomial setting. We start from a countable index set S equipped with a partial order ≤, and assume that there exists a root index 0 S ∈ S such that 0 S ≤ σ for all σ ∈ S. We assume that (B σ ) σ∈S is a basis of functions defined on [−1, 1] such that The set F is equipped with a partial order induced by its univariate counterpart: ν ≤ ν if and only if ν j ≤ ν j for all j ≥ 1. We may then define downward closed sets Λ ⊂ F in the same way as in Definition 1.1 which corresponds to the particular case S = N. We then define the associated downward closed approximation space by V Λ := V ⊗ B Λ , B Λ := span{B ν : ν ∈ Λ}, that is the space of functions of the form ν∈Λ u ν B ν with u ν ∈ V . Given a sequence T = (t σ ) σ∈S of pairwise distinct points we say that the basis (B σ ) σ∈S is hierarchical when it satisfies B σ (t σ ) = 1 and B σ (t σ ) = 0 if σ ≤ σ and σ = σ. We also define the tensorized grid y ν := (t νj ) j≥1 ∈ U. Then, if Λ ⊂ F is a downward closed set, we may define an interpolation operator I Λ onto V Λ associated to the grid Γ Λ := {y ν : ν ∈ Λ}. In a similar manner as in the polynomial case, this operator is defined inductively by I Λ u := I Λ u + α ν B ν , α ν := α ν (u) = u(y ν ) − I Λ u(y ν ), where ν / ∈ Λ and Λ is any downward closed set such that Λ = Λ ∪ {ν}. We initialize this computation with Λ 1 = {0 F }, where 0 F is the null multi-index, by defining I Λ1 u as the constant function with value u(y 0F ). Examples of relevant hierarchical systems include the classical piecewise linear hierarchical basis functions. In this case the set S is defined by S = {λ −1 , λ 1 , (0, 0)} ∪ (j, k) : −2 j−1 ≤ k ≤ 2 j−1 − 1, j = 1, 2, . . . equipped with the partial order λ −1 ≤ λ 1 ≤ (0, 0) and (j, k) ≤ (j + 1, 2k), (j, k) ≤ (j + 1, 2k + 1), (j, k) ∈ S. The set S is thus a binary tree where λ −1 is the root node, (0, 0) is a child of λ 1 which is itself a child of λ −1 , every node (j, k) has two children (j + 1, 2k) and (j + 1, 2k + 1), and the relation λ ≤ λ means that λ is a parent of λ. The index j corresponds to the level of refinement, i.e. the depth of the node in the binary tree. We associate with S the sequence T := {t λ−1 , t λ1 , t (0,0) } ∪ t (j,k) := 2k + 1 2 j : (j, k) ∈ S, j ≥ 1 , where t λ−1 = −1, t λ1 = 1 and t (0,0) = 0. The hierarchical basis of piecewise linear functions defined over [−1, 1] is then given by B λ−1 ≡ 1, B λ1 (t) = 1 + t 2 , B (j,k) (t) = H(2 j (t − t (j,k) )), (j, k) ∈ S, where H(t) := max{0, 1 − \|t\|}, is the usual hat function. In dimension d = 1, the hierarchical interpolation amounts in the following steps: start by approximating f with the constant function equal to f (−1), then with the affine function that coincides with f at −1 and 1, then with the piecewise affine function that coincides with f at −1, 0 and −1; afterwards refine the approximation in further steps by interpolating f at the midpoint of an interval between two adjacents interpolation points. Other relevant examples include piecewise polynomials, hierarchical basis functions, and more general interpolatory wavelets, see [10] for a survey. a given bounded Lipschitz domain D ⊂ R k (say with k = 1, 2 or 3), for some fixed right-hand side f ∈ L 2 (D), homogeneous Dirichlet boundary conditions u \|∂D = 0, and where a has the general form a = a(y) = a + j≥1 y j ψ j .(1.4) ν = (ν j ) j=1,...,d . outlined in the previous section, we are interested in deriving polynomial approximations of the map (1.2) acting from U = [−1, 1] d with d ∈ N or d = ∞ to the Banach space V . Our first vehicle to derive such approximations, together with precise error bounds for relevant classes of maps, consists in truncating certain polynomial expansions of (1.2) written in general form as ν∈F u ν φ ν , (2.1) Lemma 2. 1 ≤ 1Let 0 < p < q < ∞ and let (c ν ) ν∈F ∈ ℓ p (F ) be a sequence of nonnegative numbers. Then, if Λ n is a set of indices which corresponds to the n largest c ν C(n + 1) −s , C := (c ν ) ν∈F ℓ p , s B νj (y j ), and α ν = α ν (u) := u(y ν ) − I Λ u(y ν ). Proving that this bound, or any other algebraic growth bound, holds for all values of k ≥ 0 is currently an open problem.• The ℜ-Leja points: they are the real part of the Leja points defined on the complex unit disc {\|z\| ≤ 1}, taking for example e 0 Theorem 4. 1 1For any r > 0, if m and n satisfy K n ≤ κ m ln m , with κ := κ(r) Theorem 4. 2 2For any r > 0, if m and n satisfy condition (4.4) and u satisfies (4.3), then the estimator u E ∈ {u L , u T , u C } in the noisy case (4.6) satisfies Corollary 4. 1 1For any r > 0, with multivariate polynomial spaces P Λ and Λ downward closed, if m and n satisfy , if θ 1 , θ 2 ∈ N 0 , (4.9)with κ = κ(r) as in (4.4), then the same conclusions of Theorem 4.1 hold true. Theorem 4. 3 3For any r > 0, if m and n satisfy m ln m ≥ κ K n,w , with κ := κ(r) = 1 − ln 2 2 + 2r , then the same conclusions of Theorem 4.1 hold true with u L replaced by u W . Theorem 4. 4 4For any r > 0, if m and n satisfy m ln m ≥ κ n, with κ := κ(r) Definition 5.1 A monotone nonincreasing positive sequence (c ν ) ν∈F is said to be anchored if and only if l ≤ j =⇒ c ej ≤ c e l . A finite downward closed set Λ is said to be anchored if and only if e j ∈ Λ and l ≤ j =⇒ e l ∈ Λ,where e l and e j are the Kroenecker sequences with 1 at position l and j, respectively. For example, one can double the number of samples by choosing m k = 2m k−1 whenever (5.5) is broken, and keep m k = m k−1 otherwise. The sole prescription for applying Corollary 4.1 is that the samples are independent and drawn from dµ. Since all the samples at all iterations are drawn from the same measure dµ, at the kth iteration, where m k samples are needed, it is possible to use m k−1 samples from the previous iterations, thus generating only m k − m k−1 new samples. B 0S ≡ 1. We then define by tensorization a basis of functions on U = [−1, 1] d when d < ∞, or U = [−1, 1] N in the case of infinitely many variables, according to B ν (y) = j≥1 B νj (y j ), ν := (ν j ) j≥1 ∈ F , where F := S d in the case d < ∞, or F = ℓ 0 (N, S), i.e. the set of finitely supported sequences, in the case d = N. Sparse polynomial approximation of parametric elliptic PDEs. Part I: affine coefficients. M Bachmayr, A Cohen, G Migliorati, 2to appear in ESAIMBachmayr, M., Cohen, A., Migliorati, G.: Sparse polynomial approximation of parametric elliptic PDEs. Part I: affine coefficients. to appear in ESAIM:M2AN Sparse polynomial approximation of parametric elliptic PDEs. Part II: lognormal coefficients. 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[ "Quantum stabilization of a single-photon emitter in a coupled microcavity-half-cavity system", "Quantum stabilization of a single-photon emitter in a coupled microcavity-half-cavity system" ]	[ "C Y Chang \nSchool of Physics\nGeorgia Institute of Technology\n30332-0250AtlantaGeorgia, USA\n", "Loïc Lanco \nCentre de Nanosciences et de Nanotechnologies\nCNRS\nUniversit Paris-Sud\nUniversit Paris-Saclay\nC2N, 91460MarcoussisMarcoussisFrance\n\nUniversit Paris Diderot\nParis 775205Paris CEDEX 13France\n", "Pascale Senellart \nCentre de Nanosciences et de Nanotechnologies\nCNRS\nUniversit Paris-Sud\nUniversit Paris-Saclay\nC2N, 91460MarcoussisMarcoussisFrance\n", "D S Citrin \nSchool of Electrical and Computer Engineering\nGeorgia Institute of Technology\n30332-0250AtlantaGeorgia, USA\n" ]	[ "School of Physics\nGeorgia Institute of Technology\n30332-0250AtlantaGeorgia, USA", "Centre de Nanosciences et de Nanotechnologies\nCNRS\nUniversit Paris-Sud\nUniversit Paris-Saclay\nC2N, 91460MarcoussisMarcoussisFrance", "Universit Paris Diderot\nParis 775205Paris CEDEX 13France", "Centre de Nanosciences et de Nanotechnologies\nCNRS\nUniversit Paris-Sud\nUniversit Paris-Saclay\nC2N, 91460MarcoussisMarcoussisFrance", "School of Electrical and Computer Engineering\nGeorgia Institute of Technology\n30332-0250AtlantaGeorgia, USA" ]	[]	We analyze the quantum dynamics of a two-level emitter in a resonant microcavity with optical feedback provided by a distant mirror (i.e., a half-cavity) with a focus on stabilizing the emittermicrocavity subsystem. Our treatment is fully carried out in the framework of cavity quantum electrodynamics. Specifically, we focus on the dynamics of a perturbed dark state of the emitter to ascertain its stability (existence of time oscillatory solutions around the candidate state) or lack thereof. In particular, we find conditions under which multiple feedback modes of the half cavity contribute to the stability, showing certain analogies with the Lang-Kobayashi equations, which describe a laser diode subject to classical optical feedback. arXiv:1804.06734v1 [quant-ph]	10.1103/physrevb.101.024305	[ "https://arxiv.org/pdf/1804.06734v2.pdf" ]	119,281,924	1804.06734	6c67e0b7163ac5cce89f666b1febd9e2d6bd259c	Quantum stabilization of a single-photon emitter in a coupled microcavity-half-cavity system C Y Chang School of Physics Georgia Institute of Technology 30332-0250AtlantaGeorgia, USA Loïc Lanco Centre de Nanosciences et de Nanotechnologies CNRS Universit Paris-Sud Universit Paris-Saclay C2N, 91460MarcoussisMarcoussisFrance Universit Paris Diderot Paris 775205Paris CEDEX 13France Pascale Senellart Centre de Nanosciences et de Nanotechnologies CNRS Universit Paris-Sud Universit Paris-Saclay C2N, 91460MarcoussisMarcoussisFrance D S Citrin School of Electrical and Computer Engineering Georgia Institute of Technology 30332-0250AtlantaGeorgia, USA Quantum stabilization of a single-photon emitter in a coupled microcavity-half-cavity system (Dated: April 19, 2018) We analyze the quantum dynamics of a two-level emitter in a resonant microcavity with optical feedback provided by a distant mirror (i.e., a half-cavity) with a focus on stabilizing the emittermicrocavity subsystem. Our treatment is fully carried out in the framework of cavity quantum electrodynamics. Specifically, we focus on the dynamics of a perturbed dark state of the emitter to ascertain its stability (existence of time oscillatory solutions around the candidate state) or lack thereof. In particular, we find conditions under which multiple feedback modes of the half cavity contribute to the stability, showing certain analogies with the Lang-Kobayashi equations, which describe a laser diode subject to classical optical feedback. arXiv:1804.06734v1 [quant-ph] INTRODUCTION With recent advances in the fabrication of nanophotonic structures, there is an increasing ability to control and manipulate the optical properties on the singlephoton level [1][2][3]. One of the techniques central to such control is the ability to access the strong-coupling regime of cavity quantum electrodynamics (cQED). In solidstate structures, recent developments include coupling quantum emitters such as quantum dots (QD) to photonic crystals [4][5][6] and micropillar cavities [7,8]. These devices provide unidirectional photons that are of interest, for example, to improve quantum communications over long distances [9]. In addition, the ability to control and coherently manipulate photons coupled to the internal state of the QD is of great importance due to potential applications in quantum memory and quantum information processing [10]. Controlling a quantum system-in the sense of providing stabilization of quantum state-presents a nontrivial problem. The use of feedback to provide stabilization in classical systems is quite advanced, while less is known for quantum system. Two feedback-control schemes have been explored recently in quantum optics, viz. measurement-based and coherent feedback loops [11,12]. In a measurement-based feedback loop, the quantum system is monitored and the outcome of the measurement is used as classical information to manipulate the operations. A measurement-based feedback loop has been implemented in a cQED system with trapped atoms [13]. In coherent feedback, the dynamics are entirely quantum mechanical and the system interacts coherently with an ancillary subsystem in both the extraction and manipulation processes. For coherent feedback, such as Pyragas feedback [14], both processes utilize information stored in the reservoir degrees of freedom re-siding, for example, in an external cavity (EC). Given that the measurement process will unavoidably collapse the wavefunction, measurement-based feedback presents limitations when the target state is a superposition. To implement coherent quantum feedback in quantum optics, one typical setup is a half cavity with a mirror, similar to the time-delayed feedback setup used in EC semiconductor lasers (ECSL). The dynamics of ECSLs have been intensively studied based on the Lang-Kobayashi (LK) equations [15]. The LK equations describe the nonlinear dynamics of the electric-field amplitude \|E\|, carrier density n in the active region, and optical phase φ by a set of equations of motions (EOM) in the form of coupled delayed-differential equations. The dynamics are described by trajectories in phase space spanned by the three dynamical variables. The presence of time-delayed feedback is known for the ECSL to generate multiple steady-state solutions. The stable modes of the EC are called EC modes (ECM), while the unstable modes of the EC are known as antimodes [16]. Depending on the various parameters of the system (e.g., injection current, feedback strength, optical feedback phase, and EC length L), various dynamics can occur near and amongst these solutions [17,18]. In particular, oscillatory dynamics about various ECMs occur in certain parameter regimes, and are closely related to undamped relaxation oscillations [19], while more complex behavior exhibiting closed trajectories encircling one or more steady-state solutions are also observed. The existence of multiple solutions is often seen in many nonlinear systems, but not in a quantum system due to the linear nature of the Schrödinger equation. [20] However, in quantum systems with a bath of degrees of freedom (in our case, the modes of the EC), once these are integrated out, the system may exhibit what appears to be nonlinear behavior in the remaining explicitly considered degrees of freedom. Another point of view of nonlinearity in quantum systems with coherent feedback is to consider the variation in the dynamical behavior with the number of photons. The nonlinearity involving of states with a small number of photons has been observed experimentally [21,22]. These effects cannot be described with a classical electromagnetic field, i.e., the scaling of the behavior with photon number cannot be accounted for by the corresponding optical intensity. Hence, a fully quantum mechanical treatment of such time-delayed systems, including the electromagnetic field, is essential to understand the dynamics of single-photon emitters with coherent timedelayed feedback. Broadly speaking, two approaches have been used to account for the coherent feedback. In one approach, the optical feedback from a mirror has been accounted for using one-dimensional model for the electromagnetic field with a standing-wave basis in the Markovian limit. Recent theoretical studies have focused on Markovian coherent feedback in various systems, such as ensembles of atoms [23][24][25], a single atom in a cavity [26], onedimension waveguides [27,28], and photonic crystals [29][30][31]. It has been demonstrated that the standing-wave field model in a half cavity (Markovian) is equivalent to a time-delayed feedback system (non-Markovian) [25,30]. The involvement of the delay time can be a key factor in the stability or instability as well as the nonlinear dynamics in many complex systems [32,33], providing wide-range applications. These works provide a promising route to rapid convergence of states [33], enhancing entangled photon-pair generation from biexcitons [34], and to drive continuous exchanges for pure states [29]. Among these studies, stability in coherent feedback systems has been investigated by Grimsmo [26] based on linear delayed differential equations (non-Markovian). Since then, few other studies have focused on stability and its relation to time-delayed systems [36,37]. The evolution of the quantum states in these investigations is described by means of a time-varying Hamiltonian. These works are focused on specific features of the quantum system and not on the stability of the target state, which from the standpoint of experimental realization or practical application is of key importance. In view of the relative lack of understanding of how quantum feedback can provide stability in quantum systems, in this study we consider a quantum system composed of a single-photon emitter in the form of a quantum dot (QD) within a microcavity (MC) formed by a micropillar with coherent quantum feedback provided by a distant mirror. Thus, this work provides a testbed to study time-delayed feedback in a fully quantum system, as well as addresses a physical implementation of such a system of interest for nanophotonic and quantum information-science applications. Our approach presents a direct stability analysis in the Markovian dynamics of quantum feedback from a single-photon emitter. The quantum feedback is achieved via the EC, similar to previous studies [29,30]. We derive a set of EOMs for the quantum amplitudes associated with the natural basis describing the QD excitation, the MC photon, and the EC modes. The EOMs of the state vector are obtained with the input-output formalism in the Schrödinger picture instead of in the Heisenberg picture as in Ref. [31]. Our choice of performing the simulation in the statevector based picture is to simplify the numerical implementation of our program. In particular, it eases attaining a higher time-step resolution of stable/unstable state evolution compared with typical simulation methods such as the matrix product state [38]. Since photon leakage from the EC is neglected, our system conserves excitation number. Our focus is on the one-excitation subspace. In the one-excitation subspace, we find the stationary-state solutions, which we then investigate for stability. By stationary state, we mean states in which the probabilities in the noninteracting basis remain constant in time even including the interaction. In other words, stationary states are those that are simultaneously eigenstates of both the noninteracting and the interacting Hamiltonians. We explore the stabilty of the stationary states, that is, states of the composite MC/QD subsystem that are effectively decoupled from the EC. We find that the stable stationary states are represented by the MC/QD and its mirrored-self being in a singlet state [39]. Similar observations have been made that the singlet state corrresponds to Dicke subradiant state [40] in Ref. [41,42] in a system with chiral feedback from a single V-level atom. To be more specific, the singlet state would appear effectively decoupled from the waveguide, being in a dark state which is subradiant, and thus stable, whereas the triplet state is superradiant, and thus unstable, as it decay at twice the cavity damping rate from the MC, [41,42]. In the case with one single-photon emitter coupled to the EC, the subradiant (dark) state would be the only stationary state where the QD and MC are populated [40]. We determine the stability by studying the dynamics in the vicinity of the stationary state. This is done by constructing the Jacobian matrix of the linearized EOMs with the state amplitudes perturbed from the stationary state [20], a technique widely employed for classical stability analysis. We analyze the eigenvalues of the Jacobian matrix as a function of coupling strength between the MC and EC. A strictly imaginary eigenvalue indicates oscillatory dynamics about the candidate stationary state, which is thus deemed stable while a positive (negative) real part of the complex eigenvalues indicate unstable (stable) stationary states. The effects of time delay are also studied for various EC length L. We numerically verify the Jacobian analysis by perturbing the stationary state to investigate its stability, which agrees with the results obtained directly from the Jacobian of the state amplitudes. τ = 2L / c 0 L (b) (a) The remainder of the paper is organized as follows. In the next section, we outline the cQED description of the system. We next find the stationary state. Following this, we assess the stationary state's stability. In the final section, we conclude. THEORETICAL DESCRIPTION OF THE SINGLE-EMITTER, MICROCAVITY, EXTERNAL CAVITY SYSTEM We begin by introducing the system. We consider an intrinsic QD coupled to a single near-resonant mode of a high-Q micropillar MC [ Fig. 1 (a)] and coupled in turn to the EC modes shown in Fig. 1(b). The QD is characterized by interband-transition frequency, ω 0 . The QD interband transition is dipole coupled to a single mode of the micropillar MC with coupling strength g, as in Ref. [8]. This approach can yield strong coupling between the QD and the MC [34] and can generate high-purity, indistinguishable single photons [8]. We place an ideal mirror with reflection coefficient r = 1 a distance L = c 0 τ /2 from the micropillar to form the EC (half cavity), with c 0 the speed of light in vacuum and τ the EC round-trip feedback time. The conditions we choose are similar to the single QD in a MC subjected to an external mirror in a recent paper [30]. For a mirror with \|r\| < 1, the coherent feedback can be treated using an open quantum-system formalism, discussed in a recent publication by Whalem [43]. We work in the rotating-wave approximation (RWA) [44] in the Schrödinger picture as in Refs. [23,29,30]. The derivation of the interaction Hamiltonian can be found in Ref. [31,[44][45][46] (see supplementary material). For the EC, we use the free-space dispersion, ω k = c 0 \|k\|, for the photons assuming a sufficiently large value of L so that the photon modes can be considered a quasicontinuum. We assume that only the optical modes with angular frequencies near ω 0 interact strongly with the MC photon (ω k ≈ ω M C ≈ ω 0 ). We obtain the follow-ing interaction Hamiltonian describing the QD coupled to the MC mode and that mode in turn coupled to the EC modes, H (RW A) int = −ω g (σ − a † +σ + a)− ∞ −∞ dω k [G(k, t)a † b k +h.c.] (1) with ω g = g/ , a † (a) the creation (annihilation) operator for the MC photon and σ +(−) the raising (lowering) operator of the two-level QD system. The bosonic operators b k destroy a photon of frequency ω k in the EC as defined as in Ref. [31] and the coupling element, G(k, t) = G 0 sin(kL)e i(ω0−ω k )t , where G 0 = 2c 0 κ/π. Note that because H (RW A) int is quadratic in creation and annihilation operators, the Hamiltonian conserves the number of excitations. We extend the lower limit of integration to −∞ given the interaction bandwidth is narrow compared with ω 0 . A detailed discussion is presented in the supplemental material. We now concentrate on the one-excitation subspace where an arbitrary wavefunction can be written Ψ(t) = c e (t) \|e, 0, 0 +c c (t) \|g, 1, 0 + ∞ −∞ c k (t) \|g, 0, k dk. (2) Here, in kets \|a, b, k , a = e (g) denotes the QD in the excited (ground) state; b = 0 (1) denotes zero (one) MC photon; and k denotes the EC photon wavevector. Ψ(t) is thus described by the time-dependent amplitudes c e (t), c c (t), and c k (t). STATIONARY STATE Substituting Ψ(t) into the time-dependent Schrödinger's equation, we obtain the following coupled EOMs for the time-dependent amplitudes, ∂c e ∂t = iω g c c ,(3)∂c c ∂t = iω g c e + i ∞ −∞ c k G(k, t) dk,(4)∂c k ∂t = iG * (k, t) c c .(5) To find the stationary states (stationary in the more narrow sense defined above), we write c i (t) = \|c i (t)\| exp[iθ i (t)], to obtain a new set of EOMs, ∂c i ∂t = \|∂c i \| ∂t e iθi(t) + i θ i (t) ∂t \|c i \| e iθi(t) ,(6) with i = e, c, and k. We apply the constraint of stationarity ∂\|c i \|/∂t = 0 and find the solution as discussed in the supplemental materials. We denote the stationary state (with respect to the interaction Hamiltonian) in the single-excitation subspace asΨ(t) (where the bar indicates a stationary solution), Ψ(t) =c e (t) \|e, 0, 0 +c c (t) \|g, 1, 0 + ∞ −∞c k (t) \|g, 0, k dk where the noninteracting amplitudes of the stationary state arec e (t) = −αe −iωgt ,(6)c c (t) = αe −iωgt ,(7)c k (k, t) = αG 0 sin(kL) ω k − ω g e i(ω k −ωg)t .(8) The parameter α is defined in the next paragraph. Note that the minus sign results from the MC/QD being in a singlet state of the quantum system and its mirrored self [39,41]. In this state, the QD and the MC are effectively decoupled from the EC, thus being in a dark or subradiant state, as discussed in Refs. [39,41,42]. The noninteracting amplitudes associated with this stationary state can be characterized by a single parameter α = \|c c (t)\| = \|c e (t)\| = 2 + πG 2 0 L c 2 0 −1/2 = (2 + τ κ) −1/2 where recall G 0 = 2c 0 κ/π and κ is the MC photon damping rate. Since we consider 100 % reflection from the distant mirror, the damping rate characterizes both the rate into the EC and the feedback rate from the reflected photon. The stationary state is indicated in the frame rotating at ω 0 in Fig. 2. The blue arrow shows the state of the QD at a snapshot in time with the time evolution in the RWA shown by the orange arrows where we choosing initial conditionsc e (0) = α,c c (0) = −α, andc k (k, 0) = αG 0 sin(kL)/(ω k − ω g ). In this case, the EC photon is in a standing wave and its population is described by a sinc function in the k channel (of frequency ω k ) with respect to the ω 0 − ω g axis, as illustrated in Fig. 2(c). The evolution of the MC photon population \|c c (t)\| 2 is plotted in Fig. 3. The theoretical and numerical results for the stationary state are plotted in the black and yellow dotted curves, respectively. In this case, the MC photon population remains constant in time. The green curve plots the MC photon population \|c c \| 2 , however, for the initial state c e (0) = 1, c c (0) = 0, c k (0) = 0 for all k. In this case, the MC photon population varies significantly at fist and eventually mainly leaks out into the EC. Note that there is only one stationary state in our sense within the one-excitation subspace. To lend insight into the nature of the stationary state, the initial state amplitudes of the EC photons of wavenumber,c k (k, 0) are given by an even function with respect to the ω 0 − ω g axis in Fig. 2(c) and the coupling G(k, t) = 2c 0 κ/π sin(kL) exp[i(ω 0 − ω k )t] is an odd function of k. Thus, while L satisfies the commensurability condition, i.e., 2L/c 0 = 2πn/ω g with n ∈ N, the EC is in effect decoupled from the MC photon and the QD for the stationary state. As this occurs, the QD and the MC photon experience a cavity-assisted interaction at the rate of ω g , and both QD and MC-photon state are stationary when their complex amplitudes are π out of phase. The only stationary state corresponding to the interaction Hamiltonian is antisymmetric in the amplitudes c e and c c , and corresponds to the bound state in Ref.. [27] or the subradiant/dark state in the recent studies of Refs. [42,50]. STABILITY ANALYSIS In the previous section we identified a stationary state in the one-excitation subspace. In this section, we ascertain the stationary state's stability. To begin, we constructed a Markovian model to numerically simulate the evolution based on Eqs. (3)-(5), and the stationarity of Ψ is demonstrated in Fig. 3. For example, for c e (0) = 1, c c (0) = 0, and c k (0) = 0 for all k, the MC-photon population exhibits nonperiodic oscillations on the timescale of vacuum-field Rabi oscillations, shown in Ref. [34]. Next, we study the dynamics in the vicinity of the stationary state to ascertain its stability. We expect that for a stable state, the probabilities will remain near the initial values. Here, we perturb the stationary state and track the dynamics. We add small values δ c,i to the initial conditions for c e and c c with respect to the stationary stateΨ(t) in Eqs. (6)-(8), viz. δc c = [± 0.01, ± 0.02]α while the amplitude of QD state c e (0) is perturbed such that \|c e + δc e \| 2 + \|c c + δc c \| 2 = \|c e \| 2 + \|c c \| 2 . We shall see that the nature of the stable state depends crucially on the ratio R = κ/(4g) between the MC damping rate κ and the coupling strength g, the inversion of the conventional coupling strength parameter between a two-level system and a cavity, where R < 1 (R > 1) indicates the strong-(weak-) coupling regime [35]. Small R thus means the vacuum-field Rabi frequency is much larger than the MC photon leakage rate; large R indicates a relatively high MC photon leakage rate compared with the vacuum-field Rabi frequency. The effect on the stability is illustrated by plotting the probability for R = 0.5 and R = 8. The probabilities of the excited QD state, \|c e (t)\| 2 are plotted in Fig. 4(a1) and (b1) and MC photon state \|c c (t)\| 2 in Fig. 4(a2) and (b2) for R = 0.5, and 8, respectively. The feedback phase of the reflected photon, ∆φ, is 0 for both cases in Fig. 4. In Fig. 4(a1) and (a2), for small R (strong coupling), the variation of the probability from its mean value exhibits oscillations at roughly the vacuum-field Rabi frequency 2ω g with some initial damping from t = 0 to τ at rate 2κ. For t τ , the damping is arrested and oscillations at the vacuum-field Rabi frequency persist. By comparison, for R = 8 [ Fig. 4(b1) and (b2)] when vacuumfield Rabi oscillations do not have a chance to occur before photon leakage from the MC, the dynamics of both probabilities are not evidently periodic, indicating the participation of multiple frequencies. We shall see this is due to the participation of many coupled EC-like modes. In order to explore the stability of the stationary state in a rigorous fashion, we consider an analysis of the eigenvalues of the Jacobian matrix [47] to confirm the foregoing numerical simulations. The Jacobian is defined as the matrix of all first-order partial derivatives with respect to the each variable (in this case, the perturbed state amplitudes) evaluated for the stationary state [20]. In other words, the Jacobian probes the change in the stationary state with respect to an arbitrary infinitesimal perturbation. Its eigenvalues, therefore, indicate whether or not the state is stable, with a tendency to oscillate around the stationary state for eigenvalues being strictly imaginary, stable and evolve toward a stationary state for real parts of all eigenvalues being negative, or unstable and to evolve away from the stationary state (all real parts of any complex eigenvalues being positive). In addition, saddle points are also possible where the stationary state is stable against perturbations in certain directions in state space, but not in others. More specifically, the Jacobian matrix J δ (see supplementary materials for definition) satisfies the equation ∂ ∂t δc i (t) = J δ δc i (t).(10) We begin by studying the Jacobian matrix constructed with linearized EOMs of the perturbed state. The perturbed states along the i direction in state space from the stationary state will evolve according to δc i (t) = δc i (0)e λit where δc i (t) is the small perturbation along the i vector from the stationary state,c α at t = 0. Given the eigenvalues, possible cases for the steady state are are attractors, repellors, or saddle points corresponding to all negative, all positive, or some positive and some negative eigenvalue, respectively. The dynamics can be analyzed by the nature of the equilibrium state, in this case, the stationary state,Ψ(t) in Eqs. (6)- (8). The interaction of the perturbed state amplitudes can be derived using linearized EOMs near the stationary state, and the Jacobian matrix of rank (2 + N ) is derived following the small-signal model, where N is the number of EC photon states, c k , used in the simulation. Note that Jacobian, J δ , is derived from linearization at the stationary state and is found to be a rank-N + 2 skew Hermitian matrix, while the interaction Hamiltonian is a rank-2 Hermitian matrix in the one-excitation subspace. We proceed to analyze the stability of the stationary state by considering the eigenvalues, λ of J δ . We focus on the dependence of the eigenvalues of J δ on the ratio R = 4κ/g between the MC photon damping rate and the QD-MC coupling strength. In addition, we carry out this analysis adjusting two parameters, viz. the optical feedback phase ∆φ and the EC round-trip time τ . We first consider the effect of ∆φ, under conditions of constructive, destructive, and partial interference, and next consider the effect of varying τ , restricting its value to integer multiples of τ g = 2π/ω g , i.e., τ = 2L/c 0 = nτ g where n ∈ N. In the above-mentioned scenarios, we find all the eigenvalues are purely imaginary (iλ ∈ R). Imaginary eigenvalues indicate oscillatory dynamics upon perturbation about the stationary state, i.e., this indicates stability. Notably, since the perturbation of the stationary state evolves following δc c (t) = δc c (0) exp(λt), we expect to observe oscillation in the probabilities both for the QD excited state and for the MC photon \|c c (t) \| 2 = α 2 + αδc c (0) cos ω osc t + \|δc c (0)\| 2 , where ω osc = ω osc (λ) = ω 0 + ω g − iλ. The eigenvalues of the Jacobian are solutions of the determinental equation \|J δ − λI (2+N k ) \| = 0. One finds (ω osc − ω g ) 2 + κ(ω osc − ω g ) sin[τ (ω osc − ω g − ω 0 )] − ω 2 g = 0.(11) Here, we focus on finding the frequency ω osc (λ) as a function of the parameter R = κ/(4g), the feedback phase ∆φ (obtained by varying L on the lengthscale of the photon wavelength Λ 0 = 2πc 0 /ω 0 ) of the reflected photon, and and the time delay τ . Given the wide range of R investigated, the following results are plotted using a horizontal scale in log 2 R. Constructive interference: ∆φ = 0. The EC roundtrip distance is a half-integer multiple of the photon wavelength, 2L = (m + 1/2)Λ 0 . The reflected photon is assumed to undergo π phase change from an ideal mirror (\|r\| 2 = 1). For R less than a critical valueR(n, ∆φ) (depending on ∆φ and τ ), we find only one non-zero imaginary eigenvalue and it corresponds to oscillatory dynamics about the stationary state (i.e., stability) at frequency 2ω g . In this region, starting with perturbed initial conditions results in limit-cycle dynamics. An arbitrary perturbation leads to oscillatory dynamics at the Rabi frequency. For values of R >R(n, ∆φ), we find multiple emerging imaginary eigenvalues as we increase R. In this region, for large L we have in effect a quasi-continuum of EC modes, and an arbitrary perturbation leads to nonoscillatory dynamics involving the many frequencies associated with the various eigenvalues of the Jacobian. This is a region of asymptotic stability. Recall that n is the ratio of the EC round-trip time τ and the Rabi period τ g = 2π/ω g .R(n, ∆φ) is indicated by the vertical dotted red line in Figs. 5 and 6. In the case of constructive interference, for R =R(1, 0) = 0.156 (log 2 R = −2.68) as shown in Fig. 5(a), we find six solutions, indicating a possible bifurcating point in the strong-coupling regime. We also find that additional oscillatory modes emerge pairwise and their eigenfrequencies are smooth functions of R satisfying Eq. (11). In addition, the frequency separation between the additional eigenvalues in the weakcoupling regime e.g. R = 64 (log 2 R = 6), tends toward δf osc = 1/(2τ ), the EC free spectral range. The expected oscillation frequencies for n = 1 as a function of R are shown in Fig. 5(a). Destructive interference: ∆φ = π. The EC round-trip distance is an integer multiple of the photon wavelength, 2L = mΛ 0 . The reflected photon interferes destructively with the emitted photon at the MC. We plot the oscillation frequencies as a function of R in Fig. 5(b) for n = 1. Additional eigenvalues appear for R above a different critical valueR(1, π) = 0.62 (log 2R (1, π) = −0.69, lower than that for the case of constructive interference) as shown in Fig. 5(b). For R increasing above 0.62, two pairs of oscillating modes emerge (in addition to the existing solutions, ω osc = 0 and 2ω g ). The higherfrequency pair of eigenvalues emerges at approximately −i(ω 0 + ω g ) + i2.75ω g , and further splits into values sep-arated by the EC free spectral range for higher values of R (weaker coupling strength). Note that in both cases of constructive and destructive interference, the additional solutions for the frequency, ω osc , appear symmetrically with respect to the ω osc = ω g axis from Eq. (11). Partial interference: ∆φ = 0 = π. In this case, the reflected photon partially interferes with the emitted photon. The oscillation frequency of the MC photon state probability, ω osc depends also on the phase difference of the reflected photon. We plot the MC oscillation frequencies, ω osc for ∆φ = π/2 and 3π/2 in Fig. 5(c) and (d), respectively. An analogy exists between the behavior of the perturbed dynamics about the stationary state in the quantum system under consideration and the nonlinear dynamics of the ECSL. Under weak feedback, the LK equations predict the ECSL dynamics are strongly influenced by the optical feedback phase. Namely, it is indicated in Ref. [48] that the laser line does not split under the influence of out-of-phase feedback, but does alternately narrow and broaden depending on the phase of the feedback. Since weak feedback in the ECSL is analogous to R >R in the quantum system, we comment here on the optical phase of the feedback field. Specifically, the feedback phase of a single photon has a similar impact on the dynamics in the weak coupling regime for the MC/QD/EC system. These effective changes appear near R slightly above different critical values ofR under the various interference conditions shown in Fig. 5(a)-(d). We next explore the stability dependence on R for various L. We find the stationary state only exists when the time delay τ is an integer multiple of τ g = 2π/ω g , i.e., when τ = nτ g with n ∈ N, or in other words when ω osc with τ = 2nπ/ω g . We chose ∆φ = 0 in all cases. In the case of coherent quantum feedback from a single photon, one can show that the product of the time delay τ and the dimensionless ratioR(n, 0) is constant 1/(2π) for n ∈ N from Eq. (11). We find the critical valueR of R, beyond which more than two (imaginary) eigenvalues of the Jacobian matrix appear decreases as L increases shown by various vertical dotted red lines in the panels of Fig. 6. That is,R(1, 0) In the LK model for an ECSL, this product is also proportional to the dimensionless parameter C = γτ characterizing feedback strength defined in Ref. [49], where γ is the feedback parameter. Specifically, in the theoretical study of weak optical feedback (classical ECSL), C < 1 there is always one stable solution; for C > 1 there may exist more stable solutions corresponds to singlefrequency operation [49]. The frequency difference between the excess solution of ω osc scales linearly with n as one expected from the LK model. As we pointed out in the previous paragraph, the critical value,R in the single-photon limit corresponds up to a constantmultiplicative factor to the feedback parameter, γ in the classical regime. That is,R satisfies the relationship Rτ = 1/(2π). A stability analysis of the type implemented here can also be applied to the dark state existing in a quantum dimer system with a coupled cavity [42,50] or in certain spin system [51,52]. In the case with spin system, the coupling element between two interacting spins place at distance L apart is replaced by G sd (k , t) = G 0 cos(k L)e i(ω0−ω k )t [51]. By making the transformation k = k − π/2, we obtain the relationship between the eigenvalues in two cases, i.e., the eigenvalues of λ sd = λ − π/2. Following the derivation of this work, we can solve the emergent behavior of the additional solutions of ω osc,sd (R, ∆φ, n) = ω osc (R, ∆φ − π/2, n) near the stationary (dark) state. Thus, the dynamical behaviors of the interacting quantum dimer coupled through (single) photon bath should behave similarly to coherent quantum feedback. In this case, the frequencies appearing in Fig. 5 (a) and (b) must be interchanged as well as those appearing in Fig. 5 (c) and (d). DISCUSSION AND CONCLUSION The implementation of coherence feedback with fewexcitation states incorporating cQED systems can be realized in various ways. Albert, et al., have realized coherent optical feedback with microlaser, where chaotic behavior is observed with self-feedback for a few few photons ∼ 100, and studied the second-order autocor-relation function g (2) (τ ) [22]. Note that in these experiments, more realistic parameters, including the QD dephasing rate, the QD decay rate, the transmission and reflection coefficients of the EC mirror and the MC-EC coupling are needed. However, these parameters describe the loss channels and contribute to decay of the probabilities, we expect these parameters will weakly influence the oscillation frequencies provided the large R limit can be reached. In conclusion, we consider a system composed of a QD in a MC coupled to a EC and find a stationary state where the state initialized in the MC and QD degrees of freedom is stable against decay into EC photons. Specifically, we give the analytical expression for the stationary solution for such a system in the one-excitation subspace. We find this state is stable by performing stability analysis on the Jacobian of state amplitudes. The periodic solutions perturbed about the stationary state, obtained from the eigenvalues of Jacobian, indicate additional solutions arise above a critical valueR or R as experimental parameters such as the cavity damping rate, the cavity coupling strength, the feedback phase, and the external cavity length are varied. We found a strong similarity to the LK model in this behavior in terms of its dependence on ∆φ and τ . In addition, numerical simulation verifies these results, showing interesting dynamics appear in the vicinity of the stationary states. Our stability analysis may serve as a bridge between classical and quantum models for nanophotonic structures subject to optical feedback. of a two-level QD in a near-resonant MC, where g is the coupling strength and κ = πG 2 0 /(2c0) is the MC damping rate. (b) The QD-MC system coupled to an EC of length L in which a quasi-continuum of photon modes exist. τ is the delay time of the coherent feedback. FIG. 2 . 2Representation of the one-excitation stationary state in the frame rotating at the frequency ω0 (a)ce(t), (b)cc(t), and (c)c k (k, t). The blue arrows indicate the initial condition and the orange arrows indicates the time evolution. FIG. 3 . 3Time evolution of the probabilities of \|cc(t)\| 2 for various initial conditions. 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[ "PREPRINT SUBMITTED TO IEEE TRANSACTION ON NEURAL NETWORKS AND LEARNING SYSTEMS 1 Adaptive Propagation Graph Convolutional Network", "PREPRINT SUBMITTED TO IEEE TRANSACTION ON NEURAL NETWORKS AND LEARNING SYSTEMS 1 Adaptive Propagation Graph Convolutional Network" ]	[ "Student Member, IEEEIndro Spinelli ", "Simone Scardapane ", "Member, IEEEAurelio Uncini " ]	[]	[]	Graph convolutional networks (GCNs) are a family of neural network models that perform inference on graph data by interleaving vertex-wise operations and message-passing exchanges across nodes. Concerning the latter, two key questions arise: (i) how to design a differentiable exchange protocol (e.g., a 1-hop Laplacian smoothing in the original GCN), and (ii) how to characterize the trade-off in complexity with respect to the local updates. In this paper, we show that state-of-theart results can be achieved by adapting the number of communication steps independently at every node. In particular, we endow each node with a halting unit (inspired by Graves' adaptive computation time [1]) that after every exchange decides whether to continue communicating or not. We show that the proposed adaptive propagation GCN (AP-GCN) achieves superior or similar results to the best proposed models so far on a number of benchmarks, while requiring a small overhead in terms of additional parameters. We also investigate a regularization term to enforce an explicit trade-off between communication and accuracy. The code for the AP-GCN experiments is released as an open-source library.	10.1109/tnnls.2020.3025110	[ "https://arxiv.org/pdf/2002.10306v1.pdf" ]	211,258,593	2002.10306	be3c232fb65f8b0bfb6e3614c9a1f86d42ee64d1	PREPRINT SUBMITTED TO IEEE TRANSACTION ON NEURAL NETWORKS AND LEARNING SYSTEMS 1 Adaptive Propagation Graph Convolutional Network Student Member, IEEEIndro Spinelli Simone Scardapane Member, IEEEAurelio Uncini PREPRINT SUBMITTED TO IEEE TRANSACTION ON NEURAL NETWORKS AND LEARNING SYSTEMS 1 Adaptive Propagation Graph Convolutional Network Index Terms-Graph neural networkGraph dataConvolutional networkNode classification Graph convolutional networks (GCNs) are a family of neural network models that perform inference on graph data by interleaving vertex-wise operations and message-passing exchanges across nodes. Concerning the latter, two key questions arise: (i) how to design a differentiable exchange protocol (e.g., a 1-hop Laplacian smoothing in the original GCN), and (ii) how to characterize the trade-off in complexity with respect to the local updates. In this paper, we show that state-of-theart results can be achieved by adapting the number of communication steps independently at every node. In particular, we endow each node with a halting unit (inspired by Graves' adaptive computation time [1]) that after every exchange decides whether to continue communicating or not. We show that the proposed adaptive propagation GCN (AP-GCN) achieves superior or similar results to the best proposed models so far on a number of benchmarks, while requiring a small overhead in terms of additional parameters. We also investigate a regularization term to enforce an explicit trade-off between communication and accuracy. The code for the AP-GCN experiments is released as an open-source library. I. INTRODUCTION D EEP learning has achieved remarkable success on a number of high-dimensional inputs, by properly designing architectural biases that can exploit their properties. This includes images (through convolutional filters) [2], text, biomedical sequences [3], and videos [4]. A major research question, then, is how to replicate this success on other types of data, through the implementation of novel differentiable blocks adequate to them. Among the possibilities, graphs represent one of the largest sources of data in the world, ranging from recommender systems [5] to biomedical applications [6], social networks [7], computer programs [8], knowledge bases [9], and many others. In its most general form, a graph is composed by a set of vertices connected by a series of edges representing, e.g., social connections, citations, or any form of relation. Graph neural networks (GNNs) [10]- [12], then, can be designed by interleaving local operations (defined on either individual nodes or edges) with communication steps, exploiting the graph topology to combine the local outputs. These architectures can then be exploited for a variety of tasks, ranging from node classification to edge prediction and path computation. Among the different families of GNN models proposed over the last years, graph convolutional networks (GCN) [13] have become a sort of de facto standard for node and graph classification, representing one of the simplest (yet efficient) building blocks in the context of graph processing. GCN are built by interleaving vertexwise operations, implemented via a single fully-connected layer, with a communication step exploiting the so-called Laplacian matrix of the graph (see Fig. 1). In practice, a single GCN layer provides a weighted combination of information across neighbors, representing a localized 1-hop exchange of information. 1 Taking the GCN layer as a fundamental building block, several research questions have received vast attention lately, most notably: Emails: {firstname.lastname}@uniroma1.it Authors are with the Department of Information Engineering, Electronics and Telecommunications (DIET), Sapienza University of Rome, Italy 1 In the paper, we use node and vertex as synonyms, and the same for communication and propagation. (i) how to design more effective communication protocols, able to improve the accuracy of the GCN and potentially better leverage the structure of the graph [14]- [16]; and (ii) how to trade-off the amount of local (vertex-wise) operations with the communication steps [17]. While we defer a complete overview of related works to Section II, we briefly mention two key results here. Firstly, [18] showed that the use of the Laplacian (a smoothing operator) has as consequence that repeated application of standard GCN layers tend to over-smooth the data, disallowing the possibility of naively stacking GCN layer to obtain extremely deep networks. Secondly, [17] showed that state-of-the-art results can be obtained by replacing the Laplacian communication step with a PageRank variation, as long as completely separating communication between nodes from the vertex-wise operations. We exploit both of these key results later on. Contributions of the paper We note that the vast majority of proposals to improve point (i) mentioned before consists in selecting a certain maximum number of communication steps T , and iterating a simple protocol for T steps in order to diffuse the information across T -hop neighbors. In this paper, we ask the following research question: can the performance of GCN layers be improved, if the number of communication steps is allowed to vary independently for each vertex? To answer this question, we propose a variation of GCN that we call adaptive propagation GCN (AP-GCN). In the AP-GCN (see Fig. 2) every vertex is endowed with an additional unit that outputs a value controlling whether communication should continue for another step (hence combining the information from neighbors farther away), or should stop, and the final value be kept for further processing. In order to implement this adaptive unit, we leverage previous work on adaptive computation time in recurrent neural networks [1] to design a differentiable method to learn this propagation strategy. On an extensive set of comparisons and benchmarks, we show that AP-GCN can reach state-of-the-art results, while the number of communication steps can vary significantly not only across datasets but also across individual vertexes. This is achieved with an extremely small overhead in terms of computational time and additional trainable parameters. In addition, we perform an large hyper-parameter analysis, showing that our method can provide a simple way to balance accuracy of the GCN with the number of propagation steps. A. Outline of the paper The rest of the paper is structured as follows. In Section II we describe more in-depth related works from the field of GCNs and GNNs, focusing in particular on several proposals describing how to design more complex propagation steps. Then, in Section III we introduce the GCN model and the way a deep network can be composed and trained from GCN blocks. Our proposed AP-GCN is first introduced in Section IV and then tested in Section V. We conclude with some general remarks in Section VI. II. RELATED WORKS GCNs belong to the class of spectral graph neural networks, which are based on graph signal processing (GSP) tools [19]- [22]. LG] 24 Feb 2020 GSP allows to define a Fourier transform over graphs by exploiting the eigen-decomposition of the so-called graph Laplacian. The first application of this theory to graph NNs was in [23]. This approach, however, was both computationally heavy and not spatially localized, meaning that each node-wise update depended on the entire graph structure. Later proposals [24] showed that by properly restricting the class of filters applied in the frequency domain, one could obtain simpler formulation that were also spatially localized in the graph domain. Polynomial filters [24] can be implemented via T -hop exchanges on the graph, but they require to select a priori a valid T for all the vertices. The GCN, introduced in [13], showed that stateof-the-art results could be obtained even with simpler linear (i.e., 1-hop) operations. However, they failed to build deeper architectures (i.e., > 2 GCN layers) in practice. [18] formally analyzed the properties of the GCN, showing that the difficulty of building deeper networks could depend from the over-smoothing of the data due to a repeated application of the Laplacian operator. Further analyses and the need to consider higherorder structures in GNNs were provided by [25], showing that GCNs are equivalent to the so-called 1-dimensional Weisfeiler-Leman graph isomorphism heuristic. Several recent papers have proposed to avoid some of these shortcomings by using different types of propagation methods, most notably PageRank variations [16], [17]. In this paper we explore an orthogonal idea, where we hypothesize that performance can be improved not only by modifying the existing propagation method, but by allowing each node to vary the amount of communication independently from the others, in an adaptive fashion. Jumping knowledge (JK) networks [26] and GeniePath [27] achieve something similar by exploiting an additional network aggregation component (e.g., an LSTM network) after multiple diffusion steps, however, they fail to reach state-of-the-art results [16]. Finally, we underline that we focus on GCN in this paper, but alternative models for graph neural networks have been devised, including those from [28], graph attention networks [29], graph embeddings, and others. We refer to multiple recent surveys on the topic for more information [11], [12]. III. GRAPH CONVOLUTIONAL NEURAL NETWORKS A. Graph definitions Consider a generic undirected graph G = (V, E), where V = {1, . . . , n} is the set of node indexes, and V = {(i, j) \| i, j ∈ V} is the set of arcs (edges) connecting pairs of nodes. The meaning of a single node or edge depends on the application. For example, a classic setup in text classification encodes each text as a node [13], and a citation among two texts as an arc in the corresponding graph. Connectivity in the graph can be summarized in the adjacency matrix A ∈ {0, 1} n×n . From this, we can define the diagonal degree matrix D where Dii = j Aij, and the Laplacian matrix L = D−A. In the context of GNNs, the Laplacian is generally used in its normalized form L = D −1/2 LD −1/2 . As we will see, the Laplacian operators can be used to define (normalized) 1-hop communication protocols across the graph. In the context of inference over graphs, we suppose that node i is endowed with a vector xi ∈ R d of features. For tasks of node classification [13], we also know a desired label yi for a subset T ⊂ V of nodes, and we wish to infer the labels for the remaining nodes. Graph classification is easily handled by considering sets of graphs defined as above, with a single label associated to every graph, e.g., [6]. While we focus on node / graph classification in the rest of the paper, the techniques we introduce in the next section can further be extended by considering edge features vij, global graph features [30], and applied to other tasks such as edge classification [5]. We return on this later on in Section III-C. Node-wise update ( ) Propagation ( ) Ψ Fig. 1. Single update in a GCN. Active nodes for each step are shown in light red color. B. Graph convolutional networks The basic idea of GCNs is to combine local (node-wise) updates with suitable message passing across the graph, following the graph topology. In particular, consider the n×d matrix X collecting all node features for the entire graph. A generic GCN layer can be written as [13]: H = φ LXW + b ,(1) where φ is an element-wise nonlinearity (such as the ReLU φ(·) = max (0, ·)), L is the normalized Laplacian defined above, and W and b are the learnable parameters of the layer. More in general, the Laplacian matrix can be renormalized in different ways (see [13]) or substituted with any appropriate shift operator defined on the graph. The name GCN derives from an interpretation of (1) in terms of GSP [19], as described in Section II. A graph Fourier transform can be defined for the graph by considering the eigen-decomposition of the Laplacian matrix [22]. In this context, (1) can be shown to be equivalent to a graph convolution implemented with a linear filter [13]. Because its implementation requires only 1-hop exhanges across neighbours, the GCN is also an example of a message-passing neural network (MPNN) [12]. These two interpretations bring forth two classes of extensions for the basic model in (1), which we comment on to the extent that they relate to our proposed method. Firstly, under a GSP interpretation, it makes sense to substitute the linear filtering operation with a more complex filter, such as Chebyshev filters [24], rational filters [14], ARMA filters [15], and more [10]. In fact, as we described in Section II, some of these works predate the introduction of the GCN itself. Interestingly, the majority of these works require diffusion protocols going beyond simple 1-hop neighbors of every node. For example, Chebyshev filters [24] result in the following layer (omitting biases for simplicity): H = φ K k=1 T k ( L)XW k ,(2) where T k (s) is defined recursively as T k (s) = 2sT k−1 (s)−T k−2 (s), and the layer has a number of adaptable matrices {W k } K k=1 that depend on the user-defined hyper-parameter K. Setting K corresponds to selecting a 'depth' for the information being propagated. For example, setting K = 2 propagates information across 2-hop neighbors, while K = 1 simplifies to the GCN described above. This decision, however, must be made beforehand by the user, or the parameter must be fine-tuned accordingly. Under the more general interpretation of (1) as a MPNN, however, we are not restricted to considering filtering operations. In fact, the most general extension of (1) becomes (expressed for simplicity for a single node i) [12]: where Ψ is a permutation-invariant function, ψ a node-wise update, and Ni is the inclusive neighborhood of node i. Selecting ψ(x) = W T x and Ψ({ψ(xi)}) = j Lijψ(xj) recovers the previous GCN formulation. More in general, both Ψ and ψ can be implemented as generic neural networks or any other differentiable mechanism. Most notably, [17] proposes the use of PageRank protocols for the propagation step to counteract the oversmoothing effect of repeated applications of the Laplacian matrix [18], although the maximum number of propagation steps must still be selected a priori by the user. The overall setup is shown graphically in Fig. 1. hi = Ψ ({ψ(xj) \| j ∈ Ni}) ,(3) C. Designing and training deep GCNs In the spirit of classical deep networks, the basic building blocks described in the previous section can be composed to design deeper architectures. For example, a network for binary classification with a single hidden layer and one output layer, both implemented according to (1), is defined by: y = σ L · φ LXW + b v + c ,(4) where the adaptable weights are W, v, b and c. A more recent line of reasoning, popularized by [16], is to implement architectures in the form (3), making both ψ, Ψ deeper networks, but without interleaving multiple node-wise and propagation steps. We follow this design principle here, as we have found it to perform better empirically. Once a specific network f has been designed, its optimization follows the same strategies as for other deep networks. For example, for node classification (as described in Section III-A) we optimize the network with a cross-entropy loss on the known node labels: f * = arg min i∈T yi · log (f (xi)) .(5) Note, however, that differently from standard neural networks, the output of f (xi) will depend on several other nodes, depending on the specific architecture. For this reason, (5) is harder to solve efficiently in a stochastic fashion [31]. IV. PROPOSED ADAPTIVE PROPAGATION PROTOCOL In the previous sections, we analyzed the motivation for having graph modules with complex diffusion steps across the graph. However, the vast majority of proposals has considered a single, maximum number of communication steps that is shared for all the nodes in the graph (e.g., the number K in (2)). In this section we introduce a novel variation of GCN wherein (i) the number of communication steps is selected independently for every node, and (ii) this number is adapted and computed on-the-fly during training. To the best of our knowledge, our proposed Adaptive Propagation GCN (AP-GCN) is the only model in the literature combining these two properties. Our AP-GCN framework is summarized in Fig. 2. Considering the notation in (3), we separate the node-wise operations ψ from the propagation step Ψ. The former is implemented with a generic NN applied on a single node zj = ψ(xj), described on the left part of Fig. 2. This embedding is then used as the starting seed for a propagation step Ψ which is done iteratively: z 0 i = zi z 1 i = propagate( z 0 j \| j ∈ Ni ) z 2 i = propagate( z 1 j \| j ∈ Ni ) . . . Key to our proposal, the number of propagation steps depends on the index of node i and it is computed adaptively while propagating. The mechanism to implement this is inspired by the adaptive computation time in RNNs [1]. First, we endow each node with a linear binary classifier acting as a 'halting unit' for the propagation process. After the generic iteration k of propagation, we compute node-wise: h k i = σ Qz k i + q ,(6) where Q and q are trainable parameters. The value h k i describes the probability that the node should stop after the current iteration. In order to ensure that the number of propagation steps remains reasonable, following [1] we adopt two techniques. Firstly, we fix a maximum number of iterations T . Secondly, we use the running sum of the halting values to define a budget for the propagation process: Ki = min    k : k k=1 h k i >= 1 −    ,(7) where is a hyper-parameter, generally set to a small value, that ensures that the process can terminate also after a single update. Dataset Classes Features Nodes Edges Citeseer 6 3703 2110 3668 Cora-ML 7 2879 2810 7981 PubMed 3 500 19717 44324 MS-Academic 15 6805 18333 81894 Whenever k = Ki, the budget is reached and the propagation stops for node i at iteration k. The final probability is: p k i = Ri = 1 − K i −1 k=1 h k i if k = Ki or k = T K i k=1 h k i otherwise ,(8) The rationale for the last two steps is that the sequence p k i now forms a valid cumulative distribution for the halting probabilities. By exploiting it, instead of using the latest value in the propagation, we can adaptively combine the information at every time-step for free: zi = 1 Ki K i k=1 p k i z k i + (1 − p k i )z k−1 i . (9) zi is now the final output for node i. The number of propagation steps can be controlled by the definition of a propagation cost similarly to [1]: Si = Ki + Ri .(10) Denoting by L the loss term in (5), this term is added to be minimized, weighed by a propagation penalty α: L = L + α i∈V Si .(11) The propagation penalty is responsible for the trade-off between computation time and accuracy. Moreover, it regulates how 'easily' the information spreads on the graph. In practice, the optimization of the halting unit is performed in an alternate fashion once every L steps of the main network (in our experiments, L = 5). V. EXPERIMENTAL RESULTS A. Experimental setup We used the same experimental setup proposed in [17] which aims to reduce experimental bias. This setup has shown that many advantages reported by recent works vanish under this statistically rigorous evaluation. The first step in this process is the subdivision in visible and invisible sets. The invisible set will serve as a test set and will be used only once to report the final performance. The visible set is subdivided in a training set with N nodes per class and an early stopping set for model selection. A validation set containing the remaining nodes of the visible set is used for hyper-parameters tuning. These splits are determined using the same 20 seeds used in [17] and each experiment is run with 5 different initialization of the weights leading to a total of 100 experiments per dataset. The datasets, whose characteristics are summarized in Table I, are three citation graphs; Citeseer, Cora-ML, and PubMed, and a coauthorship one, MS-Academic. The nodes contain a feature vector with a bag-of-words representation. These features are normalized with an 1 norm and to conclude the preprocessing, which is the same for all the datasets, we select the largest connected component. To be in line with the evaluation of [17] we use the same number of layers (2) and hidden units (64), dropout rate (0.5) on both layers and the adjacency matrix, resampled at each propagation, and Adam optimizer [32] with learning rate 0.01. We choose instead the following hyperparameters for all the datasets: 2 regularization parameter 0.008 on the weights of the first layer, maximum steps of propagation T = 10. We adapted the propagation penalty α, controlling the distribution of the propagation steps, to each dataset. The average density distribution obtained with a fixed propagation penalty of 0.005 is different for all the datasets, as can be observed in Fig. 3. The code to test our proposed AP-GCN and replicate our experiments is available on the web. 2 B. Results and comparisons In Table II we report the average accuracy when using a training set of 20 nodes per class, with uncertainties showing the 95% confidence level calculated by bootstrapping. When not indicated with the symbol ( * ), the results are taken from [17]. In Fig. 4 we show the distribution of the steps selected by AP-GCN with a propagation cost of 0.02, 0.004, 0.001, 0.02 respectively for Citeseer, Cora-ML, PubMed, and MS-Academic. We included in our comparison ARMA [15], with a configuration compatible with the experimental setup. For the two smallest datasets AP-GCN outperforms its competitors. The advantage is reduced in PubMed, and disappears in MS-Academic where APPNP remains the state-of-the-art and AP-GCN is the second-best alternative closely followed by ARMA. Furthermore, AP-GCN shows a low variance, which ensures robustness to the choice of the splits and random initializations. In Table III we report the average training time per epoch of our implementation of a subset of algorithms of Table II using the framework introduced in [33]. Due to the higher number of propagation steps, and the presence of an additional (small) layer, AP-GCN is the slowest method on the two smallest datasets. However, AP-GCN scales better to bigger dataset with respect to more complex methods like GAT [29] and ARMA [15]. C. Sensitivity to hyper-parameters Here we would like to inspect the sensitivity of AP-GCN to the propagation penalty α. In Figure 5 we show the effect on both the average density distribution of the number of selected propagation steps and the corresponding accuracy on Cora-ML. The propagation penalty α strongly affects the performances and the behaviour of AP-GCN. For α = 0.1 and α = 0.05 higher values, AP-GCN performs mostly less than 2 propagation steps. For the latter value, the performances are comparable to the GCN reported in Table II. Decreasing the value of the propagation penalty has the effect of augmenting the receptive field of AP-GCN. This allows propagation of the information to the nodes that are far away from the labelled samples. The advantage of our method is that with the right choice of α each node will choose the K that maximizes the chances of being classified correctly. Finally, we want to analyze the performances of GCN, APPNP, and AP-GCN on Cora-ML for different dimensions of the training set. This is a crucial aspect since labelling is one of the most expensive processes in modern machine learning. Therefore a model capable of working with very few labelled samples has a great advantage over those that do not. Fig. 6(a) shows, as noticed in [17], that the higher range of APPNP and AP-GCN permits to have a great increment in performance when the label information is very sparse. The improvement of AP-GCN over APPNP, even if present for every size of the training set, behaves similarly. This suggests that a loosely labelled dataset highlights the effectiveness of a propagation protocol. In Figure 6 this information. Contrary, when the number of labelled samples increases more and more, nodes in the graph select as maximum propagation K < T , preventing the issue of over-smoothing. VI. CONCLUSION In this paper we introduced the adaptive propagation graph convolutional network (AP-GCN), a variation of GCN wherein each node selects automatically the number of propagation steps performed across the graph. We showed experimentally that the method performs favourably or better than the state-of-the-art, and that it is robust to the training set size and the hyper-parameters selection. Future work will consider extending the ideas presented here to different types of GNNs and to tasks going beyond node classification. arXiv:2002.10306v1 [cs. Fig. 2 . 2Schematics of the proposed framework. Fig. 3 . 3Average density distribution of the maximum number of propagations K selected by AP-GCN with a fixed α = 0.005. (b) we show the variation of the average density distribution of the maximum number of propagation steps selected by AP-GCN under the different training sizes. The behaviour of AP-GCN is in line with the previous observation. The sparsest the labels, the more propagation steps performed by AP-GCN, Fig. 5 . 5(a) Average density distribution of the maximum number of propagations K and (b) accuracy of AP-GCN on CORA-ML dataset with the propagation penalty α varing in the range [0.1, 0.001]. Fig. 6 . 6(a) Accuracy of GCN, APPNP and AP-GCN for different number of labeled nodes per class on Cora-ML. (b) AP-GCN relative average density distribution of the maximum number of propagations K. TABLE I DATASET ISTATISTICS. TABLE II AVERAGE IIACCURACY WITH UNCERTAINTIES SHOWING THE 95% CONFIDENCE LEVEL CALCULATED BY BOOT-STRAPPING. Fig. 4. Average density distribution of the maximum number of propagations K selected by AP-GCN in the evaluation associated toTable II.Model Citeseer Cora-ML PubMed MS-Academic V. GCN 73.51 ± 0.48 82.30 ± 0.34 77.65 ± 0.40 91.65 ± 0.09 GCN 75.40 ± 0.30 83.41 ± 0.34 78.68 ± 0.38 92.10 ± 0.08 N-GCN 75.40 ± 0.30 83.41 ± 0.39 78.68 ± 0.38 92.10 ± 0.11 GAT 75.39 ± 0.47 82.25 ± 0.30 77.46 ± 0.44 91.22 ± 0.11 JK 73.03 ± 0.47 82.69 ± 0.35 77.88 ± 0.38 91.71 ± 0.07 Bt.FP 73.03 ± 0.47 80.84 ± 0.37 72.94 ± 1.00 91.71 ± 0.10 PPNP 75.83 ± 0.27 85.29 ± 0.25 - - APPNP 75.73 ± 0.30 85.09 ± 0.25 79.73 ± 0.31 93.27 ± 0.08 ARMA* 73.56 ± 0.36 82.58 ± 0.28 76.31 ± 0.41 92.41 ± 0.07 AP-GCN* 76.11 ± 0.26 85.71 ± 0.22 79.76 ± 0.32 92.53 ± 0.08 0 2 4 6 8 10 Number of Steps 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Density Citeseer Cora-ML PubMed MS-Academic TABLE III AVERAGE IIITRAINING TIME PER EPOCH (MILLISECONDS).Dataset AP-GCN ARMA APPNP GCN GAT Citeseer 32.4 25.2 19.6 8.6 11.1 Cora-ML 36.2 27.6 22.1 7.9 13.4 PubMed 42.0 51.1 23.3 16.1 45.4 MS-Academic 100.3 121.2 86.1 56.0 110.5 https://github.com/spindro/AP-GCN Adaptive computation time for recurrent neural networks. 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[ "Estimating the sensitivity of centrality measures w.r.t. measurement errors", "Estimating the sensitivity of centrality measures w.r.t. measurement errors" ]	[ "Christoph Martin [email protected] \nLeuphana University of Lüneburg Universitätsallee 1\n21335LüneburgGermany\n", "Peter Niemeyer [email protected] \nLeuphana University of Lüneburg Universitätsallee 1\n21335LüneburgGermany\n" ]	[ "Leuphana University of Lüneburg Universitätsallee 1\n21335LüneburgGermany", "Leuphana University of Lüneburg Universitätsallee 1\n21335LüneburgGermany" ]	[]	Most network studies rely on an observed network that differs from the underlying network which is obfuscated by measurement errors. It is well known that such errors can have a severe impact on the reliability of network metrics, especially on centrality measures: a more central node in the observed network might be less central in the underlying network.We introduce a metric for the reliability of centrality measurescalled sensitivity. Given two randomly chosen nodes, the sensitivity means the probability that the more central node in the observed network is also more central in the underlying network. The sensitivity concept relies on the underlying network which is usually not accessible. Therefore, we propose two methods to approximate the sensitivity. The iterative method, which simulates possible underlying networks for the estimation and the imputation method, which uses the sensitivity of the observed network for the estimation. Both methods rely on the observed network and assumptions about the underlying type of measurement error (e.g., the percentage of missing edges or nodes).Our experiments on real-world networks and random graphs show that the iterative method performs well in many cases. In contrast, the imputation method does not yield useful estimations for networks other than Erdős-Rényi graphs.	10.1017/nws.2019.12	[ "https://arxiv.org/pdf/1704.01045v2.pdf" ]	18,076,655	1704.01045	00588f2f19495903a5f42f0a3a33286d3568207d	Estimating the sensitivity of centrality measures w.r.t. measurement errors September 20, 2017 Christoph Martin [email protected] Leuphana University of Lüneburg Universitätsallee 1 21335LüneburgGermany Peter Niemeyer [email protected] Leuphana University of Lüneburg Universitätsallee 1 21335LüneburgGermany Estimating the sensitivity of centrality measures w.r.t. measurement errors September 20, 2017 Most network studies rely on an observed network that differs from the underlying network which is obfuscated by measurement errors. It is well known that such errors can have a severe impact on the reliability of network metrics, especially on centrality measures: a more central node in the observed network might be less central in the underlying network.We introduce a metric for the reliability of centrality measurescalled sensitivity. Given two randomly chosen nodes, the sensitivity means the probability that the more central node in the observed network is also more central in the underlying network. The sensitivity concept relies on the underlying network which is usually not accessible. Therefore, we propose two methods to approximate the sensitivity. The iterative method, which simulates possible underlying networks for the estimation and the imputation method, which uses the sensitivity of the observed network for the estimation. Both methods rely on the observed network and assumptions about the underlying type of measurement error (e.g., the percentage of missing edges or nodes).Our experiments on real-world networks and random graphs show that the iterative method performs well in many cases. In contrast, the imputation method does not yield useful estimations for networks other than Erdős-Rényi graphs. Introduction Measurement errors in network data are a central problem in the field of network analysis. Virtually all empirical network data is affected by a certain kind of measurement error and previous research has shown that these errors often have a major impact on the results of network analysis methods, especially on centrality measures [36]. For example, a more central node in the observed (erroneous) network might be less central in the hidden (unobserved, error-free) network. To quantify the impact of measurement errors on centrality measures, we introduce a metric -called sensitivity. Given two randomly chosen nodes, the sensitivity means the probability that the more central node in the observed network is also more central in the underlying network. Currently, most applied network studies only report that measurement errors might have affected the data collection (e.g., due to absence of actors at the day of the survey or due to the study design). Most of the time, however, the impact of these measurement errors on centrality measures are not discussed. This might be due to the fact that there is currently no established way to estimate the sensitivity for an observed network. Some researchers have investigated the impact that different kinds of measurement errors have on the reliability of centrality measures in the case of random graphs and real-world networks [9,6,21,12,36,33,29,28,24]. For an extensive survey of previous studies about the reliability of centrality measures see Smith et al. (2017) [34]. These important studies provide guidelines for researchers on how to design future studies (e.g., what kind of measurement error might be especially harmful in a given scenario) and suggestions on which centrality measure might be more reliable in a given scenario. Unfortunately, it is difficult to identify general patterns for the reliability of centrality measures in real-world networks. As common sense suggests, centrality measures become less reliable with an increasing level of error. Additionally, the particular relationship between error level and reliability is highly dependent on the type of measurement error, the centrality measure, and the network structure. There are studies that use the observed network to reconstruct the hidden network, estimate statistics about the hidden network, or estimate the influence that measurement errors have on network analysis methods [8,18,16,20,11,35,27]. Despite their important contributions, these studies usually focus on network invariants (e.g., diameter or average path length) and do not explicitly address node invariants (e.g., the centrality values for all nodes in a network). Studies about the robustness of networks, the capacity of networks to maintain functionality when the network is modified, also usually focus on network invariants. In this paper, however, we do not study the robustness of networks. For an extensive overview about the robustness of networks see Barabási (2016) [2] and Havlin & Cohen (2010) [17]. The sensitivity concept relies on the hidden network which is usually not accessible. In this paper, we propose two methods ("imputation method" and "iterative method") that allow the researcher to estimate the sensitivity of the hidden network, given the observed network and some assumptions about the measurement error. In case of the imputation method, we try to simulate the hidden network by inverting the measurement error (e.g.: if 10% of randomly chosen edges are missing, we add 11,11% edges by linking randomly chosen pairs of non-adjacent nodes) and then compute the sensitivity with respect to the simulated hidden network. In case of the iterative method, we assume that the considered hidden network has a certain kind of self-similarity property. That is, we assume that the sensitivity of the observed network (regarding the assumed measurement error) is a good estimate of the sensitivity of the hidden network. While it is known that imputation techniques may work well [30,1,31], simulating appropriate hidden networks is complicated and calculating the estimate based on the imputation method is, therefore, difficult. In contrast, the estimate based on the iterative method is relatively easy to calculate. However, this method relies on a self-similarity property that is difficult to verify. In addition to the estimation methods, we introduce an easy-to-interpret measure for the reliability of centrality measures and a generic concept to model measurement errors. We test both estimation methods on random graphs and real-world networks. While the imputation method and the iterative method perform well on Erdős-Rényi graphs (ER graphs), the iterative method, which is easier to calculate, performs much better in case of Barabási-Albert graphs (BA graphs) and real-world networks. For realworld networks, the iterative method performs especially well in case of the PageRank. To measure the reliability of centrality measures, we introduce the necessary concepts in Section 2. The estimation methods are presented in Section 3. In Section 4, we apply these methods to real-world networks. Results are discussed in Section 5. Basic concepts Let G be an undirected, unweighted, finite graph with vertex set V (G) and edge set E(G). 1 A centrality measure c is a real-valued function that assigns centrality values to all nodes in a graph and is invariant to structurepreserving mappings, i.e., centrality values depend solely on the structure of a graph. External information (e.g., node or edge attributes) have no influence on the centrality values (cf. Koschützki et al. (2005) [22]). We denote the centrality value for node u ∈ V (G) by c G (u) and the centrality values for all nodes in G (u 1 , u 2 , . . . , u n ) by the vector c(G) := (c(u 1 ), . . . , c(u n )). The following centrality measures are used in this study: closeness centrality, betweenness centrality [13], degree centrality, eigenvector centrality [5], and the PageRank [7]. All centrality measures are calculated using the igraph library (version 0.7.1, Csardi & Nepusz [10]). Let G and G be two graphs and c a centrality measure. A pair of nodes u, v ∈ V (G) ∩ V (G ) and u = v is called concordant w.r.t. c if both nodes have distinct centrality values and the order of u and v is the same in c(G) and c(G ), i.e., either c G (u) < c G (v) and c G (u) < c G (v) or c G (u) > c G (v) and c G (u) > c G (v) . A pair of nodes is called discordant if both nodes have distinct centrality values and the order of u and v in c(G) differs from the order of u and v in c(G ), i.e., either c G (u) < c G (v) and c G (u) > c G (v) or c G (u) > c G (v) and c G (u) < c G (v). Ties are neither concordant nor discordant. A random graph consists of a finite set of graphs Ω equipped with a function P that assigns a probability to every graph in this set (cf. Bollobás & Riordan (2002) [4]). Modeling measurement errors Network data can be influenced by a variety of different measurement errors. Recently, Wang et al. [36] categorized measurement errors into six groups: false negative nodes and edges, false positive nodes and edges, and false aggregation and disaggregation. For example, when 10% of the edges are missing in the observed network data, the graph constructed from this observed data suffers from false negative edges. To describe measurement errors, we introduce the notion of an error mechanism. An error mechanism ϕ is a procedure that describes measurement errors which may occur during the data collection. For a graph G, ϕ(G) is a random graph which probability distribution for the possible graphs depends on the error procedure that describes ϕ. Hence, all graphs that could be observed when the error procedure influences the data collections are possible outcomes of this random graph. To illustrate the concept of an error mechanism, consider the graphs illustrated in Figure 1. The initial graph is denoted by H (drawn in the upper left corner). We assume that we know the error mechanism that compromises the data collection. For this example, we assume that the error mechanism ϕ is edges missing uniformly at random with an error level of 50%. All graphs in the set of possible outcomes for this random graph Ω = {G 1 , G 2 , . . . , G 6 } are also shown in Figure 1. In this example, the probability function is P (G i ) = 1 6 ; all graphs in Ω occur with the same probability. However, this concept is not limited to a uniform distribution. In general, error mechanisms can rely on node or edge attributes. In this study, we focus on four common error mechanism that do not depend on external attributes: 1. Nodes missing uniformly at random (rm nodes): A fraction of nodes (and all edges connected to these nodes) is missing in the observed network. All nodes have the same probability to be missing in the observed network. q q q q q A B C D E H q q q q q A B C D E G 1 q q q q q A B C D E G 2 q q q q q A B C D E G 3 q q q q q A B C D E G 4 q q q q q A B C D E G 5 q q q q q A B C D E G 6 Figure 1: In this example, ϕ is defined as the error mechanism "50% of all edges are missing uniformly at random". Hence, ϕ(H) is a random graph with possible outcomes Ω = {G 1 , G 2 , . . . , G 6 } and P (G i ) = 1 6 . 2. Edges missing uniformly at random (rm edges unif.): A fraction of edges is missing in the observed network. All edges have the same probability to be missing in the observed network. 3. Edges missing proportional (rm edges prop.): A fraction of edges is missing in the observed network. The probability that an edge is missing in the observed network is proportional to the sum of the degree values of the endpoints. Spurious edges (add edges): The observed network contains too many edges. Every non-existing edge has the same probability to be erroneously observed. Imputation techniques are commonly used to replace missing data with plausible estimates. Inspired by Huisman [18], we define an imputation mechanism as a procedure that aims to 'undo' the effects that measurement errors have on network data. For a given error scenario, there might be multiple or no appropriate imputation mechanisms. Moreover, the choice of the appropriate imputation mechanism depends on the type of measurement error and possibly also on the network structure. For example, if edges are missing uniformly at random, a possible imputation mechanism is to add edges randomly (uniform) to the observed network. However, the success of this procedure depends on the structure of the network. Consider the scenario that the hidden network is a star graph. In this case, it is quite likely that this imputation approach would have a negative impact on subsequent analyses because it is much more likely that the imputed edges are between leaf nodes than between the internal node and a leaf node. Hence, if the structure of the hidden network is not known (which is usually the case), it is difficult to define an appropriate imputation procedure. We use ψ to denote a specific imputation mechanism. Analogously to error mechanisms, ψ(G) denotes a random graph. All graphs that can occur when the imputation mechanism ψ is applied to graph G are possible outcomes for this random graph. In this paper, we use three generic imputation mechanisms: 1. If k edges are missing in the observed network, we choose k edges uniformly at random from the set of edges that are not in the observed network and add them to the observed network. 2. If there are k spurious edges in the observed network, we choose k edges that do exist in the observed network and delete them. 3. If k nodes are missing, we choose the degrees for k new nodes from the current degree distribution and successively add these nodes to the networks. The links between new and existing nodes are created uniformly at random. 2 Sensitivity of centrality measures Let G and G denote graphs on the same vertex set and c a centrality measure. The sensitivity of the centrality measure c w.r.t. those two graphs is the probability that two nodes with distinct centrality values, randomly chosen from the vertex set of G and G , have the same order in c(G) and c(G ), i.e., they are concordant. If this quantity is close to one, the centrality measure is considered to be robust. If it is close to zero, the centrality measure is considered to be sensitive. We calculate the sensitivity ρ for a centrality measure c with respect to G and G as follows: ρ c (G, G ) = n c n c + n d(1) With n c as the number of concordant pairs and n d as the number of discordant pairs w.r.t. the order given by c(G) and c(G ). 3 The sensitivity as defined above is closely related to Goodman and Kruskal's rank correlation coefficient γ which is the difference between concordant and discordant pairs divided by the sum of concordant and discordant pairs [15]. We can use this relationship to calculate the sensitivity as follows: ρ c (G, G ) = γ(c(G), c(G )) + 1 2 .(2) Let us apply this concept to a graph illustrated in Figure 1. Assume that we have observed the graph labeled as G 6 and that we are interested in the sensitivity of the degree centrality. Then, the degree centrality values are deg(H) = (1, 2, 3, 1, 1) and deg(G 6 ) = (1, 2, 1, 0, 0). Based on the degree values, we can calculate the sensitivity of the degree centrality with respect to G 6 and H: ρ deg (G 6 , H) = 5 6 . If we randomly choose a pair of nodes, there is a 0.83 chance that their order induced by the degree centrality is the same in both graphs. Exemplary application of the sensitivity concept In this section, we apply the concepts introduced above to analyze the sensitivity of centrality measures in ER graphs. For all combinations of centrality measures and error mechanisms introduced in Section 2 and 2.1, we perform the experiment described below 500 times. For every error mechanism, we consider two cases, a moderate scenario of 10% error level and a more intense scenario with 30% error level. The procedure for the experiment is as follows: 1. Generate an ER graph with 100 nodes and edge probability 0.2 and denote it by H. This is the error-free (hidden) graph which is not available to the researcher. 2. Choose a graph from ϕ(H) and denote it by O. This is the observed graph which is affected by measurement errors. Calculate the sensitivity ρ c (O, H). The results are shown in Figure 2. Every panel shows violin plots for the distribution of the sensitivity of the centrality measures. Despite the fact that these graphs are very homogeneous, we make interesting observations. The sensitivity differs between centrality measures. The degree centrality is, in all cases, the most robust measure. Generally, the sensitivity values for the 30% error mechanisms are much lower than for the 10% error mechanism. The variance of the sensitivity values also increases with increasing error level. Usually, the sensitivity also depends on the error mechanism. However, this is not always the case. (e.g., degree centrality the the case of 10% error mechanisms). These observations are conclusive with the results of [6]. These results show that measurement errors have severe consequences for the reliability of centrality measures, even for homogeneous networks such as ER graphs. How to estimate the sensitivity of a centrality measure In a lot of network studies, the observed network data contains sampling errors [25,32,35]. But in general, the authors of such studies have no tools to describe the impact of sampling errors on the network measures (e.g., centrality measures) they apply. In general, the assumptions made on sampling errors are mentioned in the limitations, but they are not considered as part of the network model. The sensitivity concept as introduced in Section 2.2 helps researchers to describe this impact: given the observed network O, the (unknown) hidden network H, and a centrality measure c, ρ c (O, H) measures the probability, that two randomly chosen nodes (with distinct centrality values) have the same order in c(O) and c(H). Hence, ρ c (O, H) is capable to measure the impact of the sampling error on the centrality measure c. Thus, the sensitivity can be used to measure the reliability of a centrality measure with respect to sampling errors. We call ρ c (O, H) the "true sensitivity". Unfortunately, the hidden network H is not known and thus the true sensitivity cannot be computed explicitly. In this section, we propose two methods for the estimation of the true sensitivity based on the observed network O. Moreover, we provide an example for their application and demonstrate how the estimation results can be evaluated. Methods to estimate the sensitivity In this section we propose two methods for the estimation of the true sensitivity ρ c (O, H). In addition to the observed network O and a centrality measure c, each of these methods needs an additional assumption. 4 The first method that we propose is the imputation estimate for the sensitivity of a given centrality measure ("imputation method"). Based on the observed network O and an imputation mechanism ψ, we try to "reconstruct" the hidden network from the observed network. Based on the reconstructed network and the observed network, we calculate the estimate for the true sensitivity of the hidden network H and a centrality measure c as follows: ρ imp c (O, H) := E(ρ c (O, ψ(O))).(3) Since ψ(O) is a random graph, ρ c (O, ψ(O)) is a random variable and we use the expected value of this expression as the estimate for the sensitivity. The actual form of the imputation mechanism ψ depends strongly on the error mechanism ϕ which has influenced the data collection. However, depending on the network structure and the error mechanism, it might be very difficult to define an appropriate imputation mechanism (see Section 2.1). The second method that we propose is the iterative estimate for the sensitivity of a given centrality measure ("iterative method"). If we assume that the network of interest has a self-similarity property in the sense that subgraphs of this network have the same sensitivity as the initial network, we can apply the (assumed) error mechanism ϕ to the observed network and calculate the estimate for the true sensitivity of the hidden network H and a centrality measure c as follows: ρ iter c (O, H) := E(ρ c (O, ϕ(O))).(4) Since ϕ(O) is a random graph, ρ c (O, ϕ(O)) is a random variable and we use the expected value of this expression as the estimate for the sensitivity. Our experiments indicate that this self-similarity property may exist in many cases even though it is hard to prove that such a property does exist in complex networks. Both methods do rely on assumptions that are difficult to prove (appropriate imputation mechanism and self-similarity property). However, if these assumptions hold, we should be able to make good estimates for the sensitivity. Example for method application As a fist step to verify whether the proposed methods yield useful results, we apply the four error mechanisms to ER graphs and try to predict the sensitivity using the imputation method as well as the iterative method. For all combinations of centrality measures and error mechanisms introduced in Section 2 and 2.1, we perform the experiment described below 500 times. For every error mechanism, we consider two cases, a moderate scenario of 10% error level and a more intense scenario with 30% error level. We perform the following steps to simulate erroneous data collection and to collect 3-tuples of true sensitivity, the estimate based on the iterative method, and the estimate based on the imputation method: (O, H)) is denoted byŝ iter . The results of these experiments are listed in Table 1. Values in the s columns represent the mean values of the true sensitivity for all 500 runs. The 95th percentile values of the absolute difference between the true sensitivity and the estimate are labeled withê imp for the imputation estimates andê iter for the iterative estimates. We call this value the absolute error. For example, the average true sensitivity of the betweenness centrality under the influence of the error mechanism add edges random 10% is 0.891 and the imputation estimate of the true sensitivity is in the interval [0.871, 0.911] in 95% of all runs. It can be seen from Table 1 that there is a wide range of absolute error values (ranging from 0.017 to .07). How should these values be interpreted? For example, the estimates for the sensitivity of the degree centrality and the PageRank under the influence of the add edges 10% error mechanism show approximately the same error values (ranging from 0.017 to 0.019). However, we argue that the estimate for the PageRank works better because the (average) sensitivity of the PageRank is 0.897 while the average sensitivity of the degree centrality is substantially higher (0.942). Therefore, one has to consider the magnitude of the true sensitivity when interpreting the absolute error of an estimate for the sensitivity. Evaluating estimation results The estimate for the true sensitivity is successful if it is "close" to the true sensitivity value. To determine this "closeness", we calculate the error (absolute difference between true value and estimate) relative to the magnitude of the true sensitivity. On the one hand, if the true sensitivity is low, the estimate does not need to be as accurate as if the true sensitivity is high. On the other hand, we assume that estimating the sensitivity is more difficult if the true sensitivity is relatively low. Moreover, for the evaluation of the "closeness" between the true sensitivity s and an estimate for the true sensitivityŝ, we ignore the direction of the deviation. Therefore, we define the weighted error of the estimate as weightederror(s,ŝ) := \|s −ŝ\| 1 − s for s < 1,(5) where smaller values indicate better performance. For the ultimate decision whether the estimate is close enough to the target value, we use an indicator function which takes the value one if weightederror(s,ŝ) is below a given threshold value and zero otherwise. Figure 3 illustrates the values of this indicator function combined with threshold values of 0.1, 0.3, and 0.5. The dark areas indicate combinations of true sensitivity and estimate of the true sensitivity that we consider successful with respect to the particular threshold value. For example, with a threshold value of 0.1, the estimate of the sensitivity has to be very close to the true value, even for low sensitivity values. In this study, we focus on a threshold value of 0.3. We will use this approach for the remainder of this study to evaluate the performance of the estimate. The weighted error combined with a threshold has two important advantages compared to the absolute error. It takes the variation within the experimental runs into account and requires estimates for higher sensitivity values to be more precise than estimates for lower sensitivity values. Applied to the previous example, we get the following success rates (ratio between the number of successful estimates and the total number of estimates): imputation estimate 0.938, iterative estimate 0.910 in the case of degree and imputation estimate 0.996, iterative estimate 1.000 in the case of PageRank. Using the weighted error, we notice that the estimates for the sensitivity of the degree centrality are very good and estimates for the sensitivity of PageRank are remarkable. Figure 4 illustrates the performance of the estimates for the experiments on ER graphs. In general, the performance of the estimate for the sensitivity in the context of ER graphs is remarkable. The success rate, i.e. the fraction of cases where the estimate is within the boundaries as defined in Section 3.3, is largely above 90%. Results for synthetic graphs The success rates for all 30% error mechanisms and centrality measures are illustrated in Figure 4. In most cases, there is no difference between iterative and imputation estimate except for cases that involve the closeness centrality. In those cases, the imputation estimate is better if edges or nodes are missing and the iterative method performs better if there are additional edges. This effect diminishes with increasing intensity. The success rates for betweenness centrality, eigenvector centrality, and PageRank at the same, very high, level, followed by closeness and degree centrality. The latter two show slightly lower (but still high) success rates. The different error mechanisms show similar results. When comparing the 10% and 30% error mechanism, we cannot observe that the success rates are lower for the latter. The converse seems to be the case. The success rates for cases that involve error mechanisms with 30% error level are higher than the corresponding values for 10% error mechanisms. At first, this observation seems counter-intuitive. However, we also observe that the sensitivity decreases with increasing measurement error. Since the sensitivity is lower, the interval for valid estimates becomes larger (Equation 5) and in the case of ER graphs, there is low variation and the success rates become better with increasing intensity. We also perform the experiment described in Section 3.2 with one difference in the first step: instead of an ER graph, we generate a BA graph [3], 100 nodes, parameter m = 11, undirected). Results for this experiment for cases with 30% error level are shown in Figure 5. In general, the imputation method performs worse than the iterative method. The performance of the imputation method is particularly bad in cases where the betweenness centrality has to be estimated. In contrast, the iterative method shows high success rates in most of the cases. In cases that involve the degree or closeness centrality, the performance is usually worse that in the remaining cases. There is little difference between the four error mechanisms. The results for the 10% error mechanism (Appendix A) are similar. However, in some cases, we observe lower success rates than in for the corresponding 30% error mechanism. Application to real-world networks Here, we apply our methods from Section 3.1 to real-world networks in order to investigate the suitability of these methods for practical application. We use four networks from different domains and thus different structural properties to get an impression how these methods perform on real data. Descriptive statistics for these networks are listed in Table 2. We use our proposed methods to estimate the sensitivity of five centrality [23] measures under the influence of four error mechanisms. For every error mechanism, we consider two cases, a moderate scenario of 10% error level and a more intense scenario with 30% error level. For every combination of network, centrality measure, and error mechanism, the experimental setup is as follows: 1. Due to the very nature of the hidden networks, we cannot access them. For every combination, we perform this experiment 500 times. To evaluate the results, we use the procedures described in Section 3.3. Results for real-world networks In this section, we study how our methods for estimating the sensitivity of centrality measures perform on real-world networks. Regarding the iterative estimates, we observe a fair amount of cases with high success rate. However, the results for empirical networks are more heterogeneous than the results for Erdős-Rényi and Barabási-Albert networks (Section 3.4). But since the real-world networks are more complex than graphs generated by these procedures, we expected that our estimation methods would not work as well for real-world networks compared to synthetic networks. The results for the estimate of the sensitivity of centrality measures in real-world networks are shown in Figure 6 and 7. First, we focus on the results for error mechanisms with 10% error level ( Figure 6). Comparing both estimation methods, it turns out that the iterative method is at least as good as the imputation method, except for a few cases where the former is slightly better. Hence, we first focus on the estimates of the iterative method. The iterative estimate for PageRank works in virtually all cases, regardless of the specific network or error mechanism. Among the four networks, the success rates for the Dolphins network are usually the lowest. It is reasonable to assume that this effect is due to the small size of the Dolphin network (62 nodes, 159 edges). If we focus our discussion on the three larger networks, we observe high success rates for the estimates of the sensitivity of closeness, betweenness, and eigenvector centrality if edges are missing uniformly or proportional. For these networks, the estimates for the sensitivity of the eigenvector centrality also work if the measurement errors lead to too many edges in the observed network. The results suggest, that the error mechanism missing nodes is most difficult for the estimate. There is no strong relationship between sensitivity and the success rate, higher sensitivity values are not easier to estimate. The average sensitivity values for all real-world networks can be found in Appendix A. Comparing Figure 6 and 7 shows that the success rates for the iterative method become lower with an increasing level of error. It is more difficult to estimate the sensitivity for higher levels of measurement errors. The estimate for the sensitivity of PageRank still works for error mechanisms additional edges and missing edges. If we focus on the three largest networks, the cases involving betweenness and closeness centrality show good success rates if edges are missing (uniformly and proportional). Cases involving the eigenvector centrality show good success rates if edges are missing uniformly and for the error mechanism additional edges. We observe cases with a subpar performance for 10% error level where the success rates continue to decrease. For example, the Jazz network with additional edges in combination with the closeness and betweenness centrality. There are few cases that show good performance for 10% error level but work barely for 30% error level (e.g., if edges are missing proportionally in the Jazz network and we try to estimate the sensitivity of eigenvector centrality or PageRank). The results for the imputation estimates of the sensitivity are rather different. There is no centrality measure or error mechanism where this method shows high success rates for all four networks. In some cases, the imputation estimate for error levels of 30% is better than the imputation estimate for the corresponding 10% case (e.g., PageRank and additional edges). Most of these situations occur when the error mechanism is additional edges. Discussion and conclusion Errors in network data are a ubiquitous problem in network analysis and previous studies have shown that these errors can have a severe impact on the reliability of centrality measures. Most studies that use centrality measures, however, rarely discuss the ramifications that measurement errors have on their analyses. Usually, these studies mention that the observed data might contain errors, but analyses are performed as if the data is error-free. Even though the reliability of centrality measures has been studied extensively, there is no technique that allows researchers to assess the reliability of centrality measures in the case of imperfect observed data. In the first part of this study, we introduced concepts to describe such a technique. We defined an easy-to-interpret metric, the sensitivity, to measure the reliability of centrality measures. Additionally, we presented the concept of error mechanisms, which model measurement errors as random graphs. We applied these concepts to ER graphs and the results are consistent with previous research [6]. In the main part of this study, we proposed two methods ("imputation method" and "iterative method") that allow the researcher to estimate the sensitivity of the error-free (hidden) network, given the observed network and some assumptions about the measurement error. Our experiments showed that both methods performed very well on ER graphs. In the case of BA graphs and real-world networks, the imputation method rarely worked and should therefore not be used. These findings extend those of Huisman (2009) [18], confirming that imputation methods are only useful in a few specific situations. Surprisingly, the method that is easier to calculate yielded better results. We could identify cases where the iterative method showed remarkable performance. It worked especially well for the PageRank for all error mechanisms with 10% error level. If the error level increased to 30%, the iterative method still showed good performance if edges were missing uniformly at random or if there were spurious edges. If 10% of the edges were missing uniformly at random or proportional and the network was not too small, the iterative method performed well for all centrality measures except for the degree centrality. The sensitivity values for the degree centrality were, however, relatively high. Our results provide compelling evidence that the iterative method is, in principle, a suitable technique for the estimation of the sensitivity of centrality measures. Hence, it is a promising first step that helps researchers to assess the impact of measurement errors on their observed network data. Although the iterative method works in well in many cases, there are limitations. There is a need to clarify the conditions under which the self-similarity assumption does not hold true and thus identify, based on the observed network, the cases where the iterative method should not be used. Another important question for future studies is to determine more suitable impu-tation mechanisms and thus improving the performance of the imputation method. A Additional results Figure 2 : 2Sensitivity of centrality measures in ER graphs. The violin plots in each panel show the distribution of the sensitivity of the corresponding centrality measure under the influence of different error mechanisms. Mean values are indicated by a black dot. Figure 3 : 3The dark areas in the panels above show successful combinations of true sensitivity s and estimate of the true sensitivityŝ for three threshold values (from left to right: 0.1, 0.3, and 0.5). An estimate of the true sensitivity is only considered successful if the pair (s,ŝ) is within the dark area. Figure 4 :Figure 5 : 45The success rates for the estimate (ER graphs, 30% error level) are shown in the figure above. The bar length indicates the percentage of successful cases among all trials (success rate). Grey (black) bars represent the success rates for the imputation (iterative) method. The success rates for the estimate (BA graphs, 30% error level) are shown in the figure above. The bar length indicates the percentage of successful cases among all trials (success rate). Grey (black) bars represent the success rates for the imputation (iterative) method. Hence, for the sake of our experiments, we treat the real-world network as the error-free hidden network H. (This is a common approach used in existing studies about the sensitivity of centrality measures.) 2. To simulate erroneous data collection, we choose a graph from ϕ(H) and denote it by O. This graph represents the observed network which is affected by measurement errors. For evaluation purposes, the true sensitivity ρ c (H, O) is calculated and denoted by s. 3. Based on the observed network O, two estimates for the true sensitivity are calculated. The imputation estimate (ρ imp c (O, H)) is denoted byŝ imp , the estimate calculated according to the iterative method (ρ iter c (O, H)) is denoted byŝ iter . Figure 6 : 6The success rates for the estimate (real-world networks, 10% error level) are shown in the figure above. The bar length indicates the percentage of successful cases among all trials (success rate). Grey (black) bars represent the success rates for the imputation (iterative) method. Figure 7 : 7The success rates for the estimate (real-world networks, 30% error level) are shown in the figure above. The bar length indicates the percentage of successful cases among all trials (success rate). Grey (black) bars represent the success rates for the imputation (iterative) method. Figure 8 :Figure 9 : 89The success rates for the estimate (ER graphs, 10% error level) are shown in the figure above. The bar length indicates the percentage of successful cases among all trials (success rate). Grey (black) bars represent the success rates for the imputation (iterative) method. The success rates for the estimate (BA graphs, 10% error level) are shown in the figure above. The bar length indicates the percentage of successful cases among all trials (success rate). Grey (black) bars represent the success rates for the imputation (iterative) method. 1 . 1We generate an ER graph with 100 nodes and edge probability 0.2 and denote it by H.This graph represents the (error-free) hidden network. 5 2. We choose a graph from ϕ(H) and denote it by O. This graph repre- sents the observed network which is affected by measurement errors. For evaluation purposes, the true sensitivity ρ c (H, O) is calculated and denoted by s. 3. Based on the observed network O, two estimates for the true sensitiv- ity are calculated. The imputation estimate (ρ imp c (O, H)) is denoted byŝ imp , the estimate calculated according to the iterative method (ρ iter c Table 1 : 1Results for the estimate of the sensitivity (ER graphs). For all centrality measures, average true sensitivity (column s) and 95th percentile of the absolute error for the estimate (ê imp : imputation method;ê iter : iterative method) are shown. (Values are multiplied by 100 for better readability.)betweenness closeness degree eigenvector PageRank sê impêiter sê impêiter sê impêiter sê impêiter sê impêiter Table 2 : 2Statistics of real-world networks. If the original network is not connected, we only consider the largest connected component. Network Nodes Edges Clustering Density Diameter Source Dolphins 62 159 0.3029 0.0841 8 [26] Jazz 198 2,742 0.6334 0.1406 6 [14] Protein 1,458 1,948 0.1403 0.0018 19 [19] Hamsterster 1,788 12,476 0.1655 0.0078 14 Table 3 : 3The average sensitivity values for the real-world networks are listed in the table below. (Values are multiplied by 100 for better readability.) 76.2 82.1 76.3 98.5 94.2 89.3 81.5 93.0 87.6 Hamsterster 84.5 81.3 94.4 90.5 97.7 94.1 96.4 93.3 91.7 87.7 Jazz 82.3 79.5 90.4 86.5 98.2 95.9 97.0 94.1 95.0 92.4 Protein 92.3 86.1 90.8 83.0 99.0 95.1 91.1 83.7 89.6 81.6 Remove edges (prop.) Dolphins 91.9 83.3 93.1 85.0 98.5 92.2 92.6 76.6 93.7 85.7 Hamsterster 97.1 93.4 95.7 89.5 99.5 97.6 96.3 90.7 97.6 94.Hamsterster 96.4 91.7 96.7 91.9 99.3 97.0 96.7 92.4 96.5 92.4Centrality measure bc cc dc ec pr Level of error 0.1 0.3 0.1 0.3 0.1 0.3 0.1 0.3 0.1 0.3 Error mechanism Network Add edges Dolphins 80.8 6 Jazz 96.0 91.7 95.5 90.7 98.5 95.3 97.2 94.1 97.0 93.3 Protein 95.4 89.7 85.0 67.2 99.6 96.9 80.2 63.0 93.5 85.6 Remove edges (unif.) Dolphins 91.4 82.9 93.2 85.3 98.4 93.3 93.7 85.5 93.4 86.1 Hamsterster 96.0 91.2 96.6 91.6 99.3 97.0 96.8 92.6 96.3 92.1 Jazz 95.1 89.6 96.2 91.5 98.4 95.8 97.5 94.9 96.5 93.1 Protein 95.5 90.7 88.8 75.8 99.3 95.5 87.9 76.3 92.2 82.9 Remove nodes Dolphins 90.9 82.8 92.2 84.5 98.4 93.8 90.1 80.7 93.5 87.1 Jazz 95.5 90.5 96.8 92.8 98.7 96.4 97.0 93.5 97.3 94.4 Protein 95.6 91.2 88.3 75.0 99.4 95.4 87.0 75.3 92.2 83.1 In this study, we consider unweighted and undirected graphs. However, most of our concepts can be extended to directed and weighted graphs. There are numerous possibilities how to connect the new nodes to the existing ones. For example, one could also use a preferential attachment like approach. This variety of options illustrates once again, that choosing an appropriate imputation mechanisms is challenging.3 It may occur that V (G ) = V (G). In these cases, we only consider entries in c(G) and c(G ) that correspond to nodes that are in both graphs (G and G ). This is a common approach for the comparison of graphs on different vertex sets[36]. Both methods are not limited to our definition of sensitivity. It would be interesting to see results for other metrics, for example, the estimation of the most central node (seeFrantz & Carley (2016) [11]). Our experiments have shown that the choice of p has little influence on the main results associated with this section. Hence we will only consider the case of p = 0.2. 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[ "A Reinforcement Learning Approach to Power Control and Rate Adaptation in Cellular Networks", "A Reinforcement Learning Approach to Power Control and Rate Adaptation in Cellular Networks" ]	[ "Euhanna Ghadimi \nHuawei Technologies Sweden AB, R&D Center\nKistaSweden\n", "Francesco Davide Calabrese \nHuawei Technologies Sweden AB, R&D Center\nKistaSweden\n", "Gunnar Peters \nHuawei Technologies Sweden AB, R&D Center\nKistaSweden\n", "Pablo Soldati \nHuawei Technologies Sweden AB, R&D Center\nKistaSweden\n" ]	[ "Huawei Technologies Sweden AB, R&D Center\nKistaSweden", "Huawei Technologies Sweden AB, R&D Center\nKistaSweden", "Huawei Technologies Sweden AB, R&D Center\nKistaSweden", "Huawei Technologies Sweden AB, R&D Center\nKistaSweden" ]	[]	Optimizing radio transmission power and user data rates in wireless systems via power control requires an accurate and instantaneous knowledge of the system model. While this problem has been extensively studied in the literature, an efficient solution approaching optimality with the limited information available in practical systems is still lacking. This paper presents a reinforcement learning framework for power control and rate adaptation in the downlink of a radio access network that closes this gap. We present a comprehensive design of the learning framework that includes the characterization of the system state, the design of a general reward function, and the method to learn the control policy. System level simulations show that our design can quickly learn a power control policy that brings significant energy savings and fairness across users in the system. Index Terms-Power and rate control, reinforcement learning.	10.1109/icc.2017.7997440	[ "https://arxiv.org/pdf/1611.06497v1.pdf" ]	12,898,640	1611.06497	6f285c80b5d8f6fe8896804756e7408c698104c4	A Reinforcement Learning Approach to Power Control and Rate Adaptation in Cellular Networks Euhanna Ghadimi Huawei Technologies Sweden AB, R&D Center KistaSweden Francesco Davide Calabrese Huawei Technologies Sweden AB, R&D Center KistaSweden Gunnar Peters Huawei Technologies Sweden AB, R&D Center KistaSweden Pablo Soldati Huawei Technologies Sweden AB, R&D Center KistaSweden A Reinforcement Learning Approach to Power Control and Rate Adaptation in Cellular Networks 1 Optimizing radio transmission power and user data rates in wireless systems via power control requires an accurate and instantaneous knowledge of the system model. While this problem has been extensively studied in the literature, an efficient solution approaching optimality with the limited information available in practical systems is still lacking. This paper presents a reinforcement learning framework for power control and rate adaptation in the downlink of a radio access network that closes this gap. We present a comprehensive design of the learning framework that includes the characterization of the system state, the design of a general reward function, and the method to learn the control policy. System level simulations show that our design can quickly learn a power control policy that brings significant energy savings and fairness across users in the system. Index Terms-Power and rate control, reinforcement learning. I. INTRODUCTION Radio interference can drastically degrade the performance of radio systems when not properly dealt with. Interference mitigation has thereby played a major role in radio access networks and shall continue to do so as we move toward the 5 th generation (5G) of mobile broadband systems. Compared to the 4G systems, however, interference in 5G networks is expected to have different behavior and characteristics due to an increasingly heterogeneous, multi-RAT and denser environment to the extent that conventional inter-cell interference coordination (ICIC) will become inadequate [1]. Interference mitigation can broadly be posed as a power control optimization problem which, under certain conditions, admits an optimal solution. In particular, distributed power control methods based on linear iterations to meet signalto-interference-and-noise ratio (SINR) targets were proposed in [2], [3]. An axiomatic framework for studying general power control iterations was proposed in [4] based on the so called standard interference functions, and extended in [5], [6]. Despite the elegant and insightful results available in the power control literature, their direct application to practical systems has been impaired by the dependency on some simplifying assumptions, such as the knowledge of the instantaneous channel gains to/from all user devices, full buffer traffic, etc. Recent advances in the field of machine learning vastly expanded the class of problems that can be tackled with such methods and made new techniques available that can potentially change the way radio resource management (RRM) problems are solved in wireless systems. Within this field, Reinforcement Learning (RL) is arguably the most appropriate branch in which one or more agents solve a complex control problem by interacting with an environment, issuing control actions, and utilizing the feedbacks obtained from the environment to find optimal control policies (see e.g. [7]). Only recently, learning algorithms made their appearance in the context of wireless networking. Cognitive radios presented a natural ground for the application of learning methods, for instance for efficient spectrum sensing and shaping see, e.g. [8], [9], for mitigating interference generated by multiple cognitive radios at the receivers of primary users, see e.g. [10], [11], and so on. A survey of learning methods suitable for cognitive radio networks can be found in [12] and references therein. Distributed RL was proposed in [10] to enable radio cells (i.e., the agents) to learn an efficient policy for controlling the aggregated interference generated by multiple neighboring cells on primary licensed users. The proposed algorithm exploits a Q-learning method based on a representation with tables and neural networks. Table-based Q-learning was also considered in [11] for interference mitigation in LTE heterogeneous radio cellular networks (HetNets) and in [13] for frequency-and time-domain inter-cell interference coordination in HetNets. In this paper we apply RL for distributed downlink intercell power control and rate adaptation in the downlink of a radio access network. We begin by modeling the problem as a network utility maximization and solve it to optimality via Lagrange duality theory. While technically sound, several practical limitations refrain us from considering this optimization framework for applications to real systems. To overcome the impracticalities of such approach, we propose an advanced RL framework which is particularly data-efficient. By carefully designing the features required to characterize the system state for multi-cell downlink power control and rate adaptation as well as the reward function to drive the behavior of the agent, the framework is able to quickly produce a policy for power control that brings energy saving as well as fairness across the users in the system. The validity of this approach is verified in a fully LTE-A compliant event-driven system level simulator. The results also demonstrate the flexibility of this approach which enables us to promote entirely different behaviors in terms of fairness and system performance by changing the parameters of the reward function. The rest of the paper is organized as follows: Sec. II introduces a system model for the rate and power control optimization problem solved in Sec. III. Sec. IV introduces a RL architecture for downlink power control. Finally, Sec. V and VI present simulation results and final remarks, respectively. II. SYSTEM MODEL We consider a radio cellular system with C cells, labelled c = 1, . . . , C, each serving a set of users N c . Users in the system are labelled by n = 1, . . . , N where N = c N c and N c =\| N c \|. We assume a frequency reuse-1 scheme where all cells operate within the same frequency bandwidth W with maximum downlink transmission power P max c , respectively. We assume the system bandwidth divided into K equally sized time-frequency resource blocks. Within each cell, users are orthogonally scheduled in frequency bandwidth W so that only inter-cell interference is considered. Let P c denote the downlink transmission power budget used by cell c in a given transmission time interval (TTI), P max c be the corresponding maximum power budget (i.e. P c ≤ P max c ), and p = [P 1 , . . . , P C ] T be the network-wide vector of transmission power budgets for all cells in a TTI. Assuming each cell uniformly distributes it transmission power budget P c over the time-frequency resource blocks scheduled to users in the cell, each user within a cell is served with the same power level, i.e. P c /K for all users n ∈ N c . Furthermore, we define G n,c as the channel gain between user n and cell c which takes into account for large-scale fading effects (i.e., pathloss and shadowing). Therefore, the average SINR experienced by user n can be modelled as γ n (p) = G n,c(n) P c(n) σ 2 + c =c(n) G n,c P c n = 1, . . . , N.(1) where c(n) denotes the serving cell of user n. We would like to stress that this model is applied exclusively to the optimization framework developed in Section III but it is not explicitly needed for the learning framework described in Section IV. III. OPTIMAL RATE AND POWER CONTROL We consider the problem of joint user rate and power control optimization in a multi-celluar radio network. We pose the problem as a network-wide utility maximization maximize n u n (r n ) subject to r n ≤ W n log 2 (1 + γ n (p)) n = 1, . . . , N, p p max(2) wherein W n = W/N c(n) denotes the average amount of bandwidth scheduled for user n assuming an equal share of frequency resources, and r n models the theoretically achievable user data rate according to the Shannon bound. Associated with each user n is a utility function u n (·), which describes the utility of the user to communicate at rate r n (cf. [14]). We assume that u n is increasing and strictly concave, with u n → −∞ as r n → 0 + . Therefore, problem (2) aims at optimizing the downlink transmission power of each cell so as to maximize a network-wide utility of the users in the system. A. Convexification and optimal solution Problem (2) is not convex in the variables p due to inter-cell interference in the rate expression. However, with a suitable log-transformation of both constraints and variables one can obtain an equivalent convex formulation, cf. [15]. In particular, we definer n = log(r n ) andP n = log(P n ) and rewrite (2) as maximize n u n (er n ) subject tor n ≤ log(W n ) + log(log 2 (1 + γ n (ep))) ∀n, p log(p max ), wherein constraints are obtained with a log-transformation and variable change applied to both sides of the inequalities in (2). Proposition 3.1: The transformed problem (3) is jointly convex inr andp. The optimal solution to (2) coincides with the one of (3). Proof: The transformed capacity constraints are jointly convex inp and linear inr, while the remaining power budget constraints are linear inp. Therefore, the rest of the results follows from [15,Theorem 2] and [15,Corollary 1]. An optimal and distributed solution to the rate and power control problem (2) can be found by solving (3) with standard Lagrange duality theory 1 . Examples of utility functions satisfying these conditions include the α-fair utility functions, such as sum-log utility. B. Practical considerations While theoretically sound, solving problem (2) to optimality with the signaling and time limitations of conventional radio access networks would be infeasible. Firstly, solving problem (3) to optimality requires cross-cell knowledge of channel gains G n,c (or similar message passing see e.g. [15]). While this would create large signaling overhead, the channel aging would render the information outdated before it serves its purpose. On the other hand, the user devices can in practice estimate only the channel gain G n,c for a small subset of interfering cells. Secondly, any change in the users and traffic distributions (e.g., user positions, number of users, number of resources used, etc.) would also need to be exchanged between the radio access nodes as it affects the achievable user data rate. Such changes are difficult to track, measure and expensive to be communicated in the system. These impairments have refrained the application of inter-cell interfere coordination via optimal power control in state of the art wireless systems. IV. RL FOR CELLULAR NETWORKS A. Reinforcement Learning Reinforcement Learning RL is an area of machine learning concerned with learning how a software agent should learn to behave in a given environment in order to maximize some form of cumulative reward. The ingredients used to model the interaction of the agent with the environment are the state, the actions and the reward. The state s is a tuple of values, known as features, that describes the agent in relation to the environment in a way that is relevant for the problem at hand. The action a (chosen in the set of available actions) represents the change (e.g., parameter changes) that the agent applies to the environment in order to achieve his goal of maximizing the given notion of reward. The reward r is a multi-objective scalar function which expresses what we want the agent to achieve. The agent therefore cycles through a series of transitions which consist of going from one state to the next state by applying actions to the environment and receiving rewards as a consequence of his actions. The evolution of the agent in the environment can therefore be described in terms of quadruples of the form (s t , a t , r t+1 , s t+1 ) where t represents time. The goal of RL is to derive a policy π, that is, a function that, given a state, provides the action to take in order to maximize the cumulative reward: π(s t ) = a t .(4) Unlike Supervised Learning, where input data and output labels are given and a function has to be derived from it, in RL the inputs are the state-action pairs, while the outputs are the rewards (or, more precisely, a function of the reward) that the agents receives over time. The rewards are often few and far in between and the relevant actions that lead to certain outcomes might have been taken far back in time and/or space. A central problem of RL is thus that of properly assigning the reward to the actions that lead to such reward (a notion known as credit-assignment). Another main issue of learning behavioral policies in domains that are unknown at first (especially in critical systems like a cellular network) is that of efficiently bringing the agent from a tabula-rasa state to a condition where the agent is acting as close to optimality as possible. This notion is also known as regret minimization and is closely related to the topic of trading off exploration of the environment (to sample previously unseen parts of the state-action space) with exploitation of the knowledge accumulated so far. A tabula-rasa agent that enters the world would therefore need to explore the environment by applying actions and to gather data which are as informative as possible so that a policy, encoding the knowledge accumulated so far, is derived via a credit assignment algorithm. Over time the agent will gradually move from a situation where it is mostly exploring the environment to a situation where it is mostly exploiting the accumulated knowledge. This gradual transition from an exploration strategy to an exploitation strategy is referred to as epsilon-greedy. Every time an agent has to take an action, it will take a random action with probability and a policy action with probability 1 − . To value of is gradually reduced from an initial value max to a final value min over a predefined number of actions. For a complete treatment of the subject of RL, we refer the interested reader to [7]. B. Q-learning Different classes of RL algorithms existing in literature are often classified as critic-only, actor-only and actor-critic. One of the most popular algorithm in the RL literature is that of Q-learning [16], which is a critic-only algorithm. Qlearning aims at learning an action-value function (the so called Q-function) which gives the expected utility of taking an action a in a given state s and following the policy π afterwards. Once such a function is learned, the policy can be derived from it by evaluating the Q-function for every action and choosing the action which gives the largest Q-value. The Q-function for a given policy π is defined as: Q π (s, a) = E ∞ t=0 γ t r(s t , π(s t ))\|s 0 = s, a 0 = a ,(5) The term γ ∈ [0, 1) is a discounting factor causing the value of rewards to decay exponentially over time indicating the preference for immediate rewards while making the optimization horizon for our agent finite (that is, the sum of rewards is finite). In other words the Q-function learns to predict the expected cumulative discounted reward from taking action a in state s and following the policy π afterwards. As the agent explores the environment by applying actions and receiving rewards, it collects and stores transitions (s t , a t , r t+1 , s t+1 ) in a growing batch. From the batch of transitions a training sequence of input-output pairs is formed and used to learn the Q-function. The input is represented by state-action pairs (s, a), while the output is represented by the Q-function expressed as Q(s t , a t ) = r t+1 + γ max a Q(s t+1 , a )(6) Despite the fact that the target Q-function we are trying to learn contains the Q-function itself, the presence of the reward term is sufficient for the target to be improved over time and move closer to the "real" Q-function. This approach was originally conceived in conjunction with the usage of tables to store the transitional data, but it becomes quickly unusable as the state-action space grows and the required number of data samples becomes prohibitively large. A more powerful approach consists then in combining Q-learning with function approximators which, besides accelerating the learning, offer the possibility to generalize it to previously unseen states. In our work we chose to represent the Q-function via an Artificial Neural Network (ANN). The interested reader can find a detailed description of ANNs, their characteristics and how they are trained in [17]. In the context of this paper suffices to say that we have used an ANN with 3 hidden layers trained using the R-PROP algorithm [18] to update the weights of the ANN because it is consistent with our approach of using the full growing batch of data to update the Q-function. In short, to estimate the gradient we minimize the meansquare error L = 1 2 t \|r t+1 + max a Q(s t+1 , a ) − Q(s t , a t )\| 2 (7) over the full batch of samples and then update the weights using the gradient descent with time-varying adaptive step sizes. C. Specifics of RL in the Context of Cellular Networks In the context of this paper, RL can be mapped to the entities in the network as follows: • The environment is represented by the cellular network; • The agents are the logical network nodes implementing the RL algorithms and therefore capable of extracting a control policy (in our case power control) from the collected set of transitions; • The state, in its entirety, could be represented by the type and number of terminals, their traffic, their positions, their capabilities, the type and number of cells, the different measurements and KPIs, etc. but in general a more synthetic representation is used, that is, a more limited number of elements is added to describe the state of the agent in relation to the network; • The actions are typically represented in form of parameter adjustments (e.g., power-up, power-down, power-hold); • The reward could be represented as a function (non necessarily linear) of different Key Performance Indicators (KPI)s (capacity, coverage, delay, etc.). The cellular network is usually affected by a few complications which are not present in the basic formulation of RL. 1) The state as seen by the agent is only partially observable, that is, the agent can only observe a limited part of the state; 2) Once the action a t has been applied to the environment, the transition to the next state s t+1 is stochastic and not deterministic, that is Pr(s t+1 = s \|s t , a t ) for all s ; 3) In the environment there are several agents whose actions interfere with each other. Each of these problems is more or less severe depending on how much the actions of one agent influences the environment as perceived by the other agents. If the impact of the actions of each agent in the environment is such that the reward signal is strongly corrupted then the learning will not be possible. If this is not the case, the convergence rate will still be affected (because other agents will still introduce noise into the system) but the agents will nonetheless be able to learn a policy. As we show in the results of the paper, the latter is the case for downlink power control investigated here. Each agent has access only to local information to construct the state but the reward is exchanged across nodes and a network-wide reward function is constructed by aggregation of individual reward values from the neighboring agents so as to encourage cooperation. Moreover, in order to minimize the impact of other agents into the system, they coordinate with each other by taking turns rather than acting all at the same time. In the following, a more detailed description of each of the components of the RL framework (that is, the state, action and reward) is provided. D. State In downlink power control, the state is represented by a set of features constructed based on local measurements in each cell. These features should provide concise information regarding the performance of local cells as well as good indicators about the situation of interfering cells. In particular, we employed the following features to describe the state: in the cell. It is used as part of the state so that its correlation to the global reward can be used to provide a useful information for the agent. The agent can then relate its own goal (local reward) to the welfare of society (global reward). E. Actions The true action space of downlink power control, the amount of cell power, is a continuous variable. To avoid the action space explosion, the action space is quantized. To strike a proper trade-off between the speed of learning and the quality of the derived control strategies, the action space should be reduced to a few discrete values. In downlink power control, actions are such that agents can gradually or more rapidly change their cells power level. A typical choice in our simulations was given by the set {0, ±1, ±3}dBs. F. Reward Since the objective of the RL problem is to maximize the cumulative discounted reward, the design of the reward function is the only tool we have to enforce a particular agent behavior compared to another. Given that the wireless network is a cooperative environment (cells are not competing with each other) it makes sense to define the reward in a global networkwide sense as function of some fundamental KPIs. For this work, we have considered the following general definition of α-fair resource allocation utility function [19]. (8) where α = [0, ∞) is a scalar coefficient, and h i : X → R is a transfer function. Here, x i represents a radio measurement or a performance indicator associated with the radio cell, X is the set of all radio measurements or performance indicators associated with the radio cell and used for the definition of the performance measurement, w i is a weight associated with x i , and x = [x 1 , . . . , x \|X \| ] is a vector comprising all x i ∈ X . r(x) = 1 1 − α xi∈X w i (h i (x i ) 1−α − 1), α ∈ [0, ∞), When the function h i (x i ) represent the average data throughput of user i in the cell, for instance, the reward r t (x) in (8) can be approximated for different values of α and weights w i with the following expressions: 1) The average throughput associated to the user devices in the cell, i.e., r( x) = 1/\|X \| xi∈X h i (x i ), if α = 0, w i = 1/\|X \| ∀ i. 2) The average throughput associated to the cell, i.e., r(x) = xi∈X h i (x i ), if α = 0, w i = 1 ∀ i. 3) The average log-throughput of the users in the cell, i.e., r(x) = xi∈X log(h i (x i ))/\|X \|, if α = 1, w i = 1/\|X \| ∀ i. 4) The average sum of log-throughput of the cell, i.e., Each reward expression enables the agent node to optimize a different performance metric that can either be associated to individual user devices, to radio cells, or the whole cellular radio network. r(x) = xi∈X log(h i (x i )), if α = 1, w i = 1 ∀ i. V. PERFORMANCE EVALUATION We evaluate our reinforcement learning approach for power control and rate adaptation using a LTE-A compliant eventdriven system-level simulator. Table I summarizes the main simulation parameters. A. A 2-agent validating example We first evaluate the convergence our RL framework on a 2-cell/2-agents example with 10 UEs with full-buffer traffic deployed randomly in the network and compare its performance against the optimal one. Agents take turns every 100 milliseconds to control the donwlink power budget of the corresponding cell. In this example, we model the cell state using a limited set of features comprising the cell's power, the average RSRP of the UEs in the cell, and the average interference measured at the UEs in the cell. We also evaluate the RL approach using two reward functions: the harmonicmean throughput of the network and the sum-log throughput of the network. The RSRP, interference, and reward functions are averaged with measurements collected over the period between two consecutive actions of an agent (i.e., 200ms). Starting from an exploration probability max = 0.9, each agent gradually annihilates this value until the minimum min = 0.1. 2 shows the result for a 60s simulation. After an initial phase of intensive exploration, both agents quickly converge to a power control policy that maximize the chosen aggregate network reward. We have then validated these results against the power levels optimizing both reward functions by exhaustively running a set of independent simulations with fixed power levels at both cells ranging in [10,46]dBm with stepsize 1dB 2 . Fig 3a shows that the RL algorithm converges to a power control policy optimizing (in average) the corresponding reward function 3 . Fig 3b compares the cumulative distribution function (CDF) of the user rates (averaged over the last 5s of simulation) for the case of fixed transmission power with 46dBm at both cells and for the power values achieved by the RL approach with the two reward functions. The results indicate that our design can converge to the optimal power control policy that brings energy gain and fairness among users in the system, where the design of the reward function enables to tradeoff different degrees of fairness. B. Extended simulations setup Since the gain from a dynamic downlink power allocation is assumed to be an effect of load balancing between the cells, we consider a 3-cell scenario with uneven load. In particular, three cells with relative load of 10%, 20% and 70%, and a total number of 900 UEs generated over 30 simulation drops. We use the first half of the drops for training and the second Figure 3: The left hand side shows the normalized reward functions for the two-cell example with P 1 = 46 dBm (optimal value) and P 2 ranging in [10,46] dBm with step size 1dBm. The right hand side shows the CDF of the user rates for the power values (P 1 , P 2 ) optimizing the two reward functions. half for evaluation 4 . We adopt the harmonic mean network throughput as reward as it offers fairness among the users and it strongly relates to the packet delay measure. A summary of the results is given in Table II, which shows the gain of the RL approach over an equal power allocation of 46 dBm. C. Full buffer traffic For full buffer traffic, we have simulated the training and evaluation drops for 60s and 20s, respectively, corresponding to 4000 data samples per agent. Figure 4, Figure 5 and Table I summarize the results: compared to a baseline with fixed power allocation, the RL approach driven by an harmonicmean reward brings fairness and throughput gains to low-rate users with a significant transmission power reduction. Figure 4 show that the RL algorithm enforces inter-cell user fairness by reducing data rate to users in low loaded cells through a reduction of the associated downlink transmission 4 Training is also performed in the drops used for policy evaluation power. In particular, the throughput gain ranges from 63% for 5%-tile cell edge UEs (and up to the 20%-tile UEs) to a 15% gain for the median UEs, respectively. Fairness, with full buffer traffic, comes at the price of a slight decrease in network throughput -below 10% over all evaluation drops. Figure 5 show that RL approach further achieves an average downlink power reduction of 8.54 dB corresponding to over 86% power savings compared to the baseline, while effectively maximizing the reward function in each evaluation drop. D. Bursty traffic For bursty traffic, we consider UEs downloading files of size 0.1 MB with mean reading time 100ms, corresponding to an average traffic 1 MB/s per UE. We have simulated the training and evaluation drops for 180s and 10s, respectively, corresponding to 9000 training data samples per agent. Figure 6, Figure 7 and Table I summarize the results: with bursty traffic, the RL approach successfully enforces inter-cell user fairness through a significant reduction of the cells transmit power without degrading the overall network throughput compared to a baseline with fixed transmit power. Figure 6 shows that the RL approach achieves 94% throughput gain for the 5%-tile UEs and 22% gain for median UEs. Overall, RL approach converges to a power control strategy that enables nearly 70% of UEs to enjoy a higher data rate. Figure 7 shows an average transmit power reduction of 10.57dB among the evaluation drops, i.e. over 91% power saving, as well as an average network throughout gain of 14% compared to the baseline. VI. CONCLUSIONS In this paper, we considered the problem of downlink power control for cellular systems. In particular, we provided a RL based method that adapts the power budget of cells to the dynamic conditions of the network and user traffics. With proper features and reward functions selection, we presented simulation results where tuning cell powers using our algorithm offers significant improvements over baseline for both full and bursty traffic scenarios. An interesting future direction is to further investigate the multi-agent aspects of learning framework by increasing the number of cells and making the power control action user specific rather than cell specific. Figure 1 : 1An example of a cellular network with 3 agents controlling the downlink transmission power of their own cell. 1 ) 1Cell power: One basic indicator of the state of the network is indeed the amount of cell power. Since each agent is placed in the base station, it has direct access into the values of transmit power of each controlling cell. 2) Average Reference Signal Received Power (RSRP) in the cell: this feature is calculated by averaging the RSRP measurements of users in the cell. The feature is essential since it implicitly embeds an overview of users location in the cell. Cell-edge users generally report a weak RSRP value. Thus, a low average RSRP in the cell is an indicator of having majority of users at cell edge. 3) Average interference in the cell: each user reports the received signal strength from dominant interfering cells. The average interference received from each of interfering cells is then computed by taking the average of values reported by users in the cell. Together with the average RSRP, an agent can correlate the performance of its cell to the average channel quality of its served users which in turn, is affected by the power values of other cells (and thus to the actions of other agents). 4) Cell reward: finally, the value of the reward is calculated 5 ) 5The harmonic-mean throughput of users in the cell, i.e.,r(x) = \|X \|/ x i ∈X hi(xi) −1 , if α = 2, wi = 1/\|X \| ∀ i.6) The harmonic-mean throughput of the cell, i.e., r(x) = 1/ x i ∈X hi(xi) −1 , if α = 2, wi = 1 ∀ i. Figure 2 : 2DL power level adjustments of two agents and associated normalized reward functions. Fig. Fig. 2 shows the result for a 60s simulation. After an initial phase of intensive exploration, both agents quickly converge to a power control policy that maximize the chosen aggregate network reward. We have then validated these results against the power levels optimizing both reward functions by exhaustively running a set of independent simulations with fixed power levels at both cells ranging in [10, 46]dBm with stepsize 1dB 2 . Fig 3a shows that the RL algorithm converges to a power control policy optimizing (in average) the corresponding reward function 3 . Fig 3b compares the cumulative distribution function (CDF) of the user rates (averaged over the last 5s of simulation) for the case of fixed transmission power with 46dBm at both cells and for the power values achieved by the RL approach with the two reward functions. The results indicate that our design can converge to the optimal power control policy that brings energy gain and fairness among users in the system, where the design of the reward function enables to tradeoff different degrees of fairness. Figure 4 : 4CDF of the average user throughput with full buffer traffic. Figure 5 : 5Average cell transmit power and harmonic-mean network throughput with full buffer traffic. Figure 6 :Figure 7 : 67CDF of users DL throughput for evaluation scenarios including 450 UEs with bursty traffic. Average cell transmit power and average network throughput for 15 evaluation scenarios with bursty user-traffics. Table I : ISimulations parametersParameter Value TTI 1ms Bandwidth 10 MHz Default transmit power at BS 46 dBm Traffic type Full buffer or bursty #Cells (agents) 2 or 3 Reward functions Harmonic-mean or sum-log throughput Action period 100 ms Agents scheduling Round-robin Policy update period 50 data samples ( min , max) (0.1, 0.9) γ 0.7 Table II : IISummary of simulation results for RL approach.Scenario 5% (UE rate) gain Median gain Power reduction Full buffer 63% 15% 86% Burst 94% 22% 91% The details are omitted due to space limitations but we refer to[15] for a similar approach. 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[ "Generative diffeomorphic atlas construction from brain and spinal cord MRI data", "Generative diffeomorphic atlas construction from brain and spinal cord MRI data" ]	[ "Claudia Blaiotta \nWellcome Trust Centre for Neuroimaging\nUniversity College London\nLondonUK\n", "Patrick Freund \nWellcome Trust Centre for Neuroimaging\nUniversity College London\nLondonUK\n\nSpinal Cord Injury Center Balgrist\nUniversity Hospital Zurich\nUniversity of Zurich\nZurichSwitzerland\n", "M Jorge Cardoso \nTranslational Imaging Group\nCMIC\nUniversity College London\nLondonUK\n", "John Ashburner \nWellcome Trust Centre for Neuroimaging\nUniversity College London\nLondonUK\n" ]	[ "Wellcome Trust Centre for Neuroimaging\nUniversity College London\nLondonUK", "Wellcome Trust Centre for Neuroimaging\nUniversity College London\nLondonUK", "Spinal Cord Injury Center Balgrist\nUniversity Hospital Zurich\nUniversity of Zurich\nZurichSwitzerland", "Translational Imaging Group\nCMIC\nUniversity College London\nLondonUK", "Wellcome Trust Centre for Neuroimaging\nUniversity College London\nLondonUK" ]	[]	In this paper we will focus on the potential and on the challenges associated with the development of an integrated brain and spinal cord modelling framework for processing MR neuroimaging data. The aim of the work is to explore how a hierarchical generative model of imaging data, which captures simultaneously the distribution of signal intensities and the variability of anatomical shapes across a large population of subjects, can serve to quantitatively investigate, in vivo, the morphology of the central nervous system (CNS). In fact, the generality of the proposed Bayesian approach, which extends the hierarchical structure of the segmentation method implemented in the SPM software, allows processing simultaneously information relative to different compartments of the CNS, namely the brain and the spinal cord, without having to resort to organ specific solutions (e.g. tools optimised only for the brain, or only for the spinal cord), which are inevitably harder to integrate and generalise.	null	[ "https://arxiv.org/pdf/1707.01342v1.pdf" ]	32,944,333	1707.01342	2f15b5158f5844335f4270f6abbc5ad90c07b94a	Generative diffeomorphic atlas construction from brain and spinal cord MRI data 5 Jul 2017 Claudia Blaiotta Wellcome Trust Centre for Neuroimaging University College London LondonUK Patrick Freund Wellcome Trust Centre for Neuroimaging University College London LondonUK Spinal Cord Injury Center Balgrist University Hospital Zurich University of Zurich ZurichSwitzerland M Jorge Cardoso Translational Imaging Group CMIC University College London LondonUK John Ashburner Wellcome Trust Centre for Neuroimaging University College London LondonUK Generative diffeomorphic atlas construction from brain and spinal cord MRI data 5 Jul 2017BrainSpinal cordMRIAtlasGenerative modelsSegmentation In this paper we will focus on the potential and on the challenges associated with the development of an integrated brain and spinal cord modelling framework for processing MR neuroimaging data. The aim of the work is to explore how a hierarchical generative model of imaging data, which captures simultaneously the distribution of signal intensities and the variability of anatomical shapes across a large population of subjects, can serve to quantitatively investigate, in vivo, the morphology of the central nervous system (CNS). In fact, the generality of the proposed Bayesian approach, which extends the hierarchical structure of the segmentation method implemented in the SPM software, allows processing simultaneously information relative to different compartments of the CNS, namely the brain and the spinal cord, without having to resort to organ specific solutions (e.g. tools optimised only for the brain, or only for the spinal cord), which are inevitably harder to integrate and generalise. Introduction The spinal cord is a long and thin cylindrical structure of the central nervous system, which constitutes the main pathway for transmitting information between the brain and the rest of the body. Not only is the spinal cord a major site of traumatic injury (SCI), but it can also be affected by a number of neurodegenerative diseases, such as multiple sclerosis, amyotrophic lateral sclerosis, transverse myelitis and neuromyelitis optica (Rocca et al., 2015). Indeed, the spinal cord is a clinically eloquent structure, since trauma, ischemia and inflammation can affect the cord at any level, thus resulting in impairment of motor, sensory and autonomic functions (Huber et al., 2015;Freund et al., 2013a). Understanding these degenerative processes represents a crucial step towards the development of effective therapeutic interventions, as well as towards the identification of sensitive and selective diagnostic criteria. In particular, quantification of spinal cord tissue loss (i.e. atrophy) has been regarded over the past two decades as a promising biomarker, which could potentially help in monitoring disease progression, pre-dicting clinical outcome and understanding the mechanisms underlying neurological disability (e.g. demyelination, inflammation, axonal or neuronal loss), in a number of conditions that affect the central nervous system both at the brain and spinal cord level, such as multiple sclerosis (MS) and traumatic spinal cord injury (SCI) Freund et al., 2013b,a;Grossman et al., 2000;Bakshi et al., 2005). Since the spinal canal is surrounded and protected by a thick vertebral bone layer, neuroimaging techniques, particularly MRI, represent the most effective tools to investigate non-invasively and in vivo the structure and function of the spinal cord, both in physiological and pathological conditions. Unfortunately spinal cord MRI is not immune from technical challenges. Some of them are intrinsic to MR imaging, such as the presence of intensity inhomogeneities, while others arise from the peculiar anatomy of the cord itself, for instance from its small cross-sectional area (Grossman et al., 2000;. Nevertheless, spinal cord imaging using MR techniques has improved significantly over the past few years, especially with the introduction of phased-array surface coils and fast spin-echo sequences . Further advances in the field of spinal cord MRI are encouraged by the fact that significant correlations between spinal cord atrophy measures, obtained from imaging data, and indicators of neurological impairment, such as motor or sensory function scores, have been shown and reproduced, within multiple spinal cord imaging studies (Losseff and Miller, 1998;Kidd et al., 1993;Filippi et al., 1996;Losseff et al., 1996;Freund et al., 2013b;Grabher et al., 2015). Within these type of studies, delineating the cord represents the first step for assessing atrophy or detecting any other morphometric change, or difference. This indicates that there is an urgent need not only for automated algorithmic solutions dedicated to spinal cord tissue classification and image registration (Chen et al., 2013;Van Uitert et al., 2005;Fonov et al., 2014;Levy et al., 2015;, but also for large, systematic and reproducible validation studies to objectively assess the performance of such tools (Prados et al., 2017). Not surprisingly, the first methods that appeared in the literature to perform spinal cord image segmentation and the subsequent volumetric analyses were based on semi-automated algorithms. Among these, one of the earliest is described in the work of Coulon et al. (2002), where they introduce an algorithm for fitting a cylindrical cubic B-spline surface to MR spinal cord images. Later on, a few other semi-automated solutions have been presented by Van Uitert et al. (2005) and Horsfield et al. (2010). All of these methods require that the user approximately marks the cord centre, so as to provide a reliable initialisation of the algorithms. Only very recently have fully automated spinal cord segmentation methods started to be proposed. Chen et al. (2013) introduced a fuzzy c-means algorithm with topological constraints to segment the cervical and thoracic spinal cord from MR images. Their method relies on a statistical atlas of the cord and the surrounding CSF, which is constructed from only five manual segmentations. Instead, De Leener et al. (2014) proposed a fully automated method for delineating the contour of the spinal cord, in T1-and T2-weighted MR images, by warping of a deformable cylindrical model. The first significant effort to define and introduce a standard anatomical space for spinal cord neuroimaging studies relates to the work of Fonov et al. (2014), who developed a standard stereotactic space for spinal cord imaging data, between the vertebral levels of C1 and T6 (MNI-Poly-AMU template). Their template is generated using the image registration algorithm presented in Avants et al. (2008) and includes a T2-weighted average image, together with probabilistic gray and white matter maps. Such tissue probability maps were developed by , via automated registration of manually labelled MRI scans of 15 subjects. The work of Fonov et al. (2014); Levy et al. (2015); constitutes an important step towards the development of robust and reliable tools for analyzing structural spinal cord data. Indeed, having a common anatomical framework can potentially allow the comparison of results obtained by different research groups on different data sets, thus speeding up the progress of spinal cord imaging research. The work presented here aims to provide researchers with a general and comprehensive modelling framework to interpret large data sets of MRI scans from a Bayesian generative perspective. This is achieved by building on the modelling elements introduced in Ashburner andFriston (2005, 2011); Blaiotta et al. (2016), which are further expanded here and integrated in one single algorithmic framework. The aim is to demonstrate the validity of such a generative approach, especially for the purpose of performing simultaneous brain and spinal cord morphometric analyses using MRI data sets. In doing so, a strategy is outlined on how to overcome some of the limitations of most currently available image processing tools for neuroimaging, whose performance has been optimised on the brain at the expense of the spinal cord (indeed the spinal cord is frequently neglected tout court by such tools). Methods Let us consider a population of M subjects belonging to a homogeneous group, from an anatomical point of view, and let us assume that D image volumes of different contrast are available for each subject. From a generative perspective, the image intensities X = {X i } i=1,...,M , which constitute the observed data, can be thought of as being generated by sampling from D-dimensional Gaussian mixture probability distributions, after non-linear warping of a probabilistic anatomical atlas (Evans et al., 1994). Such an atlas carries a priori anatomical knowledge, in the form of average shaped tissue probability maps. From a mathematical modelling point of view, the atlas encodes local (i.e. spatially varying) mixing proportions Θ π = {π j } j=1,...,N π of the mixture model, with j being an index set over the N π template voxels, as detailed in the following subsection. Tissue priors Each image voxel j ∈ {1, . . . , N i }, for each subject i ∈ {1, . . . , M} is considered as being drawn from K possible tissue classes. The following prior latent variable model defines the probability of finding tissue type k, at a specific location j (i.e. centre of voxel j), in image i, prior to observing the corresponding image intensity signal p(z i jk = 1\|Θ π , Θ w , Θ u ) = w ik π k (ξ i (y j )) K c=1 w ic π c (ξ i (y j )) ,(1) or equivalently p(z i j \|Θ π , Θ w , Θ u ) = K k=1       w ik π k (ξ i (y j )) K c=1 w ic π c (ξ i (y j ))       z i jk .(2) Class memberships, for each subject and each voxel, are encoded in the latent variable z i j , which is a Kdimensional binary vector. {π k } k=1,...,K are scalar functions of space π k : Ω π → R, common across the entire population, which satisfy the constraint K k=1 π k (y) = 1 , ∀y ∈ Ω π ⊂ R 3 ,(3) with y being a continuous coordinate vector field. Global weights Θ w = {w i } i=1,...,M are introduced to further compensate for individual differences in tissue composition. In equation (1), ξ i denotes a generic spatial transformation, parametrised by Θ u , which allows projecting prior anatomical information onto individual data, with ξ i : Ω i → Ω π being a continuous mapping from the domain Ω i ⊂ R 3 of image i, into the space of the tissue priors Ω π ⊂ R 3 . Since digital image data is a discrete signal, defined on a tridimensional voxel grid, each mapping ξ i needs to be discretised as well, via sampling at the centre of every voxel j ∈ {1, . . . , N i }, to give the discrete mapping {ξ i (y j )} j=1,...,N that appears in (1). As opposed to the modelling approach described in Ashburner and Friston (2005), where the tissue priors were considered as fixed and known a priori quantities, here the tissue probability maps are treated as random variables, whose point estimates or full posteriors can be inferred via model fitting (Bhatia et al., 2007;Ribbens et al., 2014). For this purpose, a finite dimensional parametrisation of {π k } k=1,...,K needs to be defined. Typically, whenever a continuous function needs to be reconstructed from a finite discrete sequence, it is possible to formulate the problem as an interpolation that makes use of a finite set of coefficients and continuous basis functions. Since the priors {π k } k=1,...,K are bounded to take values in the interval [0, 1] on the entire domain Ω π (see equation (3)), not all basis functions are well suited here. Linear basis functions, besides being quite a computationally efficient choice, have the convenient property of preserving the values of {π k } k=1,...,K in the interval [0, 1], as long as the coefficients are also in the same interval. Such coefficients belong to the discrete set Θ π = {π j } j=1,...,N π of K-dimensional vectors, with K k=1 π jk = 1, ∀ j ∈ {1, . . . , N π } .(4) They can be learned directly from the data, as it will be shown in the following section. Additionally, prior distributions on the parameters {π j } j=1,...,N can be introduced (Bishop, 2006). Dirichlet priors are the most convenient choice here, since they are conjugate to multinomial forms of the type in (2), and they can be expressed as p(π j ) = Dir(π j \|α 0 ) = C(α 0 ) K k=1 π α k −1 jk ,(5) where the normalising constant is given by C(α 0 ) = Γ(ᾱ) Γ(α 1 ) . . . Γ(α k ) ,(6) with Γ(·) being the gamma function and α = K k=1 α k .(7) Diffeomorphic image registration As anticipated in the previous sections, the generative interpretation of imaging data that this work relies on involves warping an unknown, average-shaped atlas to match a series of individual scans. Such a problem, that is to say template matching via non-rigid registration, has been largely explored in medical imaging, mainly for solving image segmentation or structural labeling problems, in an automated fashion (Ashburner and Friston, 2005;Shen and Davatzikos, 2004;Christensen, 1999;Chui et al., 2001;Bajcsy et al., 1983;Iglesias et al., 2012;Pluta et al., 2009;Warfield et al., 1999;Khan et al., 2008;Bowden et al., 1998). Indeed, the modelling of spatial mappings between different anatomies can be approached in a variety of manners, depending on the adopted model of shape and on the objective function (i.e. similarity metric and regularisation) that the optimisation is based on, thus leading to a variety of algorithms with remarkably different properties (Penney et al., 1998;Denton et al., 1999;Klein et al., 2009). The work presented here is formulated according to the Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework (Younes, 2010), where the transformations mapping between the source images and the target image are assumed to belong to a Riemannian manifold 1 of diffeomorphisms (Ashburner, 2007). A diffeomorphism φ : Ω → Ω is a smooth differentiable map (with a smooth differentiable inverse φ −1 ) defined on a compact, simply connected domain Ω ⊂ R 3 . One way of constructing transformations belonging to the diffeomorphic group Diff(Ω) is to solve the following non-stationary transport equation (Joshi and Miller, 2000) d dt φ(y, t) = u(φ(y, t), t), φ(y, 0) = y, t ∈ [0, 1] ,(8) where u(φ(y, t), t) ∈ H is a time dependent, smooth velocity vector field, in the Hilbert space 2 H. The initial map, at t = 0, is equal to the identity transform φ(y, 0) = y, while the final map, endpoint of the flow of the velocity field u, can be computed by integration on the unitary time interval t ∈ [0, 1] (Beg et al., 2005). φ(y, 1) = 1 0 u(φ(y, t), t)dt + φ(y, 0) .(12) Following from the theorems of existence and uniqueness of the solution of partial differential equations (p.d.e.), the solution of (8) is uniquely determined 1 A Riemannian manifold, in differential geometry, is a smooth manifold M equipped with a Riemannian metric (inner product). In particular, the Riemannian metric G p on the n-dimensional manifold M n defines, for every point p ∈ M, the scalar product of vectors in the tangent space T p M, in such a way that given two vectors x, y ∈ M, the inner product G p (x, y) depends smoothly on the point p. The tangent space represents the nearest approximation of the manifold by a vector space (Warner, 2013). 2 A Hilbert space H is a complete inner product space, where an inner product is a map ·, · : H × H → C , which associates each pair of vectors in the space with a scalar quantity. In particular given x, y, z ∈ H and a, b ∈ C ax + by, z = a x, z + b y, z , x, x ≥ 0, and x, x = 0 ⇔ x = 0 ,(10) x, y = y, x . An inner product naturally induces a norm by \|\|x\|\| = x, x 1/2 , therefore every inner product space is also a normed vector space (Dieudonné, 2013). by the velocity field u(φ(y, t), t) and by the initial condition φ(y, 0). A diffeomorphic path φ is not only differentiable, but also guaranteed to be a one-to-one mapping. Such a quality is highly desirable for finding morphological and functional correspondences between different anatomies without introducing tears or foldings, which would violate the conditions for topology preservation (Christensen, 1999). Additionally, the diffeomorphic framework provides metrics to quantitatively evaluate distances between anatomies or shapes. It should also be noted that diffeomorphisms are locally analogous to affine transformations (Avants et al., 2006). In practice, finding an optimal diffeomorphic transformation to align a pair, or a group, of images involves optimising an objective function (e.g. minimising a cost function), in the space H of smooth velocity vector fields defined on the domain Ω. The required smoothness is enforced by constructing the norm on the space H through a differential operator L u (Beg et al., 2005), such that a quantitative measure of smoothness can be obtained via R(u) = \|\|L u u\|\| 2 L 2 ,(13) where u is a discretised version of u. The form of the cost function will depend on how the observed data is modelled. For the work presented here, groupwise alignment is achieved via maximisation of the following variational objective function E(Θ u ) =E Z [log p( Z \|Θ π , Θ w , Θ u )] + log p(Θ u ) + const = M i=1 N i j=1 K k=1 γ i jk log       w ik π k (φ i (y j )) K c=1 w ic π c (φ i (y j ))       − 1 2 M i=1 \|\|L u u i \|\| 2 L 2 + const ,(14) where Z = {Z i } i=1,. ..,M is the set of latent variables across the entire population, {γ i j } i, j = E[z i j ] i, j are Kdimensional vectors of posterior belonging probabilities, Θ π indicates the coefficients used to parametrise the tissue priors {π k } k=1,...,K and Θ w denotes a set of individual tissue weights {w i } i=1,...,M for rescaling the tissue probability maps. The coordinate mappings {φ i } i=1,...,M are encoded in the parameter set Θ u , which consists of M vectors of coefficients {u i } i=1,...,M , containing 3 × N i elements each. Such coefficients can be used to construct continuous initial velocity fields via trilinear, or higher order, interpolation. A procedure known as geodesic shooting (Miller et al., 2006;Ashburner and Friston, 2011;Allassonnière et al., 2005;Vialard et al., 2012;Beg and Khan, 2006) is applied, within the work presented here, to compute diffeomorphic deformation fields from corresponding initial velocity fields. Such a procedures exploits the principle of conservation of momentum (Younes et al., 2009), which is given by m t = L † u L u u t , with L † u being the adjoint of the differential operator L u , to integrate the dynamical system governed by (8) without having to store an entire time series of velocity fields. The implementation adopted here relies on the work presented in Ashburner and Friston (2011). The posterior membership probabilities {γ i j } i, j that appear in (14) can be computed by combining the prior latent variable model introduced in 2.1 with a likelihood model of image intensities, which will be described in subsection 2.4, thus leading to a fully unsupervised learning scheme. Alternatively, when manual labels are available, binary posterior class probabilities can be derived directly from such categorical annotations, without performing inference from the observed image intensity data. In particular, if all input data has been manually labelled, then the resulting algorithm would implement a fully supervised learning strategy, while, if only some of the data has associated training labels, a hybrid approach can be adopted, which would fall into the category of semisupervised learning (Chapelle et al., 2006;Filipovych et al., 2011). Finally, it is also possible to take into account the uncertainty inherent in the process of manual rating. In such a case, the actual posterior probabilities can be computed by making use of the categorical output of manual labelling together with an estimate of the rater sensitivity and with a generative intensity model. Making use of Bayes rule, this gives γ i jk = p(z i jk = 1\|x i j , Θ, l i j ) = p(x i j \|z i jk = 1, Θ)p(z i jk = 1\|Θ)p(z i jk = 1\|l i j ) K c=1 p(x i j \|z i jc = 1, Θ)p(z i jc = 1\|Θ)p(z i jc = 1\|l i j ) ,(15) where Θ indicates the set of model parameters, {l i j } j=1,...,N are categorical manual labels assigned to image i and p(z i jk = 1\|l i j ) indicates the probability of voxel j in image i belonging to class k, given the manual label attributed to the same voxel. A simple model for this, is p(z i jk = 1\|l i j ) =        ζ l , if l i j = k 1−ζ l K−1 , if l i j k(16) where ζ l is the sensitivity of the rater that generated the set of labels {l i j } j=1,...,N for image i. The problem of how to evaluate the performance of a manual or automated rater is not addressed here. For instance, a probabilistic scheme, which has been widely used to assess segmentation performance in medical imaging, is presented in Warfield et al. (2004). Combining diffeomorphic with affine registration Anatomical shapes are very high dimensional objects. The diffeomorphic model described in the previous section can account for a significant amount of shape variability in the observed data. Nevertheless, it is still convenient, mainly for computational reasons, to combine such a local, high dimensional shape model with global, lower dimensional transformations, such as rigid body or affine transforms. In fact, by beginning to solve the registration problem from the coarsest deformation components (e.g. rigid body or affine), it is possible to ensure that the subsequent diffeomorphic registration starts from a good initial estimate of image alignment, that is to say closer to the desired global optimum (Lester and Arridge, 1999). This makes the optimisation problem faster to solve and at the same time it reduces significantly the chance of registration failure (Modersitzki, 2004). Indeed, it is relatively common for non-linear registration algorithms to perform poorly in the presence of a large translational or size mismatch between the reference and the target images (Jenkinson and Smith, 2001). A possible parametrisation that combines affine and diffeomorphic transformations is ξ i (y) = T i φ i (y) + t i , ∀y ∈ Ω i ,(17) where ξ i (y) is the resulting mapping from image of subject i into the template space. Such a mapping is obtained by affine transforming the diffeomorphic deformation field φ i . The transformation matrix T i encodes nine degrees of freedom (rotation, zooming and shearing) and is computed via an exponential map T i = exp(Q i (a i )) with Q i (a i ) ∈ ga(3), where ga(3) is the Lie algebra for the affine group in three dimension GA(3) and a i is a vector of nine parameters (Ashburner and Ridgway, 2013). Translations are modelled by the vector t i ∈ R 3 . The entire set of affine parameters is denoted as Θ a = {a i , t i } i=1,...,M . Intensity model From a general probabilistic perspective, classification of tissue types based on MR signal intensities requires a model of the observed data that is capable of capturing the probability of occurrence of each signal sample value x i j , provided that the true labels are known. In other words, the problem breaks down into defining suitable conditional probabilities p(x i j \|z i jk = 1), for each k = {1, . . . , K} and then applying Bayes rule to infer the posterior class probabilities. In the model adopted here, image intensity distributions are represented as Gaussian mixtures, with the unknown mean µ ik and covariance matrix Σ ik of each Gaussian component k, for subject i, being governed by Gaussian-Wishart priors (Bishop, 2006;Blaiotta et al., 2016). Correction of intensity inhomogeneities is also performed within the same modelling framework and it involves multiplying the uncorrected intensities of each image volume by a bias field, which is modelled as the exponential of a weighted sum of discrete cosine transform basis functions (Styner et al., 2000;Ashburner and Friston, 2005). Such an approach is conceptually equivalent to scaling the probability distributions of all Gaussian components by a local scale parameter, which is the bias itself, such that p(x i j \|z i jk , µ ik , Σ ik , Θ β ) = N(x i j \|μ ik ,Σ ik ) ,(18)withμ ik = diag(b i j ) −1 µ ik , Σ ik = diag(b i j ) −1 Σ ik diag(b i j ) −1 ,(19) where Θ β denotes the set of bias field parameters and b i j is a D-dimensional vector representing the bias for subject i at voxel j. Graphical model A graphical representation of the model adopted in this paper is depicted in Figure 1, while a legend of the symbols used to indicate the different variables can be found in table 1. Given such a model, it is possible to define the following variational objective function L, which constitutes a lower bound on the logarithm of the marginal joint probability p( X , Θ β , Θ a , Θ u , Θ π \|Θ w ), such that Figure 1: Graphical representation of the model adopted in this paper. Observed variables {x i j } are represented by a filled circle. Latent variables {z i j } as well as model parameters are depicted as unfilled circles. Blue solid dots correspond to fixed hyperparameters. The so called plate notation is adopted to indicated repeated variables. Symbols referring to all variables and parameters are listed in table 1. log p( X , Θ β , Θ a , Θ u , Θ π \|Θ w ) ≥ L (20) Λ ik µ ik m 0k β 0k ν 0k W 0k W 0k K π j π j z ij x ij u ij N L u Λ ik µ ik w i a i M β i β i Σ β Σ β µ β µ β α 0 Σ a Σ a µ a µ a and L =E Z,Θ µ ,Θ Σ [log p( X \| Z , Θ µ , Θ Σ , Θ β )] +E Z [log p( Z \|Θ π , Θ w , Θ u , Θ a )] +E Θ µ ,Θ Σ [log p(Θ µ , Θ Σ )] + log p(Θ π ) + log p(Θ β ) + log p(Θ a ) + log p(Θ u ) −E Z [log q( Z )] − E Θ µ ,Θ Σ [log q(Θ µ , Θ Σ )] ,(21) where the expectations indicated as E Z and E Θ µ ,Θ Σ are computed with respect to variational posterior distributions q(·) on the latent variables Z and on the Gaussian means and covariances {Θ µ , Θ Σ }, respectively. Optimisation of L, which provides optimal parameter and hyperparameter estimates, will be discussed in the following section. Model fitting The model described in the previous section can be fit to data sets of MR images by combining a variational expectation-maximisation (VBEM) algorithm with gradient based numerical optimisation techniques. Indeed, the VBEM algorithm described in Blaiotta et al. (2016) is well-suited for addressing the model estimation problem discussed here, since Symbol Meaning x i j Observed image intensity at voxel j for subject i. z i j Vector of latent class membership probabilities. π j Tissue priors at voxel j. µ ik Mean intensity of class k for subject i. Σ ik Covariance of intensities for class k and subject i. W 0k Scale matrix of Wishart prior distribution on Λ k = (Σ k ) −1 . ν 0k Degrees of freedom of Wishart prior distribution on Λ k . m 0k Mean of Gaussian prior distribution over µ k β 0k Scaling hyperparameter of Gaussian prior distribution over µ k α 0 Hyperparameter governing the Dirichlet prior on π. Θ β Bias field parameters. µ β Prior mean of bias parameters. Σ β Prior covariance matrix of bias parameters. Θ a Affine transformation parameters. µ a Prior mean of affine transformation parameters. Σ a Prior covariance matrix of affine transformation parameters. w i Weights for rescaling the tissue priors. u i j Initial velocity at voxel j for subject i. L u Differential operator to compute penalty on u i . Table 1 it allows learning posterior distributions on the Gaussian mixture parameters, under the assumption that q( Z , Θ µ , Θ Σ ) factorizes as q( Z )q(Θ µ , Θ Σ ) (Bishop, 2006), and at the same time it is able to transfer the information encoded in such posteriors to estimate empirical intensity priors for each tissue type. Additionally, the algorithm proposed in this paper loops over all subjects in the population and, for each subject, it iterates over estimating the Gaussian posteriors, the bias field, the affine parameters and the initial velocities, which are all treated as conditional optimisations. Subsequently the tissue probability maps and intensity priors are updated and the whole cycle is repeated until convergence. Estimation of the bias field parameters Θ β can be conveniently performed via non-linear optimisation techniques. Here the problem is solved using the Gauss-Newton method (Bertsekas, 1999), so as to maximise the objective function in (21) with respect to Θ β . The resulting implementation is very similar to the one described in Ashburner and Friston (2005), therefore further details are omitted here. Optimisation of the affine parameters Θ a = {a i , t i } i=1,...,M can also be carried out by means of a Gauss-Newton scheme and a brief description of the required computations can be found Appendix A. For the update of the weight parameters Θ w we adopt the same strategy outlined in Ashburner and Friston (2005); Blaiotta et al. (2016). The following sections instead will present in detail the algorithmic scheme used to learn the average shaped tissue templates Θ π = {π j } j=1,...,N π and to estimate the set of initial velocity fields Θ u = {u i } i=1,...,M . Updating the tissue priors At each main iteration of the algorithm, the tissue priors Θ π = {π j } j=1,...,N π need to be updated, given the current estimates of all the other parameters, which are kept fixed for each individual in the population. Considering only the terms in (21) that depend on Θ π gives the following objective function, which has to be maximised with respect to Θ π L π = E Z [log p( Z \|Θ π , Θ w , Θ u , Θ a )] + log p(Θ π ) + const = M i=1 K k=1 Ω i γ ik (y) log       w ik π k (ξ i (y)) K c=1 w ic π c (ξ i (y))       dy + log p(Θ π ) + const .(22) It should be noted that the parameters Θ π that need to be estimated are defined on the domain of the template Ω π , rather than on the individual spaces {Ω i } i=1,...,M . For this reason equation (22), which is a sum of integrals on the native domains, needs to be mapped to Ω π , by inverting the warps {ξ i } i=1,...,M , to give L ′ π = M i=1 K k=1 Ω π det       ∂ξ −1 i ∂y       γ ik (ξ −1 i (y)) log w ik π k (y) K c w ic π c (y) dy + log p(Θ π ) + const ,(23) where the determinants of the Jacobian matrices of the deformations are included to preserve volumes after the change of variables. Finally equation (23) is discretised on a regular voxel grid, whose centres have coordinates {y j } j=1,...,N π , to give L ′ π = M i=1 N π j=1 K k=1 det(J ξ −1 i j ) γ ik (ξ −1 i j ) log       w ik π jk K c=1 w ic π jc       + log p(Θ π ) + const ,(24) where ξ −1 i j = ξ −1 i (y)\| y=y j ,(25)det(J ξ −1 i j ) = det       ∂ξ −1 i (y) ∂y       y=y j ,(26)π jk = π k (y)\| y=y j .(27) The prior term p(Θ π ) is given by the following Dirichlet distribution p(Θ π ) = N π j=1 Dir(π j \|α 0 ) = C(α 0 ) N π j=1 K k=1 π α 0k −1 jk .(28) Maximising equation (24) is a constrained optimisation problem, subject to K k=1 π jk = 1 , ∀ j ∈ {1, . . . , N π }(29) A closed form solution could be easily found if the rescaling weights {w i } i=1,...,M were all equal to one. In such a case L ′ π = M i=1 N π j=1 K k=1 det(J i j ) γ ik (ξ −1 i j ) log π jk + N π j=1 K k=1 (α 0k − 1) log p(π jk ) + const ,(30) which could be maximised under the constraint (29), by making use of Lagrange multipliers (Falk, 1967), to give π jk = N jk + α 0k − 1 K k=1 (N jk + α 0k ) − K ,(31)with N jk = M i=1 det(J i j ) γ ik (ξ −1 i j ) . This solution would provide maximum a posteriori point estimates of Θ π = {π j } j=1,...,N π . However for this problem, it would also be possible to derive a full variational posterior distribution, which, like its prior, would take a Dirichlet form, with parameters α j = α 0 + N j . When rescaling of the tissue priors is allowed the optimisation problem becomes more complex. The strategy adopted here consists in finding an approximate solution to the unconstrained optimisation problem by setting the derivatives of the objective function in (23) to zero α 0k − 1 π jk + M i=1 det(J ξ −1 i j ) γ ik (ξ −1 i j )       1 π jk − w ik K c=1 w ic π jc       = 0 .(32) Solving with respect to π jk , under the simplifying assumption that the term K c=1 w ic π jc can be treated as a constant, gives π jk = N jk + α 0k − 1 M i=1 det(J ξ −1 i j ) γ ik (φ −1 i j )w ik K c=1 w ic π jc .(33) Such a solution is then projected onto the constraining hyperplane, by preserving tissue proportions at each voxel π jk =π jk K c=1π jc .(34) Experimental testing of this strategy indicated that it gave a constant improvement of the objective function at a relatively cheap computational cost. Alternatively, iterative constrained non-linear optimisation techniques (Powell, 1978) could have been exploited to solve the template update problem. Computing the deformation fields Groupwise image alignment is achieved by optimisation of the variational objective function defined in (21), with respect to the parameters used to compute the deformations. This is equivalent to adopting the following image matching or similarity term D = E Z [log p( Z \|Θ π , Θ w , Θ u , Θ a )] = M i=1 y∈Ω i K k=1 γ ik (y) log       w ik π k (ξ i (y)) K c=1 w ic π c (ξ i (y))       dy .(35) Additionally, working on discretised image grids, with associated voxel centres {y i j } j=1,...,N i , requires reformulating D as D = M i=1 N i j=1 K k=1 γ i jk log w ik π ′ jk K c=1 w ic π ′ jc ,(36) with π ′ jk = π k (ξ i (y))\| y=y i j . The penalty term for this groupwise image registration problem is given by R = R di f + R a f = log p(Θ u ) + log p(Θ a ) = − 1 2 M i=1 \|\|L u u i \|\| 2 L 2 + a T i Σ −1 a a i + const ,(38) with u i being a 3 × N i dimensional vector of parameters used for representing the initial velocity field of image i and a i encoding affine deformation parameters used to compute the transformation in (17). For each image i in the data set, updating the corresponding initial velocity field, given the current estimates of the templates and all the other model parameters, involves optimising the following objective function E (i) di f =D (i) + R (i) di f = N i j=1 K k=1 γ i jk log w ik π k (ξ i j ) K c=1 w ic π c (ξ i j ) − 1 2 \|\|L u u i \|\| 2 L 2 ,(39) with respect to u i , under the following deformation model ξ i j = ξ i (y i j ) = T i φ i (y i j ) + t i ,(40) where φ i is a diffeomorphism computed via geodesic shooting (Ashburner and Friston, 2011) from the corresponding initial velocity field u i . Here image registration is solved via Gauss-Newton optimisation, which requires computing both the first and second derivatives of the objective function (Hernandez and Olmos, 2008). Such derivatives can be found in Appendix B. This leads to a very high dimensional inverse problem, which unfortunately cannot be solved via numerical matrix inversion, since this would be prohibitively expensive from a computational point of view. The approach adopted in this work consists in treating this optimisation as a partial differential equation problem, which can efficiently be solved using multigrid methods (Modersitzki, 2004). In particular, we adopt the same full multigrid implementation as in Ashburner (2007). Validation and Discussion In this section we will present results obtained by applying the presented modelling framework to real brain and cervical cord MR scans acquired with different imaging protocols, as well as to synthetic MR head volumes. Both qualitative and quantitative measures will be provided to assess the behaviour of the proposed approach. Template construction As discussed in Section 2, the proposed method can serve to learn prior tissue probability maps from crosssectional imaging data sets. In this paper we mainly explore the performance of such a framework to address the quest for integrated brain and spinal cord neuromorphometric tools, even if, given the generality of the presented approach, many more applications should in principle be possible. Data The input data for training the model was obtained from three different databases, two of which are freely accessible for download, thus ensuring that the results presented here could readily be compared to those produced by competing algorithms for medical image registration or segmentation. OASIS data set. The first data set consists of thirty five T1-weighted MR scans from the OASIS (Open Access Series of Imaging Studies) database (Marcus et al., 2007). The data is freely available from the web site http://www.oasis-brains.org, where details on the population demographics and acquisition protocols are also reported. Additionally, the selected thirty five subjects are the same ones that were used within the 2012 MICCAI Multi-Atlas Labeling Challenge (Landman and Warfield, 2012). Balgrist data set. The second data set consists of brain and cervical cord scans of twenty healthy adults, acquired at University Hospital Balgrist with a 3T scanner (Siemens Magnetom Verio). Magnetisationprepared rapid acquisition gradient echo (MPRAGE) sequences, at 1 mm isotropic resolution, were used to obtain T1-weighted data, while PD-weighted images of the same subjects were acquired with a multi-echo 3D fast low-angle shot (FLASH) sequence, within a whole-brain multi-parameter mapping protocol (Weiskopf et al., 2013;Helms et al., 2008). IXI data set. The third and last data set comprises twenty five T1-, T2-and PD-weighted scans of healthy adults from the freely available IXI brain database, which were acquired at Guy's Hospital, in London, on a 1.5T system (Philips Medical Systems Gyroscan Intera). Additional information regarding the demographics of the population, as well as the acquisition protocols, can be found at http://brain-development.org/ixi-dataset. The complete data set therefore consists of eighty multispectral scans of healthy adults, obtained with fairly diverse acquisition protocols and using scanning systems produced by different vendors. Unfortunately, not all the three modalities of interest (T1-, T2-and PD-weighted) are available for all of the subjects. To circumvent the difficulties arising from the presence of missing imaging modalities, without neglecting any of the available data (indeed deletion of entries with missing data is still, in spite of its crudity, a common statistical practice), the Gaussian mixture modelling approach discussed in Blaiotta et al. (2016) was generalised by introducing an additional variational posterior distribution over the missing data points. In practice, the resulting variational EM scheme iterates over first estimating an approximated posterior distribution on the unknown image intensities, secondly updating the sufficient statistics of the complete (observed and missing) data and finally computing variational posteriors on the Gaussian mixture parameters. Additional computational details relative to this strategy are provided in Appendix C. In synthesis, it was possible to fit the generative groupwise model described in this paper to the entire data set, in spite of having different imaging modalities available from the different acquisition sites. This is indeed a very common scenario in real life medical imaging problems, therefore it should be actively addressed by processing or modelling solutions that claim to be applicable to large population data (van Tulder and de Bruijne, 2015). Manual brain labels are freely available for all images in data set one. Such labels have been generated and made public by Neuromorphometrics, Inc. (http://Neuromorphometrics.com) under academic subscription and they provide a fine parcellation of cortical and non cortical structures, for a total of 139 labels across the brain. Part of this label data was used for training of the model while the remainder was left out for testing and validation. In particular, brain labels of twenty out of the thirty five OASIS subjects were used to create gray and white matter ground truth segmentations, which were provided as training input for semisupervised model fitting. Similarly, spinal cord manual labels were created for forty subjects (twenty from data set two and twenty from data set three). Such labels were randomly split in half for training and half for subsequent test analyses. Due to the limited resolution of the data it was not possible to manually delineate gray and white matter within the spinal cord. For this reason, each voxel classified as spinal cord in the training data was allowed to be assigned either to the gray or to the white matter tissue classes, based on the fit of its intensity value to the underlying Gaussian mixture model, as outlined in equation (15). Analogously, in spite of having defined only one gray matter training label, two distinct gray matter classes were introduced in the mixture model (top two rows in Figure 2), to best capture the corresponding distribution of image intensities, which is poorly represented by a single Gaussian component, as opposed to the distribution of white matter intensities. Also in this case, membership probabilities of the labelled training data were computed based on the corresponding intensity values, by making use of equation (15). Tissue templates and intensity priors The tissue probability maps obtained by applying the modelling framework presented in this paper to the data set described above are depicted in Figure 2. The total number of tissue classes used for this experiment is equal to twelve but three classes, representing air in the background, are not shown. In particular, Figure 2 shows how one of the two gray matter classes (first row) best fits the subcortical nuclei and also includes voxels affected by partial volume effects at the interface between gray and white matter, while the second one (second row) is more representative of cortical structures, with the presence of partial volume effects generated by the juxtaposition of gray matter and CSF. The third row in Figure 2 shows the white matter class, which also includes most of the brainstem and the spinal cord. The remaining tissue classes were estimated in a purely unsupervised way. Therefore a non ambiguous anatomical interpretation is not straightforward. Tissue class four (fourth row) mainly contains CSF, even if other tissues are also present, especially in the neck area. This should be attributed to the lack of CSF training labels as well as to a poor multivariate coverage of the cervical region in the available data. In fact, data from the OASIS set is truncated around the first cervical vertebra. The T1-weighted scans of the IXI data set cover up to the C2/C3 vertebral level, but the corresponding T2-and PD-weighted scans do not extend beyond the brainstem. Indeed, only the data from the second database (Balgrist hospital) provides more than one modality covering up to around the fourth cervical vertebra. In this case though, additional difficulties arose from poor inter-modality alignment of the data, a problem that turned out to be particularly severe in the cervical region and that, given its non-linearity, could not be fully compensated for by affine inter-modality coregistration. Bone tissue is also not easily identifiable from the data available for this experiment, but it could have potentially been much better extracted by incorporating some CT scans into the training data. Fat and soft tissues are mainly represented in the last two classes (bottom two rows in Figure 2). Figure 3 illustrates orthogonal views of the gray and white matter tissue probability maps, where the gray matter map is obtained by evaluating the sum of the two top classes in Figure 2. The empirical Bayes learning procedure, introduced in Blaiotta et al. (2016) to estimate suitable prior distributions for the parameters of the Gaussian mixture model, was applied here to the same data used to construct the templates. Some of the results are summarised in figure 4, where the estimated empirical prior distributions on the mean intensity of gray and white matter are depicted, with overlaid contour plots showing some of the individual posteriors (randomly selected across the entire population). Such results indicate that the proposed empirical Bayes learning scheme can serve to capture, not only the variability of mean tissue intensity across subjects for each of the modalities of interest, but also the amount of covariance between such modalities. Information of this sort can potentially be used in a number of different frameworks, for solving problems such as tissue segmentation, pathology detection or image synthesis. Validity of groupwise registration The performance of groupwise registration achieved by the presented algorithm was assessed by computing pairwise overlap measures for all possible couples of spatially normalised test images (i.e. images whose ground truth labels were not used for training the model). The Dice score coefficient was chosen as a metric of similarity. Results are summarised in figure 5, where the accuracy of the algorithm presented here is com- (2010), whose implementation is publicly available, as part of the Advanced normalisation Tools (ANTs) package, through the web site http://stnava.github.io/ANTs/. Indeed, the symmetric diffeomorphic registration framework implemented in ANTs has established itself as the state-ofthe-art of medical image nonlinear spatial normalisation (Klein et al., 2009). A number of options can be customised within the template construction framework distributed with ANTs. The experiments, whose results are reported here, were performed with the settings recommended in the package documentation for brain MR data, which are also reported in table 2. Results of this validation analyses indicate that the method presented here, in spite of not being as accurate as ANTs for aligning some subcortical brain structures (e.g. thalamus, putamen, pallidum and brainstem), provided significantly better overlap when registering cortical regions, as assessed by means of paired t-tests with a significance threshold of 0.05 and without correcting for multiple comparisons. No statistically significant differences were found between the two methods with respect to registration of the spinal cord. Accuracy of tissue classification The accuracy of tissue classification achieved by the method presented in this paper was first evaluated on test data that was used to create the templates but without providing manual labels for training the model during atlas construction. The aim in this case was to determine to which extent the proposed method can capture relevant features of the training data, when manual labels are not provided, by learning from few annotated Figure 2: Tissue probability maps obtained by applying the presented groupwise generative model to a multispectral data set comprising brain and cervical cord scans of eighty healthy adults, from three different databases. examples. Dice scores 3 were computed to compare the automated segmentations produced via semisupervised groupwise model fitting, with the ground truth, obtained by merging all the gray and white matter brain structures (labels) into two tissue classes respectively, and by considering the spinal cord as a third separate class. The probabilistic gray and white matter segmentations of the brain were thresholded at 0.5, in order to obtain binary label maps, directly comparable to the ground truth. To derive binary cord segmentations instead, the sum of gray and white matter posterior belonging probabilities was first computed in a subvolume containing the neck only, and then thresholded at 0.5. Results are summarised in Figure 6, which shows the distributions of Dice scores obtained for brain gray matter, brain white matter and spinal cord. Such results were then compared to those produced by the brain segmentation algorithm implemented in SPM12, using the standard tissue probability maps distributed with the SPM software. Results of these analyses, which are summarised in Figure 6, indicate that the population specific atlases constructed with the method presented here enable higher tissue classification accuracy, at least for test data drawn from the same population that the model was trained on but whose labels were not exploited for training. A potential source of bias in the results of this experiment is the fact that the test data was actually employed for constructing the atlases, even if the corresponding labels were not seen by the algorithm. However a more cautious k-fold cross-validation, which would have required constructing multiple templates, was not practical in this case due to the expensive computational cost of groupwise model fitting. Such results however seem to suggest that the model presented in this paper could potentially be useful to create templates with the purpose of capturing the peculiar anatomical features of those populations that are poorly represented by standard anatomical atlases (Tang et al., 2010;Fillmore et al., 2015), such as young or elderly populations, diseased populations, or individuals belonging to different ethnic groups. This would not only lead to more accurate segmentation results, but as a direct consequence, also increase the reliability of subsequent data analyses, which build models of the segmented data to infer or predict clinically meaningful information. Figure 5: Accuracy of groupwise registration achieved by the presented method, compared to the performance of ANTs, for different neural regions. Stars indicate statistically significant differences between the two methods, assessed by means of paired t-tests without correcting for multiple comparisons. Modelling unseen data Further validation experiments were performed to quantify the accuracy of the framework described in this paper to model unseen data, that is to say data that was not included in the atlas generation process. Such experiments were performed on synthetic T1weighted brain MR scans from the Brainweb database (http://brainweb.bic.mni.mcgill.ca/), generated using a healthy anatomical model. Accuracy of bias correction A healthy adult brain MR model was processed by means of the algorithm discussed here, using the head and neck templates previously constructed as tissue priors. Different noise and bias field levels were added to the uncorrupted synthetic data, to test the behaviour of the proposed modelling scheme in different noise (1%, 3%, 7%) and bias conditions (20% and 40%). The noise in these simulated images has Rayleigh statistics in the background and Rician statistics in the signal regions and its level is computed as a percent standard deviation ratio, relative to the MR signal, for a reference tissue (Cocosco et al., 1997). Regarding the bias field instead, 20% bias is modelled as a smooth field in the range [0.9, 1.1] while 40% bias is obtained by rescaling of the 20% field, so as to range between 0.8 and 1.2 . Table 3 reports the Pearson product-moment correlation coefficients between the ground truth and the estimated bias fields, for the different bias ranges and noise levels. Results indicate that the similarity between the estimated and true bias decreases for more intense nonuniformity fields and higher noise levels. Indeed this is not surprising, as the penalty term, which enforces smoothness of the bias field, has a greater impact in determining the shape of the estimated bias when the non-uniformity fields have a larger dynamic range. Nevertheless, results reported in the following section will show how this increased mismatch between the estimated and true bias, for higher nonuniformities, does not seem to affect the accuracy of tissue segmentation. On the other hand, the accuracy of bias correction is directly related to the amount of noise corrupting the data, mainly due to how this affects the precision associated with estimation of the Gaussian mixture parameters. For a comparison of these results with the performance of SPM12 bias correction on simulated T1-weighted scans from the Brainweb database see Blaiotta et al. (2016). Accuracy of tissue classification For the same data the accuracy of tissue classification was also evaluated, by comparing the similarity between the estimated gray and white matter segmentations and the underlying anatomical model. Results are reported in Figure 7, which shows the Dice score coefficients obtained under different bias and noise conditions. The Brainweb database has been extensively used in the neuroimaging community to validate MR image processing algorithms. Therefore the results reported here should be directly comparable to the performance of many brain segmentation techniques present in the literature. Conclusions This paper presented a comprehensive generative framework for modelling cross-sectional MR data sets, which is intended to enable simultaneous morphometric analyses of brain and cervical spinal cord data. From a theoretical perspective, such a framework relies on variational probability density estimation tech- Figure 7: Dice scores between the estimated and ground truth segmentations for brain white matter and brain gray matter, under different noise and bias conditions, for synthetic T1weighted data. niques to model the observed data (i.e. MR signal intensities). Additionally, a hierarchical modelling perspective is proposed, where observations from a population of subjects are used to construct empirical intensity priors, which can then serve to inform models of new data. Shape modelling is performed via groupwise diffeomorphic registration, thus ensuring bijective (i.e. one-to-one) differentiable mappings between anatomical configurations (Miller, 2004). Such an approach enables a rigorous mathematical encoding of anatomical shapes via deformable template matching (Christensen et al., 1996), therefore providing a quantitative framework for the analysis of shape variation and covariation. Data for training the method was collected from three different databases, two of which are publicly accessible to the research community. Results of validation experiments performed both on training and unseen test data indicate that the presented framework is suitable to perform integrated brain and cervical cord computational morphometrics. Thus, the proposed algorithm represents a concrete solution to extract volumetric and morphometric information from large structural neuroimaging data sets, in a fully automated manner. At the same time it provides outputs that could be readily interpreted, for instance via statistical hypothesis testing, with the ultimate goal of comparing different populations, treatment effects etc. (Ashburner and Friston, 2000). Appendix A. Derivatives of the lower bound with respect to the affine parameters The affine parameters, for each subject i, can be estimated (i.e. optimised) in a Gauss-Newton fashion, so as to maximise of the following objective function E (i) a f = D (i) + R (i) a f = N i j=1 K k=1 γ i jk log w ik π k (ξ i j ) K c=1 w ic π c (ξ i j ) − 1 2 a T i Σ −1 a a i , (A.1) with respect to a i . The gradients and Hessians, which are useful to solve this problem are reported below. In particular, for the matching term, the following derivatives need to be computed ∂D (i) ∂a i = N i j=1 K k=1       γ i jk − w ik π k (ξ i j ) K c=1 w ic π c (ξ i j )       g π jk , (A.2) where g π jk is defined as Optimisation of the initial velocities, for each image i, requires maximising the following objective function g π jk = B T i [φ i j , 1] ⊗ ∇ log π k (ξ i j ) , (A.3) with B T i = ∂S i ∂a i , (A.4) and S i = T i t i 0 1 . (A.5) ∂ 2 D (i) ∂a 2 i = N i j=1        K k=1 w ik π k (ξ i j ) K c=1 w ic π c (ξ i j ) g π jk        ×         K k=1 w ik π k (ξ i j ) K c=1 w ic π k (ξ i j ) g π jk         T − N i j=1 K k=1 w ik π k (ξ i j ) K c=1 w ic π c (ξ i j ) g π jk g πE (i) di f = D (i) + R (i) di f = N i j=1 K k=1 γ i jk log w ik π k (ξ i j ) K c=1 w ic π c (ξ i j ) − 1 2 M i=1 \|\|L u u i \|\| 2 L 2 , (B.1) with respect to u i . Here, we report the first and second derivatives of this objective function, which are useful to solve the registration problem using gradient-based techniques, such as the Gauss-Newton algorithm. The gradient of the matching term D (i) with respect to u i is given by ∂D (i) ∂u i = K k=1 γ i jk ∂ ∂u i log w ik π k (ξ i ) K c=1 w ic π c (ξ i ) = K k=1 γ i jk        g π k − K c=1 w ic π c (ξ i ) K c=1 w ic π c (ξ i ) g π c        , (B.2) which, making use of K k=1 γ i jk = 1 , can be rewritten as ∂D (i) ∂u i = K k=1       γ ik − w ik π k (ξ i ) K c=1 w ic π c (ξ i )       g π k , (B.3) where g π k is computed, at each voxel j, by g π jk = T i , J ξ i j T ∇ log π k (ξ i j ) , (B.4) and J ξ i indicates the Jacobian matrix of ξ i j . An approximated positive semidefinite Hessian of D can instead be computed by discarding the second derivatives of the logarithm of tissue priors ∂ 2 ∂y 2 log       w ik (π k (ξ i (y))) K c=1 w ic (π c (ξ i (y)))       = 0 , ∀y ∈ Ω i , (B.5) to give ∂ 2 D (i) ∂u i 2 =        K k=1 w ik π k (ξ i ) K c=1 w ic π c (ξ i ) g π k        ×        K k=1 w ik (π k (ξ i )) K c=1 w ic (π k (ξ i )) g π k        T − K k=1 w ik π k (ξ i ) K c=1 w ic π c (ξ i ) g π k g π k T . (B.6) Finally, the first and second derivatives of the penalty term R, which are also required to optimise (B.1), can be computed by ∂R (i) di f ∂u i = −L u † L u u i , (B.7) ∂ 2 R (i) di f ∂u i 2 = −L u † L u . (B.8) Appendix C. Variational Gaussian mixtures: inference of missing data The variational Bayes EM algorithm for fitting Gaussian mixture models, described in Blaiotta et al. (2016), can be generalised to handle the case where some components of the D-dimensional observation x j are missing. Having denoted x j = o j h j , (C.1) with o j being the observed data and h j the missing data, the Gaussian likelihood p(x j \|z jk = 1, µ k , Σ k ) can be expressed as p(x j \|z jk = 1, µ k , Λ k ) = N o j h j µ o k µ h k , Λ o,o k Λ o,h k Λ o,h k Λ h,h k , (C.2) by making use of block matrix notation to partition the mean vector µ k and the precision matrix Λ k . In this case h j is treated as an unobserved random variable. Thus, in a variational Bayes setting, an additional posterior factor can be introduced for each missing data point h j to give q(H, Z, Θ µ , Θ Σ ) =q(H)q(Z)q(Θ µ , Θ Σ ) =q(Z)q(Θ µ , Θ Σ ) N j=1 q(h j ) . (C.3) Making use of the general result qŝ(Θŝ) ∝ exp(E s ŝ [log p(X, Θ)]) (Bishop, 2006), an approximated posterior on the missing data point h j can be computed by log q(h j ) =E Z,Θ µ ,Θ Σ log p(x j , z j , Θ µ , Θ Σ \|Θ π ) + const =E Z,Θ µ ,Θ Σ log p(x j \|z j , Θ µ , Θ Σ ) + E Z log p(z j \|Θ π ) + E Θ µ ,Θ Σ log p(Θ µ , Θ Σ ) + const , (C.4) where Θ π denotes the mixing proportion parameter set, treated here via maximum likelihood, and p(Θ µ , Θ Σ ) is a conjugate Gaussian-Wishart prior on the means and covariances of the model. Ignoring the terms independent from h j , equation (C.4) can be rewritten as log q(h j ) = K k=1 γ jk E Θ µ ,Θ Σ log N(x j \|µ k , Σ k ) + const = 1 2 K k=1 γ jk h T j E Θ µ ,Θ Σ Λ h,h k h j + K k=1 γ jk h T j E Θ µ ,Θ Σ Λ o,h k o j − E Θ µ ,Θ Σ µ o k − K k=1 γ jk h T j E Θ µ ,Θ Σ Λ h,h k E Θ µ ,Θ Σ µ h k + const . (C.5) The previous equation indicates that the unobserved value h j is drawn from a Gaussian mixture distribution with mixing proportions equal to the posterior (after having observed o j ) membership probabilities {γ jk } k=1,...,K , while the Gaussian means {n jk } k=1,...,K and covariances {P jk } k=1,...,K are given by n jk =E Θ µ ,Θ Σ µ h k + E Θ µ ,Θ Σ Λ h,h k −1 × E Θ µ ,Θ Σ Λ o,h k (E Θ µ ,Θ Σ µ o k − o j ) , (C.6) P k = E Θ µ ,Θ Σ Λ h,h k . (C.7) Given the posteriors q(Z) and q(H), the following sufficient statistics of X can be computed s 1k = N j=1 γ jk o j N j=1 γ jk n jk , (C.8) S 2k =        N j=1 γ jk o j o T j N j=1 γ jk o j n T jk N j=1 γ jk n jk o T j N j=1 γ jk n k n T jk + (P k ) −1        . (C.9) Once such sufficient statistics have been evluated, they can be used to update the Gaussian-Wishart posteriors q(Θ µ , Θ Σ ) in the exact same way as in Blaiotta et al. (2016). Such posteriors are in turn used to compute the expectations that appear in equations (C.6) and (C.7), in an iterative EM fashion. Figure 3 :Figure 4 : 34Brain and spinal cord tissue probability maps of gray (a) and white (b) matter, constructed as described in this paper. Prior distribution over the mean intensity of gray and white matter, in T1-and PD-weighted data. Figure 6 : 6Brain and spinal cord segmentation accuracy of the presented method. Table 2 : 2Options selected to perform group- wise registration with ANTs, using the antsMultivariateTemplateConstruction script provided with the ANTS package. Option Value Similarity Metric Cross-correlation (CC) Transformation model Greedy SyN (GR) Initial rigid body yes N4 Bias Correction yes Number of resolution levels 4 Number of iterations 100 × 70 × 50 × 10 Gradient step 0.2 Number of template updates 4 pared to that achieved by the method described in Avants et al. Table 3 : 3Pearson's correlation coefficients between the ground truth bias fields and those estimated by the presented algorithm, for simulated T1-weighted data.Noise 1% 3% 7% 20% 0.86 0.86 0.70 Bias 40% 0.72 0.72 0.51 The Dice score over two sets A and B is defined as DS C = 2 \|A∩B\| \|A\|+\|B\| . AcknowledgmentsClaudia Blaiotta is co-funded by UCL and Zurich Balgrist Hospital, as part of the UCL 'Impact' award scheme. This research was supported by Wings for Life -Spinal Cord Research Foundation. The Wellcome Trust Centre for Neuroimaging is supported by core funding from the Wellcome Trust (091593/Z/10/Z). The OASIS project was funded by the National Institutes of Health grants P50 AG05681, P01 AG03991, R01 AG021910, P50 MH071616, U24 RR021382, R01 MH56584. The IXI project was supported by the EP-SRC grant GR/S21533/02. Geodesic shooting and diffeomorphic matching via textured meshes. 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[ "Predicting wind pressures around circular cylinders using machine learning techniques A PREPRINT", "Predicting wind pressures around circular cylinders using machine learning techniques A PREPRINT" ]	[ "Gang Hu [email protected] \nCentre for Wind, Waves and Water\nSchool of Civil Engineering\nThe University of Sydney\n2006SydneyNSWAustralia\n", "K C S Kwok \nCentre for Wind, Waves and Water\nSchool of Civil Engineering\nThe University of Sydney\n2006SydneyNSWAustralia\n" ]	[ "Centre for Wind, Waves and Water\nSchool of Civil Engineering\nThe University of Sydney\n2006SydneyNSWAustralia", "Centre for Wind, Waves and Water\nSchool of Civil Engineering\nThe University of Sydney\n2006SydneyNSWAustralia" ]	[]	Numerous studies have been carried out to measure wind pressures around circular cylinders since the early 20 th century due to its engineering significance. Consequently, a large amount of wind pressure data sets have accumulated, which presents an excellent opportunity for using machine learning (ML) techniques to train models to predict wind pressures around circular cylinders. Wind pressures around smooth circular cylinders are a function of mainly the Reynolds number (Re), turbulence intensity (Ti) of the incident wind, and circumferential angle of the cylinder. Considering these three parameters as the inputs, this study trained two ML models to predict mean and fluctuating pressures respectively. Three machine learning algorithms including decision tree regressor, random forest, and gradient boosting regression trees (GBRT) were tested. The GBRT models exhibited the best performance for predicting both mean and fluctuating pressures, and they are capable of making accurate predictions for Re ranging from 10 4 to 10 6 and Ti ranging from 0% to 15%. It is believed that the GBRT models provide very efficient and economical alternative to traditional wind tunnel tests and computational fluid dynamic simulations for determining wind pressures around smooth circular cylinders within the studied Re and Ti range.	10.1016/j.jweia.2020.104099	[ "https://arxiv.org/pdf/1901.06752v1.pdf" ]	58,981,552	1901.06752	75424921eaf0602c32a2d0f22200ca3dc121ba59	Predicting wind pressures around circular cylinders using machine learning techniques A PREPRINT January 20, 2019 Gang Hu [email protected] Centre for Wind, Waves and Water School of Civil Engineering The University of Sydney 2006SydneyNSWAustralia K C S Kwok Centre for Wind, Waves and Water School of Civil Engineering The University of Sydney 2006SydneyNSWAustralia Predicting wind pressures around circular cylinders using machine learning techniques A PREPRINT January 20, 20191circular cylinderwind pressuremachine learningrandom forestgradient boosting regression trees Numerous studies have been carried out to measure wind pressures around circular cylinders since the early 20 th century due to its engineering significance. Consequently, a large amount of wind pressure data sets have accumulated, which presents an excellent opportunity for using machine learning (ML) techniques to train models to predict wind pressures around circular cylinders. Wind pressures around smooth circular cylinders are a function of mainly the Reynolds number (Re), turbulence intensity (Ti) of the incident wind, and circumferential angle of the cylinder. Considering these three parameters as the inputs, this study trained two ML models to predict mean and fluctuating pressures respectively. Three machine learning algorithms including decision tree regressor, random forest, and gradient boosting regression trees (GBRT) were tested. The GBRT models exhibited the best performance for predicting both mean and fluctuating pressures, and they are capable of making accurate predictions for Re ranging from 10 4 to 10 6 and Ti ranging from 0% to 15%. It is believed that the GBRT models provide very efficient and economical alternative to traditional wind tunnel tests and computational fluid dynamic simulations for determining wind pressures around smooth circular cylinders within the studied Re and Ti range. Introduction Circular cylinders can be classified as a type of bodies in between streamlined bodies such as airfoils, and non-streamlined bodies with sharp edges such as rectangular prisms. Subjected to an oncoming flow, streamlined bodies avoid flow separation, while non-streamlined bodies exhibit flow separations at fixed points (e.g. the sharp corners). For circular cylinders, the flow separation is much more complicated. The separation position varies according to a number of parameters, including incident flow velocity, turbulence, body geometry, and surface roughness (Niemann and Hölscher, 1990). Consequently, the flow around a circular cylinder is one of the most challenging problems in fluid dynamics. However, understanding the flow regime and aerodynamic characteristics of circular cylinders is crucial for ensuring the safety of engineering structures ranging from heat exchangers to bridge cables, silos, industrial chimney, and cooling towers, as shown in Fig. 1. Since the early 20 th century, a considerable amount of research has been devoted to study flow around circular cylinders (e.g. Thom, 1933;Zdravkovich, 1997aZdravkovich, , 1997b. The research methodologies adopted by these studies can be categorized into wind tunnel experiments (Ke et al., 2017;Zou et al., 2018), field measurements (Zhao et al., 2017, and computational fluid dynamic simulations (Catalano et al., 2003;Moussaed et al., 2014;Yeon et al., 2016;Zhao and Cheng, 2011). Although substantial efforts have been made to acquire the pressure distributions around circular cylinders with various dimensions under various incident flow conditions, timeconsuming and expensive wind tunnel tests and/or computational fluid dynamic simulations remain indispensable for determining the pressure around a circular cylinder with a particular dimension immersed in a particular flow field. Fortunately, the research endeavors in the past have accumulated a large amount of wind pressure data sets for circular cylinders. These data sets form a basis for using machine learning (ML) techniques to build models for predicting pressures around circular cylinders. ML is a powerful tool of data mining that automates analytical model building. It is a branch of artificial intelligence based on the idea that systems can learn from data and make decisions or predictions with minimal human intervention. There are a number of ML techniques such as supervised ML (Kotsiantis, 2007), semi-supervised ML, and unsupervised ML (Hastie et al., 2009a). Supervised ML relies on an algorithm to generate a hypothesis or model to predict future values based on input values. Supervised ML requires labelling of data. This means that data would need to be separated into two categories called labels and features. For unsupervised ML, none of the data is labeled. The supervised ML algorithms mainly include k-Nearest Neighbors, decision trees, naï ve Bayes, logistic regression, support vector machines, and neural network (Harrington, 2012;Hastie et al., 2009b). The general process of applying supervised ML to a problem is given by Kotsiantis (2007) (Figueiredo et al., 2011;Ni et al., 2005;Nick et al., 2009;Santos et al., 2016;Worden and Manson, 2007), construction materials (Cheng et al., 2012;Chou et al., 2014;Sonebi et al., 2016), wind energy (Becker and Thrä n, 2017;Clifton et al., 2013;Heinermann and Kramer, 2016), and transportation engineering (Liu et al., 2018a(Liu et al., , 2018b. However, the application of ML techniques in wind engineering is still in its infancy. Wu and Kareem (2011) utilized an artificial neural network framework with the embedded cellular automata scheme to develop a new promising approach to model aerodynamic nonlinearities in the time domain. Aruljayachandran et al. (2016) assessed the fullscale acceleration data obtained from field measurements at six levels of the world's tallest building, Burj Khalifa, by using ML techniques. Li et al. (2018) adopted a data-driven approach using a ML scheme to model vortex-induced vibrations (VIVs) of a suspension bridge based on a database of field measured VIVs of the bridge over six years. A decision tree algorithm was adopted to train the VIV mode classification model and a support vector regression (SVR) algorithm was used to model the VIV response of the bridge deck. The classification and regression models can accurately identify and predict the VIV response for various modes of the bridge. Another interesting study done by Jin et al. (2018) built a data-driven model for predicting the velocity field around a circular cylinder via fusion convolutional neural networks based on measured pressure field on the cylinder. The model was proven to be accurate when compared with CFD results and furthermore it successfully learned the underlying flow regimes of the cylinder. This study aims to build ML regression models for predicting mean and fluctuating pressures around smooth circular cylinders under various combinations of Reynolds number and turbulence intensity in the range with sufficient training data sets. Section 2 provides a literature review on aerodynamic characteristics of circular cylinders and introduces data collected from previous studies. In section 3, ML algorithms adopted in this study are detailed and the prediction models based on these algorithms are built. Performances of the ML models are compared in section 4. Additionally, predictions on mean and fluctuating pressures by using the best models are demonstrated in this section. Discussions and conclusions are given in sections 5 and 6 respectively. Literature review and data collection As mentioned above, flow around circular cylinders is very complex due to the variable separation point, and hence the surface pressure is complex. Overall, the surface pressure around a smooth cylinder is a function of mainly Reynolds number, turbulence intensity of incident wind, and circumferential angle of circular cylinder. Effects of Reynolds number Reynolds number (Re) is a dimensionless parameter defining the ratio of fluid inertia force to viscous forces. It is an important parameter to describe the flow patterns around circular cylinders and also dominate aerodynamic characteristics and surface pressures of the cylinders. The effects of Re on the force coefficients acting on circular cylinders are summarized and depicted in Fig. 3. Evidently, the effects of Re on these coefficients are very complex. Due to the importance of Re in the flow around a circular cylinder, it attracts many studies. In 1930s, Thom (1933) adopted an arithmetical method to solve the equations of viscous flow past a circular cylinder at Re =20, and obtained its drag and pressure distribution. Meanwhile, the surface pressure of the cylinder at very low Re ranging from 3.5 to 240 were measured. To understand the aerodynamic behaviors of a circular cylinder at very high Re from 10 6 to 10 7 , Roshko (1961) measured the pressure distribution, force coefficients, and Strouhal numbers of a large circular cylinder in a pressurized wind tunnel. Another set of experiments at high Re flow around a circular cylinder was also conducted in a pressurized wind tunnel by Achenbach (1968). The drag coefficient, pressure distribution, and separation point position were determined in the Re range from 6×10 4 to 5×10 6 in that study. Jones et al. (1969) measured the force coefficients acting on and surface pressures around a circular cylinder in a two-dimensional flow at Re from 3.6×10 5 to 1.87×10 7 in a transonic wind tunnel. For the critical Re flow around a circular cylinder, Farell and Blessmann (1983) experimentally studied the pressure distribution around and the velocity fluctuation in the wake of the cylinder. Two substates were identified in the critical or lower transition: the first state features symmetric pressure distributions while the second one characterizes by intense flow oscillations. A review of the flow past circular cylinder at different Re was presented by Niemann and Hölscher (1990). Apart from the wind tunnel experimental approach, field measurement on realistic structures with a circular cross section is another effective way to study high Re flow past circular cylinder. Ruscheweyh (1975) measured the wind pressures around a tapered reinforced-concrete television tower at Re as large as 1.8×10 7 . Melbourne et al. (1983) made measurements of the wind-induced responses of and surface pressures around a 265 m high reinforced concrete stack at Re = 2×10 7 . Waldeck (1989) measured mean pressure distribution around a 300 m concrete chimney. A series of field measurements on wind pressures around cooling towers were conducted by Zhao et al. (2017). Effects of turbulence intensity of incident wind Turbulence intensity (Ti) is the ratio of fluctuating component to the associated mean component of wind velocity, which quantifies fluctuation of wind flow. As reported by Norberg (1987), the flow around a circular cylinder is insensitive to Ti at Re lower than about 1000, while the influence of Ti is significant at Re higher than about 1000. Surry (1972) experimentally examined the effects of high Ti on the flow past a circular cylinder at the subcritical Re range. Batham (1973) reported that at Re = 1.11×10 5 and 2.35×10 5 , the presence of turbulence plays a role on suppressing the coherent vortex shedding and leads to a complex pressure field independent of Re. For a similar Re range, from 0.8×10 5 to 6×10 5 , Bruun and Davies (1975) found that the pressure fluctuation magnitudes and correlations on the cylinder windward face are strongly associated with free-stream turbulence. However, on the leeward side, the pressure fluctuations are almost independent of the turbulence. Another study concentrated on the effect of the turbulence on the flow past a circular cylinder at subcritical Re ranging from 5.2×10 4 to 2.09×10 5 was conducted by Sadeh and Saharon (1982). They concluded that the effect of turbulence in this Re range is equivalent to an increase in the Re. That is to say, the existence of incident turbulence alters the surface pressure distribution, shifts the separation point position, and reduces mean drag coefficient. Cheung and Melbourne (1983) studied the effect of turbulence at Re up to 10 6 . Their experimental results showed that the turbulence effects in the subcritical and supercritical Re ranges are opposite. For example, in the subcritical Re range, the base pressure increases while the minimum pressure, mean and fluctuating drag and fluctuating lift decrease with turbulence. However, the effects are opposite in the supercritical Re range. Furthermore, they found that the turbulence effect not only is equivalent to an increase in the Re, but also enhances interactions between the separated shear layers and the wake. Effects of circumferential angle of circular cylinder Flow past a circular cylinder experiences a complicated evolution process along the circumference of the cylinder, which causes a complex pressure distribution around the cylinder surface. When flow approaches the cylinder, it first comes to rest at the front of the cylinder, i.e., the stagnation point with the maximum pressure as shown in Fig. 4. Further downstream, the flow accelerates, and the surface pressure drops and reaches the minima at a certain circumferential angle θm, as shown in Fig. 5. After passing this location with the minimum pressure, the flow slows down and the pressure increases. The point with the local maximum pressure corresponds to the separation point θs. The region downstream the separation point is immersed in the wake region that has a nearly constant pressure. Overall, as depicted in Fig. 5, the mean pressure coefficient (mean Cp) on the cylinder surface decreases from the maxima at the stagnation point, reaches the minima at θm, then increases to the local maxima at θs, and keeps nearly invariant in the wake region. Thus, the effect of the circumferential angle on the mean pressure is significant. Likewise, its effect on the fluctuating pressure coefficient (RMS Cp) cannot be ignored, as can be seen in Fig. 5. Data collection of surface pressures around circular cylinders Data collected from the published high-quality experimental studies were used as data set in training ML models for predicting the mean and fluctuating pressures around circular cylinders. The relevant information of these studies used in the present study is summarized in Table 1. Generally, both the mean and fluctuating pressure at two sides of a cross-section of a circular cylinder is symmetrical with respect to the incident wind direction. In this study, therefore, pressure on only half circle is studied. All studies summarized in Table 1 measured pressures on a half circle or the whole circle of the cylinder section. Therefore, the circumferential angle θ in these studies covers the range from 0 to 180˚. To demonstrate the distribution of the data collected, two scatter plots of the mean pressure and fluctuating pressure data with various combinations of Re and Ti are given in Fig. 6. The red triangular markers in each plot correspond to the data that will be used to validate the final ML models in section 4.2. They were randomly selected and set aside from the training data set. It can be seen that in the Reynolds number range from 10 4 to 10 6 and turbulence intensity range from 0% to 15%, the data sets broadly scatter over the whole range for both mean and fluctuating pressure coefficients. Therefore, it is anticipated that the ML models can cover the above ranges in this study. Machine learning algorithms and prediction model training In this study, three ML algorithms including the decision tree regressor (DTR), random forest (RF), and gradient boosting regression trees (GBRT) were employed to train models for predicting the mean and fluctuating pressure around smooth circular cylinders. The implementation of these algorithms relied on the open-source library scikit-learn. Performance evaluation methods k-fold cross-validation There are a number of methods that have been developed to evaluate the performance of ML models, such as re-substitution, hold-out, crossvalidation (CV), and bootstrap (Reich and Barai, 1999). Ideally, if there is enough data, it is desirable to set aside a validation data set and use this data set to evaluate the performance of the prediction model, which is hold-out validation (Refaeilzadeh et al., 2009). However, this is often impossible since data are usually scarce. To finesse this problem, the CV method is often used to minimize the bias associated with random sampling of training and hold out data samples. The CV method includes leave-one-out and k-fold CV. Most empirical studies found k-fold CV method to be reasonably unbiased and with reasonable variability (Reich and Barai, 1999). The k-fold CV method splits the data into k roughly equal-sized parts (Hastie et al., 2009b). In each of k rounds of model training and validation, it chooses a different data subset for testing and trains the model with the remaining k-1 data subsets, as shown in Fig. 7. The performance of the trained model is evaluated by the test data subset. The accuracy of the ML algorithm is then expressed as average accuracy of the k models in k validation rounds. An appropriate value for k is crucial for the performance of k-fold CV method. Although increasing k leads to more performance estimates and large training set size, the overlap between training sets also increases. Considering these competing factors, Refaeilzadeh et al. (2009) suggested that k = 10 is a decent compromise. Therefore, 10-fold CV method was adopted to evaluate the performance of the ML algorithms in this study. 3-stage evaluation process It is very important to achieve the performance as best as possible, out of the data set when training ML models for practical problems. Therefore, optimizing the hyperparameters of ML algorithms is indispensable during the model training process. Considering the aforementioned k-fold CV method and this optimization procedure, a 3-stage evaluation process called training-testingvalidation (TTV) was proposed by Reich and Barai (1999). This TTV process comprises of three steps. The first step subdivides the data into data for model training and testing. In this study, 90% of the whole data set were used to train the ML model and the rest were used to test the model. The second step selects the best ML algorithms and tunes hyperparameters using the k-fold CV method. The last step builds a model based on the entire training data set by using the best ML algorithms and best hyperparameters. This model is then validated by the testing data set, i.e. 10% of the whole data set. This TTV process and the k-fold CV method are summarized in Fig. 7. Machine learning algorithms and model trainings Decision tree regression The decision tree method trains a supervised ML model whereby the local region is identified in a sequence of recursive splits in a smaller number of steps (Alpaydin, 2014). A decision tree consists of root node, internal decision nodes and terminal leaves (see Fig. 9). Given an input, a test is applied at each node, and one of the branches is taken depending on the outcome. This process starts at the root and is repeated recursively until a leaf node is hit, at which point the value written in the leaf constitutes the output. The classification and regression tree (CART) algorithm, one of the algorithms of implementing decision trees (Breiman, 1984), was utilized to construct decision trees in this study. This algorithm can handle both categorical and numerical dependent variables (Loh, 2014(Loh, , 2011. That is to say, it is able to construct both classification trees and regression trees depending on the variable type, either categorical or numerical. The Gini index is usually used as its impurity function (Loh, 2011). The Gini index is defined as ( ) = 1 − ∑ [ ( \| )] 2 −1 −0 (1) where p(i\|t) denotes the fraction of records belonging to class i at a given node t; c is the number of classes. During the model training process in the present study, effects of the hyperparameters, including the maximum depth of the tree (max_depth), the maximum leaf nodes, and the minimum number of samples required to be at a leaf node (min_samples_leaf), on the performance of the model for the data set of mean pressure coefficients were evaluated as shown in Fig. 8. It can be seen that when the maximum depth and the maximum leaf nodes exceeds exceed 20 and 1250 respectively, and the min_samples_leaf equals 2, the 10-fold mean squared error (MSE) reaches the minimum. Therefore, 20, 1250, and 2 were chosen for the above three parameters k folds k subsets 1 subsets for testing model k-1 subsets for training model Data for model training Data for model testing 1. Divide data into training and testing subsets 2. Select the best ML algorithm and best parameters: a). Test each combination of ML algorithm and its parameters with a k-fold CV test; b). Select the combination that leads to the best CV performance. 3. Assessment of best prediction model: a). Build a model from all training data using the best ML algorithm and best parameters; b). Test the prediction model on the testing subset. respectively for the decision tress regressor (DTR) for the mean pressure coefficients in this study. The same hyperparameter optimization process was performed on the data set of fluctuating pressure coefficients. It was found 20, 1500, and 2 are the optimal values for the three parameters respectively. An exemplary regression tree for demonstrating the usage of DTR to predict the mean pressure coefficients of circular cylinder is given in Fig. 9. Fig. 8 Variations of 10-fold MSE with hyperparameters of DTR for the data set of mean pressure coefficients. Fig. 9 Exemplary regression tree for demonstrating usage of DTR to predict mean pressure coefficients of circular cylinder. θ: circumferential angle; MSE: mean squared error; Re: Reynolds number; Ti: turbulence intensity. Ensemble methods It has been recognized that a single learner (e.g., DTR) is sensitive to training data and it is not robust. This can be overcome by aggregating multiple weak learners, which is a so-called ensemble method (Zhou, 2012). The ensemble method constructs a set of weak learners from training data and performs regression by averaging the predictions by each weak learner, as shown in Fig. 10. According to how the weak learners are generated, there are two ensemble methods: parallel ensemble method and sequential ensemble method (Zhou, 2012). In parallel ensemble method with Bagging as a representative, the weak learners are generated in parallel, while in sequential ensemble method with Boosting as a representative, the weak learners are generated sequentially. It has been proven that an ensemble is usually much more accurate and reliable than a single learner (Lahouar and Ben Hadj Slama, 2017;Ma and Cheng, 2016;Natekin and Knoll, 2013;Wang et al., 2018;Zhou, 2012). (a) Bagging regression trees: Random Forest (RF) Bagging, short for bootstrap aggregating, was introduced by Breiman (1996) and can be applied to regression trees, called bagging regression trees. In the bagging regression trees, a regression tree is trained for each subsamples extracted from samples by using bootstrap sampling method (Hastie et al., 2009b). A prediction is made by averaging the predictions of the trees. This prediction exhibits higher accuracy than that made by a single regression tree. Unfortunately, the trees produced in this approach exhibit a similar tree structure, which is called tree correlation. The higher tree correlation causes a lower prediction performance in the bagging regression trees (Breiman, 2001). Random forest (RF) is an extension of bagging regression trees by incorporating randomized feature selection for reducing the correlation between the trees, as sketched in Fig. 11. RF selects n (a predefined number) features among the total m features for the split in each node. The RF algorithm tries to find the best split among only the n features. The number n is set the same for all prediction trees, and it is recommended to be 1/3m or √2 m (Breiman, 2001). The remainder of the algorithm is similar to the CART algorithm. In Fig. 12, variations of 10-fold MSE with number of grown trees, number of selected features, and the maximum depth of trees are presented for the data set of mean pressure coefficients. As can be seen, MSE drops quickly with increasing maximum depth of trees up to 20. As the maximum depth reaches 20 or 30, MSE decreases with reducing the number of selected features. Furthermore, MSE is generally stable as the number of trees exceeds 125. Therefore, the number of grown trees, the number of selected features, and the maximum depth were set as 150, 1, and 20 respectively for the data set of mean pressure coefficients. Similarly, it was found that 150, 2, and 20 are the optimal values for the three parameters respectively for the data set of fluctuating pressure coefficients. (b) Boosting: Gradient Boosting Regression Trees (GBRT) The main feature of boosting is to add new weak learners to the ensemble sequentially. At each iteration, the ensemble fits a new learner to the difference between the observed response and the aggregated prediction made by all learners grown so far. Boosting regression trees incorporate the strengths of both regression trees, handling various types of predictor variables and accommodating missing data, thus boosting and improving predictive performance by combining many simple learners (Elith et al., 2008). Gradient boosting regression trees (GBRT) is one of the most widely used boosting regression trees and has been recognized as a powerful and successful ML technique in a wide range of practical applications (Natekin and Knoll, 2013;Persson et al., 2017). In GBRT, a gradient descent algorithm is used to minimize the squared error loss function. As mentioned above, a new regression tree is fitted to the current residuals at each boosting iteration. After adding enough regression trees to the ensemble, the training error reaches a minimum. To avoid overfitting, the contribution of each regression tree is scaled by a factor, called the learning rate (Lr) that is substantially less than 1. In other words, the learning rate determines the contribution of each tree to the final model. The prediction values by the final model are computed as the sum of all trees multiplied by the learning rate (Elith et al., 2008). Another important parameter in GBRT is the maximum tree depth (Td) that controls whether interactions are fitted. Lr and Td determines the number of regression trees (Nt) required for the optimal model. Generally, smaller Lr increases regression trees for converge. It has been proven by numerous studies that smaller learning rates result in less test error (Natekin and Knoll, 2013;Persson et al., 2017;Zhang and Haghani, 2015), although it increases computational time. For a given Lr, fitting more trees with larger Td leads to less trees being required for minimum error. Therefore, as Td is increased, Lr must be decreased if sufficient trees are to be fitted (Elith et al., 2008). In addition, to improve the generation capability of the training model, a subsampling procedure was introduced by Friedman (2002). Specifically, a subsample of the training data, quantified by subsampling fraction Fs, is drawn randomly from the full data set for fitting the base learner. Consequently, GBRT optimization involves joint optimization of the above four parameters, i.e., Lr, Td, Nt, and Fs. In this study, the influences of all these four parameters on the performance of the GBRT model were evaluated via a 10-fold CV method. The evaluation results with Fs =0.3 for the data set of mean pressure coefficients are given in Fig. 14 Results and analyses Performance evaluations of ML models In the above section, three ML algorithms (DTR, RF and GBRT) have been tested to train models for predicting pressure coefficients around circular cylinders. Meanwhile, their hyperparameters have been optimized to minimize mean squared error (MSE) of the predictions by using 10-fold CV method. So far, the first two stages of the 3-stage evaluation process in Fig. 7 have been accomplished. In this section, three ML models using the three algorithms with their optimal hyperparameters based on the entire training data set are built and tested on the testing data subset. Comparisons between the predicted mean pressure coefficients and the testing data are given in Fig. 15. Comparisons on a random segment with 100 mean pressure coefficients from the testing data set are shown in Fig. 15(a). In general, all the three ML models are capable of making good predictions. The R-squared (R 2 ) scores of the three models on the test data set in Fig. 15(b) to (d) quantify their performances. Evidently, the two ensemble methods, i.e., RF and GBRT, exhibit a better performance than the single regression tree model (DTR). The GBRT model has the highest score, which implies that this model has the best performance on predicting the mean pressure around circular cylinders. This is further proven by the MSE shown in Fig. 17(a). Likewise, the performances of the three ML models on fluctuating pressure coefficients are compared in Fig. 16 and Fig. 17(b). Similarly, the GBRT model presents the best performance. Therefore, the GBRT models with the optimized hyperparameters are chosen as the ML model for predicting both the mean and fluctuating pressure coefficients. Final ML models for predicting mean and fluctuating pressure coefficients The GBRT model with its optimized hyperparameters has been identified as the best model for predicting both mean and fluctuating pressure coefficients in this study. The final prediction models for mean and fluctuating pressure coefficients are built using the GBRT with its optimized hyperparameters based on the whole dataset including both training dataset and testing dataset, i.e., all the blue circle markers in Fig. 6. To further demonstrate the capability of these two final ML models, predictions on the mean and fluctuating pressures around a smooth circular cylinder for their respective 6 sets of Re and Ti were made and given in Fig. 18 and Fig. 19 respectively. It should be mentioned that the 12 sets of Re and Ti, i.e. the red triangular markers in Fig. 6, were randomly selected and set aside from the training and testing data set at beginning. Therefore, these data sets are eligible to be used to validate the final ML models. Fig. 18 Prediction of mean pressure coefficients around circular cylinders by using final ML model. Experimental data are from: (a) Igarashi (1986); (b) Kiya et al. (1982); (c) Sadeh and Saharon (1982); (d) Sadeh and Saharon (1982); (e) Sun et al. (1992); (f) Basu (1986). As can be seen, the GBRT models accurately predict both the mean and fluctuating pressures under their respective 6 sets of Re and Ti, although minor discrepancies can be found in some cases such as those shown in Fig. 18(a) and Fig. 19(d). Therefore, the two GBRT models are capable of providing accurate predictions of the mean and fluctuating wind pressures around smooth circular cylinder under various combination of Re ranging from 10 4 to 10 6 and Ti ranging from 0% to 15%. It is believed that the GBRT models provide a very efficient and economical alternative to traditional wind tunnel tests and computational fluid dynamic simulations for predicting pressures around smooth circular cylinders in the studied Re and Ti range. Discussions Machine learning technique has been proven successful in predicting the wind pressures around smooth circular cylinders for Re ranging from 10 4 to 10 6 and Ti ranging from 0% to 15% in this study, in spite of minor discrepancies in some cases. It is anticipated that feeding more data to upgrade these models will be able to significantly diminish these discrepancies. Although it has been recognized that surface roughness has a significant effect on the wind pressures around circular cylinders (Güven et al., 1980;Nakamura, 1982), insufficient data addressing this aspect prevents the inclusion of this feature in the ML models. Similarly, the Re ranging from 10 6 to 10 8 is of great interest to wind engineers since many practical wind-sensitive structures, such as buildings with circular features, industrial chimneys and cooling towers experience such a high Re flow (Ke et al., 2017;Zhao et al., 2017;Zou et al., 2018). However, the two ML models are unable to cover this Re range due to a limitation of available data. Nevertheless, it is believed that the two ML models can be readily upgraded to address the high Re flow around circular cylinders when relevant data become available in future. Concluding remarks This study has used machine learning (ML) techniques to predict the wind pressures around circular cylinders. The mean and fluctuating pressure data were collected from previous experimental studies presented in literature. Three ML algorithms, including decision tree regressor, random forest, and gradient boosting regression trees (GBRT), have been tested on the data set. It was found the GBRT models exhibit the best performance for predicting both the mean and fluctuating pressure coefficients. Furthermore, the GBRT models are capable of providing accurate predictions on the mean and fluctuating pressure coefficients around smooth circular cylinders under various combinations of Reynolds number (Re) ranging from 10 4 to 10 6 and flow turbulence intensity (Ti) ranging from 0% to 15%. Thus, the GBRT models provide a very efficient and economical alternative to traditional wind tunnel tests and computational fluid dynamic simulations for predicting the wind pressures around smooth circular cylinder in the studied Re and Ti range. Although ML technique has been successfully used in various engineering fields and has been proven to have a huge potential in solving practical engineering problems, its application in wind engineering is still in the infant stage. This study has employed ML to predict the mean and fluctuating pressure around a smooth cylinder within specific range of Re and Ti, and demonstrated the potential applying ML to wind engineering. Therefore, it is worth devoting more efforts to employ ML techniques to address intransigent issues in the wind engineering field, such as high Re flow and the effects of surface roughness on the pressure distribution around circular cylinder. Fig. 1 1Engineering structures with circular cross sections: (a) cooling tower, (b) cables of bridge, (c) industrial chimney, (d) underwater cables. Fig. 3 3Effects of Reynolds number on force coefficients acting on circular cylinders; -mean drag coefficient, rms -fluctuating lift coefficient; Llaminar in all regions of flow, TrWtransition in wake, laminar elsewhere, TrSLtransition in free shear layers, wake turbulent, TrBLtransition in boundary layers, Tturbulent in all regions of flow (AfterZdravkovich, 1990). Fig. 4 Fig. 5 45Detailed flow pattern around a circular cylinder (from unknown source). Variations of mean and fluctuating pressure coefficients with circumferential angle. θm represents the minimum pressure angular position; θs denotes the separation angle. Mean Cp:Roshko (1961); fluctuating Cp:Batham (1973). Fig. 7 k 7-fold CV method and 3-stage evaluation process (AfterReich and Barai, 1999). Fig. 10 A 10common architecture of ensemble method. of 10-fold MSE with hyperparameters of RF for the data set of mean pressure coefficients. Fig. 13 13General procedure of gradient boosting regression trees. Fig. 14 14Variations of 10-fold MSE with hyperparameters of GBRT for the data set of mean pressure coefficients. Fig. 15 15Comparisons between test mean Cp and mean Cp predicted by ML models. Fig. 16 16Comparisons between test fluctuating Cp and fluctuating Cp predicted by ML models. (a) Mean Cp (b) RMS Cp Fig. 17 17Comparison of mean square errors of ML models for predicting pressure coefficients. Fig. 19 19Prediction of fluctuating pressure coefficients around circular cylinders by using final ML model. Experimental data are from: (a) West and Apelt (1993); (b)Norberg (1986); (c)West and Apelt (1993); (d)Yokuda and Ramaprian (1990); (e)Eaddy (2004); (f)Eaddy (2004). as shown in Fig. 2.Fig. 2 General process of applying supervised machine learning to a problem (After Kotsiantis, 2007).Problem Identification of required data Data pre-processing Selecting training set Algorithm selection Training Evaluation with test set Ok? Parameters tuning Final model Yes No No ML techniques have shown a huge potential of application in various engineering fields, such as structural health monitoring Table 1 1Test details of the studies used in the present studyNo. Experimental studies Reynolds number Turbulence intensity (%) Mean pressure Fluctuating pressure 1 Lockwood and McKinney (1960) 5.02×10 5~1 .064×10 6 0 √ × 2 Roshko (1961) 1.1×10 5 ~ 8.4×10 6 0 √ × 3 Tani (1964) 1.06×10 5~4 .65×10 5 0 √ × 4 Achenbach (1968) 1.0×10 5 ~ 3.6×10 6 0.7 √ × 5 Jones et al. (1969) 5.2×10 5 ~ 1.78×10 7 0.2 √ × 6 Surry (1972) 3.38×10 4 ~ 4.42×10 4 2.5~14.7 √ √ 7 Batham (1973) 1.11×10 5 ~ 2.39×10 5 0.5; 12.9 √ √ 8 Bruun and Davies (1975) 8.0×10 4 ~ 4.8×10 5 0.2~11.0 × √ 9 Güven et al. (1980) 4.1×10 5 0.2 √ × 10 So and Savkar (1981) 2.62×10 4~ 8.21×10 5 0; 10 √ × 11 Sadeh and Saharon (1982) 5.2×10 4 ~ 2.14×10 5 0 √ × 12 Kiya et al. (1982) 2.64×10 4 ~ 3.98×10 4 1.4 ~ 12.8 √ √ 13 Arie et al. 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[ "Practical Inexact Proximal Quasi-Newton Method with Global Complexity Analysis", "Practical Inexact Proximal Quasi-Newton Method with Global Complexity Analysis" ]	[ "Katya Scheinberg ", "Xiaocheng Tang " ]	[]	[]	Recently several methods were proposed for sparse optimization which make careful use of second-order information[11,30,17,3]to improve local convergence rates. These methods construct a composite quadratic approximation using Hessian information, optimize this approximation using a first-order method, such as coordinate descent and employ a line search to ensure sufficient descent. Here we propose a general framework, which includes slightly modified versions of existing algorithms and also a new algorithm, which uses limited memory BFGS Hessian approximations, and provide a global convergence rate analysis in the spirit of proximal gradient methods, which includes analysis of method based on coordinate descent.	10.1007/s10107-016-0997-3	[ "https://arxiv.org/pdf/1311.6547v3.pdf" ]	16,986,924	1311.6547	fa2cb6638bbab2f12e8b26ae59e3e62f13f516c6	Practical Inexact Proximal Quasi-Newton Method with Global Complexity Analysis March 19, 2014 Katya Scheinberg Xiaocheng Tang Practical Inexact Proximal Quasi-Newton Method with Global Complexity Analysis March 19, 2014 Recently several methods were proposed for sparse optimization which make careful use of second-order information[11,30,17,3]to improve local convergence rates. These methods construct a composite quadratic approximation using Hessian information, optimize this approximation using a first-order method, such as coordinate descent and employ a line search to ensure sufficient descent. Here we propose a general framework, which includes slightly modified versions of existing algorithms and also a new algorithm, which uses limited memory BFGS Hessian approximations, and provide a global convergence rate analysis in the spirit of proximal gradient methods, which includes analysis of method based on coordinate descent. Introduction In this paper, we are interested in the following convex optimization problem: min x∈R n F (x) ≡ f (x) + g(x) (1.1) where f, g : R n → R are both convex functions such that ∇f (x) is assumed to be Lipschitz continuous with Lipschitz constant L(f ), i.e., ∇f (x) − ∇f (y) 2 ≤ L(f ) x − y 2 , ∀x, y ∈ R n , and g(x) is convex and has some structure that can be exploited. In particular, in much of the research on first order methods for problem (1.1) g(x) is considered to be such that the following problem has a closed form solution for any z ∈ R n : min x∈R n g(x) + 1 2 x − z 2 . Here our general requirement on g(x) is slightly different -we assume that the following problem is computationally inexpensive to solve approximately, relative to minimizing F (x) for any z ∈ R n and some class of positive definite matrices H, : min x∈R n g(x) + 1 2 x − z 2 H . (1.2) Here y 2 H denotes y Hy. Clearly, the computational cost of approximately solving (1.2) depends on the choice of matrix H and the solution approach. We are particularly interested in the case of sparse optimization, where g(x) = λ x 1 . While the theory we present here applies to the general form (1.1), the efficient method for solving (1.2) that we consider in this paper is designed with g(x) = λ x 1 example in mind. In this case problem (1.2) takes a form of an unconstrained Lasso problem [25]. We consider matrices H which are a sum of a diagonal matrix and a low-rank matrix and we apply randomized coordinate descent to solve (1.2) approximately. An extension to the group sparsity term g(x) = λ x i 2 [18], is rather straightforward. Problems of the form (1.1) with g(x) = λ x 1 have been the focus of much research lately in the fields of signal processing and machine learning. This form encompasses a variety of machine learning models, in which feature selection is desirable, such as sparse logistic regression [29,30,24], sparse inverse covariance selection [11,17,21] and unconstrained Lasso [25], etc. These settings often present common difficulties to optimization algorithms due to their large scale. During the past decade most optimization effort aimed at these problems focused on development of efficient first-order methods, such as accelerated proximal gradient methods [15,2,27], block coordinate descent methods [30,10,9,21] and alternating directions methods [20]. These methods enjoy low per-iteration complexity, but typically have slow local convergence rates. Their performance is often hampered by small step sizes. This, of course, has been known about first-oder methods for a long time, however, due to the very large size of these problems, second-order methods are often not a practical alternative. In particular, constructing and storing a Hessian matrix, let alone inverting it, is prohibitively expensive for values of n larger than 10000, which often makes the use of the Hessian in large-scale problems prohibitive, regardless of the benefits of fast local convergence rate. Nevertheless, recently several new methods were proposed for sparse optimization which make careful use of second-order information [11,30,17,3]. These new methods are designed to exploit the special structure of the Hessian of specific functions to improve efficiency of solving (1.2). Several successful methods employ coordinate descent to approximately solve (1.2). While other approaches to solve Lasso subproblem were considered in [3], none generally outperform coordinate descent, which is well suited when special structure of the Hessian approximation, H, can be exploited and when low accuracy of the subproblem solutions is sufficient. In particular, [30] proposes a specialized GLMNET [10] implementation for sparse logistic regression, where coordinate descent method is applied to the unconstrained Lasso subproblem constructed using the Hessian of f (x). The special structure of the Hessian is used to reduce the complexity cost of each coordinate step so that it is linear in the number of training instances, and a two-level shrinking scheme proposed to focus the minimization on smaller subproblems. Similar ideas are used in [11] in a specialized algorithm called QUIC for sparse inverse covariance selection, where the Hessian of f (x) also has a favorable structure for solving Lasso subproblems. Another specialized method for graphical Markov random fields was recently proposed in [28]. This method also exploits special Hessian structure to improve coordinate descent efficiency. There are other common features shared by the methods described above. These methods are often referred to as proximal Newton-type methods. The overall algorithmic framework can be described as follows: • At each iteration k the smooth function f (x) is approximated near the current iterate x k by a convex quadratic function q k (x). • A working subset of coordinates (elements) of x is selected for subproblem optimization. • Then l(k) passes of coordinate descent are applied to optimize (approximately) the function q k (x) + g(x) over the working set, which results in a trial point. Here l(k) is some linear function of k. • The trial point is accepted as the new iterate if it satisfies some sufficient decrease condition (to be specified). • Otherwise, a line search is applied to compute a new trial point. In this paper we do not include the theoretical analysis of various working set selection strategies. Some of these have been analyzed in the prior literature (e.g., see [13]). Combining such existing analysis with the rate of convergence results in this paper is a subject of a future study. This paper contains the following three main results. 1. We discuss the theoretical properties of the above framework in terms of global convergence rates. In particular, we show that if we replace the line search by a prox-parameter update mechanism, we can derive sublinear global convergence results for the above methods under mild assumptions on Hessian approximation matrices, which can include diagonal, quasi-Newton and limited memory quasi-Newton approximations. We also provide the convergence rate for the case of inexact subproblem optimization. 2. The heuristic of applying l(k) passes of coordinate descent to the subproblem is very useful in practice, but has not yet been theoretically justified, due to the lack of known complexity estimates. Here we use probabilistic complexity bounds of randomized coordinate descent to show that this heuristic is indeed well justified theoretically. In particular, it guarantees the sufficiently rapid decrease of the expectation of the error in the subproblems and hence allows for sublinear global convergence rate to hold for the entire algorithm (again, in expectation). This gives us the first complete global convergence rate result for the algorithmic schemes for practical proximal Newton-type methods. 3. Finally, we propose an efficient general purpose algorithm that uses the same theoretical framework, but which does not rely on the special structure of the Hessian, and yet in our tests compares favorably with the state-of-the-art, specialized methods such as QUIC and GLMNET. We replace the exact Hessian computation by the limited memory BFGS Hessian approximations [16] (LBFGS) and exploit their special structure within a coordinate descent approach to solve the subproblems. Let us elaborate a bit further on the new approaches and results developed in this paper and discuss related prior work. In [4] Byrd et al. propose that the methods in the framework described above should be referred to as sequential quadratic approximation (SQA) instead of proximal Newton methods. They reason that there is no proximal operator or proximal term involved in this framework. This is indeed the case, if a line search is used to ensure sufficient decrease. Here we propose to consider a prox term as a part of the quadratic approximation. Instead of a line search procedure, we update the prox term of our quadratic model, which allows us to extend global convergence bounds of proximal gradient methods to the case of proximal (quasi-)Newton methods. The criteria for accepting a new iteration is based on sufficient decrease condition (much like in trust region methods, and unlike that in proximal gradient methods). We show that our mechanism of updating the prox parameter, based on sufficient decrease condition, leads to an improvement in performance and robustness of the algorithm compared to the line search approach as well as enabling us to develop global convergence rates. Convergence results for, so-called, proximal Newton method have been shown in [12] and more recently in [4] (with the same sufficient decrease condition as ours, but applied within a line search). These results apply to our framework when exact Hessian of f (x) is used to construct q(x). But they do not apply in the case of LBFGS Hessian approximations, moreover they do not provide global convergence rates. To provide such rates we use techniques similar to those in [2] and [23] for the proof of convergence rates of the (inexact) proximal gradient method, where we replace the diagonal Hessian approximation with a general positive definite Hessian approximation matrix. We extend the results in [2] and [23] to accept iterates based on sufficient decrease condition instead of full decrease, which allows more flexibility in the algorithm. Finally, we use the complexity analysis of randomized coordinate descent in [19] to provide a simple and efficient stopping criterion for the subproblems and thus derive the total complexity of proximal (quasi-)Newton methods based on randomized coordinate descent to solve Lasso subproblems. Our theory can be easily extended to the case of inexact gradient computation, in addition to inexact subproblem optimization, similarly to the theory developed in [23]. Another very relevant work [8] was brought to our attention. In that paper the authors analyze global convergence rates of an accelerated proximal quasi-Newton method, as an extension of FISTA method [2]. The convergence rate they obtain match that of accelerated proximal gradient methods, hence it is a faster rate than that of our method presented here. However, as in the case of many accelerated gradient methods, they have to impose much stricter conditions on the Hessian approximation matrix, in particular they require that the difference between any two consecutive Hessian approximations (i.e., H k+1 −H k ) is positive semidefinite. This is a natural extensions of the FISTA's requirement that the prox parameter is never increased. Such a condition is fairly restrictive in practice and in particular would not apply to our simple LBFGS approximation strategy. Additionally, the convergence rate dependence on the error in the subproblem minimization is more complex for the method in [8] and it is unclear whether the use of randomized coordinate descent will maintain the convergence rate of this method, as it does for ours. Investigating accelerated version of our approach without restrictive assumptions on the Hessian approximations and with the use of randomized coordinate descent is a subject of future research. The paper is organized as follows: in Section 2 we describe the algorithmic framework. Then, in Section 3 we present the convergence rate for the exact method using sufficient decrease condition, and address the inexact case in Section 4. We show the convergence analysis for randomized coordinate descent in Section 5. Brief description of the details of our proposed algorithm are in Section 6 and computational results validating the theory are presented in Section 7. Basic algorithmic framework and theoretical analysis The following function is used throughout the paper as an approximation of the objective function F (x). Q(H, u, v) := f (v) + ∇f (v), u − v + 1 2 (u − v), H(u − v) + g(u). For a fixed pointx, the function Q(H, x,x) serves as an approximation of F (x) aroundx. Matrix H controls the quality of this approximation. In particular, if f (x) is smooth and H = 1 µ I, then Q(H, x,x) is a sum of the prox-gradient approximation of f (x) atx and g(x). This particular form of H plays a key role in the design and analysis of proximal gradient methods (e.g., see [2]) and alternating direction augmented Lagrangian methods (e.g, see [20]). If H = ∇ 2 f (x), then Q(H, x,x) is a second order approximation of F (x) [12,22]. In this paper we assume that H is a positive definite matrix such that M I H σI for some positive constants M and σ. Minimizing the function Q(H, u, v) over u reduces to solving problem (1.2). We will use the following notation to denote the accurate and approximate solutions of (1.2). (2.1) p H (v) := arg min u Q(H, u, v), and (2.2) p H,φ (v) is a vector such that : Q(H, p H,φ (v), v) ≤ Q(H, v, v), and Q(H, p H,φ (v), v) ≤ Q(H, p H (v), v) + φ. The method that we consider in this paper computes iterates by (approximately) optimizing Q(H, u, v) with respect to u using some particular H which is chosen at each iteration. The basic algorithm is described in Algorithms 1 and 2. Algorithm 1: Proximal Quasi-Newton method 1 Choose 0 < ρ ≤ 1 and x 0 ; 2 for k = 0, 1, 2, · · · do 3 Choose 0 <μ k , θ k > 0, G k 0; 4 Find H k = G k + 1 2µ k I and x k+1 := p H k (x k ) 5 by applying Prox Parameter Update (μ k , G k , x k , ρ);(p H k (x)) − F (x) ≤ ρ(Q(H k , p H k (x), x) − F (x)) , for a given 0 < ρ < 1, -a step acceptance condition which is a relaxation of conditions used in [2] and [23]. An inexact version of Algorithm 1 is obtained by simply replacing p H k by p H k ,φ k in both Algorithms 1 and 2 for some sequence of φ k values. Sufficient decrease condition and convergence rate In the next three sections we present the analysis of convergence rate of Algorithm 1. Recall that we assume that f (x) is convex and smooth, in other words ∇f (x) − ∇f (y) ≤ L(f ) x − y for all x and y in the domain of interest, while g(x) is simply convex. In Section 5 we assume that g(x) = λ x 1 . Note that we do not assume that f (x) is strongly convex or that it is twice continuously differentiable, because we do not rely on any accurate second order information in our framework. We only assume that the Hessian approximations are positive definite and bounded, but their accuracy can be arbitrary, as long as sufficient decrease condition holds. Hence we only achieve sublinear rate of convergence. To achieve higher local rates of convergence stronger assumptions on f (x) and on the Hessian approximations have to be made, see for instance, [4] and [12] for related local convergence analysis. First we present a helpful lemma which is a simple extension of Lemma 2 in [23] to the case of general positive definite Hessian estimate. This lemma establishes some simple properties of an -optimal solution to the proximal problem (1.2). It uses the concept of the -subdifferential of a convex function a at x, ∂ a(x), which is defined as the set of vectors y such that a(x) − y T x ≤ a(t) − y T t + for all t. Lemma 1 Given > 0, a p.d. matrix H and v ∈ R n , let p (v) denote the -optimal solution to the proximal problem (1.2) in the sense that g(p (v)) + 1 2 p (v) − z 2 H ≤ + min x∈R n g(x) + 1 2 x − z 2 H (3.1) where z = v − H −1 ∇f (v). Then there exists η such that 1 2 η 2 H −1 ≤ and H(v − p (v)) − λ − η ∈ ∂ g(p (v)) (3.2) Proof. (3.1) indicates that p (v) is an -minimizer of the convex function a(x) := g(x) + 1 2 x − z 2 H . If we let a 1 (x) = 1 2 x − z 2 H and a 2 (x) = g(x) , then this is equivalent to 0 ⊂ ∂ a(p (v)) ⊂ ∂ a 1 (p (v)) + ∂ a 2 (p (v)) (3.3) Hence, ∂ a 1 (p (v)) = y ∈ R n \| 1 2 y + H(z − p (v)) 2 H −1 ≤ = y ∈ R n , y = η − H(z − p (v)) \| 1 2 η 2 H −1 ≤ From (3.3) we have H(z − p (v)) − η ∈ ∂ g(p (v)) with 1 2 η 2 H −1 ≤ (3.4) Then (3.2) follows using z = v − H −1 ∇f (v). The following lemma, is a generalization of Lemma 2.3 in [2] and of a similar lemma in [23]. This lemma serves to provide a bound on the change in the objective function F (x). Lemma 2 Given , φ and H such that F (p φ (v)) ≤ Q(H, p φ (v), v) + (3.5) Q(H, p φ (v), v) ≤ min x∈R n Q(H, x, v) + φ where p φ (v) is the φ-approximate minimizer of Q(H, x, v) , then for any u and η such that 1 2 η 2 H −1 ≤ φ 2(F (u) − F (p φ (v))) ≥ p φ (v) − u 2 H − v − u 2 H − 2 − 2φ − 2 η, u − p φ (v) Proof. The proof closely follows that in [2]. Recall that p φ (v) denotes p H,φ (·, v). From (3.5), we have F (u) − F (p φ (v)) ≥ F (u) − Q(µ, p φ (v), v) − = F (u) − (f (v) + g(p φ (v)) + ∇f (v), p φ (v) − v + 1 2 p φ (v) − v 2 H ) − . (3.6) Also g(u) ≥ g(p φ (v)) + u − p φ (v), γ g (p φ (v)) − φ (3.7) by the definition of φ-subgradient, and f (u) ≥ f (v) + u − v, ∇f (v) , (3.8) due to the convexity of f . Here γ g (·) is any subgradient of g(·) and γ g (p φ (v)) satisfies the first-order optimality conditions for φ-approximate minimizer from Lemma 1, i.e., γ g (p φ (v)) = H(v − p φ (v)) − ∇f (v) − η, with 1 2 η 2 H −1 ≤ φ (3.9) Summing (3.7) and (3.8) yields F (u) ≥ g(p φ (v)) + u − p φ (v), γ g (p φ (v)) − φ + f (v) + u − v, ∇f (v) . (3.10) Therefore, from (3.6), (3.9) and (3.10) it follows that F (u) − F (p φ (v)) ≥ ∇f (v) + γ g (p φ (v)), u − p φ (v) − 1 2 p φ (v) − v 2 H − − φ = −H(p φ (v) − v) − η, u − p φ (v) − 1 2 p φ (v) − v 2 H − − φ = 1 2 p φ (v) − u 2 H − 1 2 v − u 2 H − − φ − η, u − p φ (v) . Note that if φ = 0, that is the subproblems are solved accurately, then we have 2( F (u) − F (p(v))) ≥ p(v) − u 2 H − v − u 2 H − 2 . If the sufficient decrease condition (F (x k+1 ) − F (x k )) ≤ ρ(Q(H k , x k+1 , x k ) − F (x k )) (3.11) is satisfied, then F (x k+1 ) ≤ Q(H k , x k+1 , x k ) − (1 − ρ) Q(H k , x k+1 , x k ) − F (x k ) ≤ Q(H k , x k+1 , x k ) − 1 − ρ ρ (F (x k+1 ) − F (x k )) and Lemma 2 holds at each iteration k of Algorithm 1 with = − 1−ρ ρ (F (x k+1 ) − F (x k )) . ISTA [2] is a particular case of Algorithm 1 with G k = 0, for all k, and ρ = 1. In this case, the value of µ k is chosen so that the conditions of Lemma 2 hold with = 0. In other words, the reduction achieved in the objective function F (x) is at least the amount of reduction achieved in the model Q(µ k , p(x k ), x k ). It is well known that as long as µ k ≤ 1/L(f ) (recall that L(f ) is the Lipschitz constant of the gradient) then condition (3.11) holds with ρ = 1. Relaxing condition (3.11) by using ρ < 1 allows us to accept larger values of µ k , which in turn implies larger steps taken by the algorithm. This basic idea is the cornerstone of step size selection in most nonlinear optimization algorithms. Instead of insisting on achieving "full" predicted reduction of the objective function (even when possible), a fraction of this reduction is usually sufficient. In our experiments small values of ρ provided much better performance than values close to 1. We now derive the complexity bound for the exact version of Algorithm 1. Theorem 3 In Algorithm 1, suppose that at iteration k, H k , is chosen so that the sufficient decrease condition (3.11) holds. Then the iterates {x k } in Algorithm 1 satisfy F (x k ) − F (x * ) ≤ x 0 − x * 2 H 2k + (1 − ρ)(F (x 0 ) − F (x * )) 2kρ + k−1 i=0 M i+1 x i − x i+1 2 2k , ∀k, (3.12) where x * is an optimal solution of (1.1) and M i+1 = H i+1 − H i . Thus, the sequence {F (x k )} produced by Algorithm 1 converges to F (x * ) if k−1 i=0 M i+1 x i − x i+1 2 k → 0 as k → 0. Moreover, the number of iterations needed to obtain an -optimal solution is at most O( 1 ), if (3.13) k−1 i=0 M i+1 x i − x i+1 2 ≤ K, for some K > 0 ∀k. The proof of this result is included as a special case in the proof of the convergence rate of the inexact method, which is presented below. Here we will discuss the existence of the uniform bound on k−1 i=0 M i+1 x i − x i+1 2 . First, from the assumption M I H k σI for all k, we have H k − H k+1 = M k+1 ≤ M ≤ M, ∀k, (3.14) where M is the bound on the norm change of the Hessian approximations. We know that µ k , computed in Algorithm 2, is bounded from below by β/L(f ) for any positive definite G k , hence as long as G k are bounded, then (3.14) is satisfied by our algorithm. Moreover, the changes in the Hessian estimates from one iteration to the next ideally are expected to decrease (as they would, if full memory BFGS method is used). While we do not assume that M k → 0, using M i+1 on the right hand side of (3.12) explicitly, instead of the upper bound M , allows us to demonstrate the fact that diminishing change in Hessian approximations will reduce the constants in the bound (3.12). For that same reason, in what follows we will use M instead of M as an upper bound on M i+1 . Recall now our assumption that H k σI for some σ > 0 for all k. This assumption is also easy to enforce, by simply enforcing an upper bound onμ k in Algorithm 1. We note the following simple result from [12]. p(x k ) − x k 2 H k ≤ 2(Q(H k , x k , x k ) − Q(H k , x k+1 , x k )). (3.15) Using (3.14), H k σI and (3.15) we have k−1 i=0 M i+1 x i+1 − x i 2 ≤ M σ k−1 i=0 x i+1 − x i 2 H i ≤ 2M σ k−1 i=0 (Q(H i , x i , x i ) − Q(H i , x i+1 , x i )) ≤ 2M σρ k−1 i=0 (F (x i ) − F (x i+1 )) ≤ 2M σρ (F (x 0 ) − F (x * )). (3.16) Hence we have established bound (3.13) and a sublinear convergence rate of Algorithm 1. Corollary 4 In Algorithm 1, suppose that at each iteration k, H k , is chosen so that the sufficient decrease condition (3.11) holds and M I H k σI for some M, σ > 0. Then the iterates {x k } in Algorithm 1 satisfy F (x k ) − F (x * ) ≤ C k ∀k, (3.17) where x * is an optimal solution of (1.1) and C = x 0 − x * 2 H 0 2 + (1 − ρ) 2ρ + M σρ (F (x 0 ) − F (x * )). Note that ρ, M and σ are parameters that can be controlled (i.e., reduced or increased) at a possible expense of making shorter steps or obtaining less accurate Hessian approximations. In particular, it is possible to set ρ = 1 and M = 0 by reverting the simple proximal gradient algorithm with µ k = 1/L(f ), thus recovering to the standard proximal gradient complexity bound. Analysis of inexact proximal Quasi-Newton method We now follow the theory proposed in [23] to analyze Algorithm 1 in the case when the computation of p H (v) is performed inexactly. In other words, we consider the version of Algorithm 1 (and 2) where we compute x k+1 := p H k ,φ k (x k ). As in the exact case, we need to establish bound (3.13) to obtain global sublinear rate of convergence. We first present a lemma below, which establishes an upper bound on the step size, similar to (3.15), but in the case where the step is not required to be the exact minimizer of the quadratic function Q(·). In fact, the bound holds when the step can be arbitrarily far from the optimal point with the error characterized by φ defined in (3.5). Lemma 5 Consider some vector v ∈ R n , ascalar φ > 0 and a matrix H σI. Let p φ (v) ≡ p H,φ (v) be defined as in (2.2). Then \|\|v − p φ (v)\|\| 2 ≤ 8 σ (Q(H, v, v) − Q(H, p φ (v), v) + φ) (4.1) Proof. We use (3.15) with H = H k and the definition of σ to obtain the upper bounds on the distances between v and p(v) (the exact minimizer of Q(H, u, v)) and between p φ (v) and p(v), respectively, by \|\|v − p(v)\|\| 2 ≤ 2 σ (Q(H, v, v) − Q(H, p(v), v)) \|\|p φ (v) − p(v)\|\| 2 ≤ 2 σ (Q(H, p φ (v), v) − Q(H, p(v), v)) Taking square root on both sides of the above inequalities, we have \|\|v − p(v)\|\| ≤ 2 σ (Q(H, v, v) − Q(H, p(v), v)) \|\|p φ (v) − p(v)\|\| ≤ 2 σ (Q(H, p φ (v), v) − Q(H, p(v), v)) Then, \|\|v − p φ (v)\|\| ≤ \|\|v − p(v)\|\| + \|\|p φ (v) − p(v)\|\| ≤ 2 σ (Q(H, v, v) − Q(H, p(v), v)) + 2 σ (Q(H, p φ (v), v) − Q(H, p(v), v)) ≤ 2 2 σ (Q(H, v, v) − Q(H, p(v), v)) ≤ 2 2 σ (Q(H, v, v) − Q(H, p φ (v), v) + Q(H, p φ (v), v) − Q(H, p(v), v)) ≤ 2 2 σ (Q(H, v, v) − Q(H, p φ (v), v) + φ) where the third and the last inequalities follow from (3.5) the definition of p φ (v). Hence, (4.1) follows by taking square on both sides of the above inequality. Applying (4.1) instead of (3.15) within the derivation of (3.16) and replacing v and p φ (v) in (4.1) with x i , x i+1 , respectively, we have k−1 i=0 M i+1 x i+1 − x i 2 (4.2) ≤ M k−1 i=0 x i+1 − x i 2 ≤ 8M σ k−1 i=0 (Q(H i , x i , x i ) − Q(H i , x i+1 , x i ) + φ i ) ≤ 8M σρ k−1 i=0 (F (x i ) − F (x i+1 )) + 8M σ k−1 i=0 φ i ≤ 8M σρ (F (x 0 ) − F (x * )) + 8M σ k−1 i=0 φ i . Hence bound (3.13) holds with appropriately chosen K if the errors φ i 's are summable. Before presenting the main result, we state the following auxiliary lemma from [23], Lemma 6 Assume that the nonnegative sequence {u k } satisfies the following recursion for all k ≥ 1 u 2 k ≤ S k + k i=1 ν i u i with {S k } an increasing sequence, S 0 ≥ u 2 0 and ν i ≥ 0 for all i. Then, for all k ≥ 1, then u k ≤ 1 2 k i=1 ν i +   S k + 1 2 k i=1 ν i 2   1/2 We can now state the general convergence rate result for the inexact version of Algorithm 1. Theorem 7 Assume that for all iterates {x k } of inexact Algorithm 1 (4.1) holds for some θ > 0, then F (x k ) − F (x * ) ≤ 1 k σ(B k + C) 2 + (2A k + √ C + B k )A k (4.3) with A k = k−1 i=0 2M φ i , C = 1 σ x 0 − x * 2 H 0 + 2(M σ + 1 − ρ) ρσ (F (x 0 ) − F (x * )), B k = 2(M σ + 1) σ k−1 i=0 φ i where x * is an optimal solution of (1.1), M σ = 4M /σ and M and σ are bounds of the smallest and largest eigenvalues of H k . Proof. As is done in [2] and [23], we apply Lemma 2, sequentially, with u = x * and p φ (v) = x i for i = 1, . . . , k + 1. Adding up resulting inequalities, we obtain k−1 i=0 F (x * ) − k−1 i=0 F (x i+1 ) (4.4) ≥ 1 2 k−1 i=0 x i+1 − x * 2 H i − x i − x * 2 H i − k−1 i=0 i − k−1 i=0 φ i − k−1 i=0 η i , x * − x i+1 = 1 2 x k − x * 2 H k − x 0 − x * 2 H 0 − k−1 i=0 M i+1 x i+1 − x i 2 − k−1 i=0 i − k−1 i=0 φ i − k−1 i=0 η i , x * − x i+1 (4.5) ≥ 1 2 − x 0 − x * 2 H 0 − k−1 i=0 M i+1 x i+1 − x i 2 − (1 − ρ)(F (x 0 ) − F (x * )) ρ − k−1 i=0 φ i − k−1 i=0 η i · x * − x i+1 . Here we used the already established bound k−1 i=0 i ≤ (1−ρ)(F (x 0 )−F (x * )) ρ . We can now apply the bound (4.2) with M σ = 4M /σ and obtain k−1 i=0 F (x i+1 ) − k−1 i=0 F (x * ) ≤ 1 2 x 0 − x * 2 H 0 + M σ + 1 − ρ ρ (F (x 0 ) − F (x * )) + (M σ + 1) k−1 i=0 φ i + k−1 i=0 η i · x * − x i+1 . Note that η i ≤ √ 2M φ i . We now need to establish a global bound on x k − x * . This bound, while it still holds, is not needed in the case when all subproblems are solved accurately. Following [23] we use Lemma 6 to bound x k − x * 2 . First, observe in the inequality (4.5) that its left-hand side is always less than zero since F (x * )−F (x i+1 ) ≤ 0 holds in every iteration. Setting the left-hand side to zero and moving \|\|x k − x * \|\| 2 H k to the left, we obtain x k − x * 2 H k ≤ x 0 − x * 2 H 0 + k−1 i=0 M i+1 x i+1 − x i 2 + 2 k−1 i=0 i + 2 k−1 i=0 φ i + 2 k−1 i=0 2M φ i · x * − x i+1 ≤ x 0 − x * 2 H 0 + 2M σ ρ (F (x 0 ) − F (x * )) + 2M σ k−1 i=0 φ i + 2 k−1 i=0 i + 2 k−1 i=0 φ i + 2 k−1 i=0 2M φ i · x * − x i+1 hence, x k − x * 2 ≤ 1 σ x k − x * 2 H k ≤ 1 σ x 0 − x * 2 H 0 + 2M σ ρ (F (x 0 ) − F (x * )) + 2M σ k−1 i=0 φ i + 2 k−1 i=0 i + 2 k−1 i=0 φ i + 2 k−1 i=0 2M φ i · x * − x i+1 Now using Lemma 6 with S k = 1 σ x 0 − x * 2 H 0 + 2Mσ ρ (F (x 0 ) − F (x * )) + 2M σ k−1 i=0 φ i + 2 k−1 i=0 ( i + φ i ) , ν i = 2 √ 2M φ i ) and i = 1−ρ ρ (F (x i ) − F (x i+1 )) we obtain x k − x * ≤ k−1 i=0 2M φ i + 1 σ ( x 0 − x * 2 H 0 + 2(M σ + 1 − ρ) ρ (F (x 0 ) − F (x * )) + 2M σ k−1 i=0 φ i + 2 k−1 i=0 φ i ) + ( k−1 i=0 2M φ i ) 2 Denoting A k = k−1 i=0 √ 2M φ i , C = 1 σ x 0 −x * 2 H 0 + 2(Mσ+1−ρ) ρσ (F (x 0 )−F (x * )) and B k = 2(Mσ+1) σ k−1 i=0 φ i , we obtain x k − x * ≤ A k + C + B k + A 2 k 1/2 Since A i and B i are increasing sequences, we have, for i < k x i − x * ≤ A i + 1 σ ( x 0 − x * 2 H 0 + 2(M σ + 1 − ρ) ρ (F (x 0 ) − F (x * )) + B i + A 2 i 1/2 ≤ A k + 1 σ ( x 0 − x * 2 H 0 + 2(M σ + 1 − ρ) ρ (F (x 0 ) − F (x * )) + B k + A 2 k 1/2 ≤ A k + √ C + B k + A k Using the bound on x i − x * above we can now derive the main complexity bound using k−1 i=0 F (x i+1 ) − k−1 i=0 F (x * ) ≤ 1 2 x 0 − x * 2 H 0 + M σ + 1 − ρ ρ (F (x 0 ) − F (x * )) + (M σ + 1) k−1 i=0 φ i + k−1 i=0 η i · x * − x i+1 ≤ σ(B k + C) 2 + (2A k + √ C + B k )A k . Hence, F (x k ) − F (x * ) ≤ 1 k ( k−1 i=0 F (x i+1 ) − k−1 i=0 F (x * )) ≤ 1 k σ(B k + C) 2 + (2A k + √ C + B k )A k As in [23] it follows that the inexact version of Algorithm 1 converges to the optimal value if ( k−1 i=0 √ φ i )/k → 0 and it has sublinear convergence rate if ∞ i=0 √ φ i is bounded. To ensure such a bound, it is sufficient to require that φ i ≤ α i , for some α ∈ (0, 1) for all i. We are hence interested in practical approaches to subproblem optimization which can be terminated upon reducing the gap in the function value of the subproblem to α i , where 0 < α < 1 is a fixed number and i = 1, 2, . . . is the index of the corresponding iteration of Algorithm 1. The question now is: what method and what stopping criterion should we use for subproblem optimization, so that sufficient accuracy is achieved and no excessive computations are performed, in other word, how can we guarantee the bound on φ i , while maintaining efficiency of the subproblem optimization? It is possible to consider terminating the optimization of the i-th subproblem once the duality gap is smaller than α i . However, checking duality gap can be computationally very expensive. Alternatively one can use an algorithm with a known convergence rate. This way it can be determined apriori how many iterations of such an algorithm should be applied to the i-th subproblem to achieve α i accuracy. In particular, we note that the objective functions in our subproblems are all σ-strongly convex, so a simple proximal gradient method, or some of its accelerated versions, will enjoy linear convergence rates when applied to these subproblems. Hence, after i iterations of optimizing Q i , such a method will achieve accuracy α i , for some fixed α ∈ (0, 1). Under this trivial termination criterion, the subproblem is solved more and more accurately as the outer iteration approaches optimality, which is a common condition required in classic inexact Newton method analysis [6]. Note that the same property holds for any linearly convergent method, such as the proximal gradient or a semi-smooth Newton method, as discussed above. As we pointed out in the introduction, the most efficient practical approach to subproblem optimization, in the cases when g(x) = λ x 1 , seems to be the coordinate descent method. One iteration of a coordinate descent step can be a lot less expensive than that of a proximal gradient or a Newton method. In particular, if matrix H is constructed via the LBFGS approach, then one step of a coordinate decent takes a constant number of operations, m (the memory size of LBFGS, which is typically [10][11][12][13][14][15][16][17][18][19][20]. On the other hand, one step of proximal gradient takes O(mn) operations and Newton method takes O(nm 2 ). Unfortunately, cyclic (Gauss-Seidel) coordinate descent, does not have deterministic complexity bounds, hence it is not possible to know when the work on a particular subproblem can be terminated to guarantee the desired (i.e., α i ) level of accuracy. However, a randomized coordinate descent has probabilistic complexity bounds, which can be used to demonstrate the linear rate of convergence in expectation. Hence we will terminate the randomized coordinate descent after it performs a number of iterations which is a linear function of i (it is possible to simply use i for the number of iteration, but other linear functions of i are more practical, as we will discuss below). In the next section we bring practice and theory together by showing that randomized coordinate descent can guarantee sufficient accuracy for subproblem solutions and hence maintain the sub linear convergence rate (in expectation) of Algorithm 1. Moreover, we show in Section 7, that the randomized coordinate descent is as efficient in practice as the cyclic one. p(x) ← p(x) + z * e j ; 6 Return p(x). Our analysis is based on Richtarik and Takac's results on iteration complexity of randomized coordinate descent [19]. In particular, we make use of Theorem 7 in [19], which we restate below without proof, while adapting it to our context. Lemma 8 Let v be the initial point and Q * := min u∈R n Q(H, u, v). If v i is the random point generated by applying i randomized coordinate descent steps to a strongly convex function Q, then for some constant 0 < α < 1 (dependent only on n and σ -the bound on the smallest eigenvalue of H) we have E[Q(H, v i , v) − Q * ] ≤ α i (Q(H, v, v) − Q * ) (5.1) Next, we establish a bound on the maximal possible reduction of the model function Q(·) objective function value, for any positive definite matrix H 0, and fixed point v ∈ R n . Lemma 9 Assume that for any v ∈ R n such that F (v) ≤ F (x 0 ), all of the the subgradients of F at v are bounded in norm by a constant κ, i.e. ∇f (v) + γ ≤ κ for all γ ∈ ∂g(v). Then the maximum function reduction for Q(·) is uniformly bounded from above by Q(H, v, v) − Q * ≤ R, with R = M κ 2 2σ 2 (5.2) where M and σ are respectively the bounds on the largest and smallest eigenvalues of H and Q * := min u∈R n Q(H, u, v). Proof. Let v * = arg min u∈R n Q(H, u, v) and let γ g (v * ) be any subgradient of g(·) at v * . From the first-order optimality conditions H(v * − v) + ∇f (v) + γ g = 0 (5.3) we can obtain an upper bound on v * − v v * − v = H −1 · ∇f (v) + γ g ≤ κ/σ (5.4) Now, we bound the reduction in the objective function in terms of \|\|v * − v\|\|. From the convexity of g, g(v) − g(v * ) ≤ γ g , v − v * (5.5) Multiplying (5.3) with v * − v, we obtain − ∇f (v), v * − v = v * − v 2 H + γ g , v * − v (5.6) From (5.6), (5.3), (5.5) and the definition of Q, we have Q(H, v, v) − Q * = g(v) − g(v * ) − ∇f (v), v * − v − 1 2 v * − v 2 H ≤ γ g , v − v * + v * − v 2 H + γ g , v * − v − 1 2 v * − v 2 H = 1 2 v * − v 2 H ≤ M κ 2 2σ 2 , which concludes the proof of the lemma. It follows immediately from Lemma 9 that the subproblem optimization error φ i is also bounded for all i, say by Φ, and that Φ ≤ R. Using this bound we now present the auxiliary result that derives the bounds on separate terms that appear on the right hand side of (4.3) and involve φ i . Lemma 10 Assume that φ i are nonnegative bounded independent random variables whose value lies in an interval [0, Φ]. Assume also that E[φ i ] ≤Rᾱ i for all i, and some constants 0 <ᾱ < 1 andR > 0. Then the following inequalities hold E[ k i=1 φ i ] ≤ √Rᾱ 1 − √ᾱ (5.7) E[( k i=1 φ i ) 2 ] ≤ 2Rᾱ (1 − √ᾱ ) 2 (5.8) E[ k i=1 φ i k i=1 φ i ] ≤R √ᾱ √ 1 −ᾱ(1 − √ᾱ ) (5.9) Proof. First we note that E[ φ i φ j ] = E[ φ i ]E[ φ j ] (5.10) due to independence of φ i 's, and E[ φ i ] ≤ E[φ i ] (5.11) due to Jensen inequality and the fact that the square root function is concave and φ i ≥ 0. Then, given (5.10) and (5.11), we derive (5.7) using a bound on E[φ i ] E[ k i=1 φ i ] = k i=1 E[ φ i ] ≤ k i=1 E[φ i ] ≤ k i=1 Rᾱ i/2 ≤ √Rᾱ 1 − √ᾱ . Similarly, we establish (5.8). Observe: E[( k i=1 φ i ) 2 ] = E[ k i,j=1,i =j φ i φ j + k i=1 φ i ] = k i,j=1,i =j E[ φ i φ j ] + k i=1 E[φ i ] = k i,j=1,i =j E[ φ i ]E[ φ j ] + k i=1 E[φ i ] ≤ k i,j=1,i =j E[φ i ] E[φ j ] + k i=1 E[φ i ] ≤ k j=1 Rᾱ j/2 k i=1 Rᾱ i/2 + k i=1Rᾱ i ≤ ( Rᾱ 1 − √ᾱ k 1 − √ᾱ ) 2 +Rᾱ 1 −ᾱ k 1 −ᾱ ≤Rᾱ (1 − √ᾱ ) 2 +Rᾱ 1 −ᾱ ≤ 2Rᾱ (1 − √ᾱ ) 2 To bound E[ k i=1 φ i k i=1 √ φ i ] and hence establish (5.9), we again use Jensen inequality -E(C) 2 ≤ E(C 2 ) for any random variable C. Hence, (E[ k i=1 φ i k i=1 φ i ]) 2 ≤ E[ k l=1 φ l ( k i=1 φ i ) 2 ] ≤ E[ k l=1 φ l k i=1 k j=1 φ j φ i ] (5.12) We now consider three cases, and derive a bound for each, respectively. (i) For any fixed l, consider the case when i = j = l, E[φ l k i =j =l;i,j=1 φ j φ i ] ≤ E[φ l ] k i =j =l;i,j=1 E[φ j ] E[φ i ]] ≤ᾱ lR2 k i =j =l;i,j=1ᾱ i/2+j/2 ≤ᾱ lR2 k i,j=1ᾱ i/2+j/2 ≤ᾱ lR2 ( √ᾱ (1 −ᾱ k/2 ) 1 − √ᾱ ) 2 ≤ᾱ lᾱR 2 (1 − √ᾱ ) 2 (ii) Next, we consider i = j = l or j = i = l, in which case we recall that φ is bounded and lies in an interval [0, Φ], hence E[φ 3/2 i ] ≤ Φ 1/2 E[φ i ], and it follows that E[2 l−1 i=1 φ 3/2 l φ i + 2 k i=l+1 φ 3/2 l φ i ] = 2E[φ 3/2 l ]E[ l−1 i=1 φ i + k i=l+1 φ i ] ≤ 2Φ 1/2 E[φ l ]( l−1 i=1 E[φ i ] + k i=l+1 E[φ i ]) ≤ 2Φ 1/2ᾱlR R ( l−1 i=1ᾱ i/2 + k i=l+1ᾱ i/2 ) ≤ 2Φ 1/2ᾱlR R √ᾱ (1 −ᾱ k/2 ) 1 − √ᾱ ≤ᾱ l 2R √ᾱ ΦR 1 − √ᾱ (iii) Finally, we consider the case when i = j = l. Again, we note E[φ 2 i ] ≤ ΦE[φ i ] from the fact that φ is bounded from above by Φ. Then, E[φ 2 l ] ≤ ΦE[φ l ] ≤ ΦRᾱ l Hence, accounting for all three cases in one sum, we have E[ k l=1 φ l k i=1 k j=1 φ j φ i ] = k l=1 E[φ l k i=1 k j=1 φ j φ i ] ≤R k l=1ᾱ l Φ + 2 √ᾱ ΦR 1 − √ᾱ +ᾱR (1 − √ᾱ ) 2 ≤Rᾱ 1 −ᾱ Φ + 2 √ᾱ ΦR 1 − √ᾱ +ᾱR (1 − √ᾱ ) 2 (5.13) From (5.12) and (5.13) it follows that E[ k i=1 φ i k i=1 φ i ] ≤ E[ k l=1 φ l k i=1 k j=1 φ j φ i ] ≤ Rᾱ 1 −ᾱ Φ + 2 √ᾱ ΦR 1 − √ᾱ +ᾱR (1 − √ᾱ ) 2 since Φ ≤R ≤ R2ᾱ 1 −ᾱ 1 + 2 √ᾱ 1 − √ᾱ +ᾱ (1 − √ᾱ ) 2 . =R √ᾱ √ 1 −ᾱ(1 − √ᾱ ) This completes the proof by establishing inequality (5.9). Complexity of inexact proximal Quasi-Newton method, based on randomized coordinate descent . The key result in this section is showing that, to provide convergence rates developed in the previous sections, the number of the coordinate descent steps should increase as a linear function of the index of the outer iteration. Here, we assume that some linear function of k, l(k) = ak + b is chosen to indicate the number of coordinate steps for the k-th subproblem. In our computational experiments we use a step function which adds n k coordinate decent steps to subproblem optimization after each m iterations, where m is the size of L-BFGS memory and n k is the size of the working set at the k-th iteration -in other words it is the number of variables in the k-th subproblem. While this function is more complicated than a simple linear function of k it can always be bounded from below by such a linear function, hence for simplicity of analysis we consider only simple linear functions l(k). Let us now state the version of Algorithms 1 and 2 which combines the ideas from the previous sections and for which we will be able to derive complexity in expectation. Algorithm 4: Proximal Quasi-Newton method using randomized coordinate descent Else µ = β iμ ; 1 Choose 0 < ρ ≤ 1, a, b > 0 and x 0 ; 2 for k = 0, 1, 2, · · · do 3 Choose 0 <μ k , θ k > 0, G k 0.; 4 Find H k = G k + 1 2µ k I and x k+1 := p H k ,φ k (x k ) Note that φ k and φ used in in the p H,φ k (x) and p H,φ (x) in Algorithms 4 and 5 are used to indicate that the inexact subproblem optimization is performed. The algorithm does not select the values of φ k , the value of each φ k is a consequence of applying ak + b RCD iterations to the subproblem. By Lemma 8 we know that there exists a constant α ∈ (0, 1) such that, if ak + b iterations of RCD are applied for the k-th subproblem, then E(φ k ) ≤ Rα ak+b =Rᾱ k , forR = Rα b andᾱ = α a . Applying now Lemma 10 we can bound each element in (4.3) and derive the following theorem. Theorem 11 Assume that for each iterate {x k } of Algorithm 4 the following iterate {x k+1 } is generated by applying ak + b steps of Algorithm 3, then E[F (x k )] − F (x * ) ≤ ζ k where x * is an optimal solution of (1.1) and ζ is a constant. Proof. Taking expectation on both sides of (4.3) in Theorem 7, we have E[F (x k )] − F (x * ) ≤ 1 k σ 2 E[B k ] + σ 2 C + 2E[A 2 k ] + √ CE[A k ] + E[ B k A k ] (5.14) Next, we show how to bound E[B k ], E[A k ], E[A 2 k ] and E[ √ B k A k ]. We first note a key observation, that, as a result of Lemma 8, Lemma 9 and Algorithm 3, the expectation of subproblem optimization error φ i can be upper bounded by a geometric progression such that E[φ i ] ≤ Rα ai+b =Rᾱ i where 0 < α < 1 and R are specified respectively in Lemma 8 and Lemma 9 andᾱ = α a and R = Rα b . It immediately follows that E[B k ] = 2(M σ + 1) σ k−1 i=0 E[φ i ] ≤ 2(M σ + 1) σ k−1 i=0Rᾱ ≤ 2(M σ + 1)R σ 1 1 −ᾱ Recalling Lemma 10, we then obtain E[A k ] = √ 2M E[ k−1 i=0 φ i ] ≤ √ 2MR 1 − √ᾱ E[A 2 k ] = E[( k i=1 2M φ i ) 2 ] = 2M E[( k−1 i=0 φ i ) 2 ] ≤ 4MR (1 − √ᾱ ) 2 E[ B k A k ] = E[ 2(M σ + 1) σ k i=1 φ i k i=1 2M φ i ] = 4M (M σ + 1) σ E[ k i=1 φ i k i=1 φ i ] ≤ 4M (M σ + 1) σR √ 1 −ᾱ(1 − √ᾱ ) and ζ is equal to ζ =R (M σ + 1) 1 −ᾱ + σ 2 C + 8MR (1 − √ᾱ ) 2 + √ C √ 2MR 1 − √ᾱ + 4M (M σ + 1) σR √ 1 −ᾱ(1 − √ᾱ ) Remark 12 Note that the choice of the number of coordinates decent steps in each applications of RCD, in other words, the choice of constants a and b has a direct affect on the constants in the complexity bound. Clearly taking more RCD steps leads to fewer iterations of the main algorithm, as larger values of a and b lead to smaller value of ζ. On the other hand, each iterations becomes more expensive. We believe that our practical approach described in Section 7 strikes a good balance in this trade-off. Optimization Algorithm In this section we briefly describe the specifics of the general purpose algorithm that we propose within the framework of Algorithms 4, 5 and 3 and that takes advantage of approximate second order information while maintaining low complexity of subproblem optimization steps. The algorithm is designed to solve problems of the form (1.1) with g(x) = λ x 1 , but it does not use any special structure of the smooth part of the objective, f (x). At iteration k a step d k is obtained, approximately, as follows d k = arg min d {∇f (x k ) T d + d T H k d + λ x k + d 1 ; s.t. d i = 0, ∀i ∈ A k } with H k = G k + 1 2µ k I -a positive definite matrix and A k -a set of coordinates fixed at the current iteration. The positive definite matrix G k is computed by a limited memory BFGS approach. In particular, we use a specific form of Hessian estimate, (see e.g. [5,16]), (6.1) G k = γ k I − QRQ T = γ k I − QQ withQ = RQ T , where Q, γ k and R are defined below, Q = γ k S k T k , R = γ k S T k S k M k M T k −D k −1 , γ k = t T k−1 t k−1 t T k−1 s k−1 . (6.2) Note that there is low-rank structure present in G k , the matrix given by QQ, which we can exploit, but G k itself by definition is always positive definite. Let m be a small integer which defines the number of latest BFGS updates that are "remembered" at any given iteration (we used 10 − 20). Then S k and T k are the p × m matrices with columns defined by vector pairs {s i , t i } k−1 i=k−m that satisfy s T i t i > 0, s i = x i+1 − x i and t i = ∇f (x i+1 ) − ∇f (x i ), M k and D k are the k × k matrices (M k ) i,j = s T i−1 t j−1 if i > j 0 otherwise, D k = diag[s T k−m t k−m , ..., s T k−1 t k−1 ]. The particular choice of γ k is meant to promote well-scaled quasi-Newton steps, so that less time is spent on line search or updating of prox parameter µ k [16]. In fact instead of updating and maintaining µ k , exactly as described in Algorithm 2 we simply double γ k in (6.1) at each backtracking step. This can be viewed as choosing µ k = ∞ the first step of backtracking, and µ k = 1/(2 i−1 −1)γ k for the i-th backtracking step, when i > 1. As long as G K in (6.1), is positive definite, with smallest eigenvalue bounded by σ > 0, our theory applies to this particular backtracking procedure. Greedy Active-set Selection A k (I k ) An active-set selection strategy maintains a sequence of sets of indices A k that iteratively estimates the optimal active set A * which contains indices of zero entries in the optimal solution x * of (1.1). We introduce this strategy as a heuristic aiming to improve the efficiency of the implementation and to make it comparable with state-of-the-art methods, which also use active set strategies. A theoretical analysis of the effects of these strategies is a subject of future study. The complement set of A k is I k = {i ∈ P \| i / ∈ A k }. Let (∂F (x k )) i be the i-th component of a subgradient of F (x) at x k . We define two sets, I (1) k = {i ∈ P \| (∂F (x k )) i = 0}, I (2) k = {i ∈ P \| (x k ) i = 0} (6.3) As is done in [30] and [11] we select I k to include the entire set I (2) k and the entire set I (1) k . We also tested a strategy which includes only a small subset of indices from I (1) k for which the corresponding elements \|(∂F (x k )) i \| are the largest. This strategy resulted in a smaller size of subproblems (6.1) at the early stages of the algorithm, but did not appear to improve the overall performance of the algorithm. Solving the inner problem via coordinate descent We apply coordinate descent method to the piecewise quadratic subproblem (6.1) to obtain the direction d k and exploit the special structure of H k . Suppose j-th coordinate in d is updated, hence d = d + ze j (e j is the j-th vector of the identity). Then z is obtained by solving the following one-dimensional problem min z (H k ) jj z 2 + ((∇f (x k )) j + (2H k d) j )z + λ\|(x k ) j + d j + z\| which has a simple closed-form solution [7,11]. The most costly step of an iteration of the coordinate descent method is computing or maintaining vector H k d. Naively, or in the case of general H k , this step takes O(n) flops, since the vector needs to be updated at the end of each iteration, when one of the coordinates of vector d changes. The special form of G k in H k = G k + 1 µ I = γ k I − QQ provides us an opportunity to accelerate this step, reducing the complexity from problem-dependent O(n) to O(m) with m chosen as a small constant. In particular we only store the diagonal elements of G k , (G k ) ii = γ k − q T iq i , where q i is the ith row of the matrix Q andq i is the ith column vector of the matrixQ. We compute (G k d) i , whenever it is needed, by maintaining a 2m dimensional vector v :=Qd, which takes O(2m) flops, and using ( G k d) i = γ k d i − q T i v. After each coordinate step v is updated by v ← v + z iqi , which costs O(m). We also need to use extra memory for cachingQ andd which takes O(2mp + 2m) space. With the other O(2p + 2mn) space for storing the diagonal of G k , Q and d, altogether we need O(4mp + 2n + 2m) space, which is essentially O(4mn) when n m. Computational experiments The aim of this section is to provide validation for our general purpose algorithm, but not to conduct extensive comparison of various inexact proximal Newton approaches. In particular, we aim to demonstrate a) that using the exact Hessian is not necessary in these methods, b) that backtracking using prox parameter, based on sufficient decrease condition, which our theory uses, does in fact work well in practice and c) that randomized coordinate descent is at least as effective as the cyclic one, which is standardly used by other methods. LHAC, for Low rank Hessian Approximation in Active-set Coordinate descent, is a C/C++ package that implements Algorithms 3-4 for solving general 1 regularization problems. We conduct experiments on two of the most well-known 1 regularized models -Sparse Inverse Covariance Selection (SICS) and Sparse Logistic Regression (SLR). The following two specialized C/C++ solvers are included in our comparisons: • QUIC: the quadratic inverse covariance algorithm for solving SICS described in [11]. • LIBLINEAR: an improved version of GLMNET for solving SLR described in [10,30]. Note that both of these packages have been shown to be the state-of-the-art solvers in their respective categories (see e.g. [30,29,11,17]). Both QUIC and LIBLINEAR adopt line search to ensure function reduction. We have implemented line search in LHAC as well to see how it compares to the updating of prox parameter proposed in Algorithm 2. In all the experiments presented below use the following notation. • LHAC: Algorithm 4 with backtracking on prox parameter. • LHAC-L: Algorithm 4 with Armijo line search procedure described below in (7.4). Experimental Settings For all of the experiments we choose the initial point x 0 = 0, and we report running time results in seconds, plotted against log-scale relative objective function decrease given by log( F (x) − F * F * ) (7.1) where F * is the optimal function value. Since F * is not available, we compute an approximation by setting a small optimality tolerance, specifically 10 −7 , in QUIC and LIBLINEAR. All the experiments are executed through the MATLAB mex interface. We also modify the source code of LIBLINEAR in both its optimization routine and mex gateway function to obtain the records of function values and the running time. We note that we simply store, in a double array, and pass the function values which the algorithm already computes, so this adds little to nothing to LIBLINEAR's computational costs. We also adds a function call of clock() at every iteration to all the tested algorithms, except QUIC, which includes a "trace" mode that returns automatically the track of function values and running time, by calling clock() iteratively. For both QUIC and LIBLINEAR we downloaded the latest versions of the publicly available source code from their official websites, compiled and built the software on the machine on which all experiments were executed, and which uses 2.4GHz quad-core Intel Core i7 processor, 16G RAM and Mac OS. The optimal objective values F * obtained approximately by QUIC and LIBLINEAR are later plugged in LHAC and LHAC-L to terminate the algorithm when the following condition is satisfied F (x) − F * F * ≤ 10 −8 (7.2) In LHAC we choseμ = 1, β = 1/2 and ρ = 0.01 for sufficient decrease (see Algorithm 5), and for LBFGS we use m = 10. When solving the subproblems, we terminate the RCD procedure whenever the number of coordinate steps exceeds (1 + k m )\|I k \| (7.3) where \|I k \| denotes the number of coordinates in the current working set. Condition (7.3) indicates that we expect to update each coordinate in I k only once when k < m, and that when k > m we increase the number of expected passes through I l by 1 every m iterations, i.e., after LBFGS receives a full update. The idea is not only to avoid spending too much time on the subproblem especially at the beginning of the algorithm when the Hessian approximations computed by LBFGS are often fairly coarse, but also to solve the subproblem more accurately as the iterate moves closer to the optimality. Note that in practice when \|I k \| is large, the value of (7.3) almost always dominates k, hence it can be lower bounded by l(k) = ak + b with some reasonably large values of a and b, which, as we analyzed in Section 5, guarantees the sub linear convergence rate. We also find that (7.3) works quite well in practice in preventing from "over-solving" the subproblems, particularly for LBFGS type algorithms. In 2 and 4 we plot the data with respect to the number of RCD iterations. In particular Figures 2(a) and 4(a) show the number of RCD steps taken at the k-th iteration, as a function of k. Figures 2(b) and 4(b) show convergence of the objective function to its optimal value as a function of the total number of RCD steps taken so far (both values are plotted in logarithmic scale). Note that RCD steps are not the only component of the CPU time of the algorithms, since gradient computation has to be performed at least once per iteration. In LHAC-L, a line search procedure is employed, as is done in QUIC and LIBLINEAR, for the convergence to follow from the framework by [26]. In particular, the Armijo rule chooses the step size α k to be the largest element from {β 0 , β 1 , β 2 , ...} satisfying F (x k + α k d k ) ≤ F (x k ) + α k σ∆ k (7.4) where 0 < β < 1, 0 < σ < 1, and ∆ k := ∇f T k d k + λ x k + d k 1 − λ x k 1 . In all the experiments we chose β = 0.5, σ = 0.001 for LHAC-L. Sparse Inverse Covariance Selection The sparse inverse covariance selection problem is defined by min X 0 F (X) = − log det X + tr(SX) + λ\|\|X\|\| 1 (7.5) where the input S ∈ R p×p is the sample covariance matrix and the optimization is over a symmetric matrix X ∈ R p×p that is required to be positive definite. For SICS we report results on four real world data sets, denoted as ER 692, Arabidopsis, Leukemia and hereditarybc, which are preprocessed from breast cancer data and gene expression networks. We refer to [14] for detailed information about those data sets. We set the regularization parameter λ = 0.5 for all experiments as suggested in [14]. The plots presented in Figure 1 show that LHAC and LHAC-L is almost twice as fast as QUIC, in the two largest data sets Leukemia and hereditarybc (see Figure 1(c) and 1(d)). In the other two smaller data sets the results are less clear-cut, but all of the methods solve the problems very fast and the performance of LHAC is comparable to that of QUIC. The performances of LHAC and LHAC-L are fairly similar in all experiments. Again we should note that with the sufficient decrease condition proposed in Algorithm 2 we are able to establish the global convergence rate, which has not been shown in the case of Armijo line search. Sparse Logistic Regression The objective function of sparse logistic regression is given by F (w) = λ w 1 + 1 N N n=1 log(1 + exp(−y n · w T x n )) where L(w) = 1 N N n=1 log(1+exp(−y n ·w T x n )) is the average logistic loss function and {(x n , y n )} N n=1 ∈ (R p × {−1, 1}) is the training set. The number of instances in the training set and the number of features are denoted by N and p respectively. Note that the evaluation of F requires O(pN ) flops and to compute the Hessian requires O(N p 2 ) flops. Hence, we chose such training sets for our experiment with N and p large enough to test the scalability of the algorithms and yet small enough to be completed on a workstation. We report results of SLR on four data sets downloaded from UCI Machine Learning repository [1], whose statistics are summarized in Table 1. In particular, the first data set is the well-known UCI Adult benchmark set a9a used for income classification, determining whether a person makes over $50K/yr or not, based on census data; the second one we use in the experiments is called epsilon, an artificial data set for PASCAL large scale learning challenge in 2008; the third one, slices, contains features extracted from CT images and is often used for predicting the relative location of CT slices on the human body; and finally we consider gisette, a handwritten digit recognition problem from NIPS 2003 feature selection challenge, with the feature set of size 5000 constructed in order to discriminate between two confusable handwritten digits: the four and the nine. The results are shown in Figure 3. In most cases LHAC and LHAC-L outperform LIBLINEAR. On data set slice, LIBLINEAR experiences difficulty in convergence which results in LHAC being faster by an order of magnitude. On the largest data set epsilon, LHAC and LHAC-L is faster than LIBLINEAR by about one third and reaches the same precision. Finally we note that the memory usage of LIBLINEAR is more than doubled compared to that of LHAC and LHAC-L, as we observed in all the experiments and is particularly notable on the largest data set epsilon. Conclusion In this paper we presented analysis of global convergence rate of inexact proximal quasi-Newton framework, and showed that randomized coordinate descent can be used effectively to find inexact quasi-Newton directions, which guarantee sublinear convergence rate of the algorithm, in expecta-tion. This is the first global convergence rate result for an algorithm that uses coordinate descent to inexactly optimize subproblems at each iteration. Our framework does not rely or exploit the accuracy of second order information, and hence we do not obtain fast local convergence rates. We also do not assume strong convexity of our objective function, hence a sublinear conference rate is the best global rate we can hope to obtain. In [8] an accelerated scheme related to our framework is studied and an optimal sublinear convergence rate is shown, but the assumptions on the Hessian approximations are a lot stronger in [8] than in our paper, hence the accelerated method is not as widely applicable. The framework studied by us in this paper covers several existing efficient algorithms for large scale sparse optimization. However, to provide convergence rates we had to depart from some standard techniques, such as line-search, replacing it instead by a prox-parameter updating mechanism with a trust-region-like sufficient decrease condition for acceptance of iterates. We also use randomized coordinate descent instead of a cyclic one. We demonstrated that this modified framework is, nevertheless, very effective in practice and is competitive with state-of-the-art specialized methods. Algorithm 2 : 2Prox Parameter Update (μ, G, x, ρ) 1 Select 0 < β < 1 and set µ =μ; 2 for i = 1, 2, · · · do 3 Define H = G + 1 2µ I and compute p(x) := p H (x); 4 If F (p(x)) − F (x) ≤ ρ(Q(H, p(x), x) − F (x)), then output H and p(x), Exit ; 5 Else µ = β iμ ; Algorithm 2 chooses Hessian approximations of the form H k = 1 µ k I + G k . However, it is possible to consider any procedure of choosing positive definite H k which ensures M I H k σI and F by applying Prox Parameter Update with RCD (μ k , G k , x k , ρ, a, b);Algorithm 5: Prox Parameter Update with RCD (μ, G, x, ρ, a, b) 1 Select 0 < β < 1 and set µ =μ; 2 for i = 1, 2, · · · do 3 Define H = G + 1 2µ I, and compute p(x) := p H,φ (x) 4 by applying RCD (Q(H, v, x), x, ak + b ); 5 If F (p(x)) − F (x) ≤ ρ(Q(H, p(x), x) − F (x)), then output H and p(x), Exit ;6 Figure 1 : 1Convergence plots on SICS (the y-axes on log scale). Both axes are in log scale. Change of objective w.r.t. the number of coordinate descent steps. Figure 2 : 2RCD step count of LHAC on different SICS data sets. Figure 3 : 3Convergence plots on SLR (the y-axes on log scale). axes are in log scale. Change of objective w.r.t. the number of coordinate descent steps. Figure 4 : 4RCD step count of LHAC on different SLR data sets. Table 1 : 1Data statistics in sparse logistic regression experiments. 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[ "Optimal Discrimination of Quantum States on a Two-Dimensional Hilbert Space by Local Operations and Classical Communication", "Optimal Discrimination of Quantum States on a Two-Dimensional Hilbert Space by Local Operations and Classical Communication" ]	[ "Kenji Nakahira \nYokohama Research Laboratory\nHitachi, Ltd\n244-0817YokohamaKanagawaJapan\n\nQuantum Information Science Research Center\nICT Research Institute\nTamagawa University\n194-8610MachidaQuantum, TokyoJapan\n", "Tsuyoshi Sasaki Usuda \nSchool of Information Science and Technology\nAichi Prefectural University\n480-1198NagakuteAichiJapan\n\nQuantum Information Science Research Center\nICT Research Institute\nTamagawa University\n194-8610MachidaQuantum, TokyoJapan\n" ]	[ "Yokohama Research Laboratory\nHitachi, Ltd\n244-0817YokohamaKanagawaJapan", "Quantum Information Science Research Center\nICT Research Institute\nTamagawa University\n194-8610MachidaQuantum, TokyoJapan", "School of Information Science and Technology\nAichi Prefectural University\n480-1198NagakuteAichiJapan", "Quantum Information Science Research Center\nICT Research Institute\nTamagawa University\n194-8610MachidaQuantum, TokyoJapan" ]	[]	We study the discrimination of multipartite quantum states by local operations and classical communication. We derive that any optimal discrimination of quantum states spanning a twodimensional Hilbert space in which each party's space is finite dimensional is possible by local operations and one-way classical communication, regardless of the optimality criterion used and how entangled the states are.	10.1109/tit.2016.2549994	[ "https://arxiv.org/pdf/1501.06222v1.pdf" ]	119,212,981	1501.06222	d6d551cfcb0adc45e08d8db7aecb2eafe80bbdd6	Optimal Discrimination of Quantum States on a Two-Dimensional Hilbert Space by Local Operations and Classical Communication 26 Jan 2015 Kenji Nakahira Yokohama Research Laboratory Hitachi, Ltd 244-0817YokohamaKanagawaJapan Quantum Information Science Research Center ICT Research Institute Tamagawa University 194-8610MachidaQuantum, TokyoJapan Tsuyoshi Sasaki Usuda School of Information Science and Technology Aichi Prefectural University 480-1198NagakuteAichiJapan Quantum Information Science Research Center ICT Research Institute Tamagawa University 194-8610MachidaQuantum, TokyoJapan Optimal Discrimination of Quantum States on a Two-Dimensional Hilbert Space by Local Operations and Classical Communication 26 Jan 2015(Dated: January 27, 2015) We study the discrimination of multipartite quantum states by local operations and classical communication. We derive that any optimal discrimination of quantum states spanning a twodimensional Hilbert space in which each party's space is finite dimensional is possible by local operations and one-way classical communication, regardless of the optimality criterion used and how entangled the states are. I. INTRODUCTION If two or more physically separated parties cannot communicate quantum information, their possibilities of measuring quantum states are severely restricted. Intuitively, product states seem to be able to be optimally distinguished using only local operations and classical communication (LOCC), while entangled states seem to be indistinguishable. However, Bennett et al. found that orthogonal pure product states exist that cannot be perfectly distinguished by LOCC [1]. Later, Walgate et al. proved that any two pure orthogonal states in finitedimensional systems can be distinguished with certainty using local operations and one-way classical communication (one-way LOCC) no matter how entangled they are [2]. These results encourage further investigations on the distinguishability of quantum states by LOCC, and several important results have been reported in the case of orthogonal states [3][4][5][6][7][8]. In this paper, we consider only finite-dimensional systems. The problem of LOCC discrimination for nonorthogonal states is much more complicated. One of the main reasons is that perfect discrimination between them is impossible, even without LOCC restriction. Instead, optimal discrimination can be sought. Walgate et al. [2] posed the question: "Can any non-orthogonal states on a two-dimensional (2D) Hilbert space be optimally distinguished by LOCC?" To definitively answer this question, we must consider all optimality criteria. Various optimality criteria have been suggested, such as the Bayesian criterion, the Neyman-Pearson criterion, and the mutual information criterion, but the above question is not answered except for very special cases, such as an optimal error-free measurement for two non-orthogonal pure states [9,10]. Another reason is that optimal discrimination for non-orthogonal states often requires a nonprojective measurement on the space spanned by the given states, while any orthogonal states can be perfectly distinguished by projective measurement. A positive operator-valued measure (POVM) is the most general formulation of a measurement permitted by quantum mechanics and is commonly adopted in quantum infor-mation theory [11]. We denote a measurement on a 2D Hilbert space as a 2D measurement. Some important examples of 2D non-projective measurements are a measurement maximizing the success rate for more than two states on a 2D Hilbert space and a measurement giving the result "don't know" with non-zero probability, such as an inconclusive measurement [12][13][14]. Let H ex be a composite Hilbert space and H sub be a subspace of H ex . For simplicity, we say that a measurement described by the POVM {Π m } on H sub can be realized by LOCC (or one-way LOCC) if there exists an LOCC measurement (or a one-way LOCC measurement) described by the POVM {E m } on H ex such that Π m = P sub E m P sub for any index m, where P sub is the orthogonal projection operator onto H sub . If any measurement on H sub can be realized by LOCC, then any quantum states on H sub can be optimally distinguished using only LOCC. Walgate et al.'s question can be rephrased as "Can any measurement on a 2D Hilbert space be realized by LOCC?" We emphasize that this question would be quite difficult to answer. Instead of a 2D non-projective measurement, one might consider realizing a corresponding projective measurement, which is obtained by Naimark's theorem [15], by LOCC. According to Naimark's theorem, any non-projective measurement can be realized by a projective measurement on an extended Hilbert space. However, if a 2D non-projective measurement has more than two POVM operators, then so does the corresponding projective measurement, and such a measurement often cannot be realized by LOCC [16][17][18]. Thus, this approach cannot directly answer the question. Alternatively, one might try to decompose a given 2D nonprojective measurement into several 2D projective measurements. It is known that there exist "decomposable" measurements, which statistically give the same results as randomly choosing among measurements each of which has fewer POVM operators than the original one [19]. If a 2D measurement can be decomposed into 2D projective measurements, then from [2], it can obviously be realized by LOCC. However, only a few 2D non-projective measurements are decomposable [19]. In this paper, we show that any 2D measurement can be realized by one-way LOCC no matter how many POVM operators it has. Our result answers the above question: A global measurement is not needed for a 2D measurement in finite-dimensional systems, regardless of the optimality criterion used. It is worth noting that the problem of realizing a measurement by one-way LOCC is closely related to realizing a quantum receiver. Realization of an optimal or suboptimal receiver for optical states using linear optical feedback (or feedforward) and photon counting has been widely studied both theoretically and experimentally [21][22][23][24][25][26][27][28][29][30]. This type of receiver performs an individual measurement on each temporal or spatial slot. A measurement can be decomposed into such individual measurements if it can be realized by one-way LOCC; thus, our result indicates that any 2D measurement can be decomposed into individual measurements, at least in finite-dimensional systems. It is often important to investigate whether a measurement can be realized by one-way LOCC to check whether it can be implemented using only feasible resources when the whole system is spatially or temporally separated. In Section II, we present some necessary preliminaries, where we show that any 2D measurement can be realized by one-way LOCC if any measurement with finite rank-one POVM operators on any 2D bipartite Hilbert space in which Alice's subspace is two-dimensional can be realized by one-way LOCC. In Section III, we recall the idea of Walgate et al. [2], which provides a method for realizing a 2D projective measurement by one-way LOCC. In Section IV, we consider realizing a 2D nonprojective measurement by one-way LOCC. We show that, by extending Walgate et al.'s idea, any measurement with finite rank-one POVM operators on any 2D bipartite Hilbert space in which Alice's subspace is twodimensional can be realized by one-way LOCC (Propositions 6 and 8; also Theorem 2). We conclude the paper in Section V. II. PRELIMINARIES We first consider a bipartite system. Let \|ψ and \|φ be two linearly independent quantum states shared by Alice and Bob. We can write, in general form, dim H B . We consider finite-dimensional systems; N A and N B are finite. Assume that Alice and Bob share one of a known collection of L quantum states represented by density operators {ρ l } L l=1 on H and want to optimally discriminate between them in a certain optimality criterion. Our main result is that any 2D measurement (in finite-dimensional systems) can be realized by one-way LOCC (see Corollary 3), which indicates that any optimal discrimination of {ρ l } can be realized by one-way LOCC. \|ψ = n \|p n A \|q n B , \|φ = n \|p n A \|r n B ,(1) We can easily extend our result to multipartite systems in a way similar to [2]. Here, let us imagine a tripartite system: Alice, Bob, and Charlie share two linearly independent quantum states \|ψ and \|φ , which can be represented by \|ψ = n \|p ′ n A \|q ′ n BC , \|φ = n \|p ′ n A \|r ′ n BC ,(2) instead of (1). In (2), Bob and Charlie are first grouped as one party. Asuume that our main result, i.e., Corollary 3, holds in a bipartite system; then, we can show that any measurement on any tripartite 2D Hilbert space can also be realized by one-way LOCC. Indeed, Alice performs a measurement on her system according to the bipartite one-way LOCC protocol that we will propose in this paper and tells the result to Bob and Charlie. Then, Bob and Charlie can again use the same protocol. This argument can easily be extended to any multipartite system, and thus, in the rest of paper, we consider only bipartite systems. First, we show that our main reulst can be reduced to a simpler one. For example, from [20], any quantum measurement with a continuous set of outcomes (including the discrete outcomes) on a finite-dimensional Hilbert space is equivalent to a continuous random choice of measurements with finite outcomes. Thus, it suffices to show that any 2D measurement with finite outcomes can be realized by one-way LOCC. We show the following lemma: Lemma 1 If any measurement with finite rank-one POVM operators on a 2D (2, N )-space (N is finite integer) can be realized by one-way LOCC, then any 2D measurement (in finite-dimensional systems) can be realized by one-way LOCC. Proof Assume that any measurement with finite rankone POVM operators on a 2D (2, N )-space (denoted by H 2 ) can be realized by one-way LOCC. First, we show that any measurement on H 2 can be realized by one-way LOCC. From [20], any quantum measurement, even if with a continuous set of outcomes, on H 2 can always be realized as a random choice of extremal measurements on H 2 , where an extremal measurement is an extremal point of the set of all possible POVMs, which is a convex set. Moreover, from [19], an extremal measurement on H 2 must be made of finite rank-one POVM operators, apart from the trivial POVM {Π 1 = I H2 } (I H2 is the identity operator on H 2 ). The trivial POVM is obviously realized by one-way LOCC; thus, we consider only a nontrivial POVM. Next, we show that a 2D measurement {Π m } (on a 2D (N A , N B )-space) can be realized by one-way LOCC. Let H A and H B be Alice's and Bob's Hilbert spaces, respectively. The case of N A ≤ 2 is trivial; assume that N A > 2. Suppose without loss of generality that N A is even; otherwise, expand Alice's system into a (N A + 1)-dimensional Hilbert space. Alice's system can be represented on the tensor product of two-and (N A /2)-dimensional Hilbert spaces, denoted as H A1 and H A2 , respectively. Since H A ⊗ H B = H A1 ⊗ (H A2 ⊗ H B ) and dim H A1 = 2, {Π m } can be realized by one-way LOCC between H A1 and H A2 ⊗ H B . Thus, it suffices to show that a measurement on a 2D subspace of H A2 ⊗ H B can be realized by one-way LOCC. By repeating this procedure, the problem of realizing {Π m } by one-way LOCC is reduced to the problem of realizing a measurement on a 2D (2, N )-space. Therefore, by the assumption, {Π m } can be realized by one-way LOCC. In this paper, we will prove the following theorem: Theorem 2 Any measurement with finite rank-one POVM operators on a 2D (2, N )-space can be realized by one-way LOCC. From Lemma 1 and Theorem 2, we can easily obtain the following corollary (proof omitted): Corollary 3 Any 2D measurement (in finitedimensional systems) can be realized by one-way LOCC. III. REALIZATION OF 2D PROJECTIVE MEASUREMENT BY ONE-WAY LOCC In this section, using an example, we recall the idea of Walgate et al. [2], which provides a way to realize a 2D projective measurement by one-way LOCC. Let \|ψ = \|S and \|φ = \|T 0 , where \|S = \|+ A \|− B − \|− A \|+ B √ 2 , \|T 0 = \|+ A \|− B + \|− A \|+ B √ 2 ,(3) and {\|+ α , \|− α } (α ∈ {A, B}) is an orthonormal basis (ONB) in H α . In this example, H = span(\|S , \|T 0 ) ⊆ H A ⊗ H B holds. We can easily see that \|S and \|T 0 are orthogonal. If \|+ α and \|− α are the spin-up and spin-down states of a spin-1/2 particle, then \|S and \|T 0 are, respectively, singlet and triplet states of two particles. Suppose that Alice and Bob are spatially separated from each other and share a pair of particles in a state of either \|S or \|T 0 . They want to perfectly discriminate between the orthogonal states \|S and \|T 0 by one-way LOCC. This problem is identical to the problem of realizing the projective measurement {\|S S\| , \|T 0 T 0 \|} on H by one-way LOCC. If Alice simply performs a measurement in the ONB {\|+ A , \|− A }, then Bob cannot discriminate between \|S and \|T 0 ; for example, if the outcome of Alice's measurement is \|+ A , then Bob's state is transformed into \|− B , regardless of whether they share \|S or \|T 0 . Thus, Alice needs to use a proper ONB. \|S and \|T 0 are rewritten as \|S = − \|0 A \|1 B + \|1 A \|0 B √ 2 , \|T 0 = \|0 A \|0 B − \|1 A \|1 B √ 2 ,(4) where {\|0 α = (\|+ α + \|− α )/ √ 2, \|1 α = (\|+ α − \|− α )/ √ 2} (α ∈ {A, B}) is the ONB in H α . Alice may just perform a measurement in the ONB {\|0 A , \|1 A } and tell the result to Bob, and he can then find out which state they share by discriminating between \|0 B and \|1 B . From [2], for any 2D (2, N )-space, H, any ONB {\|π , \|π ⊥ } in H can be represented as the following form in Alice's proper ONB {\|0 A , \|1 A }: \|π = \|0 A \|η 0 B + \|1 A \|η 1 B , \|π ⊥ = \|0 A \|ν 0 B + \|1 A \|ν 1 B ,(5) where \|η k B and \|ν k B are orthogonal for each k ∈ {0, 1} but not necessarily normalized. We can see that (4) is a special form of (5). Similar to the above example, the projective measurement {\|π π\| , \|π ⊥ π ⊥ \|} can be realized by one-way LOCC if Alice measures her side of the system in the ONB {\|0 A , \|1 A } and Bob discriminates between \|η k B and \|ν k B . IV. REALIZATION OF ANY 2D MEASUREMENT BY ONE-WAY LOCC Now, we consider realizing a non-projective measurement {Π m } M m=1 with finite rank-one POVM operators on a 2D (2, N )-space H by one-way LOCC. Let us represent Π 1 as Π 1 = γ 1 \|π π\| ,(6)η k = η k \|η k B , ν k = ν k \|ν k B , k ∈ {0, 1}.(7) From (5), we have η 0 + η 1 = π\|π = 1, ν 0 + ν 1 = π ⊥ \|π ⊥ = 1.(8) Thus, we can assume without loss of generality (by suitably permuting \|0 A and \|1 A ) that η 0 ≥ ν 0 . A. Simple sufficient condition for realization by one-way LOCC In this subsection, we consider the case in which there exist Bob's measurements {Φ Π m = P \|0 0\| A ⊗ Φ (0) m + \|1 1\| A ⊗ Φ (1) m P.(9)= \|0 0\| A ⊗ Φ (0) m + \|1 1\| A ⊗ Φ (1) m .(10) From (9), Π m = P Ω m P holds for any m with 1 ≤ m ≤ M , which means that {Π m } can be realized by one-way LOCC. We can derive a necessary and sufficient condition that there exist Bob's measurements {Φ m } M m=1 satisfying (9) is that {c m } M m=1 exists such that 0 ≤ c m ≤ 1, 1 ≤ m ≤ M, M m=1 c m Π m = Z 0 ,(1) where Z 0 = P (\|0 0\| A ⊗ I B )P.(12) In particular, setting c m = c 2 for any m ≥ 3 in Lemma 5 gives the following proposition (proof in Appendix B). Proposition 6 Any measurement with finite rank-one POVM operators on a 2D (2, N )-space can be realized by one-way LOCC if γ 1 ≥ η 0 − (1 − γ 1 )ν 0 .(13) Note that if {Π m } is a projective measurement, then since γ 1 = 1 holds, (13) always holds. As an example, we consider {Π m } M m=1 (M ≥ 3) on H = span(\|S , \|T 0 ), where Π m = 0 (i.e., γ 1 > 0). For example, a measurement minimizing the average error probability for the M quantum states {α m \|S + β m \|T 0 } M m=1 with \|α m \| 2 + \|β m \| 2 = 1 can often be written as this form. \|π and \|π ⊥ can be written as \|π = x \|S + y \|T 0 , \|π ⊥ = −y * \|S + x * \|T 0 ,(14) with some complex values x and y with \|x\| 2 + \|y\| 2 = 1, where * denotes the complex conjugate. Indeed, we can easily verify that {\|π , \|π ⊥ } is an ONB in H. Substituting (4) into (14), we can represent \|π and \|π ⊥ in the form of (5) as \|π = \|0 A y \|0 B − x \|1 B √ 2 + \|1 A x \|0 B − y \|1 B √ 2 , \|π ⊥ = \|0 A x * \|0 B − y * \|1 B √ 2 − \|1 A y * \|0 B − x * \|1 B √ 2 .(15) From (15), η 0 = ν 0 = 1/2 holds, and thus (13) always holds regardless of γ 1 , x, and y. Therefore, from Proposition 6, {Π m } can be realized by one-way LOCC. Unfortunately, (13) does not always hold. For example, consider the measurement {Π m = \|π m π m \|} 3 m=1 on H = span(\|S , \|T + ) with \|π 1 = 2 3 \|T + , \|π 2 = − 1 6 \|T + + 1 2 \|S , \|π 3 = − 1 6 \|T + − 1 2 \|S ,(16) where \|T + = \|+ A \|+ B . After some algebra, we have η 0 = 1, ν 0 = 1/2, and γ 1 = 2/3, and thus (13) From Proposition 6, all we have to do now to prove Theorem 2 is to show that a measurement {Π m } can be realized by one-way LOCC when (13) does not hold. We here consider making Alice's subsystem interact properly with her auxiliary system. Let H S be Alice's 2D auxiliary system and {\|s 0 , \|s 1 } be an ONB in H S . Also, let L(A) = U (\|s 0 s 0 \| ⊗ A)U † , U = U SA ⊗ I B ,(17) with an operator A on H, where U SA is a unitary operator on H S ⊗ H A . Also, letP = L(P ),ρ l = L(ρ l ), and H = span({ρ l } L m=1 ). We can easily see thatP is the orthogonal projection operator ontoH. We consider the following one-way LOCC measurement: Alice prepares the auxiliary system in a state \|s 0 and transforms ρ l intoρ l = L(ρ l ) using U SA . Then, Alice and Bob perform a measurement (17), for any l with 1 ≤ l ≤ L and m with 1 ≤ m ≤ M , we have {Π m } M m=1 , wherẽ Π m = L(Π m ). Since {Π m } is on H, it follows that {Π m } M m=1 is a 2D measurement onH. FromTr(ρ lΠm ) = Tr((\|s 0 s 0 \| ⊗ ρ l )(\|s 0 s 0 \| ⊗ Π m )) = Tr(ρ l Π m ),(18)Π m = s 0 \|U †Π m U \|s 0 .(19) We consider the case in which there exist measurements {Φ From (20),Π m =PΩ mP holds for any m with 1 ≤ m ≤ M , which means that {Π m } can be realized by one-way LOCC. Using Lemma 7, we can show the following proposition. Proposition 8 Any measurement with finite rank-one POVM operators on a 2D (2, N )-space can be realized by one-way LOCC if γ 1 < η 0 − (1 − γ 1 )ν 0 .(22) Proof Let {Π m } M m=1 be a measurement with rank-one POVM operators on a 2D (2, N ) (20). Also, we showΦ (1) 1 = 0. We choose U SA such that U SA \|s 0 \|0 A = (sin θ \|s 0 + cos θ \|s 1 ) \|0 A , U SA \|s 0 \|1 A = \|s 1 \|1 A(23) for some real number θ. Such U SA is not uniquely determined; we can choose any U SA satisfying (23). Let Z 0 =P (\|s 0 s 0 \| ⊗ I AB )P ,(24) where I AB is the identity operator on H A ⊗ H B . Using Lemma 5 with replacing H A by H S , H B by H A ⊗ H B , and \|0 A by \|s 0 , we find that if {c m } M m=1 exists such that 0 ≤c m ≤ 1, 1 ≤ m ≤ M, M m=1c mΠm =Z 0 ,(25)sin 2 θ = γ 1 η 0 − (1 − γ 1 )ν 0(26) holds (see Appendix C). Note that from (22), the righthand side of (26) does not exeed 1, and thus there exists θ satisfying (26). Next, we show that such measurements {Φ m = \|0 0\| A ⊗ Ψ m ,(27) where {Ψ m } M m=1 is a POVM on Bob's side of the system. Thus, in this case, Bob may simply perform the measurement {Ψ m }. If k = 1, then Alice and Bob have to perform the 2D measurement {Φ m } can be realized by one-way LOCC, since any 2D measurement with less than three non-zero POVM operators can obviously be realized by one-way LOCC [2]. Proof of Theorem 2 Obvious from Propositions 6 and 8. C. Schematic diagram for realizing 2D measurement A schematic diagram of our measurement process in the case of a measurement with finite rank-one POVM operators on a 2D (2, N )-space is sketched in Fig. 1. Alice first determines whether she performs a binary measurement on H A or makes her system interact with her auxiliary system H S followed by performing a binary measurement on H S . The decision rule is given by (13). Then, in the former case, Alice tells the result k to Bob, and he performs a measurement on H (k) B . In the latter case, whether Alice or Bob performs a measurement is determined by the result of Alice's measurement in the ONB {\|s 0 , \|s 1 }. Alice repeats the above sequence the necessary number of times. This procedure stops after a finite number of steps. Bob may perform a measurement only once at an appropriate time. The entire algorithm for realizing such a measurement is found in the following pseudocode: 1: Input: a quantum state ρ l and a POVM {Π m } M m=1 with finite rank-one POVM operators on a 2D (2, N )space. 2: repeat 3: Compute γ 1 , ν 0 , and η 0 from (6) m } M m=1 (Φ (k) m is obtained from (A12)). 8: else 9: Compute U SA such that (23) holds (θ is obtained from (26)). 10: Alice prepares the auxiliary system in a state \|s 0 and transforms ρ l intoρ l = L(ρ l ). V. CONCLUSION In conclusion, we have proved that any 2D measurement in finite-dimensional multipartite systems can be realized by one-way LOCC. This implies that multipartite quantum states on a 2D Hilbert space can always be optimally distinguished by one-way LOCC no matter which optimality criterion is applied. This also means that in a 2D case, any entangled information of quantum states obtained by a global measurement can also be obtained only by one-way LOCC, at least in finitedimensional systems. Since η k η − k and ν k ν − k are 0 or 1, P k is the orthogonal projection operator onto span(η k \|π , ν k \|π ⊥ ) (note that if η k = 0 and ν k = 0, then P k = P ). Also, for any k ∈ {0, 1}, we have T k S k = η − k \|η k η k \| B + ν − k \|ν k ν k \| B = I (k) B , (A4) where the second equality follows since \|η k and \|ν k are orthogonal vectors of H (k) B , and (7) holds. Moreover, for any operator X on H B and k ∈ {0, 1}, we have P (\|k k\| A ⊗ X)P = S k XS † k .(A5) Indeed, from P = \|π π\| + \|π ⊥ π ⊥ \|, we have P (\|k A ⊗ I B ) = (\|π π\| + \|π ⊥ π ⊥ \|)(\|k A ⊗ I B ) = S k ,(A6) where the second line follows from (5). Thus, since \|k k\| A ⊗ X = (\|k A ⊗ I B )X( k\| A ⊗ I B ), (A5) holds. We also define Z k = P (\|k k\| A ⊗ I B )P, k ∈ {0, 1},(A7) which includes the definition of Z 0 in (12). We can easily obtain S k I B = S k from (A1); thus, from (A5), we have Z k = S k I B S † k = S k S † k . (A8) Substituting (A1) into (A8) yields Z k = η k \|π π\| + ν k \|π ⊥ π ⊥ \| .(A9) Necessity Here, we prove the necessity. Since Π m is a rank-one operator, to satisfy (9), there must exist {c m } M m=1 with 0 ≤ c m ≤ 1 such that for any m with 1 ≤ m ≤ M , P \|0 0\| A ⊗ Φ (0) m P = c m Π m , P \|1 1\| A ⊗ Φ (1) m P = (1 − c m )Π m .(A10) In contrast, since {Φ (0) m } is a POVM on H (0) B , M m=1 Φ (0) m = I (0) B holds, where I (k) B is the identity op- erator on H (k) B . Thus, from (A10), we have M m=1 c m Π m = P \|0 0\| A ⊗ M m=1 Φ (0) m P = P (\|0 0\| A ⊗ I (0) B )P = S 0 I (0) B S † 0 = S 0 S † 0 = Z 0 ,(A11) where the third and fifth lines follow from (A5) and (A8), respectively. Therefore, {c m } satisfies (11). Sufficiency Here, we prove the sufficiency. Assume that there exists {c m } M m=1 satisfying (11). It is sufficient to show that POVMs {Φ (0) m } and {Φ (1) m } exist such that (A10) holds. Indeed, in this case (9) is obtained from the sum of the first and second lines of (A10). Let Φ (k) m = c (k) m T k Π m T † k , 1 ≤ m ≤ M, k ∈ {0, 1}, c (0) m = c m , 1 ≤ m ≤ M, c (1) m = 1 − c m , 1 ≤ m ≤ M. (A12) Φ (k) m is obviously a positive semidefinite operator on H c (k) m Π m = Z k , k ∈ {0, 1}. (A13) Thus, from (A12), for any k ∈ {0, 1}, we have M m=1 Φ (k) m = T k Z k T † k = T k S k S † k T † k = I (k) B , (A14) where the second and third equalities follow from (A8) and (A4), respectively. Therefore, {Φ P \|k k\| A ⊗ Φ (k) m P = S k Φ (k) m S † k = c (k) m S k T k Π m T † k S † k = c (k) m P k Π m P k .(A15) Thus, to prove (A10), it suffices to show c (k) m P k Π m P k = c (k) m Π m . Since P k ≥ Z k holds from (A3) and (A9) (A ≥ B denotes that A − B is positive semi-definite), we have P k ≥ Z k ≥ c (k) m Π m , k ∈ {0, 1},(A16) where the second inequality follows from (A13). Thus, since P k is the orthogonal projection operator, c (k) m P k Π m P k = c (k) m Π m holds. Therefore, (A10) holds. We can see that 0 ≤ c m ≤ 1 for any m with 1 ≤ m ≤ M . Indeed, since 0 ≤ ν 0 ≤ 1, 0 ≤ c m ≤ 1 holds for any m ≥ 2. Since η 0 ≥ ν 0 ≥ (1 − γ 1 )ν 0 , which follows from γ 1 > 0, c 1 ≥ 0 holds. Moreover, c 1 ≤ 1 holds from (13). From (B1), we obtain M m=1 c m Π m = c 1 Π 1 + ν 0 (P − Π 1 ) = (c 1 − ν 0 )Π 1 + ν 0 P = (η 0 − ν 0 ) \|π π\| + ν 0 (\|π π\| + \|π ⊥ π ⊥ \|) = η 0 \|π π\| + ν 0 \|π ⊥ π ⊥ \| = Z 0 , where the third line follows from Π 1 = γ 1 \|π π\| and P = \|π π\| + \|π ⊥ π ⊥ \|, and the last line follows from (A9). Therefore, {c m } of (B1) satisfies (11). Appendix C: Supplement of (25) and (26) Assume (26); we will show that there exists {c m } such that (25) andΦ Let P s0 = \|s 0 s 0 \| ⊗ I AB ; then, from (24),Z 0 =P P s0P holds. Thus, we have U †Z 0 U = U †P P s0P U = D † D,(C3) where D = P s0P U. The second equation of (C3) follows from P s0 = P 2 s0 . In contrast, fromP = L(P ) = U (\|s 0 s 0 \| ⊗ P )U † , (17) = (1 −c 2 )Π 1 +c 2 P = (γ 1 + (1 − γ 1 )c 2 ) \|π π\| +c 2 \|π ⊥ π ⊥ \| ,(C7) where the last line follows from Π 1 = γ 1 \|π π\| and P = \|π π\| + \|π ⊥ π ⊥ \|. Thus, from (A9), (C1) (i.e., (25)) is equivalent to γ 1 + (1 − γ 1 )c 2 = η 0 sin 2 θ, c 2 = ν 0 sin 2 θ, so we letc 2 = ν 0 sin 2 θ. 0 ≤c 2 ≤ 1 obviously holds. We can see from (26) that (C8) holds; therefore, (25) holds. where {\|p n A } are quantum states of Alice, and {\|q n B } and {\|r n B } are quantum states of Bob. {\|p n A }, {\|q n B }, and {\|r n B } are generally unnormalized and non-orthogonal. Let H A = span({\|p n A }) and H B = span({\|q n B }, {\|r n B }). Also, let H be a 2D Hilbert space spanned by \|ψ and \|φ . We denote such H as a 2D (N A , N B )-space, where N A = dim H A and N B = B such that for any m with 1 ≤ m ≤ M , Π m is expressed by that satisfy (9) as given in the following lemma (proof in Appendix A).Lemma 5 Let {Π m } M m=1be a 2D measurement with rank-one POVM operators on a 2D (2, N )-space. A necessary and sufficient condition that there exist Bob' m such that, for any m with 1 ≤ m ≤ M ,Π m is expressed bỹ Π m =P \|s 0 s 0 \| ⊗Φ (0) m + \|s 1 s 1 \| ⊗Φ (1) m P . (20)The following lemma states that {Π m } can be realized by one-way LOCC if {Φ } can be realized by one-way LOCC.Lemma 7 Any measurement {Π m } M m=1 with rank-one POVM operators on a 2D (2, N )-space can be realized by one-way LOCC if a unitary operator U SA on H S ⊗ H A exists such that there exist measurements {Φ that can be realized by one-way LOCC and satisfy(20), whereΠ m = L(Π m ).Proof As described above, if {Π m } can be realized by one-way LOCC, then {Π m } can also be realized by oneway LOCC. We consider the following one-way LOCC measurement for {ρ l } (denoted by {Ω m } M m=1 ): Alice first performs a measurement on H S in the ONB {\|s 0 , \|s 1 }. Let k ∈ {0, 1} be its result. Alice and Bob then perform a measurement {Φ (k) m } and regard its result as the result of {Ω m }.Ω m is obviously expressed bỹ Ω m = \|s 0 s 0 \| ⊗Φ (0) m + \|s 1 s 1 \| ⊗Φ (1) m . m } such that(20) holds. We can show that there exists {c m } such that ( can be realized by one-way LOCC. Let k be the outcome of the measurement in the ONB {\|s 0 , \|s 1 }. If k = 0, then, from (23), the state of H A is always projected onto \|0 A , which indicates thatΦ m } has less than M non-zero POVM operators. Therefore, the problem of realizing {Π m } with M POVM diagram for realizing a measurement with finite rank-one POVM operators on a 2D (2, N )-space H by one-way LOCC. Diamonds represent decisions. Rectangles represent measurements on HS, HA, or H(k) B (k ∈ {0, 1}). Each measurement on HS is performed after Alice's state interacts with her auxiliary system. Values in the brackets show the number of measurement outcomes. operators by one-way LOCC is reduced to the problem of realizing {Φ (1) 1 } with M ′ POVM operators by one-way LOCC, where M ′ < M . Therefore, by iteratively performing the procedure stated in this paper, {Φ 11 : 11Alice performs a measurement in the ONB {\|s 0 , \|s 1 } (denote its result as k). 12: if k = 0 then 13: Bob performs a measurement {Ψ m } M m=1 satisfying (27). 14: else 15: Regardρ l and {Φ (1) m } M m=1 as ρ l and {Π m } M m=1 , respectively. 16: end if 17: end if 18: until Bob performs a measurement. 19: Output: the outcome of Bob's measurement. are POVMs satisfying (A10). Since Z 0 + Z 1 = P holds from(8)and (A9), M m=1 (1 − c m )Π m = P − Z 0 = Z 1 holds from (11), which gives M m=1 are POVMs. From (A5) and (A12), for any k ∈ {0, 1}, we have Let {Π m } M m=1 be a measurement with rank-one POVM operators on a 2D (2, N )-space H. Assume that {Π m } satisfies(13). From Lemmas 4 and 5, it suffices to show that there exists {c m } M m=1 satisfying(11). Letc 1 = η 0 − (1 − γ 1 )ν 0 γ 1 ,c m = ν 0 , m ∈ {2, 3, · · · , M }. m = (sin 2 θ)Z 0 . (C1)Premultiplying and postmultiplying both sides of (25) by U † and U , respectively, yield \|s 0 s D = P s0 U (\|s 0 s 0 \| ⊗ P ) = P s0 (sin θ \|s 0 s 0 \| ⊗ \|0 0\| A ⊗ I B )(\|s 0 s 0 \| ⊗ P ) = sin θ \|s 0 s 0 \| ⊗ (I AB (\|0 0\| A ⊗ I B )P ) = sin θ \|s 0 s 0 \| ⊗ ((\|0 0\| A ⊗ I B )P ). (C5) (C3) and (C5) yield U †Z 0 U = sin 2 θ \|s 0 s 0 \| ⊗ (P (\|0 0\| A ⊗ I B )P ) = sin 2 θ \|s 0 s 0 \| ⊗ Z 0 ,(C6)where the second line follows from (A7). From (C2) and (C6),(25) is equivalent to (C1). Now, we show that there exists {c m } such that (. Letc 1 = 1 andc m =c 2 for any m ≥ 3. As shown in the proof of Lemma 5, c m = 1 (i.e.m = Π 1 +c 2 (P − Π 1 ) with 0 < γ 1 ≤ 1 and π\|π = 1. Let \|π ⊥ ∈ H be a normalized vector perpendicular to \|π so that {\|π , \|π ⊥ } is an ONB in H. We choose an ONB {\|0 A , \|1 A } in H A such that \|π and \|π ⊥ are expressed in the form of (5).obviously holds. Also, let P be the orthogonal projection operator onto H and I B be the identity operator on H B . LetLet H (k) B = span(\|η k B , \|ν k B ); then, H B = H (0) B ∪ H (1) B In this case, {Π m } is realized by one-way LOCC when Alice measures her side of the system in the ONB{\|0 A , \|1 A }, as shown in the following lemma. N )-space can be realized by one-way LOCC if there exist Bob's measurements {Φ Proof We consider the following one-way LOCC measurement: Alice measures her side in the ONB {\|0 A , \|1 A } and reports the result k ∈ {0, 1} to Bob, and he then performs a corresponding measurement {ΦLemma 4 Any measurement {Π m } M m=1 with rank-one POVM operators on a 2D (2, (0) m } M m=1 and {Φ (1) m } M m=1 that satisfy (9). (k) m } M m=1 . They regard Bob's result m as the mea- surement outcome. This measurement can obviously be expressed by the POVM {Ω m } M m=1 , where Ω m which means that the measurement {Π m } for {ρ l } is intrinsically equivalent to the measurement {Π m } for {ρ l }. Thus, to show that {Π m } can be realized by one-way LOCC, it suffices to find U SA such that {Π m } can be realized by one-way LOCC. Note that sinceΠ m = L(Π m ), for any m with 1 ≤ m ≤ M , Π m can be expressed by -space H. Assume that {Π m } satisfies (22). From Lemma 7, it suffices to show that a unitary operator U SA on H S ⊗H A exists such that there exist measurements {Φ(0) m } M m=1 and {Φ (1) m } M m=1 that can be realized by one-way LOCC and satisfy (20). First, we show a unitary operator U SA and measure- ments {Φ (0) m } M m=1 and {Φ (1) m } M m=1 that satisfy Compute \|0 A and \|1 A such that (5) holds. 5: if (13) holds then 6: Alice performs a measurement in the ONB {\|0 A , \|1 A } and reports her result k ∈ {0, 1} to Bob.and (7). 4: 7: Bob performs a measurement {Φ (k) ACKNOWLEDGMENTSWe thank O. Hirota and K. Kato of Tamagawa University for the useful discussions we had with them. T. S. U. was supported (in part) by JSPS KAKENHI (Grant No. 24360151).Appendix A: Proof of Lemma 5PreparationsFirst, we define some operators. Let B , respectively. Let P k = S k T k ; then, from (A1) and (A2), for any k ∈ {0, 1}, we have Quantum nonlocality without entanglement. C H Bennett, D P Divincenzo, C A Fuchs, T Mor, E Rains, P W Shor, J A Smolin, W K Wootters, Phys. Rev. A. 5921070C. H. Bennett, D. P. DiVincenzo, C. A. Fuchs, T. Mor, E. Rains, P. W. Shor, J. A. Smolin, and W. K. Wootters, "Quantum nonlocality without entanglement," Phys. Rev. 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[ "Coherent control of low loss surface polaritons", "Coherent control of low loss surface polaritons" ]	[ "Ali Kamli \nInstitute for Quantum Information Science\nUniversity of Calgary\nT2N 1N4CalgaryAlbertaCanada\n\nDepartment of Physics\nKing Khalid University\nP O Box 900361314AbhaSaudi Arabia\n\nThe National Centre for Mathematics and Physics\nKACST\nRiyadhSaudi Arabia\n", "Sergey A Moiseev \nInstitute for Quantum Information Science\nUniversity of Calgary\nT2N 1N4CalgaryAlbertaCanada\n\nKazan Physical-Technical Institute of the Russian Academy of Sciences\n10/7 Sibirsky Trakt420029KazanRussia\n", "Barry C Sanders \nInstitute for Quantum Information Science\nUniversity of Calgary\nT2N 1N4CalgaryAlbertaCanada\n" ]	[ "Institute for Quantum Information Science\nUniversity of Calgary\nT2N 1N4CalgaryAlbertaCanada", "Department of Physics\nKing Khalid University\nP O Box 900361314AbhaSaudi Arabia", "The National Centre for Mathematics and Physics\nKACST\nRiyadhSaudi Arabia", "Institute for Quantum Information Science\nUniversity of Calgary\nT2N 1N4CalgaryAlbertaCanada", "Kazan Physical-Technical Institute of the Russian Academy of Sciences\n10/7 Sibirsky Trakt420029KazanRussia", "Institute for Quantum Information Science\nUniversity of Calgary\nT2N 1N4CalgaryAlbertaCanada" ]	[]	We propose fast all-optical control of surface polaritons (SPs) by placing an electromagnetically induced transparency (EIT) medium at an interface between two materials. EIT provides longitudinal compression and a slow group velocity while matching properties of the two materials at the interface provides strong transverse confinement. In particular we show that an EIT medium near the interface between a dielectric and a negative-index metamaterial can establish tight longitudinal and transverse confinement plus extreme slowing of SPs, in both transverse electric and transverse magnetic polarizations, while simultaneously avoiding losses.	10.1103/physrevlett.101.263601	[ "https://arxiv.org/pdf/0805.1739v2.pdf" ]	39,121,680	0805.1739	b8da877c22d9025e2dfc10457f7c52e9454771e2	Coherent control of low loss surface polaritons 14 Oct 2008 (Dated: October 14, 2008) Ali Kamli Institute for Quantum Information Science University of Calgary T2N 1N4CalgaryAlbertaCanada Department of Physics King Khalid University P O Box 900361314AbhaSaudi Arabia The National Centre for Mathematics and Physics KACST RiyadhSaudi Arabia Sergey A Moiseev Institute for Quantum Information Science University of Calgary T2N 1N4CalgaryAlbertaCanada Kazan Physical-Technical Institute of the Russian Academy of Sciences 10/7 Sibirsky Trakt420029KazanRussia Barry C Sanders Institute for Quantum Information Science University of Calgary T2N 1N4CalgaryAlbertaCanada Coherent control of low loss surface polaritons 14 Oct 2008 (Dated: October 14, 2008) We propose fast all-optical control of surface polaritons (SPs) by placing an electromagnetically induced transparency (EIT) medium at an interface between two materials. EIT provides longitudinal compression and a slow group velocity while matching properties of the two materials at the interface provides strong transverse confinement. In particular we show that an EIT medium near the interface between a dielectric and a negative-index metamaterial can establish tight longitudinal and transverse confinement plus extreme slowing of SPs, in both transverse electric and transverse magnetic polarizations, while simultaneously avoiding losses. Introduction:-All-optical rapid guidance, processing, and control of light in nanophotonic and quantum information applications is important but limited by weak nonlinearities in typical materials, the fast speed of light, and physical limitations to confinement such as enhanced absorption and the narrow spectral width of light. Strategies to overcome these speed and confinement weaknesses include exploiting photonic crystals with defect structures to trap light [1], surface polaritons (SPs) to confine light to wavelength dimensions [2] and use nonlinear interactions [3,4], and electromagnetically induced transparency (EIT) to slow and compress light in the longitudinal propagation direction [5], which are realized in solid state [6], Bose-Einstein condensates [7], and Mott insulators [8]. We propose placing an EIT medium at the interface of two materials. This arrangement benefits from the combination of transverse confinement of surface polaritons with the longitudinal compression and slowing of pulses due to EIT. We derive an analytical solution to the problem, which provides an elegant picture of EIT at a material interface, including hitherto unsuspected properties for the interface between a dielectric and a negative-index meta material (NIMM) [9]: our two key results for the dielectric-NIMM interface are (i) that this interface simultaneously supports both transverse electric (TE) and transverse magnetic (TM) polarizations whereas dielectricdielectric and dielectric-metal interfaces only supports TM, and (ii) SP loss can be made arbitrarily small for the dielectric-NIMM interface but not for the other cases. Our analysis applies to general interfaces, but the dielectric-NIMM interface is especially intriguing and suggests possibilities of low-loss, fast control of both TM and TE polarized SPs. For our scheme in Fig. 1, we suggest end-fire coupling excitation, which has been demonstrated experimentally with a high efficiency of 0.7 [10]. Our approach is to create two SP fields by directing two laser beams at the in- . Two incoming beams from the left waveguide create two SP pulses above the interface between media 1 and 2 in half-spaces z > 0 and z < 0, respectively: control (blue line) and signal probe (red filled). The SPs interact with a collection of Λ medium shown above the interface as a shaded green layer of thickness z0. terface between two materials with permittivities ε i and permeabilities µ i (top: i = 1; bottom: i = 2), and these two SP fields propagate toward the EIT medium comprising three-level Λ atoms (3LA), quantum dots, nitrogenvalence centers in diamond, rare-earth ions in crystals, or similar (henceforth the 'Λ medium'); the three levels are designated \|ℓ for ℓ ∈ {1, 2, 3}, and the transition frequency ω ℓℓ ′ corresponds to \|ℓ ↔ \|ℓ ′ . Our analysis does not restrict the signs of ε and µ hence accommodates dielectrics, surface plasmons at a dielectric-metal interface, and also NIMMs where both ε, µ < 0. Absorption and dispersion of SP fields:-The SP fields, which propagate in the positive x-direction, are characterized by the ω-dependent complex wave vector k + iκ. For k 2 j = k 2 − ω 2 ε j µ j /c 2 , the wave vector com-ponent normal to the interface, k 1 ς 2 = −k 2 ς 1 where ς ≡ µ for the TE mode and ς ≡ ε for the TM mode; both polarizations coexist only if both conditions are simultaneously satisfied. We analyze the SP modes at the interface between a dielectric and a NIMM media. Although the dielectric-NIMM interface is technically challenging, rapid progress is bringing metamaterials to the optical domain [11] thereby opening possibilities for optical storage and control [12]. The dielectric-NIMM interface is especially attractive because both TE and TM polarization modes can co-exist and, as we show below, complete suppression of SP losses is possible in principle thereby admitting novel opportunities in SP field control. Optical properties of the NIMM can be modeled with complex dielectric permittivity and magnetic permeability given by [9,10]: ς 2 (ω) = ς r + iς i = 1 − ω 2 f ω(ω + iγ f )(1) for the two cases ς ≡ ε, f ≡ e and ς ≡ µ, f ≡ m. Here ω e,m are the electric and magnetic plasma frequencies of the NIMM and γ e,m the loss rates, respectively. Accounting for complex ε and µ in the surface boundary conditions of SP wave vectors, we find (TM case) (2) and the TE case is similar. The real (imaginary) part of Eq. (2) yields SP TM mode dispersion (loss). We study the system numerically at room temperature and at optical frequencies to determine its features. For metal, we use the values for Ag [13]: ω e = 1.37 × 10 16 s −1 , γ e = 2.73×10 13 s −1 and ε 1 = 1.3, µ 1 = 1. As NIMM technology is embryonic, we consider a wide range of magnetic plasmon frequency such that ω m ≤ 0.5ω e [10], and γ m is between 10 −5 γ e and γ e itself. k (ω) + iκ(ω) = ω c ε 1 ε 2 (ε 2 µ 1 − ǫ 1 µ 2 )/(ε 2 2 − ε 2 1 ), Numerical results for κ(ω) are shown in Fig. 2 and reveal a deep abyss for all ω; the abyss frequency ω 0 corresponds to a specific ratio of magnetic and electric loss for each ω, and Fig. 2 reveals κ(ω 0 ) ∼ 0, i.e. a complete cancelation of losses. From Eq. (2) ω 0 is determined with high accuracy from µ i ε i = µ r (ε 2 r + ε 2 1 ) − 2ε r ε 1 ε r (ε 2 r − ε 2 1 ) ,(3) which cannot be satisfied for an interface between a dielectric and metal because µ 1 , µ 2 > 0. Figure 3 compares SP losses at a dielectric-NIMM interface for γ m = 10 11 s −1 and ω m = 0.5ω p and surface plasmon polaritons at a dielectric-metal interface. We observe that NIMMs enable absorption losses to be drastically reduced in a narrow frequency band, as studied for freely propagating fields [14]. Here we predict a similar effect but for SP fields using a typical NIMM interface. Equation (3) shows that ε r , µ r < 0 implies destructive interference between electric and magnetic absorption responses, which explains the high suppression of losses around ω 0 . The relative weakness of decay rates makes the frequency ω 0 sensitive to electric and magnetic decoherence rates as is evident from Fig. (2). Henceforth we demonstrate the possibility of coherent control of slow SP within frequency range around ω 0 of complete reduction of their losses in basic materials. EIT control of SP modes:-Here we give a general analysis of EIT based coherent control of SP pulse interacting with a Λ medium near the surface. We assume that the probe field in the TM mode has a frequency ω 31 and control field frequency equal to ω 32 . The transition \|1 ↔ \|2 is dipole-forbidden. The SP electric field near the surface is obtained from field quantization [15] in a dispersive medium [16]. In the plane wave expansion over modes indexed by λ, E(x, z) = λ dk E 0λ k , z â λ k e ik x + hc , (4) with â λ k ,â † λ ′ (k ′ ) = 2πδ λλ ′ δ k − k ′ .(5) We develop the theory for TM; the TE case is then a straightforward generalization. Dropping λ, we obtain E 0 k , z = e x + ie z k /k 1 E 0 k e −k1z , z > 0, e x − ie z k /k 2 E 0 k e k2z , z < 0 .(6) Here e x , e z are unit vectors along the x, z directions, with electric field amplitude E 0 k = ω k /2πε o L y L z (ω, ε, µ),(7) and transverse quantization length L z = D + ω 2 (k ) c 2 S with D = ∂ ∂ω (ωε 1 ) k 2 1 + k 2 k 3 1 + ∂ ∂ω (ωε 2 ) k 2 2 + k 2 k 3 2 , S = ∂ ∂ω (ωµ 1 ) ε 2 1 k 3 1 + ∂ ∂ω (ωµ 2 ) ε 2 2 k 3 2 .(8) These quantities depend on the interface parameters and on the SP mode dispersion relation ω k of Eq. (2). Adopting the usual EIT approximations [4,17] for the evolution of a SP interacting with a Λ medium, we find the Fourier SP field equation (∂/∂x − iν/v 0 )Â(ν, x) = − [α(ν) + κ(ω 31 )]Â(ν, x), (9) on the surface:Â(ν, x) = (2π) −1 dte iνtÂ (t, x, z = 0), whereÂ(t, x, z = 0) = dk E 0 k â(k , t)e ik s x . By solving Eq. (9) we obtain the electric field over the surfacê A(t, x, z > 0) = e x + ie z k (ω 31 )/k s 1 e −k s 1 z dν e −iνt+[i ν v 0 −α(ν)−κ(ω31)]xÂ (ν, 0)(10) where ν ≡ ω k − ω 31 is the SP field detuning from the central frequency ω 31 , which is assumed to be close to ω 0 . Here v 0 = ∂ω/∂k is the SP group velocity without a 3LA Λ medium at ω = ω 31 , and α(ν) 2π = \|g\| 2 v 0 ∞ 0 Ly 0 dydz n(r)(γ 21 − iν)e −2k s 1 z \|Ω c (r)\| 2 − (ν + iγ 21 ) (ν + iΓ 31 )(11) yields dispersion and absorption of the SP field, with Γ 31 a linewidth, Ω c (r) the control field Rabi frequency, g = d · e x + ie z k /k 1 E 0 k (ω 31 ) / the SP-Λ coupling constant, d the atomic dipole moment, and n(r) the Λ medium density. These parameters can be optimized for SP field control. Also the control field, which yields Ω c , can be a freely propagating mode or a SP TE or TM field. Henceforth we assume \|Ω c (r)\| 2 = \|Ω\| 2 e −2k c 1 z , where k s 1 and k c 1 are the probe and control wave vectors in the z-direction for medium 1 as in Eq. (6). Absorption and dispersion of the SP-field in the presence of a 3LA medium is given by α(ν) + κ(ω 31 ) with inhomogeneous broadening at resonance transition h = ∆ w /π(∆ 2 + ∆ 2 w ) and inhomogeneous broadening width ∆ w . Thus Γ 31 = ∆ w + γ 31 where we have assumed that all atoms share the same decay constants γ mn ; this assumption is reasonable for solid-state systems at liquid He temperatures because γ 31 /∆ w is negligible [4]. From Eq. (11), it follows that the field amplitude effectively is bounded by z ≤ min{1/k s 1 , 1/k c 1 }, and we assume confinement of the Λ medium to this height above the interface. Eq. (11) is integrable for constant density n(r) ≡ n(0 < z < z o ) = n, yields α(ν, z 0 ) = α 0 (ω 31 )G (k s 1 , k c 1 , z 0 , β(ν))(12) with G(k s 1 ,k c 1 , z 0 , β) = iΓ 31 ν + iΓ 31 2 F 1 1, k s 1 k c 1 , k s 1 + k c 1 k c 1 , 1 β(ν) − e −2k s 1 z0 2 F 1 1, k s 1 k c 1 , k s 1 + k c 1 k c 1 , e −2k c 1 z0 β(ν)(13) a spectral function, 2 F 1 the hypergeometric function, and β ≡ (ν+iγ 21 )(ν+iΓ 31 )/\|Ω\| 2 . Function (13) is a maximum G = 1 for ν = 0 and depends on ∆ w , k s 1 /k c 1 , z 0 , and Ω c , which provides rich opportunities for spectral and spatial control of the SP field. As the formula is integrable, the resonant absorption coefficient for the Λ medium at ω 31 is expressed in a simple form if the control field is off and z 0 ≫ 1/k s 1 : α 0 (ω 31 ) = πnL y \|g\| 2 /k s 1 v 0 (ω 31 )Γ 31(14) with coupling constant \|g\| 2 ∼ 1/L z . An appropriate choice of materials can lead to small L z hence considerably enhance the SP field amplitude and increase interaction coupling between the SP field and the 3LA Λ medium. Numerical example:-Using our solution we demonstrate the exciting possibility of EIT control for the low loss SP modes, which opens opportunities to exploit spatial confinement and temporal control of SPs via their interaction with the Λ medium. For L y = 2.5µm, 0.4087ω e < ω 31 < 0.4097ω e , we find κ(ω 31 ) < 0.01κ 0 , v 0 (ω 31 ) ≈ 0.6c, k s 1 (ω 31 ) ≈ 1µm −1 . From Eqs. (7) and (14) for resonant optical transitions of rare earth ions in crystals, e.g. Pr +3 -doped Y 2 SiO 5 (demonstrated for EIT experiments [6]) with density n ≈ 10 24 m −3 , Γ 31 ∼ = 10 9 rad/s, γ −1 21 ≫ 1µs, ε 1 ∼ = 1.3, and \|d\| ≈ −ea 0 (with e the electron charge and a 0 the Bohr radius), we find α 0 (ω 31 ) ≈ 10µm −1 . Using Eq. (10) we compute time delay and group velocity for a Gaussian amplitude envelope of the SP probe input pulse exp −(t/δt) 2 /2 in the medium at x = 0. Input and output pulse profiles are presented in Figs. 4, 5 for input pulse duration δt = 100ns, for media lengths x 1 = 1mm and x 2 = 3mm (< L = 1/κ). As seen in Fig. 4 the time delay decreases with the control field amplitude as Ω −2 . Fig. 5 shows the pulse profile for Ω = Γ 31 when it propagates a distance x 1 = 1mm and x 2 = 3mm. The time delays are t delay = 2δt, 6δt respectively, whereas the amplitude has decreased only by factors 0.85 and 0.65, respectively. Thus the propagation length increased by more than 500/α 0 (ω 31 ) due to EIT of the SP field. Using these results we estimate the group velocity v g ≈ 5000m/s and a compressed longitudinal envelope of the SP pulse l SP = v g δt = 0.5mm, i.e. much smaller than the medium size. Thus the SP pulse can be successfully stored in the long-lived atomic coherence ρ 12 (t, x) and subsequently retrieved in accordance with an EIT quantum memory protocol [17]. Conclusion:-We have demonstrated low loss SPs at the interface between two media with quite general electromagnetic properties, including dielectrics, metals and metamaterials, and derived a closed form solution that provides deep insight into the system and control of spatially confined slow SP fields. We show that light pulses can be stored at the interface of two media exploiting EIT and low loss SP fields near a NIMM-dielectric interface. We gratefully acknowledge financial support from iCORE, NSERC, CIFAR, KACST (Saudi Arabia), and RFBR grant # 06-02-16822. FIG . 1: (Color online) End-fire coupling scheme for coherent control of surface polaritons (SPs) FIG. 2 :FIG. 3 : 23(Color online) Absorption loss for surface polaritons as a function of frequency ω/ωe and magnetic decoherence rate γm/γe , where κ0 = 10 4 m −1 , γe = 2.73 × 10 13 s −1 , ωm = 0.5ωe and ε1 = 1.3, µ1 = 1. (Color online) Comparison of SP losses at a dielectric-NIMM interface (red curve) and losses of surface plasmons at a dielectric-metal interface (blue curve) for γm = 10 11 s −1 , γe = 2.73 × 10 13 s −1 and ωm = 0.5ωe, ωe = 1.37 × 10 16 s −1 , ω0 ∼ = 0.4092ωe. online) 3D graph of pulse propagation profile as a function of time tΓ31 and control field amplitude Ω/Γ31, Γ31 = 10 9 rad/s, κ(ω31) = 0.01κ0 = 0.1 × 10 3 m −1 . online) The pulse propagation profile as a function of time, tΓ31, at different locations near the interface: blue (solid) at x = 0, red (dashed) at x1 = 1mm, and brown (dotted) at x2 = 3mm. * Electronic address: [email protected] † Electronic address: [email protected]. * Electronic address: [email protected] † Electronic address: [email protected] S G Johnson, J D Joannopoulos, Photonic Crystals: The Road from Theory to Practice. KluwerDordrechtS. G. Johnson and J. D. Joannopoulos, Photonic Crys- tals: The Road from Theory to Practice (Kluwer, Dor- drecht, 2002). 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[ "X-ray clusters of galaxies in conformal gravity", "X-ray clusters of galaxies in conformal gravity" ]	[ "Antonaldo Diaferio \nDipartimento di Fisica Generale \"Amedeo Avogadro\"\nUniversità degli Studi di Torino\nVia P. Giuria 1I-10125TorinoItaly\n\nIstituto Nazionale di Fisica Nucleare (INFN)\nSezione di Torino\nVia P. Giuria 1I-10125TorinoItaly\n", "Luisa Ostorero \nDipartimento di Fisica Generale \"Amedeo Avogadro\"\nUniversità degli Studi di Torino\nVia P. Giuria 1I-10125TorinoItaly\n\nIstituto Nazionale di Fisica Nucleare (INFN)\nSezione di Torino\nVia P. Giuria 1I-10125TorinoItaly\n" ]	[ "Dipartimento di Fisica Generale \"Amedeo Avogadro\"\nUniversità degli Studi di Torino\nVia P. Giuria 1I-10125TorinoItaly", "Istituto Nazionale di Fisica Nucleare (INFN)\nSezione di Torino\nVia P. Giuria 1I-10125TorinoItaly", "Dipartimento di Fisica Generale \"Amedeo Avogadro\"\nUniversità degli Studi di Torino\nVia P. Giuria 1I-10125TorinoItaly", "Istituto Nazionale di Fisica Nucleare (INFN)\nSezione di Torino\nVia P. Giuria 1I-10125TorinoItaly" ]	[ "Mon. Not. R. Astron. Soc" ]	We run adiabatic N -body/hydrodynamical simulations of isolated self-gravitating gas clouds to test whether conformal gravity, an alternative theory to General Relativity, is able to explain the properties of X-ray galaxy clusters without resorting to dark matter. We show that the gas clouds rapidly reach equilibrium with a density profile which is well fit by a β-model whose normalization and slope are in approximate agreement with observations. However, conformal gravity fails to yield the observed thermal properties of the gas cloud: (i) the mean temperature is at least an order of magnitude larger than observed; (ii) the temperature profiles increase with the square of the distance from the cluster center, in clear disagreement with real X-ray clusters. These results depend on a gravitational potential whose parameters reproduce the velocity rotation curves of spiral galaxies. However, this parametrization stands on an arbitrarily chosen conformal factor. It remains to be seen whether a different conformal factor, specified by a spontaneous breaking of the conformal symmetry, can reconcile this theory with observations.	10.1111/j.1365-2966.2008.14205.x	[ "https://arxiv.org/pdf/0808.3707v2.pdf" ]	14,517,171	0808.3707	5824db23f4e77e842fea8a57275e5d5bce90a0a9	X-ray clusters of galaxies in conformal gravity 6 Nov 2008 6 November 2008 Antonaldo Diaferio Dipartimento di Fisica Generale "Amedeo Avogadro" Università degli Studi di Torino Via P. Giuria 1I-10125TorinoItaly Istituto Nazionale di Fisica Nucleare (INFN) Sezione di Torino Via P. Giuria 1I-10125TorinoItaly Luisa Ostorero Dipartimento di Fisica Generale "Amedeo Avogadro" Università degli Studi di Torino Via P. Giuria 1I-10125TorinoItaly Istituto Nazionale di Fisica Nucleare (INFN) Sezione di Torino Via P. Giuria 1I-10125TorinoItaly X-ray clusters of galaxies in conformal gravity Mon. Not. R. Astron. Soc 00000006 Nov 2008 6 November 2008Printed 6 November 2008(MN L A T E X style file v2.2)gravitation -methods: N -body simulations -galaxies: clusters: general -dark matter -X-ray: galaxies: clusters We run adiabatic N -body/hydrodynamical simulations of isolated self-gravitating gas clouds to test whether conformal gravity, an alternative theory to General Relativity, is able to explain the properties of X-ray galaxy clusters without resorting to dark matter. We show that the gas clouds rapidly reach equilibrium with a density profile which is well fit by a β-model whose normalization and slope are in approximate agreement with observations. However, conformal gravity fails to yield the observed thermal properties of the gas cloud: (i) the mean temperature is at least an order of magnitude larger than observed; (ii) the temperature profiles increase with the square of the distance from the cluster center, in clear disagreement with real X-ray clusters. These results depend on a gravitational potential whose parameters reproduce the velocity rotation curves of spiral galaxies. However, this parametrization stands on an arbitrarily chosen conformal factor. It remains to be seen whether a different conformal factor, specified by a spontaneous breaking of the conformal symmetry, can reconcile this theory with observations. INTRODUCTION On the scale of individual galaxies and larger scales, General Relativity requires large amounts of dark matter to describe the dynamics of cosmic structure. Moreover, the late-time acceleration of the Hubble expansion implies the existence of a cosmological constant, a special case of a dark energy fluid which is suggested by more sophisticated models (see Copeland et al. 2006, for a review). In principle, we can avoid the dark matter and dark energy solutions to the puzzles posed by the astrophysical data by adopting an alternative theory of gravity, which reduces to General Relativity in the appropriate limit. Independently of the dark matter and dark energy problems, a modification of General Relativity is also highly desirable if we ultimately wish to unify gravity with the other fundamental interactions. Possible modifications of the Einstein-Hilbert action proposed in the literature are, among others, (i) the introduction of additional scalar and/or vector fields (e.g., Fujii & Maeda 2003;Bekenstein 2006); (ii) the assumption of arbitrary functions f (R) of the Ricci scalar R (e.g., Capozziello & Francaviglia 2008;Nojiri & Odintsov 2008); (iii) the introduction of additional dimensions to the four di-⋆ E-mail: [email protected] (AD); [email protected] (LO) mensions of the General Relativity spacetime manifold (e.g., Maartens 2004). A different approach was suggested by Mannheim (1990) who revived Weyl's theory (Weyl 1918(Weyl , 1919(Weyl , 1920 as a possible candidate to solve the dark matter and dark energy problems. When the geometry is kept Riemannian, with a null covariant derivative of the metric tensor, we can obtain a milder version of Weyl's gravity, known as conformal gravity. In this theory, we impose a local conformal invariance on the gravitational field action in the curved four-dimensional spacetime. The Einstein-Hilbert Lagrangian density for the gravitational field is chosen on the requirement that the theory of gravity is a second-order derivative theory. In conformal gravity, the Lagrangian density is chosen on the principle of local conformal symmetry which is uniquely satisfied by the action IW = −α d 4 x √ −gC µνκλ C µνκλ , where C µνκλ is the Weyl tensor, α is a coupling constant, and g is the determinant of the metric tensor gµν . Conformal symmetry is garanteed by the invariance of the Weyl tensor to local conformal transformations gµν(x) → Ω 2 (x)gµν(x), where Ω 2 (x) is an arbitrary conformal factor that can be specified by a spontaneous breaking of the conformal invariance (Edery et al. 2006;Mannheim 2008b). The theory of gravity implied by the action IW is a fourthorder derivative theory. The vacuum exterior solution for a static and spherically symmetric spacetime contains the c 0000 RAS Schwarzschild solution (Mannheim & Kazanas 1989). The weak-field limit is consistent with the Solar System observational data (Mannheim 2007), unlike claimed in previous investigations (Barabash & Shtanov 1999;Flanagan 2006;Barabash & Pyatkovska 2007). Mannheim (1993Mannheim ( , 1997 shows that conformal gravity can reproduce the rotation curves of disc galaxies without dark matter. Moreover, conformal gravity can explain the current accelerated expansion of the universe without resorting to a fine-tuned cosmological constant or to the existence of dark energy (Mannheim 2001(Mannheim , 2003Varieschi 2008). Unlike the standard theory, where the universe starts accelerating at redshift z 1, conformal gravity predicts that the universe is accelerating at all times. Therefore, observational data probing the Hubble plot at very high redshift (e.g., Navia et al. 2008;Wei & Zhang 2008) can be a decisive test. More recently, conformal theory has been proposed as a valid candidate for building a theory of quantum gravity (Mannheim 2008a); in fact, theories based on fourthorder derivative equations of motion have had the longlasting problem of suffering from the presence of ghosts. Bender & Mannheim (2008) have recently shown that this erroneous belief is the result of considering the canonical conjugates p of the generic dynamical variables q, when the q's are real, to be Hermitian operators; however, this assumption is incorrect, and when the non-Hermiticity property of the Hamiltonian of these higher-order quantum field theories is correctly taken into account, the states with negative norm disappear. From an astrophysical perspective, however, the form of conformal gravity that has been proposed in the literature currently has two main shortcomings: the abundance of light elements, and the gravitational lensing phenomenology. Conformal gravity nicely avoids the requirement of an initial Big Bang singularity, but it still predicts an early universe sufficiently dense and hot to ignite the light element nucleosynthesis. However, conformal gravity predicts a too slow initial expansion rate. This rate favours the destruction of most of the deuterium produced in the early universe (Knox & Kosowsky 1993;Elizondo & Yepes 1994) and poses conformal gravity in serious difficulties compared to the standard Big Bang nucleosynthesis. Conformal gravity necessarily requires astrophysical processes for the production of the deuterium currently observed, for example neutron radiative capture on protons in the atmospheres of active stars (Mullan & Linsky 1999) or gamma-ray bursts (Inoue et al. 2003). However, the processes investigated to date do not seem to be efficient enough. For example, significant production of deuterium in the Galaxy seems to be ruled out (Prodanovic & Fields 2003). The second open issue is the conformal gravity prediction of gravitational lensing. Early investigations of gravitational lensing in conformal gravity (Walker 1994;Edery & Paranjape 1998, 1999Edery 1999) show that, in the weak-field limit, the deflection angle due to a point mass M is ∆α = 4GM/c 2 r − γr, where r is the radius of the photon closest approach; ∆α contains the additional term γr when compared to the General Relativity result. The constant γ has to be positive to fit the galaxy rotation curves, thus implying a repulsive effect in gravitational lensing. It was later realized (Edery et al. 2001) that the geodesics of photons are independent of the conformal factor Ω 2 (x) and one can choose an appropriate conformal factor and a radial coordinate transformation to yield attractive geodesics for both massive and massless particles. However, in the strongfield limit, the light deflection might still be divergent or even impossible (Pireaux 2004a,b). Until these open questions are completely settled, conformal gravity cannot yet be considered ruled out by observations. Moreover, conformal invariance plays a crucial role in elementary particle physics and a viable theory of gravity that includes this property can at least be suggestive of a relevant route towards the unification of the fundamental interactions. From the astrophysical point of view, conformal gravity can be further tested by investigating the formation of cosmic structure. To date, nobody has yet explored how the large-scale structure forms in conformal gravity. If structures form by gravitational instability, as in the standard theory, the development of a cosmological perturbation theory, which is still lacking, becomes inevitable. This theory would enable the comparison of conformal gravity with the spectrum of the Cosmic Microwave Background anisotropies and would provide initial conditions for the simulation of the structure evolution into the non-linear regime. Before building up such a theory, however, it is useful to check whether conformal gravity is able to reproduce the equilibrium configuration of cosmic structures, other than galaxies, without dark matter. Horne (2006) has already shown that, if we interpret the observational data of the intracluster medium of X-ray clusters by assuming hydrostatic equilibrium, conformal gravity requires a factor of ten less baryonic mass than inferred from the X-ray surface brightness measures. Here, we extend this analysis by performing hydrodynamical simulations of self-gravitating gas clouds. We modify a standard Tree+SPH code to perform numerical simulations of self-gravitating systems in conformal gravity. The numerical tool we create is extremely relevant if we eventually wish to investigate the formation of the largescale structure, because we will massively need to resort to numerical integrations when the evolution of the density perturbations reaches the non-linear regime. In Section 2, we review the basic steps leading to the gravitational potential of a static point source in conformal gravity. In Section 3, inspired by the analysis of Horne (2006), we compute the gravitational potential energy of a spherical system, and in Section 4 we use this result to compute the expected mean temperature and the temperature profile of the intracluster medium. In Section 5, we derive the same results with hydrodynamical simulations of selfgravitating gas clouds. CONFORMAL GRAVITY Conformal gravity is a theory of gravity based on the action IW = −α d 4 x √ −gC µνκλ C µνκλ . For any finite, positive, non-vanishing, continuous, real function Ω 2 (x) of the four spacetime coordinates x, this theory is invariant for any conformal transformation gµν(x) → Ω 2 (x)gµν(x), because the Weyl tensor is invariant for these transformations. The variational principle applied to the action leads to fourth-order field equations, whose solution for the vacuum exterior to a static point source leads to the line element ds 2 = Ω 2 (x) 1 + 2φ(r) c 2 c 2 dt 2 − 1 + 2φ(r) c 2 −1 dr 2 − r 2 (dθ 2 + sin 2 θdφ 2 ) (1) where c is the speed of light, φ(r) c 2 = − β 2 (2 − 3βγ) r − 3 2 βγ + γ 2 r − k 2 r 2 ,(2) and β, γ and k are three integration constants. The arbitrary function Ω 2 (x) can be specified by a mechanism which breaks the conformal symmetry. Mannheim & Kazanas (1989) implicitly assume Ω 2 (x) = 1. According to this assumption, when γ = k = 0, the metric (2) reduces to the usual Schwarzschild metric with β = GM/c 2 , M the gravitational mass of the point source and G the gravitational constant. The term kr 2 /2 corresponds to a cosmological solution which is conformal to a Robertson-Walker background; therefore, k can be chosen small enough that it can be neglected on the scales of galaxies and galaxy clusters, and we will not consider this term hereafter. Moreover, the rotation velocities of spiral galaxies suggest a value of γ sufficiently small that the terms proportional to βγ ∝ γ/c 2 can be safely ignored (Mannheim 1993). Equation (2) thus reduces to φ(r) c 2 = − β r + γ 2 r .(3) Mannheim (1997) uses the observed rotation curves of eleven spiral galaxies of widely different luminosities to constrain γ. It turns out that γ also depends on the point source mass M . Mannheim (1997) suggests the parametrization γ = γ0 + M γ * , where γ0 = 3.06 × 10 −28 m −1 and γ * = 5.42 × 10 −39 m −1 M −1 ⊙ ; γ0 and γ * should be two universal physical constants. In the following, we adopt a more convenient form of the potential (3) suggested by Horne (2006): φ(r) = − GM r + GM R 2 0 r + GM0 R 2 0 r (4) where M0 = (γ0/γ * )M⊙ = 5.6 × 10 10 M⊙ and R0 = (2GM⊙/γ * c 2 ) 1/2 = 24 kpc. GRAVITATIONAL POTENTIAL ENERGY OF A SPHERICAL SYSTEM The gravitational potential of a point source (equation 4) φps(r) = φN (r) + φCM (r) + φCC(r) ,(5) with φN (r) = −GM/r the usual Newtonian potential, φCM (r) = GM r/R 2 0 , and φCC (r) = GM0r/R 2 0 , generates the acceleration gps(r) = −∇φps = − GM r 2 − GM R 2 0 − GM0 R 2 0 .(6) The last term is a constant acceleration independent of the mass of the source of the gravitational field. The physical origin of this term is very subtle. Unlike in Newtonian gravity, the presence of the linear term φCM (r) clearly prevents us from neglecting the contribution of distant objects to the acceleration of any body in the universe. According to Mannheim (1997Mannheim ( , 2006, in conformal gravity the net effect of all the mass in the universe is to contribute a constant acceleration in addition to the acceleration due to a local source. Mannheim (1997) shows that, with an appropriate coordinate transformation allowed by the conformal invariance, the mass in the universe exactly generates the additional linear potential φCC (r), that we name the curvature potential. This argument implies that, to compute the gravitational potential energy of an extended source, we need to treat the curvature potential φCC(r) and the potential φN (r) + φCM (r), originated by the local field source, separately. For the time being, let us consider this latter gravitational potential alone. Consider an extended source. The gravitational potential energy of two of its mass elements with separation r is dWAB = dmAdmBf (r), where f (r) is the part of the potential φ(r) that depends on r only. When we sum over all the mass elements, we get the total gravitational potential energy W = (1/2) dmA dmBf (r), namely W = 1 2 φ(r)ρ(r)d 3 r ,(7) where ρ(r) is the mass density distribution of the source and φ(r) = φN (r) + φCM (r) is the gravitational potential generated by its self-gravity alone. The total potential energy W can also be derived with a different argument which provides an alternative expression to equation (7). For a system of N particles with position ri and mass mi, each feeling a force F(ri), we can define the virial i ri · F(ri); the force is F(ri) = −mi j ∇r ij φ(rij) where rij = ri − rj . Now, mi∇r ij φ(rij ) = −mj∇r ji φ(rji) for Newton's third law, and we can write i ri · F(ri) = − i mi j<i ri · ∇r ij φ(rij)+ + i mi j>i rj · ∇r ij φ(rij ) = − i mi j<i rij · ∇r ij φ(rij ) ,(8) since ∇r ij φ(rij ) = 0 when j = i. Homogeneous functions of order λ of M variables (x1, · · · , xM ) are defined by the relation f (αx1, · · · , αxM ) = α λ f (x1, · · · , xM )(9) for any non-null α; they satisfy Euler's theorem i xi · ∇x i f (x1, · · · , xM ) = λf (x1, . . . , xM ) .(10) In the potential φ = φN + φCM , φN and φCM are homogeneous functions of order λN = −1 and λCM = 1 respectively. By applying Euler's theorem for M = 1, we can thus write i ri · F(ri) = − i j<i mi[−φN (rij ) + φCM (rij )] = WN − WCM .(11) Now, W = WN + WCM and we obtain W = 2WN − i ri · F(ri). In the continuous limit To compute the gravitational potential φ(r) of a selfgravitating spherical system, we follow Horne (2006) and we first consider a homogeneous spherical shell of density ρ, radius R and mass m = 4πρR 2 dR. At a generic point in space of coordinate r, each mass element δm = ρR 2 sin θdRdθdϕ of the shell generates the potential δφ(r) given by equation (5). By setting the coordinate system such that r = (0, 0, r) without loss of generality, the element δm has coordinates R(cos ϕ sin θ, sin ϕ sin θ, cos θ) and generates the potential at r W = 2WN + ρ(r)r · ∇φ(r)d 3 r .(12)δφ(r) = GρR 2 sin θdRdθdϕ − 1 x + x R 2 0 ,(13) where x 2 = R 2 + r 2 − 2rR cos θ. By integrating over dθdϕ, we find the potential of the shell φ sh (r) = Gm    − 1 R + 1 R 2 0 r 2 3R + R r < R − 1 r + 1 R 2 0 R 2 3r + r r > R .(14) The gravitational potential of a self-gravitating sphere is thus φ(r) G = − I0(r) r − E−1(r) + 1 R 2 0 I2(r) 3r + rI0(r) + r 2 3 E−1(r) + E1(r)(15) where In(r) = 4π r 0 ρ(x)x n+2 dx and En(r) = 4π +∞ r ρ(x)x n+2 dx, and the acceleration is − ∇φ(r) G = − I0(r) r 2 + 1 R 2 0 I2(r) 3r 2 − 2 3 rE−1(r) − I0(r) .(16) The total gravitational potential W of a sphere can now be computed with either equation (7) or equation (12). For a power-law density profile ρ(r) = ρ0(r/a) −α of a system truncated at radius a with total mass M = 4πρ0a 3 /(3 − α), with α = 3, we find I0(r) = M (r/a) 3−α , I2(r) = [(3 − α)/(5 − α)]M a 2 (r/a) 5−α , E−1(r) = [(3 − α)/(2 − α)](M/a)[1 − (r/a) 2−α ], and E1(r) = M a(3 − α)/(4 − α)[1 − (r/a) 4−α ] , and the gravitational potential energy is W = WN 1 − a R0 2 2(5 − 2α)(9 − 2α) 3(7 − 2α)(5 − α) ≡ WN [1 − h(α)] ;(17) here WN = − GM 2 a 3 − α 5 − 2α(18) is the potential energy in Newtonian gravity, that is recovered in the limit R0 → ∞. To compute the contribution Wcurv of the curvature acceleration −GM0/R 2 0 to be added to W , we can use either equation (12) or equation (7) without the factor (1/2), because the curvature mass M0 is not accelerated by the mass ρ(r)d 3 r. We find Wcurv = GM M0 a R 2 0 3 − α 4 − α = −WN a R0 2 M0 M 5 − 2α 4 − α .(19) The left panel of Figure 1 shows the total gravitational potential energy Wtot = WN (α)[1 − h(α)] + Wcurv(α)(20) of a spherical system as a function of α and its total mass M . When α → 2.5, 3.5, 4 and 5, Wtot → ±∞. In Figure 1, the pole of WN at α = 2.5 seems to have different properties than the other poles, but it is only a graphic artifact. In fact, the contribution of conformal gravity to Wtot is proportional to (a/R0) 2 ∼ 10 3 , and it always dominates the newtonian WN : we must have \|α − 2.5\| ≪ 10 −3 to see Wtot → ±∞. Finally, the ratio between the curvature potential and the conformal gravity potential due to the sphere self-gravity is ∼ M0/M ; therefore, Wcurv is negligible when M ≫ M0, as in our case, where M0/M ∼ 10 −2 , unless, of course, α → 4. X-RAY CLUSTERS A system of N particles of position ri experiencing the force Fi is in virial equilibrium when it satisfies the virial theorem relation 2K + ri · Fi = 0, where K is the total kinetic energy of the system. We saw above (equation 11) that ri · Fi = 2WN − Wtot in conformal gravity. Note that we replaced WCM with Wtot to include the curvature potential. Observations indicate that in X-ray clusters the total mass of galaxies is roughly ten times smaller than the total mass of the intracluster medium (e.g., White et al. 1993). We can therefore model X-ray clusters as individual clouds of hot gas with no galaxies and, of course, with no dark matter. This approximation is partly inappropriate for cD clusters, where the contribution of the cD galaxy to the gravitational potential in the cluster center is relevant; however, cD clusters are ∼ 20% of the population of nearby clusters (Rood & Sastry 1971) and our analysis should be valid for most clusters. If the gas bulk flows and turbulent motions are negligible, the total kinetic energy is K = (3/2)(γ − 1)U = (3/2)M k T /µmp, where U = [M/(γ − 1)]k T /µmp is the internal energy of the gas of density ρ(r), M is the gas total mass, T = ρ(r)T (r)d 3 r/ ρ(r)d 3 r is its mass-weighted mean temperature, k the Boltzmann constant, mp the proton mass, µ the mean molecular weight, and γ the adiabatic index. The virial theorem 1 thus yields Figure 1 shows the mean temperatures k T of spherical systems with a power-law density profile as a function of α and its total mass M . Consider an isothermal sphere with α = 2, radius a = 1 Mpc, and mass M = 10 13 M⊙. Newtonian gravity predicts k T = 0.09 keV. Conformal gravity predicts k T = 58.21 keV, namely a temperature (a/R0) 2 ∼ 10 3 larger. To explain a more typical temperature of, say, k T ∼ 6 keV without resorting to dark matter, the gas mass should be M ∼ 10 12 M⊙, a factor of ten lower than the mass of the intracluster medium measured from the X-ray surface brightness. k T = − µmp 3M (2WN − Wtot)(21) This result agrees with the claim of Horne (2006) that the acceleration provided by the conformal gravity potential is too intense. We now show that the disagreement with observations extends to the temperature profile, besides its normalization. To compute the temperature profile of a self-gravitating spherical system in equilibrium, we need to solve the Boltzmann equation coupled to the Poisson equation. In conformal gravity, the Poisson equation ∇ 2 φ = 4πGρ is replaced by a fourth order equation ∇ 4 φ ∝ ρ (Mannheim & Kazanas 1994). Following this approach is not a trivial task, either analytically or numerically. In the next section, we resort to a smoothed-particle-hydrodynamical simulation to obtain a self-consistent solution to this problem. In this section, to illustrate qualitatively the expected result, we can consider the first moment of the Boltzmann equation, namely the Euler equation, which, in hydrostatic equilibrium, reduces to the relation ∇p = −ρ∇φ, where p is the gas pressure. Provided that the density profile ρ(r) is known, for an ideal gas with equation of state p = ρkT /µmp, this equation immediately integrates to kT (r) µmp = − 1 ρ(r) ρ(r)∇φ(r)dr + A0(22) where A0 is an integration constant. At large radii, real clusters usually have the intracluster medium density profile which decreases more rapidly than the temperature profile. This constraint requires that the term A0/ρ(r) should disappear at large radii and implies A0 = 0. For a power-law density profile, we find T (r) = TN (r) 1 + α − 1 (2 − α) 2 a R0 2 2 (5 − α) r a 2 − 4(3 − α) 3 r a α + GM0 R0 1 1 − α r R0 (23) where kTN (r) µmp = GM 2a 1 α − 1 r a 2−α .(24) Note that for an isothermal sphere (α = 2) in Newtonian gravity, this equation yields kTN /µmp = GM/2a. This temperature disagrees with the correct virial theorem result kTN /µmp = GM/3a (equation 21). This discrepancy is a simple consequence of the inconsistent approach we used to obtain equation (23). However, we can use equation (23) to predict that, at large radii, the temperature profile increases at least as r 2 . We thus conclude that the values of the conformal physical constants R0 = 24 kpc and M0 = 5.6 × 10 10 M⊙ implied by the galaxy rotation curves provide X-ray clusters with average temperatures a factor of ten too large, and rising temperature profiles, clearly at odds with real clusters. NUMERICAL SOLUTION To solve properly the coupled Boltzmann and Poisson equations and predict the correct X-ray properties of a galaxy cluster in conformal gravity, we run hydrodynamical simulations of isolated spheres of gas with a Smoothed-Particle-Hydrodynamic (SPH) code. We use a modified version of the publicly available GADGET-1.1 code (Springel et al. 2001;Springel 2005). The original version of GADGET-1.1 is a Tree+SPH code which integrates the equations of motion of N particles which interact gravitationally. The N particles are a discrete representation of either a collisionless or a collisional fluid; traditional simulations contain both collisionless and collisional particles which represent the dark matter and the baryonic fluids, respectively. In our simulations, we only require the presence of the baryonic fluid and we only have collisional particles. Moreover, in Newtonian gravity, the potential between two particles decreases with the interparticle distance, and GADGET-1.1 arranges the particles in a tree structure in order to average the contribution of distant particles to the local acceleration. This technique substantially decreases the time required to compute the acceleration of individual particles. Unfortunately, in conformal gravity this technique can not be applied, because the gravitational potential contains a term that increases linearly with the interparticle distance, and the contribution of distant particles to the gravitational potential must be computed individually. We will therefore use GADGET-1.1 as a direct N -body integrator. The computational cost thus increases with N 2 rather than N ln N . In appendix A, we describe in detail the modification we introduced in GADGET-1.1 to compute the interparticle forces in conformal gravity. We simulate the evolution of isolated spherical clouds of gas with vacuum boundary conditions. We neglect radiative cooling. At the end of this section we discuss how simulating the gas physics more realistically can actually be relevant to find a way to alleviate the discrepancy between conformal gravity and the observed thermal properties of X-ray clusters. A gas cloud is represented by N = 4096 gas particles. The chosen N is substantially smaller than state-of-the-art simulations of galaxy clusters and is due to the N 2 scaling of the computational cost. We do not actually need a larger N , because the aim of our simulations is to test the general viability of conformal gravity rather than the detailed physics of the intracluster medium, which would require higher spatial and mass resolutions. In fact, the current form of the conformal gravitational potential yields clusters so different from real clusters that running costly simulations is unwarranted. The gas cloud initially has a β-model density profile ρ(r) = ρ0 1 + r r0 2 −3β/2(25) and null thermal and kinetic energies. Each gas particle has gravitational softening length ǫ = 0.1 kpc. Each gas particle also has an individual SPH softening parameter h used to estimate the thermodynamical quantities, as mentioned below; the sphere with radius h centered on the particle contains a fixed number N b of neighbor particles. We set N b = 40 in our simulations. In the following, we show the result of a typical simulation: a cloud of total mass M = 2.7 × 10 13 M⊙, and initial values of the β-model β = 0.636, r0 = 134 kpc, and ρ0 = 1.52 × 10 −26 g cm −3 . The upper left panel of Figure 2 shows that the total energy is conserved to better then 0.1%. The evolution of the gravitational potential, kinetic, thermal and total energies (upper right panel) shows that in half a billion years the system reaches equilibrium and remains confined; in fact, 90% of the mass is within 500 kpc (Figure 3). It follows that in conformal gravity a cloud of gas can easily remain confined without the need of dark matter. The bottom right panel of Figure 2 shows the evolution of the virial ratio: the sum K of the thermal and kinetic energy over the total gravitational potential energy Wtot. At equilibrium we must have 2K + ri · Fi = 0, where ri · Fi = 2WN − Wtot (Section 4). Therefore K/Wtot = (1/2) − (WN /Wtot). From quation 20 we see that Wtot is positive and Wtot ≫ \|WN \| because (a/R0) 2 ≫ 1; therefore, unlike Newtonian gravity where K/Wtot = −1/2 at equilibrium, here the virial ratio K/Wtot is slightly larger than 1/2. In the Tree+SPH code each gas particles of mass mi and density ρi carries the electron number density ne i , and the internal energy ui. By assuming an optically thin gas of primordial cosmic abundance X = 0.76 and Y = 0.24, we can estimate the particle temperature Ti; we also estimate the ion number density nH i + nHe i that we use below. The bottom left panel of Figure 2 shows that the final mass-weighted mean temperature of the system is 90 keV, as expected from the argument provided in Section 4. Figure 4 shows the density and temperature profiles of the final configuration. As in real clusters, the β−model is still an excellent approximation to the final density profile. The best fit yields β = 1.65, r0 = 142 kpc and ρ0 = 1.6 × 10 −25 g cm −3 . These parameters roughly agree with typical values of real clusters, although β is ∼ 2 − 3 times larger than typical values (see, e.g., Markevitch et al. 1999). The result in strong disagreement with observations is the rising temperature profile shown in the right panel of Figure 4. At large radii, we see that T ∝ r 2 , as anticipated in Section 4. For completeness, we show the surface brightness and the temperature maps in Figure 5. Each map is an equally spaced Np × Np grid, with Np = 128, corresponding to a length resolution ≈ 15.6 kpc. In the Tree+SPH code, each gas particle has a smoothing length hi and the thermodynamical quantities it carries are distributed within the sphere of radius hi according to the compact kernel W (r; hi), which has the same functional form of the gravitational kernel (see Appendix A), where r is the distance to the particle center. The X-ray surface brightness S jk on the grid point {j, k} is S jk = 1 d 2 p ne i (nH i + nHe i )Λ(Ti)widVi(26) where d 2 p is the pixel area, the sum runs over all the particles, and wi ∝ W (x)dl is the weight proportional to the fraction of the particle volume dVi = mi/ρi which contributes to the grid point {j, k}. For each particle, the weights w k are normalized to satisfy the relation w k = 1 where the sum is now over the grid points within the particle circle. When hi is so small that the circle contains no grid point, the particle quantity is fully assigned to the closest grid point. We use the cooling function Λ(T ) of Sutherland & Dopita (1993). The total X-ray luminosity is LX = ne i (nH i + nHe i )Λ(Ti)dVi .(27) Given the large equilibrium temperature, the cluster shown in Figure 5 has LX = 3.86 × 10 46 erg s −1 at equilibrium, an order of magnitude larger than the most luminous X-ray clusters. The temperature whose map is shown in Figure 5 is the emission-weighted temperature T jk = ne i (nH i + nHe i )Λ(Ti)TiwidVi ne i (nH i + nHe i )Λ(Ti)widVi .(28) This map dramatically shows how the thermal properties of the intracluster medium totally disagree with real clusters. At this point, we cannot draw our conclusions without the following relevant cautionary tale. The results we show here derive from simulations that treat the gas adiabatically. In other words, we neglect radiative cooling and the gas heating processes due to astrophysical sources, as supernovae explosions, energy injection from active galactic nuclei, or galactic and stellar winds. These heating processes act at different times and with different effectiveness, depending on the detailed history of star and galaxy formation. This history is currently totally unexplored in conformal gravity, and the detailed mechanisms of the heating processes themselves are still poorly understood (e.g., Borgani et al. 2008). Therefore, the appropriate inclusion of these cooling and heating processes in our simulations is beyond the illustrative purpose of this paper. However, an appropriate interplay between cooling and heating processes could in principle provide a possible route to reconcile conformal gravity with the properties of X-ray clusters. In fact, since our results show that, for the typical gas density of real systems, the gas temperature is an order of magnitude larger than observed, radiative cooling would be very efficient in conformal clusters and the entire intracluster medium could cool in less than 1 Gyr. Effective heating processes would thus be necessary to reheat the intracluster medium to the observed X-ray temperatures. These heating processes must be more efficient in the cluster center than in the cluster outskirts, because the cooling time increases with the clustrocentric radius. However, although we cannot exclude that the various complex physical processes could eventually conspire to reconcile conformal gravity with observations, it is plausible that, in order to provide intracluster media with the observed thermal properties, the cooling and heating processes should be severely fine-tuned during the formation and evolution of clusters and cluster galaxies. CONCLUSION Conformal gravity can explain the rotation curves of disk galaxies and the current accelerated expansion of the universe without resorting to dark matter and dark energy. We have modified a Tree+SPH code to run hydrodynamical simulations of isolated X-ray galaxy clusters to show that conformal gravity does not share the same success on the scales of clusters. These simulations confirm our simple analytic estimates that show that gas clouds with mass ∼ 10 12 − 10 13 M⊙, which are typical values of the total mass of the hot gas present in real clusters, remain confined with an equilibrium mean temperature ∼ 10−100 keV, ten times larger than the observed temperatures; more dramatically, because of the presence of a linear term in the gravitational potential, at large clustrocentric radius r, the gas temperature increases with r 2 , rather than decreasing as in real systems. Our analysis totally neglects radiative cooling and gas heating from astrophysical sources, as supernovae or active galactic nuclei. The interplay between these processes can in principle provide a way to reconcile conformal gravity with observations. It is however unclear if and how much these processes should be fine-tuned to provide X-ray clusters in agreement with observations. In addition to this topic, we can see two more open issues whose solution might also reconcile conformal gravity with observations: (i) In conformal gravity, all the matter in the universe is expected to affect the local dynamics. The net effect is to contribute a constant inward acceleration −GM0/R0 in addition to the gravitational acceleration generated by local sources. Because we included this constant acceleration in our simulations, we could neglect the rest of the universe and impose vacuum boundary conditions. It might be possible that assimilating the gravitational influence of the nearby matter surrounding the X-ray cluster in the constant "universe" acceleration −GM0/R0 is inappropriate: in fact, nearby external matter might decrease the gravitational attraction of the interior matter and hopefully reduce the thermal energy of the gas. To appropriately investigate this effect, we should simulate the dynamics of largescale structure within a full cosmological context. However, this task is not trivial just because the gravitational field is highly non-local. This investigation would also benefit from the implementation into the numerical simulation of the yet unavailable theory of structure formation. (ii) The gravitationl potential we implemented in our simulations derives from a metric where the conformal invariance is broken by an arbitrary choice of the conformal factor Ω 2 (x). It is unclear whether this choice provides a coordinate system whose physics describes the real world or it is an artifact of the reference frame. It also remains to be seen whether a spontaneous breaking of the conformal invariance, in theories where matter and gravity are conformally coupled (Edery et al. 2006), can provide a metric, and thus a gravitational potential, where the observed thermal properties of the intracluster medium can be reproduced. In Section 1 we mentioned that the nucleosynthesis of light elements and the phenomenology of gravitational lensing are two open issues that need to be solved before accepting conformal gravity as a viable alternative theory of gravity and cosmology. Here, we have shown that the thermodynamics of X-ray clusters poses a third challenge to this theory. APPENDIX A: THE SOFTENING KERNEL The publicly available code GADGET-1.1 (Springel et al. 2001;Springel 2005) is a Tree+SPH code which integrates the equations of motion of N particles which interact gravitationally. The particles are a discrete representation of either a collisionless or a collisional fluid. Here, we consider only adiabatic processes and neglect the possibility of radiative cooling of the collisional fluid. The only modification to the code we need is the computation of the gravitational potential φ and its corresponding acceleration. At position r, N particles of mass mi at position xi generates the potential φ(r) = G N i=1 −migN (yi) + mi R 2 0 gCM (yi) + M0 R 2 0 gCC (yi) (A1) where yi = \|r − xi\|. For point sources, gN (y) = 1/y and gCM (y) = gCC (y) = y. The acceleration a(r) = −∇φ(r) is a(r) = G N i=1 yi mig 1 N (yi) − mi R 2 0 g 1 CM (yi) − M0 R 2 0 g 1 CC (yi) (A2) where g 1 N (y)y = dgN /dy, and analogously for g 1 CM and g 1 CC . GADGET-1.1 treats particles as extended spherical objects with mass m and density profile ρ(r) = mW (r; h) where W (r; h) = 8 πh 3 1 − 6x 2 + 6x 3 0 x < 1 2 2(1 − x) 3 1 2 x < 1 0 x 1 , where x = r/h. W (r; h) is a spline kernel which avoids unrealistic divergences of the Newtonian acceleration for arbitrary small particle separations. The spline kernel implies that each particle is not a point source of gravitational potential. Rather, its gravitational potential is correctly computed as an extended source of density W (r; h). According to equation (15) u 1 , gCC = y, and g 1 CC = 1/y. Figure 1 . 1Left panel: Total gravitational potential energy of a spherical system with a power-law density profile ρ ∝ r −α , radius a = 1 Mpc, and mass M = 2 × 10 12 M ⊙ (solid line), 4 × 10 12 M ⊙ (dotted line), 6 × 10 12 M ⊙ (dashed line), 8 × 10 12 M ⊙ (dot-dashed line), 10 13 M ⊙ (dot-dot-dot-dashed line); see Section 3 for details. Right panel: The mean gas temperature of the systems shown in the left panel according to the virial theorem; see Section 4 for details. Figure 2 . 2Upper left panel: Fluctuations of the total energy. Upper right panel: Evolution of the kinetic (dotted line), thermal (dashed line), gravitational potential (dot-dashed line), and total (solid line) energy. Lower left panel: Evolution of the mass-weighted mean X-ray temperature. Lower right panel: Evolution of the virial ratio. Figure 3 . 3Evolution of the size of the gas cloud: From bottom to top, lines are for radii containing 10% to 90% of the total cloud mass. Figure 4 . 4Mass density (left panel) and temperature (right panel) profiles of the final configuration of the gas cloud. The solid line in the left panel is the best-fit β-model profile. The straight line in the right panel shows the T ∝ r 2 slope. Figure 5 . 5X-ray surface brightness (SB) (left panel) and emission weighted (EW) temperature (right panel) maps of the final configuration of the gas cloud. Each panel shows three orthogonal projections. Each map is 2 Mpc on a side. we find, with u = y/h, gN (y; h) = 1 h gCM (y; h) = h     14 5 − 16 3 u 2 + 48 5 u 4 − 32 5 u 5 0 u < 1 2 − 1 15u + 16 5 − 32 3 u 2 + 16u 3 − 48 5 u 4 + 32 15 u 5 1 2 u < 1 1 u u 1 , g 1 N (y; h) = 1 h 3      − 32 3 + 192 5 u 2 − 32u 3 0 u < 1 2 1 15u 3 − 64 3 + 48u − 192 5 u 2 + 32 3 u 3 1 2 u < 1 − 1 u 3 u 1 ,              31 70 + 14 15 u 2 − 8 15 u 4 + 16 35 u 6 − 8 35 u 7 0 u < 1 2 − 1 840u + 16 35 − u 15 + 16 15 u 2 − 16 15 u 4 + 16 15 u 5 − 16 35 u 6 + 8 105 u 7 1 2 u < 1 u + 3 40u u 1 , g 1 CM (y; h) = 1 h      28 15 − 32 15 u 2 + 96 35 u 4 − 8 5 u 5 0 u < 1 2 1 840u 3 − 1 15u + 32 15 − 64 15 u 2 + 16 3 u 3 − 96 35 u 4 + 8 15 u 5 1 2 u < 1 1 u − 3 40u 3 c 0000 RAS, MNRAS 000, 000-000 The virial theorem can also be generalized in f (R) gravities(Böhmer et al. 2008). c 0000 RAS, MNRAS 000, 000-000 ACKNOWLEDGMENTSWe thank Volker Springel for making public his superb Tree+SPH code and the referee whose careful report prompted us to clarify some aspects of our results. 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[ "Asymptotic normalization of mirror states and the effect of couplings", "Asymptotic normalization of mirror states and the effect of couplings" ]	[ "L J Titus \nNational Superconducting Cyclotron Laboratory\nMichigan State University\n48824East LansingMIUSA\n\nDepartment of Physics and Astronomy\nMichigan State University\n48824-1321East LansingMI\n", "P Capel \nNational Superconducting Cyclotron Laboratory\nMichigan State University\n48824East LansingMIUSA\n\nHelmholtz-Insitut Mainz\nJohannes Gutenberg-Universität Mainz\nD-55128MainzGermany\n", "F M Nunes \nNational Superconducting Cyclotron Laboratory\nMichigan State University\n48824East LansingMIUSA\n\nDepartment of Physics and Astronomy\nMichigan State University\n48824-1321East LansingMI\n" ]	[ "National Superconducting Cyclotron Laboratory\nMichigan State University\n48824East LansingMIUSA", "Department of Physics and Astronomy\nMichigan State University\n48824-1321East LansingMI", "National Superconducting Cyclotron Laboratory\nMichigan State University\n48824East LansingMIUSA", "Helmholtz-Insitut Mainz\nJohannes Gutenberg-Universität Mainz\nD-55128MainzGermany", "National Superconducting Cyclotron Laboratory\nMichigan State University\n48824East LansingMIUSA", "Department of Physics and Astronomy\nMichigan State University\n48824-1321East LansingMI" ]	[]	Assuming that the ratio between asymptotic normalization coefficients of mirror states is model independent, charge symmetry can be used to indirectly extract astrophysically relevant proton capture reactions on proton-rich nuclei based on information on stable isotopes. The assumption has been tested for light nuclei within the microscopic cluster model. In this work we explore the Hamiltonian independence of the ratio between asymptotic normalization coefficients of mirror states when deformation and core excitation is introduced in the system. For this purpose we consider a phenomenological rotor + N model where the valence nucleon is subject to a deformed mean field and the core is allowed to excite. We apply the model to 8 Li/ 8 B, 13 C/ 13 N, 17 O/ 17 F, 23 Ne/ 23 Al, and 27 Mg/ 27 P. Our results show that for most studied cases, the ratio between asymptotic normalization coefficients of mirror states is independent of the strength and multipolarity of the couplings induced. The exception is for cases in which there is an s-wave coupled to the ground state of the core, the proton system is loosely bound, and the states have large admixture with other configurations. We discuss the implications of our results for novae.	10.1103/physrevc.84.035805	[ "https://arxiv.org/pdf/1108.3292v1.pdf" ]	108,283,813	1108.3292	1c0f7bac15bee9ab890fb8b9eb7b0d160e64565c	Asymptotic normalization of mirror states and the effect of couplings 16 Aug 2011 L J Titus National Superconducting Cyclotron Laboratory Michigan State University 48824East LansingMIUSA Department of Physics and Astronomy Michigan State University 48824-1321East LansingMI P Capel National Superconducting Cyclotron Laboratory Michigan State University 48824East LansingMIUSA Helmholtz-Insitut Mainz Johannes Gutenberg-Universität Mainz D-55128MainzGermany F M Nunes National Superconducting Cyclotron Laboratory Michigan State University 48824East LansingMIUSA Department of Physics and Astronomy Michigan State University 48824-1321East LansingMI Asymptotic normalization of mirror states and the effect of couplings 16 Aug 2011(Dated: April 28, 2013)numbers: 2110Jx2160Ev2560Tv Keywords: asymptotic normalization coefficientspectroscopic factorsmirror symmetryrotational modelradiative capture Assuming that the ratio between asymptotic normalization coefficients of mirror states is model independent, charge symmetry can be used to indirectly extract astrophysically relevant proton capture reactions on proton-rich nuclei based on information on stable isotopes. The assumption has been tested for light nuclei within the microscopic cluster model. In this work we explore the Hamiltonian independence of the ratio between asymptotic normalization coefficients of mirror states when deformation and core excitation is introduced in the system. For this purpose we consider a phenomenological rotor + N model where the valence nucleon is subject to a deformed mean field and the core is allowed to excite. We apply the model to 8 Li/ 8 B, 13 C/ 13 N, 17 O/ 17 F, 23 Ne/ 23 Al, and 27 Mg/ 27 P. Our results show that for most studied cases, the ratio between asymptotic normalization coefficients of mirror states is independent of the strength and multipolarity of the couplings induced. The exception is for cases in which there is an s-wave coupled to the ground state of the core, the proton system is loosely bound, and the states have large admixture with other configurations. We discuss the implications of our results for novae. I. INTRODUCTION Novae explosions are a consequence of a thermonuclear runaway on the accreting disk of a white dwarf within a binary system. The rp-process which takes place in novae, involves reactions with proton-rich nuclei close to (or at) the proton dripline [1][2][3]. Measuring the corresponding cross sections is particularly challenging, not only due to the hindrance caused by the Coulomb barrier, but also due to the fact that they involve rare isotopes (see e.g. Ref. [4]). In many cases, the capture process occurs through specific resonances which need to be well known [5]. However, even in these cases, it is important to understand the role of direct capture. Direct proton captures at low relative energies needed for astrophysics are always peripheral reactions due to the Coulomb barrier. At the limit of E → 0 these reactions are uniquely determined by the asymptotic normalization coefficient (ANC) of the single proton overlap function of the final nucleus [6]. Based on this realization, the ANC method [6] has been put forth as an indirect way of extracting proton radiative-capture cross sections from ANCs inferred from measurements of nuclear reactions, such as transfer or breakup. Another indirect technique [7] uses information on the mirror system. The idea introduced in Ref. [7] is that charge symmetry can be used to relate the ANCs of the proton and neutron overlap functions in mirror nuclei. In this way, while proton capture may require the knowledge of reactions involving a proton-rich radioactive beam, the neutron counterpart can be performed with stable beams and thus with much higher accuracy [8,9]. In Refs. [7][8][9] the ratio R of the proton to neutron ANCs squared is determined for a wide range of light nuclei within a microscopic cluster model (MCM). This ratio R is shown to be independent of the choice for the NN interaction within a few percent. An analytic derivation of the ratio, R 0 , is also presented [7]. The ratio obtained from microscopic calculations is in fair agreement with that predicted by the analytic formula [8,9]. Since the original idea was introduced, it has been generalized to resonant states [10] and to α cluster states [11]. In this work, we want to explore the effects of couplings induced by deformation and core excitation in the system. One might wonder why not calculate the ANC theoretically, instead of relying on charge symmetry approximations. The reason for not doing so is the large uncertainty related to the theoretical prediction of ANCs. The microscopic calculations presented in Refs. [7][8][9] are strongly dependent on the effective NN interactions used. Abinitio calculations for light nuclei are increasingly gaining predictive power, but for the last decade it has been a true challenge to produce ab-initio overlap functions with a reliable asymptotic behavior for various technical reasons. The many-body community has put remarkable efforts into extensions of the traditional methods to enable a good description of the asymptotic behavior. Examples include i) the coupling of the resonating group method techniques with the no-core shell model (NCSM) [12], ii) expanding the coupled cluster wavefunction in a Breggren basis [13], and iii) using a Green's function method to extract ANCs from Green's function Monte Carlo (GFMC) overlap functions, which have poor asymptotic behavior [14]. To our knowledge, the work in Ref. [14] consists of the first and only ab-initio ANC calculations for light nuclei up to A=9, to date. While ab-initio efforts show promising results, their limitations are hard set: only light nuclei for NCSM and GFMC and only nuclei around closed shells for the coupled cluster method. Many nuclei of interest in the rpprocess are mid-shell nuclei with mass A > 20 and may have multi-configuration states. It is interesting to explore the effect of couplings induced by core excitation in such systems. Effects of including explicitly excited states of the core were studied within the MCM in Refs. [8,9]. It was shown that deviations from the analytic formula increased. A simple framework of including multiconfiguration and excitation in the single nucleon overlap functions is provided by the core + N phenomenological model [15][16][17]. In the nineties, this model was applied to a number of light nuclei, including the one-neutron halos 11 Be [15] and 19 C [18]. Starting from a two-body Hamiltonian with an effective deformed core + N interaction which is adjusted to reproduce the energy levels of the system, one arrives at a coupled-channels equation. The resulting coupled-channels wavefunction has fragmentation of strength from the original single particle component to other components involving possible excited states of the core. Recently, this model was used to explore the connection between the asymptotic properties of the wavefunction and spectroscopic factors [19]. In the present work, we use the model to study the asymptotic normalization of mirror states and their ratio. The paper is organized in the following way. In Sec. II we briefly describe the model. Results are presented and discussed in Sec. III, starting with numerical details in Sec. III A, some specific applications to mirror partners in Sec. III B, and further exploration of the parameter space in Sec. III C. Finally in Sec. IV conclusions are drawn. II. THEORETICAL CONSIDERATIONS The A = B + x model introduced in Ref. [15] starts from an effective Hamiltonian representing the motion of the valence nucleon (x = n, p) relative to a core B: H A = T r + H B + V Bx (r, ξ),(1) where T r is the relative kinetic energy operator and H B is the internal Hamiltonian of the core. The effective interaction between the core and the valence nucleon depends on the B-x relative coordinate r but also on the internal degrees of freedom of the core ξ. In this model [15] V Bx is taken to be a deformed Woods-Saxon potential: V Bx (r) = −V ws 1 + exp r − R(θ, φ) a −1 ,(2) in which the depth V ws may depend on the B-x orbital angular momentum l. Motivated by a deformed shape, the radius R is angle dependent: R(θ, φ) = R ws [1 + Q q=2 β q Y q0 (θ, φ)],(3) where β q characterizes the deformation of the core and consequently the strength of the coupling between various B + x configurations. As usual, we set R ws = r ws A 1/3 , with A the mass number of the B+x system. In addition we also include an undeformed spin-orbit coupling term: V SO (r) = l · s V so 1 r d dr 1 + exp r − R ws a −1 ,(4) where s is the spin of the valence nucleon x. When x = p, a point-sphere central Coulomb interaction is also included. The B + x wavefunction is expanded in eigenstates of the core Φ I π B , with spin I, parity π B and eigenenergy ǫ I π B : Ψ J π = nljIπB ψ nlj (r)Y lj (r)Φ I π B (ξ).(5) Here we factorize the radial part ψ nlj and the spinangular Y lj part for convenience. The quantum numbers n and j correspond respectively to the principal quantum number and the angular momentum obtained from the coupling of the orbital angular momentum l and the spin s. Replacing the expansion (5) into the Schrödinger equation, one arrives at a coupled-channel equation [15]: T l r + V ii (r) ψ i (r) + j =i V ij (r)ψ j (r) = (ε x J π − ǫ i )ψ i (r)(6) where i represents all possible (nljIπ B ) combinations, ε x J π is the relative energy in the A = B + x system (i.e. same magnitude and opposite sign of the one-neutron or one-proton separation energy), T l r is the radial part of the B-x kinetic energy operator, and the potential matrix elements V ij are V ij (r) = Φ i (ξ)Y i (r)\|V Bx (r, ξ)\|Y j (r)Φ j (ξ) .(7) We take Φ i directly from the rotational model although parameters are fixed phenomenologically. Solutions of Eq. (6) are found imposing bound-state boundary conditions and normalizing Ψ J π to unity. For more details we refer to Refs. [15,19]. In this model, the norm of ψ i relates directly to a spectroscopic factor: S x i = ∞ 0 \|ψ i \| 2 r 2 dr,(8) and the ANC C x i is determined from the asymptotic behavior of ψ i : ψ i (r) −→ r→∞ C x i W −η x i ,l+1/2 (2κ i r)(9) with κ i = 2µ Bx \|ε J π − ǫ i \|/ 2 and µ Bx the reduced mass. The mass of a particle is given by its mass number times m N = 938.9 MeV/c 2 . In Eq. (9), W is the Whittaker function with η x i the B-x Sommerfeld parameter in channel i [20]. To illustrate this model, let us consider the particular mirror pair 17 O/ 17 F. The core of both nuclei is 16 O, which has, apart from the ground state 0 + , two low lying states, 2 + and 3 − , coupling strongly to the ground state through E2 and E3 transitions, respectively. If one includes in the model space 16 O(0 + , 2 + ), the ground state of 17 O/ 17 F (5/2 + ) would not only contain a 1d 5/2 valence nucleon coupled to the ground state 16 O(0 + ) but also for example a 2s 1/2 nucleon coupled to the excited state 16 O(2 + ). A model space containing 16 O(0 + , 3 − ), would instead have a 1f 5/2 valence nucleon coupled to the excited state 16 O(3 − ), amongst other orbitals with odd angular momentum. The main difference between both mirror nuclei is the B-x Coulomb interaction. We should stress that in this work our approach is strongly phenomenological. Because we are interested in ANCs and these depend strongly on the energy of the system relative to threshold [21], it is essential that we reproduce the experimental separation energies exactly. Thus, although the initial proton and neutron Hamiltonians only differ by the Coulomb interaction, in our calculations there may be small differences in the adjusted depths of V Bn and V Bp to reproduce exactly the corresponding binding energies. As proposed in Ref. [7], we compare proton ANCs C p i with neutron ANCs C n i for mirror states through their ratio R = C p i C n i 2(10) In Refs. [7,11], a useful analytical approximation of this ratio was derived R 0 = F l (iκ p i R N ) κ p i R N j l (iκ n i R N ) 2 ,(11) with F l and j l being the regular Coulomb function and the regular Bessel function, respectively [20]. The approximation R 0 is not strongly dependent on the radius of the nuclear interior, R N [7,8]. We will compare our results with the value obtained from this relation. III. RESULTS AND DISCUSSION A. Numerical details We consider the same cases as in Refs. [8,9], and here present all details concerning the model parameters. First, it is important to keep in mind that it is not our aim to reproduce all the properties of these nuclei with our simple model [15], since in principle microscopic models are much better suited. Here our aim is to use the B + x model to explore to what extent core degrees of freedom can modify the picture presented in Refs. [7][8][9]. As the B-x interaction is completely phenomenological, it is essential to have energy levels to constrain the interaction. Below we provide details of the fitting for each case. Core excitation energies are taken from the database of the National Nuclear Data Center [22]. It is the deformation that introduces tensor components in the interaction and that allows for configuration admixture between various core states. Values for the deformation parameters for each case, as well as the states to be considered in the coupled channel equation, are given. The geometry for the Woods-Saxon interaction and the strength of the spin-orbit force V so are fixed at constant values (see Secs. III B and III C). The depth of the central potential V ws is then adjusted to reproduce the B-x separation energy (shown in Table I). In some cases we fit more than one state per nucleus. This introduces an l-dependence in V ws . All calculations are performed with the program face [23]. a. 8 Li/ 8 B: The B+x description of these mirror nuclei corresponds to 7 Li+n for 8 Li and 7 Be+p for 8 B. The respective B-x relative energies are ε n 2 + = −2.032 MeV and ε p 2 + = −0.1375 MeV. The 2 + ground state of these nuclei is described as a dominant 1p 3/2 nucleon bound to the 3/2 − ground state of the core. The 1/2 − state of the core is also considered (ǫ 1/2 − ( 7 Li) = 0.478 MeV and ǫ 1/2 − ( 7 Be) = 0.429 MeV). The quadrupole deformation that couples both core states of 7 Li is β 2 = 0.34. That of 7 Be, being predicted to be around 0.3-0.4 [24], is chosen equal to that of 7 Li. b. 13 C/ 13 N: In this mirror pair the core is 12 C for both nuclei. The dominant configuration of the 1/2 − ground state is a 1p 1/2 nucleon coupled to the 0 + ground state of 12 C. The relative 12 C-x energies are ε n 1/2 − = −4.946 MeV and ε p 1/2 − = −1.944 MeV for 13 C and 13 N respectively. The 2 + excited state of 12 C at ǫ 2 + = 4.439 MeV is also considered with the coupling β 2 = −0.6 [25]. c. 17 O/ 17 F: For this 16 O+x mirror pair, our model reproduces both the 5/2 + (ε n 5/2 + = −4.144 MeV or ε p 5/2 + = −0.601 MeV) and 1/2 + (ε n 1/2 + = −3.273 MeV or ε p 1/2 + = −0.106 MeV) bound states as predominantly 1d 5/2 and 2s 1/2 valence nucleons coupled to the 0 + ground state of 16 O. For 16 O we consider the effect of the coupling between the 0 + ground state and either the 2 + excited state at ǫ 2 + = 6.917 MeV or the 3 − excited state at ǫ 3 − = 6.129 MeV. The corresponding quadrupole and octopole deformations are β 2 = 0.36 [26] and β 3 = 0.75 [27], respectively. To adjust both 5/2 + and 1/2 + states, V ws in the s and d wave differ slightly. When considering the coupling to the 3 − excited state of 16 O, we set the depths of the potential in the negative-parity partial waves according to V ws (l = 1) = V ws (l = 2) and V ws (l = 3) = V ws (l = 0). In this way, the partial waves corresponding to the dominant configurations in the 5/2 + and 1/2 + states have the same potential depth. d. 23 Ne / ε n 1/2 + = −4.184 MeV. The configuration of the ground state is dominated by a 1d 5/2 nucleon bound to the 0 + ground state of the core. The excited state of 23 Ne is mostly a 2s 1/2 neutron bound to 22 Ne(0 + ). We consider couplings between the lowest 0 + , 2 + and 4 + core states, with excitation energies ǫ 2 + = 1.274 MeV and ǫ 4 + = 3.357 MeV for 22 Ne and ǫ 2 + = 1.247 MeV and ǫ 4 + = 3.308 MeV for 22 Mg. These three states are described as the first three levels of one rotational band with deformation parameters β 2 = 0.58 [26] and β 2 = 0.562 [26] for 22 Mg and 22 Ne, respectively. To reproduce the two energy levels in 23 Ne, we need to consider a slight difference between V ws (l = 0) and V ws (l = 2). The same value for V ws (l = 0) is used in 23 Al with small adjustments made to V ws (l = 2) to reproduce the binding energy exactly. e. 27 Mg/ 27 P: In these 26 Mg+n and 26 Si+p mirror systems, we reproduce the 1/2 + ground states as a dominant 2s 1/2 nucleon bound to the 0 + ground state of the core by ε n 1/2 + = −6.443 MeV or ε p 1/2 + = −0.861 MeV. For the neutron system, we also consider the excited state 3/2 + with ε n 3/2 + = −5.459 MeV to pin down the d-wave potential as its configuration is dominated by a 1d 3/2 neutron bound to 26 Mg(0 + ). Here, we consider couplings between the first 0 + , 2 + and 4 + core states, with excitation energy ǫ 2 + = 1.8 MeV for both cores and ǫ 4 + = 4.32 MeV for 26 Mg and ǫ 4 + = 4.18 MeV for 26 Si. Deformation parameters are β 2 = 0.482 [26] and β 2 = 0.446 [26] for 26 Mg and 26 Si, respectively. To reproduce the energy levels in 27 Mg, different depths V ws are taken for l = 0 and l = 2. For 27 P, the same V ws (l = 2) is used as for 27 Mg but small adjustments are made to V ws (l = 0) to obtain the correct binding energy. B. Ratio for specific mirror partners For comparison with previous works, we fix the deformation of the core, adjust the depth V ws of the interaction to reproduce binding energies as detailed in Section III A, and solve the coupled channels equation. To evaluate the sensitivity of our calculations to the choice of the B-x potential, we consider two geometries for the mean field, namely radius r ws = 1.2 fm and diffuseness a = 0.5 fm and radius r ws = 1.25 fm and diffuseness a = 0.65 fm. We first fix V so = 6 MeV with the same geometry as the Woods-Saxon potential, but repeat the calculations for the choice of V so = 8 MeV. The depths V ws obtained for each of the cases listed in Sec. III A are given in Table I. From the resulting proton and neutron wavefunctions, we determine ANCs and the ratio R (10). The ratio R for the dominant component for each case is shown in Table II and corresponds to r ws = 1.25 fm, a = 0.65 fm and V so = 6 MeV. The uncertainty reflects the range obtained with the other geometry and spin-orbit strength. Our values for R are compared to the values obtained from the analytic formula R 0 (11) (using the experimental binding energies and R N = 1.25A 1/3 ) and those obtained within the MCM, assuming two clusters and taking the Minnesota interaction R MCM [8,9]. For the first three cases studied, namely 8 (11) and the results of the microscopic two-cluster calculations RMCM [8,9] including the Minnesota interaction. The uncertainty in R account for the sensitivity to the parameters of VBx. in 17 O/ 17 F(e.s) the core in the neutron and proton systems are the same, in the last two cases the core β 2 differs slightly. The deviations with the analytic formula and MCM are not caused by this difference. For 23 Ne/ 23 Al, it is important to note that in our calculations we impose realistic binding energies whereas in the MCM results, binding energies can sometimes differ significantly. Since R depends strongly on the binding energies, this can be the cause for the large difference between our values and those of Ref. [9]). The values of R 0 presented in Table II also assume the experimental binding energies and therefore differences between R and R 0 must be related to the failure of the simple analytical relation. One could presume that the examples for which our model predicts significantly different ratio than the analytic prediction and the MCM are those in which the admixture with core excited configurations are largest. This is not the case: large admixture, or small spectroscopic factors, alone are not sufficient to cause a deviation from R 0 or previously calculated R MCM . Spectroscopic factors are around: 0.9 for 8 Li/ 8 B, 0.3 for 13 C/ 13 N, 0.6-0.9 for 17 O/ 17 F(g.s.), 0.7-0.9 for 17 O/ 17 F(e.s.), 0.7 for 23 Ne/ 23 Al and 0.5 for 27 Mg/ 27 P. What can be remarked is that the largest discrepancies appear for the cases in which the proton is very loosely bound. Another remarkable point is that our predicted ratio is always smaller than the analytical estimate. This feature is further investigated in the following section. C. Exploring the parameter space In this subsection, we use the deformation parameter as a free variable to explore different physical situations beyond the particular nuclei used as test cases. The configurations of the 23 Ne/ 23 Al and 27 Mg/ 27 P pairs being very similar to those of the 17 O/ 17 F systems in its ground state and excited state, respectively, we concentrate on the three lighter cases. Given the range of values for the deformation parameters, we vary the deformation between 0 and 0.7. For each deformation parameter, energies for the proton and neutron systems were refitted by small adjustments of V ws to eliminate erroneous variations of the ANC due to changes in the binding energies: overall V p ws ≈ V n ws . We fix the geometry: the standard r ws = 1.25 fm and a = 0.65 fm for 8 Li/ 8 B [28], r ws = 1.14 fm and a = 0.5 fm for 13 C/ 13 N [15], and r ws = 1.2 fm and a = 0.64 fm for 17 O/ 17 F [29]. The geometry for the spin-orbit force is taken to be the same as for the nuclear force, and the depth is fixed at around 6 MeV, for all cases. We find no significant difference in the ratio R for both 8 Li/ 8 B and 13 C/ 13 N mirror pairs. In these cases the main components of the wavefunction are p waves, even in the configurations including core excitation. For \|β 2 \| = 0.0-0.7 the resulting range of values for R are: (1.038-1.044) for 8 Li/ 8 B and (1.201-1.251) for 13 C/ 13 N. This constancy is obtained even though the variation in β leads to significant changes in the spectroscopic factor: S x 1p 3/2 goes from 1 to 0.75 for 8 Li/ 8 B, while S x 1p 1/2 decreases down to 0.32 for 13 C/ 13 N. Even if the system is made artificially less bound, the variation of R remains small and within the uncertainties of the geometry parameters for the interaction. The significant stability of R with such large changes in both deformation and admixture of different configurations suggests a universality of the mirror technique developed in Ref. [7]. The situation for 17 O/ 17 F is different. In this case core excitation introduces different orbital angular momenta in the wavefunction. We consider the separate effect of including the 3 − state and the 2 + state. Let us first consider the inclusion of 16 O(0 + , 3 − ). For each β 3 , energies for the two lowest states in 17 O and 17 F were refitted by small adjustments of V ws (l = 0) and V ws (l = 2). As mentioned in Sec. III A, the depth of the potential in the negative-parity partial waves is set to V ws (l = 1) = V ws (l = 2), and V ws (l = 3) = V ws (l = 0). In this way, all the depths were constrained phenomenologically. Here again the variations in R are small. Even though over 30% of the 5/2 + ground-state wave function is in a core-excited configuration at β 3 = 0.7, the change in R is less than 2%. For this β 3 , the 1/2 + excited-state wave function is almost exclusively in the 16 O(0 + )⊗2s 1/2 configuration (S x 2s 1/2 ≈ 95%). Expectedly, the change in the corresponding ratio is limited to less than 1%. Next we consider the inclusion of 16 O(0 + , 2 + ). In this case the d 5/2 ground state admixes with an s 1/2 component with the core in its excited state, while in the 1/2 + state, the s 1/2 coupled to the g.s. core admixes with d components with the core in its 2 + state. Again, energies for the two lowest states in 17 O and 17 F were refitted by simultaneously adjusting V ws (l = 0) and V ws (l = 2) for each β 2 . For both 5/2 + and 1/2 + states, the spectroscopic factor (8) of the dominant component (which has the core in its ground state) suffers a large reduction at large β 2 , as shown in Fig. 1. While for the ground state, the proton and neutron spectroscopic factors vary together (Fig. 1a), for the excited state it becomes clear that the admixture in the neutron system is larger than in the proton system (Fig. 1b). This is then reflected in a different behavior of the ANC ratios. In Fig. 2 we present the ratio R (10), as well as a modified ratio compensating for the changes in spectroscopic factors R * = RS n /S p . The analytical prediction R 0 (11) is also shown (horizontal dashed lines). For the 5/2 + ground state, neither R nor R * deviate much from the value at β 2 = 0, corresponding to the single particle prediction (Fig. 2a). They are also very close to the analytical prediction, R 0 . On the contrary, for the 1/2 + excited state, R shows a large variation, mainly, but not only, caused by the difference between neutron and proton spectroscopic factors (Fig. 2b), as expected from the results of Ref. [19]. This can be deduced from their relative variations across the considered β 2 range: while R varies by 22%, R * varies by less than 3%. They also differ more from R 0 . As noted in the MCM studies [8,9], the ratio R at the realistic deformation of the 16 O core (i.e. β 2 = 0.36) is well approximated by the average between R 0 and the single-particle ratio, i.e. R at β 2 = 0. Since this result is strongly dependent on the value of the deformation, we do not believe it can be safely generalized to other systems. The features illustrated in Fig. 2 can be directly extrapolated to 23 Al and 27 P. As mentioned before, the former has a structure very similar to that of 17 F(g.s.), while the latter exhibits the same components as 17 F(e.s.). In Refs. [8,9] core excitation is explored within the MCM. Already then there was growing disagreement between R MCM and R 0 as more core states were explicitly included in the model space. This was understood in terms of the long range Coulomb quadrupole term added to the Hamiltonian in the proton case, a term not considered in the derivation of R 0 , nor in our present calculations. Here however, we not only see a deviation from R 0 , but also a strong dependence on the deformation parameter for particular cases. Therefore we conclude the source for deviations from R 0 and the break down of the constant ratio concept is induced by the nuclear quadrupole term, which is present in both neutron and proton systems. The surprising results for the 1/2 + mirror states led to several additional tests which isolated the cause for the large coupling dependence in R. There are three essential ingredients: low binding, the existence of an swave component coupled to the ground state of the core, and a significant admixture with other configurations. It appears that when all three conditions are met, the differences between the neutron and proton wavefunctions increase around the surface, exactly where the nuclear quadrupole interaction peaks. This results in a stronger effect of coupling on the neutron system compared to the proton system, inducing differences in S n relative to S p , which reflect on a coupling dependence in R. Our tests show that the effect is independent on whether the wavefunctions have a node. IV. CONCLUSIONS A proposed indirect method for extracting proton capture rates from neutron mirror partners relies on the ratio between asymptotic normalization coefficients of the mirror states being model independent. In this work, we test this idea against core deformation and excitation. We consider a core + N model where the core is deformed and allowed to excite and apply it to a variety of mirror pairs ( 8 Li/ 8 B, 13 C/ 13 N, 17 O/ 17 F, 23 Ne/ 23 Al, and 27 Mg/ 27 P.). We stress that our approach is strongly phenomenological: for each case we always fit the neutron and proton binding energies exactly. This is not the approach followed in previous works [8][9][10]. Imposing instead equal nuclear interactions V n = V p in our model would lead to a strong and erroneous deformation dependence of R due to unequal changes in the neutron and proton binding energies. In that case, even R 0 would become model dependent. We explored how the mirror states evolve as a function of deformation (coupling strength). For most cases the ratio of the ANC of mirror states was found to be independent of the deformation. From our investigations we conclude that there are three conditions that need to be met for the idea of a model-independent ratio to break down with deformation or core excitation: i) the proton system should have very low binding, ii) the main configuration should be an s-wave component coupled to the ground state of the core, and iii) there should be significant admixture with other configurations. This has implications for the application of the indirect method based on the ANC ratio to reactions relevant to novae, namely pertaining the direct capture component of 26 Si(p,γ) 27 P. In connecting the ANC of 27 Mg and 27 P one should be careful with coupling between different configurations. An analytic formula for the ratio R 0 was derived [7] using a single particle configuration for neutron and proton states. In [8] it is suggested that differences between R 0 and R calculated within MCM arose due to the quadrupole Coulomb interaction, which is not included in the proton state, when deriving R 0 , but of course is included in the MCM calculations. We do not include this term in our calculations and yet still find deviations between our R and R 0 . These can only be due to the nuclear quadrupole term. When an incoming s-wave neutron is involved one should choose an adequate probe to measure it. While s-wave proton capture (usually to a bound p-state) is a peripheral process for the low relative energies of astrophysical interest, the s-wave neutron capture is not and generally depends on the whole overlap function. Nevertheless, in principle one can extract ANCs for the neutron system from peripheral nuclear reactions (transfer or breakup) using an appropriate choice of kinematic conditions. That ANC would then relate to the astrophysically relevant proton ANC. We are grateful to Natasha Timofeyuk for suggesting this project and providing important feedback on the work and we thank Ron Johnson for many useful discussions. This work was partially supported by the National Science Foundation grant PHY-0555893, the Department of Energy through grant DE-FG52-08NA28552 and the TORUS collaboration de-sc0004087. 23 Al: The cores in this mirror pair are 22 Ne and 22 Mg. Our model reproduces the 5/2 + ground state of both nuclei with ε n 5/2 + = −5.200 MeV or ε p 5/2 + = −0.122 MeV, and the 1/2 + excited state of 23 Ne with Li/ 8 B, 13 C/ 13 N, and 17 O/ 17 F(g.s.) our ratios are very close to the values obtained with the analytical formula and those obtained within the MCM. Larger deviations are found for 17 O/ 17 F(e.s), 23 Ne/ 23 Al and 27 Mg/ 27 P. While online) Neutron and proton spectroscopic factors for17 O and 17 F, respectively, considering the 16 O core in its 0 + ground state and 2 + first excited state: (a) 5/2 + ground state and (b) 1/2 + first excited state. online) Ratio of proton and neutron ANCs for 17 O and 17 F, respectively, including 16 O(0 + , 2 + ): (a) 5/2 + ground state and (b) 1/2 + first excited state. TABLE I : IDepths Vws of the central potential for the various cases listed in Sec. III A (values are given in MeV). The first number is the depth for the neutron case, and the second number is for the proton case.rws = 1.2 fm a = 0.5 fm rws = 1.25 fm a = 0.65 fm I π B Vso = 6 MeV Vso = 8 MeV Vso = 6 MeV Vso = 8 MeV 8 Li/ 8 B 3/2 − , 1/2 − Vws 58.9168/59.6080 61.5840/62.3011 42.6722/42.7479 41.5371/42.0051 13 C/ 13 N 0 + , 2 + Vws 59.6504/60.4455 59.6635/60.4436 56.4058/56.8085 56.3261/56.7085 17 O/ 17 F 0 + , 3 − Vws (l = 0 and 3) 51.6320/51.8248 51.5582/51.7569 47.6517/47.2511 47.6102/47.2151 Vws (l = 1 and 2) 60.3527/61.1595 59.5078/60.3983 57.0489/57.3955 56.4297/56.8736 17 O/ 17 F 0 + , 2 + Vws (l = 0 and 3) 52.3626/53.1800 52.9035/53.6801 48.5662/48.6508 48.9239/48.9721 Vws (l = 1 and 2) 53.8238/54.2036 52.1111/52.4642 51.2895/51.2182 49.8253/49.7276 23 Ne/ 23 Al 0 + , 2 + , 4 + Vws (l = 0 and 3) 54.4839/54.4839 54.6976/54.6976 49.0477/49.0477 49.3356/49.3356 Vws (l = 1 and 2) 55.4659/56.2811 53.9916/54.7628 52.6434/52.7345 51.3534/51.4028 27 Mg/ 27 P 0 + , 2 + , 4 + Vws (l = 0 and 3) 52.5428/53.2944 51.2411/51.9676 48.9469/48.2467 47.7296/46.9657 Vws (l = 1 and 2) 56.4027/56.4027 56.9672/56.9672 53.6536/53.6536 54.1691/54.1691 TABLE II : IIRatio of proton to neutron ANCs for the dominant component: comparison of this work R with the results of the analytic formula R0 . 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[ "Coulomb Field Scattering in Born-Infeld Electrodynamics", "Coulomb Field Scattering in Born-Infeld Electrodynamics" ]	[ "Daniel Tennant \nDepartment of Physics\nAustin Community College\n78758\n" ]	[ "Department of Physics\nAustin Community College\n78758" ]	[]	In the context of Born-Infeld electrodynamics, the electromagnetic fields interact with each other via their non-linear couplings. A calculation will be performed where an incoming electromagnetic plane wave scatters off a Coulomb Field in the geometrical optics approximation. In addition to finding the first order angle of deflection, exact solutions for the trajectory will also be found. The possibility of electromagnetic bound states will be discussed.	10.1103/physrevd.83.047701	[ "https://arxiv.org/pdf/1011.6137v1.pdf" ]	118,684,564	1011.6137	6191876f8995bbf3b12038d825d5ce640cbca716	Coulomb Field Scattering in Born-Infeld Electrodynamics 29 Nov 2010 (Dated: November 30, 2010) Daniel Tennant Department of Physics Austin Community College 78758 Coulomb Field Scattering in Born-Infeld Electrodynamics 29 Nov 2010 (Dated: November 30, 2010) In the context of Born-Infeld electrodynamics, the electromagnetic fields interact with each other via their non-linear couplings. A calculation will be performed where an incoming electromagnetic plane wave scatters off a Coulomb Field in the geometrical optics approximation. In addition to finding the first order angle of deflection, exact solutions for the trajectory will also be found. The possibility of electromagnetic bound states will be discussed. I. INTRODUCTION Non-linear electrodynamics has been a subject of research for many years. That quantum electrodynamics predicts that the electromagnetic field behaves non-linearly through the presence of virtual charged particles 1 was first discussed by Heisenberg and Euler. This was rapidly followed by the proposal of Born and Infeld (BI) 2 of a new classical non-linear theory of electromagnetism which possesses a maximum field strength, β. The theory of BI has found a resurgence of attention due to its reproduction through string theory given certain boundary conditions 3 . The BI theory also contains many symmetries common to the Maxwell theory despite its non-linearity. The pioneering work of Plebanski 4 and Adler 5 amongst others 67 showed the BI theory to be unique among non-linear theories of electrodynamics in that it is the only possible theory that is absent of birefringence; both polarization modes propagate at the same velocity. More recently, it was observed that the BI theory has another common characteristic with (source-free) Maxwell. BI is the only possible non-linear version of electrodynamics that is invariant under electromagnetic duality transformations 8 . Note that there is no physical demand that electrodynamics must be a linear field theory. Therefore, the nonlinear extensions of electrodynamics which respect the same symmetries must be taken into consideration. Unfortunately, the fact that it is invisible to optical rotation experiments such as PVLAS 9 makes it difficult to find experimental signatures of BI. Recent experimental proposals include using waveguides 11 and ring lasers 10 . In this article, I will propose using a strong electric field to "trap" an incoming electromagnetic wave. Our incoming ray will behave as if it is in a novel space-time geometry. In this article, the mechanism for establishing the non-trivial metric is considering electrodynamics to be a non-linear theory. An incoming electromagnetic wave will experience a spatial and frequency dependant index of refraction which will cause a deflection in trajectory. This index of refraction will have the equivalent effect as the incoming ray being in a warped space-time geometry. The geometrization of space-time is due to the universality of gravity; it couples to all known forms of matter/energy. Recently different analogue models of general relativity have appeared in hopes of detecting Hawking Radiation 12 . In this article, another example of an effective geometrization of space-time will be presented. Recall that what I mean by effective geometrization is that it is not a universal effect. In our case, this effective geometry will be perceived only by electromagnetic plane waves. Other forms of matter and energy will behave as in their usual space-time geometry. The special case we will consider is a Coulomb Field in the context of Born-Infeld electrodynamics. This case actually has many similarities with the Schwarzschild solution of General Relativity. We will find that they share a potential energy with some common characteristics. The most interesting is the existence of a point of unstable equilibrium about which rays can be momentarily trapped. II. WAVE PROPAGATION Consider an incoming electromagnetic wave propagating through a background electromagnetic field. One of the main difficulties with non-linear theories is that, in general, it is impossible to disentangle fields into separate parts (this part of the total field is due to this source, that part of the field is due to that source, etc.). The ability to superimpose solutions to differential equations is a property of linear differential equations only. It is necessary to impose some conditions on our example to observe the behaviour of the incoming wave separate from the background field. First, the incoming disturbance, f ab , is "weak" compared to the background field, F ab . This guarantees that the background field will not be affected by the incoming field. A similar situation arises in General Relativity where one usually assumes that the mass moving in a background gravitational field does not affect that field. Second, we can only consider incoming waves whose wavelengths that are sufficiently small. To be precise, the background field cannot change appreciably over a wavelength. This allows us to consider the background field to be approximately constant when compared to the incoming wave so that when we find the rate of change of the total field, it effectively isolates the wave portion, ∂F T ∼ ∂f . Once these approximations are made, it is possible to write an expression for the disturbance propagating through a background field. Combining the dynamical "Maxwell" equations, ∂ b ∂L ∂F ab = 0,(1) with the kinematic equations, ∂ a f bc + ∂ c f ab + ∂ b f ca = 0,(2) one arrives at ∂ 2 f ab + Λ rs ab f rs = 0. Λ rs ab = 1 2L F [L F F F [a, m F rs + L F G (F [a, mF rs +F [a, m F rs ) + L GGF[a, mF rs ]∂ b] ∂ m(3) Brackets in the subscript denote the antisymmetric combination. Terms such as L F G are to be understood as ∂ 2 L ∂F ∂G . Note that this expression holds for any Lagrangian density and any slowly-varying background field. Following the usual prescription to arrive at the geometrical optics regime 13 , the incoming wave is of the form f ∼ e iS , where higher derivatives of the phase are negligible, \| ∂S \| 2 >> \| ∂ 2 S \|. This allows us to identify the first partial derivatives of the field as ∂ a S → k a . III. BACKGROUND COULOMB FIELD By the method of effective geometries laid out by Novello 14 and others, I will describe a ray passing through a background Coulomb field, µ r 2r . Due to the effect on the ray by the background field, the ray will no longer follow null geodesics in Minkowski space, but in another, effective geometry; g ab k a k b = 1 1 − ( α r 2 ) 2 (−k 2 t + k 2 r ) + r 2 k 2 φ = 0 α = µ β .(4) Our coordinate system is arranged such that the incoming ray falls in the x-y plane. Due to the symmetries inherent in this metric, one can expect to find a conserved energy, ω, and angular momentum, δ. The effective metric can be expressed as a sum of the ray's kinetic and potential energies. z 2 + U(z) = 0 U(z) = ω z 2 (1 − ( γ z 2 ) 2 )(1 − z 2 + ( γ z ) 2 ) γ = α b 2 = E(b) β (5) The coordinate z is the radial length in terms of the scattering parameter, b = δ ω . There is one local maximum for this potential. By analysing the potential, we can ascertain what types of orbits are possible. For the most physically realistic case, γ << 1, the potential is slightly repulsive. However, at approximately γ ≃ 0.87, the incoming ray passes close enough to cross the potential maximum such that attractive and repulsive forces cancel and the ray continues on its original trajectory. For gamma values greater than this, the ray is bent inward. We will see below that for γ = 1 the ray is actually caught in a bound, unstable orbit. z o = − 2 3 [2γ 2 + (a 2 + b 2 ) 1/6 cos( φ+2π 3 )] a = (2γ) 6 b = (3γ) 3 [1 + 3 4 (1 + 32 9 γ 2 ) 2 ] 1/2 tanφ = b a(6) Exact solutions for the trajectory in this effective geometry are attainable in terms of elliptical functions. It is necessary to make some coordinate transformations; x = r 2 α and φ = 1 2 ϕ. Equation (4) becomes ( dx dϕ ) 2 = γ(x − 1)(x + 1)(x − 1/γ) ≡ f (x).(7) Physically possible trajectories correspond to positive values of f (x). Again, the most physically relevant case is that of the incoming ray with γ << 1. In this case, x approaches 1/γ from infinity and then returns out to infinity. This corresponds to the ray approaching from infinity, reaching b, the impact parameter, and continuing on. The solution to equation (7) for γ ≤ 1 is r 2 = b 2 1 + γ(1 − sn 2 (a, k)) sn 2 (a, k) a = 1 + γφ k 2 = 2γ 1 + γ .(8) Note for this case, this trajectory is an exact expression as long as previous assumptions, namely a slowly varying metric, still hold. In the limit β → ∞, γ → 0, k → 0, sn → sin, r = b sin(φ) ,(9) For γ approaching one from zero, the effective potential begins slightly repulsive, and then becomes more and more attractive until the ray is caught in circular orbit. This orbit, however, is unstable. IV. DISCUSSION This is a situation where closed time-like photon orbits can occur. The Planck Field Strength is of the order 10 62 V /m, but there is some discussion that possibly the string scale is much lower, possibly even to 10 46 V /m 16 . This is still an absurdly large value, much larger than the critical field strength of QED, 10 18 V /m. Recall that this is the electric field strength such that there is an energy density of m e c 2 per electron Compton wavelength cubed, λ 3 e . At these energy densities, the electric field is primarily expected to decay into electron-positron pairs. The behaviour of strong field QED needs to be better understood in order to know whether these field strengths can even be attained. This calculation shows behaviour of light that is usually only considered to arise from its interaction with gravity. The Schwarchild solution reveals a similar unstable equilibrium in its potential function. Only this time, it is not a gravitational force that is responsible for creating the closed orbit, but a non-linear electromagnetic mechanism. This article presents another case where cosmological curiosities, such as black holes, event horizons, etc. that are usually confined to the study of gravity can be found in new places. V. ACKNOWLEDGEMENTS The author wishes to thank Dr. Gerardo Munoz and Dr. Emil Akhmedov for their helpful discussion and suggestions. ranges from approaching zero in the limit of no non-linear coupling to z o ≃ 1.23 in the opposite limit, γ → 1. FIG. 3 : 3Various Trajectories just the equation of a strait line. It is strait forward to calculate the first order angle of deflection 13 . * Electronic address: dtennant@austincc. * Electronic address: [email protected] . W Heisenberg, H Euler, Z. Phys. 98W. Heisenberg and H. Euler, Z. Phys. 98 (1936) . M Born, L Infeld, Proc. R. Soc. London, Ser A. 144M. Born and L. Infeld, Proc. R. Soc. London, Ser A 144, 425-451 (1934) . E S Fradkin, A A Tseytlin, Phys. Lett. B. 163E. S. Fradkin and A. A. Tseytlin, Phys. Lett. B 163 123-130 (1985) . J Plebanski, Lecture Notes on Nonlinear Electrodynamics. NorditaJ. Plebanski, Lecture Notes on Nonlinear Electrodynamics, Nordita, (1970) . S Adler, Ann. Phys. (N.Y.). 67599S. Adler, Ann. Phys. (N.Y.) 67, 599 (1971) . G Boillat, J. Math Phys. 2G. Boillat, J. Math Phys, Vol. 2, Number 3, March (1970) . Z Bialynicka-Birula, I Bialynicka-Birula, Phys. Rev. D. 2Z. Bialynicka-Birula and I. Bialynicka-Birula, Phys. Rev. D, Vol. 2, Number 10, Nov. (1970) . Rasheed Gibbons, Nucl. Phys. B. 454185Gibbons and Rasheed, Nucl. Phys. B, 454, 185 (1995) . Pvlas Coll, PRL. 99129901PVLAS Coll., PRL 99, 129901 (2007) . V Denisov, Phys. Rev. D. 61V. Denisov, Phys. Rev. D, Vol. 61, Jan. (2000) . R Ferraro, Phys. Rev. Lett. 99230401R. Ferraro, Phys. Rev. Lett. 99, 230401 (2007) . M Visser, Phys. Rev. Lett. 807M. Visser, Phys. Rev. Lett. 80, 3436 (1998) 7 & Landau, Lifchitz, The Classical Theory of Fields. Oxford, New York Pergamon PressLandau & Lifchitz, The Classical Theory of Fields, Oxford, New York Pergamon Press (1974) . M Novello, J M Salim, V A De Lorenci, E Elbaz, Physical Review D. M. Novello, J.M. Salim, V.A. De Lorenci, E. Elbaz, Physical Review D, (2001) D F Lawden, Elliptic Functions and Applications. New YorkSpringer-Verlag126D. F. Lawden, Elliptic Functions and Applications, Springer-Verlag, New York (1989) p. 126 I Antoniadis, arXiv:hep-th/0710.4267v2Topics on String Phenomenology, lectures given at the Les Houches 2007 Summer School String Theory and the Real World From particle physics to astrophysics. I. Antoniadis, Topics on String Phenomenology, lectures given at the Les Houches 2007 Summer School String Theory and the Real World From particle physics to astrophysics, 2-27 July (2007), arXiv:hep-th/0710.4267v2	[]
[ "Detecting the periodicity of highly irregularly sampled light-curves with Gaussian processes: the case of SDSS J025214.67-002813.7", "Detecting the periodicity of highly irregularly sampled light-curves with Gaussian processes: the case of SDSS J025214.67-002813.7" ]	[ "Stefano Covino \nINAF / Brera Astronomical Observatory\nvia Bianchi 4623807Merate (LC)Italy\n", "Felipe Tobar \nInitiative for Data & Artificial Intelligence\nUniversidad de Chile\nSantiago de ChileChile\n", "Aldo Treves \nINAF / Brera Astronomical Observatory\nvia Bianchi 4623807Merate (LC)Italy\n\nUniversità degli Studi dell'Insubria\nVia Valleggio 1122100ComoItaly\n" ]	[ "INAF / Brera Astronomical Observatory\nvia Bianchi 4623807Merate (LC)Italy", "Initiative for Data & Artificial Intelligence\nUniversidad de Chile\nSantiago de ChileChile", "INAF / Brera Astronomical Observatory\nvia Bianchi 4623807Merate (LC)Italy", "Università degli Studi dell'Insubria\nVia Valleggio 1122100ComoItaly" ]	[ "MNRAS" ]	Based on a 20-year-long multiband observation of its light-curve, it was conjectured that the quasar SDSS J025214.67-002813.7 has a periodicity of ∼ 4.4 years. These observations were acquired at a highly irregular sampling rate and feature long intervals of missing data. In this setting, the inference over the light-curve's spectral content requires, in addition to classic Fourier methods, a proper model of the probability distribution of the missing observations. In this article, we address the detection of the periodicity of a light-curve from partial and irregularly-sampled observations using Gaussian processes, a Bayesian nonparametric model for time series. This methodology allows us to evaluate the veracity of the claimed periodicity of the abovementioned quasar and also to estimate its power spectral density. Our main contribution is the confirmation that considering periodic component definitely improves the modeling of the data, although being the source originally selected by a large sample of objects, the possibility that this is a chance result cannot be ruled out.	10.1093/mnras/stac596	[ "https://arxiv.org/pdf/2203.03614v1.pdf" ]	247,257,625	2203.03614	805ccec181aa153149f8dee8e7239fb400d3343b	Detecting the periodicity of highly irregularly sampled light-curves with Gaussian processes: the case of SDSS J025214.67-002813.7 2021 Stefano Covino INAF / Brera Astronomical Observatory via Bianchi 4623807Merate (LC)Italy Felipe Tobar Initiative for Data & Artificial Intelligence Universidad de Chile Santiago de ChileChile Aldo Treves INAF / Brera Astronomical Observatory via Bianchi 4623807Merate (LC)Italy Università degli Studi dell'Insubria Via Valleggio 1122100ComoItaly Detecting the periodicity of highly irregularly sampled light-curves with Gaussian processes: the case of SDSS J025214.67-002813.7 MNRAS 0002021Accepted XXX. Received YYY; in original form ZZZPreprint 8 March 2022 Compiled using MNRAS L A T E X style file v3.0quasars: individual: SDSS J02521467-0028137 -quasars: supermassive black holes -methods: statistical Based on a 20-year-long multiband observation of its light-curve, it was conjectured that the quasar SDSS J025214.67-002813.7 has a periodicity of ∼ 4.4 years. These observations were acquired at a highly irregular sampling rate and feature long intervals of missing data. In this setting, the inference over the light-curve's spectral content requires, in addition to classic Fourier methods, a proper model of the probability distribution of the missing observations. In this article, we address the detection of the periodicity of a light-curve from partial and irregularly-sampled observations using Gaussian processes, a Bayesian nonparametric model for time series. This methodology allows us to evaluate the veracity of the claimed periodicity of the abovementioned quasar and also to estimate its power spectral density. Our main contribution is the confirmation that considering periodic component definitely improves the modeling of the data, although being the source originally selected by a large sample of objects, the possibility that this is a chance result cannot be ruled out. INTRODUCTION Since the seminal paper of Begelman et al. (1980), the search of Binary Supermassive Black Holes (BSBH) has attracted the interest of a vast group of astronomers; this is because BSBH are regarded as the parent population of the strongest gravitational wave (GW) signals, due to the final merging of the two black holes. Though such events have not been detected yet, current developments associated to, e.g., i) the discovery of GW bursts associated to collapsed star merging detected by the LIGO/Virgo Collaboration (Abbott et al. 2020), ii) the advancement of detection techniques through the Pulsar Timing Array (PTA) experiment (Verbiest et al. 2021), and iii) future space borne instrumentation like LISA (Vitale 2014;Sesana 2021), have strongly increased the development of the field in recent years. Among the various approaches to finding BSBH, a promising possibility has been that of detecting periodicities in the light curves of Active Galactic Nuclei (AGN). In particular, it is known that if the mass of the holes are of the order of 10 8 and the period is somehow related to the orbital one, with ∼ 0.01 pc separations, the periodicity should be of the order of years. Two main types of AGN have been considered thus far: Blazars (i.e. AGN dominated by a relativistic jet) and Quasars. Blazars largely populate the gamma-ray sky, and most efforts related to the detection of their periodicities have been performed examining data from the Fermi Gamma-Ray Space Telescope (Atwood et al. 2009), which covers the entire sky every three hours and has been collecting data for ∼ 11 years. For Quasars, main observational data are in the form of large collections of optical frames partly obtained from blind searches of transients. ★ E-mail: [email protected] Data sources that are worth noticing in this regard are the Catalina Real-Time Transient Survey (Djorgovski et al. 2011) and the Palomar Transient Factory (Law et al. 2009). Though there are thousands of Fermi gamma-ray blazars, those which are bright enough for constructing a light curve suitable for a temporal analysis are only a dozen. Careful examination of these blazars, performed by several researchers, has been possible due to the free availability of the data (e.g. Covino et al. 2019, and references therein). In general, the search of a period of about 1 year is a challenging endeavour because: 1) due to uneven sampling, the required observation time needs to be about 10 times larger than the target period; 2) blazars are strongly variable sources, and the signal should be searched above a frequency dependent (red) noise; 3) the appearance of periodicity could be episodical, that is, the signal may be identifiable on a time scale of a few years, but not of a decade or longer. However, one advantage of the Fermi light curves is that the sampling is homogeneous, without seasonal gaps. Various methods have been used for the periodicity search. Some are more apt for sinusoidal signals, whereas others are essentially independent of the shape of signal, and often, unsurprisingly, results are widely distinct under the consideration of different methods (Rieger 2019). The resulting picture is therefore still controversial. Arguably, the only source where the hypothesis of periodicity is supported by sufficient evidence is PG 1553+113. The period ∼ 2.2 years in the Fermi data was first proposed by Ackermann et al. (2015), and then confirmed by Tavani et al. (2018). Note, however, that its significance was challenged by Ait Benkhali et al. (2019) and Covino et al. (2019), essentially due to the short number of epochs covered by the Fermi observations. As for quasars, more than a hundred periodic candidates have been proposed in the last decade (e.g. Graham et al. 2015;Charisi et al. 2016), but the significance of some of these periods has been challenged because of the incorrect treatment of the multiple-trial effect due to the large number of considered sources (Vaughan et al. 2016). Finally, we have also to quote the case of OJ 287, which is a Blazar, but its periodicity ∼ 12 yr was identified from optical data using a century of archived plates (Carrasco et al. 1985;Dey et al. 2019). In this case, some concerns on the significance of the periodicity have also been raised (e.g. Stothers & Sillanpää 1997;Butuzova & Pushkarev 2021). The methods for detecting the periodicity in blazars and quasars have proceeded practically independently, with little interactions between the communities cultivating the two fields. See, however, Holgado et al. (2018), where the contributions of the two populations to the GW background, constrained by PTA observations, are compared and Charisi et al. (2021) for future perspective of the field. Recently, Chen et al. (2020) presented a systematic search of periodicity of 625 quasars with a median redshift ∼ 1.8 and with multiband photometry available for a time scale of ∼ 20 years. They found 5 candidates with periodicity of 3-5 years. Liao et al. (2021), using the same dataset, considered very carefully the most interesting case, SDSS J025214.67-002813.7 (z=1.53), discussing the significance of the proposed ∼ 4.4 year (∼ 1607 days). Chen et al. (2021) focused on the possible physical interpretation for the identified periodicity. In this paper we re-examine the available optical data for SDSS J025214.67-002813.7 (hereinfater J0252) using the same dataset as in Liao et al. (2021). Our analysis is carried out modeling the J0252 data by means of Gaussian processes (GPs) extending the procedure proposed in Covino et al. (2020) for the case of blazars. The main goal of our work is to assess the statistical significance of the proposed periodicity and to determine the Power Spectral Density (PDS) of the target by means of a Bayesian non-parametric analysis. Data about J0252 are described in Section 2, the GP-based analysis methods are shortly described in Section 3. Results are reported in Section 4 and a discussion is developed in Section 5. Finally, conclusions are given in Section 6. Software libraries used for the computations carried out in this paper are listed in Appendix A. SDSS J025214.67-002813.7 J0252 was selected from a set of 625 quasars with spectroscopic identification . For these objects, multi-band photometry was available from the Dark Energy Survey (DES, Dark Energy Survey Collaboration et al. 2016) and from the Sloan Digital Sky Survey (SDSS, Ivezić et al. 2007) covering a range of about 20 years of observations, with irregular samplings, gaps, etc. We refer the reader to Liao et al. (2021) for any detail about data analysis. The light curves in the are shown in Fig. 1. The median spacing is about 5-6 days, although the longest interruptions lasted more than about 1000 days for all the light curves. In total the dataset consists of about 230-280 epochs. Liao et al. (2021) carried out a detailed analysis of the optical light-curves of J0252 and reported an interesting periodicity at = 1607 ± 7 days with a significance of 99.95% in the band, moreover the significance is larger than 99.43% in any band. Given the length of the monitoring this implies that less than 5 complete cycles of variations have been covered. The short duration light-curve, in terms of the time scale under analysis, makes any periodicity claim intrinsically delicate. Liao et al. (2021) identified the periodicity applying different analysis tools. A spectral analysis was carried out by the Lomb-Scargle algorithm (Lomb 1976;Scargle 1982;Zech-meister & Kürster 2009;VanderPlas 2018) and the significance of the maximum power in the derived periodogram compared to the null hypothesis that it is due to noise was assessed by simulating a large number of artificial light curves with the same sampling of the original one. The analyses were also carried out in the temporal domain, fitting a sine curve or physically motivated models to the data, and modeling the auto-correlation function (ACF) of the data as a damped periodic oscillation (Alexander 1997;Graham et al. 2015). J0252 turned out to be the most interesting case (i.e. with the lowest false alert probability) in a small set of five objects showing periodicities larger than 99.74% in at least one band. The significance of the periodicity appears substantially stronger if all the bands are analysed together. In Fig. 2 we show the Lomb-Scargle periodograms for the bands, i.e. the convolution of the periodogram of the observed window together with that of the data (e.g., van der Klis 1989). The maxima of the periodograms are of course consistent with those reported in Liao et al. (2021) and very similar in each band. However, the periodograms also show the typical signatures of correlated noise (e.g., Vaughan 2010), with a flat region at shorter periods, where the periodograms are probably dominated by Poissonian noise, and a raising trend going to the longer (shorter) periods (frequencies). The presence of correlated noise and the short duration of the monitoring, compared to the proposed periodicities, make any claim about their significance somewhat ambiguous (see also, e.g., Krishnan et al. 2021). The Lomb-Scargle algorithm can also be obtained by a Bayesian analysis under the prior assumption that data periodicity are modeled by a sinusoidal functional form (Jaynes & Bretthorst 2003;Bretthorst 2003;Mortier et al. 2015). This scenario allows one to obtain the probability density function (PDF) for the frequency given the stated assumptions and the uncertainties associated to the maxima in the periodograms. In our case the PDF maxima and their 1 uncertainties are: 1648 ± 69, 1657 ± 97, 1663 ± 87 and 1650 ± 145 days, for the bands, respectively. GAUSSIAN PROCESSES Gaussian processes (see Rasmussen & Williams 2006, for a comprenhensive introduction) are probabilistic generative models for time series , which can be considered as the infinitedimensional extension of the multivariate Normal distribution. As such, they provide a Bayesian nonparametric approach to smoothing, interpolation and forecasting (Hogg & Villar 2021). Being defined on the real line rather than a finite, user-defined, discrete grid, GPs are particularly suited for (Bayesian) regression of irregularly-sampled data, possibly with large gaps, while providing sound error bars. The hyperparameters of the GP model are the mean and covariance functions. The former is usually set to zero or to linear/quadratic functions, while the latter is a positive semidefinite function. The covariance kernel is chosen based on a priori knowledge of the dynamic behavior of the data, for instance, the existence of periodicities or the requirement for differentiability. In addition to the well-known collection of off-the-shelf covariance functions (Rasmussen & Williams 2006), the choice of a covariance function can rest upon the Wiener-Khinchin theorem (Brockwell & Davis 2016), which states that -in the stationary case-the covariance kernel and the power spectral density (PSD) of the data are Fourier pairs. Periodicity assessment GP techniques have been applied to identify possible periodicities in time series Tobar et al. 2015;Durrande et al. 2016;Littlefair et al. 2017;Angus et al. 2018;Tobar 2018;Covino et al. 2020;Corani et al. 2020;Zhang et al. 2021). Here, we implement a procedure originally suggested by Littlefair et al. (2017) and then further developed by Covino et al. (2020). The idea is essentially to model the light curve of interest either with a stationary covariance function or with the same kernel but multiplied by a periodic (e.g., cosine) covariance function. Then, the problem of determining the presence of a periodic component hidden in the (often correlated) noise affecting an astronomical time series can be addressed by means of Bayesian model selection (Kass & Raftery 1995 Wilkins (2019) and Griffiths et al. (2021) carried out extensive simulations verifying the capability of some of the most common kernel functions (e.g. the squared exponential or radial basis, Matérn and rational quadratic kernels, Rasmussen & Williams 2006) to accurately modeling the data PSD. Results show that, depending on the specific sampling pattern and gap duration, either the Matérn and rational quadratic covariance functions typically outperform the squared exponential kernel. The Matérn kernel family is characterized by a parameter, , that drives the degree of "smoothness" of the kernel. Time-series models corresponding to autoregressive processes of the first order are discrete time equivalents of GP models with Matérn covariance functions with = 1/2, which correspond, in one dimension, to the Ornstein-Uhlenbeck process (e.g. Roberts et al. 2012). The Ornstein-Uhlenbeck process is frequently invoked in modeling the statistical behavior of AGN and blazars (e.g. Takata et al. 2018;Burd et al. 2021). Given that the product and the sum of covariance kernels are again legitimate (i.e., symmetric and positive definite) covariance kernels, we can build a periodic covariance function by multiplying a stationary kernel with the so-called "cosine" kernel (Covino et al. 2020). Since a GP models are naturally suited for a Bayesian treatment, we can compute the probability favoring the more complex (i.e. periodic) model compared to a simpler one (non-periodic) by the Bayes factor. A remarkable by-product of this kind of analysis is that the PSD for the period can be easily obtained marginalizing out (i.e integrating) the posterior probability distribution of the parameters obtained from the regression. Spectral estimation by means of GP regression is an active research subject, which had a considerable boost after the seminal paper by Wilson & Adams (2013) where the PSD is modeled by a Gaussian mixture in the Fourier domain. This way, the resulting covariance function (via the Wiener-Khinchin theorem) is a spectral mixture of sub-kernels that allows one to approximate any stationary covariance kernel to arbitrary precision, given enough mixture components. The multioutput extension of the spectral mixture kernel was developed by Parra & Tobar (2017);de Wolff et al. (2021). Other approaches have also been proposed. In the recent astrophysical literature, a mixture of the celerite covariance functions (Foreman-Mackey et al. 2017), a family of physically motivated kernels developed also for allowing a very fast computation, was applied to derive the PSD from Fermi-LAT observations for a sample of blazars with published claims of quasi-periodicities. Another method, termed Bayesian nonparametric spectral estimation (BNSE), was proposed by Tobar (2018); BNSE is the limit of the well known Lomb-Scargle algorithm when an infinite number of components is considered with a Gaussian prior over the weights. RESULTS Periodic vs non-periodic kernels We first modeled the light curves with the standard stationary kernels mentioned in Sect.3, i.e., the Square Exponential (SE), the Matérn with = 1 2 aka absolute exponential (AE) and the rational quadratic (RQ) (Rasmussen & Williams 2006). For all kernels, we denote = ( − ) the temporal difference between data points. The SE kernel is given by: = exp − 2 2 2 ,(1) where > 0 is the amplitude and > 0 is the lengthscale of the exponential decay. The AE kernel (or, in one dimension, the kernel of the Ornstein-Uhlenbeck process) is given by: = exp − .(2) Lastly, the RQ kernel 1 is: = 1 + 2 2 2 − ,(3) with > 0. This kernel can be obtained by a scale mixture (i.e. an infinite sum) of SE covariance functions with different characteristic lengthscales drawn from a gamma distribution. The limit of the RQ covariance for → +∞ is, again, the SE covariance function. In our regression experiments, the mean function of the GPs was set to the (constant) empirical mean of the observations. We also adopted flat priors for the kernel hyper-parameters as reported in Table 1. We considered both maximum likelihood implemented via L-BFGS-B (Byrd et al. 1995) Vousden et al. 2016). We started the Markov chains from small Gaussian balls centered on the best fit values. Then, we thinned the chain by discarding a fraction of the samples corresponding to a few times (typically 4 or 5) the auto-correlation length of the computed chains and we checked that a stationary distribution was reached (Sharma 2017). In most cases the posterior distribution of the parameters (hereinafter just "posterior") is single-peaked (unimodal) and model comparison could be quickly implemented by means of, e.g., the Bayesian Information Criterion (BIC, Schwarz 1978). However, in the general case of periodic behavior assessment, we are interested in deriving the posterior PDF of the periods and therefore we carried out model comparison by a full computation of the Bayes factors (Kass & Raftery 1995;Trotta 2017 Table 3. Bayes factors (expressed as dB) computed for the light curves in all the four available bands for a periodic kernel over the best-fit stationary one. interpolating) our datasets (e.g. Covino et al. 2020), yet Bayes factors show very different results for the adopted kernels ( Joachimi et al. 2021), therefore, only very large values unambiguously identify the most suitable hypothesis, given the data and the prior assumptions. Quite interestingly, results are different for each band, likely due to the different sampling and long-term coverage. The and band data are better described by the RQ kernel, while the and band data by the "AE" kernel. The SE kernel is generally disfavored although for the band curve it behaves almost comparably to the RQ kernel. A possible interpretation of this behavior depends on the presence of meaningful correlation for very long lags that, in case, can more effectively modeled by the RQ kernel (Rasmussen & Williams 2006). The different results in different bands are probably due to the more or less effective coverage for long lags and the quality (i.e. the associated uncertainties) of the data. Perhaps the simplest periodic covariance function is the cosine kernel: = cos(2 / ),(4) where > 0 is the amplitude, is the period and is the temporal difference as defined above. Given that the product of two kernels is still a valid positive semidefinite kernel, we modeled our data with a kernel constructed by multiplying the the best performing kernel in the previous experiment (Table 2) with the cosine kernel. We also adopted a flat prior distribution for the logarithm of the period as reported in Table 1. With this new kernel, we witnessed a relevant improvement as quantified by the Bayes factors shown in Table 3. Bayes factors, for ease of visualization, can be directly converted into probabilities conditioned on the data (e.g., Trotta 2007;Covino et al. 2020) obtaining factors of 99.2, 99.4, 97.6, and 92.1% for the , , and bands, respectively. In addition, the square root of the variances associated to each band, the parameter in Eq.(s) 1-4, can be interpreted as modulation amplitudes giving: ∼ 0.23, ∼ 0.14, ∼ 0.14 and ∼ 0.13 mag, respectively. These values are close to those reported in Liao et al. (2021, their A direct byproduct of performing GP regression is the conditional covariance of the process given the observations, which can be compared to the ACF of the data. Computing the ACF for data affected by a very irregular sampling and long gaps is not an easy task. We followed here the discrete correlation function (DCF) algorithm proposed by Edelson & Krolik (1988), although we stress that the DCF computed in this paper are shown only for illustration purposes. Fig. 3 show the (rather) noisy DCF compared to the GP regression best-fit parameters. As mentioned already, the preference for the RQ or the AE kernel depends on the presence of correlation for long lags. The data, however, clearly show an oscillatory behavior, yet the highlyirregular sampling makes it difficult to assess whether we have a true periodicity or a simply a recurrent behavior with changing time scale. Within the limits of our models, we can expect that the periods are better constrained when the RQ kernel is preferred, since the AE kernel cannot support correlation on long lags. This is indeed the case, as we can see in Fig. 4. As expected, the PDF for the and partly for the bands are more peaked while the range of possible periods for the and bands are larger. Periods maximising the GP regression posterior PDF tend to be shorter that those maximising the LS periodograms. A similar behavior was also obtained by Liao et al. (2021) (their Table 2) comparing results from LS and time domain analyses. Single-band study does not actually make the best use of all the available information. Having data collected in four different bands allows us to obtain a more solid inference modeling the , , and band data together and asking that the period be same at all frequencies. This is an important step that is getting more and more significant given the common availability of multi-band information from modern and future monitoring facilities (see, e.g., Huĳse et al. 2012;VanderPlas & Ivezić 2015;Mondrik et al. 2015;Saha & Vivas 2017;Huĳse et al. 2018;Hu & Tak 2020;Elorrieta et al. 2021). However, analyzing multi-band time-series requires some care since we may want to consider the, typically large, degree of correlation among the different bands. This is a basic topic in the GP literature (Pinheiro & Bates 1996;Roberts et al. 2012;Osborne et al. 2012;Parra & Tobar 2017) due to the clear interest in deriving information from multiple inputs in many sectors from research to industry . In this work we have applied a simple technique known as "intrinsic model of coregionalization" (ICM, Bonilla et al. 2008;Álvarez et al. 2012) although many other, often more sophisticated, approaches are possible (e.g. van der Wilk et al. 2020;de Wolff et al. 2021). Assuming we have different input data sets (or bands in our case) the kernel for our problem can be defined as: coreg = ( , ) · [ , ],(5) where and are observations from any input data sets, ( , ) is a valid covariance function, as those introduced before, and is a × positive-definite matrix. coreg , therefore, turns out to be: coreg =        11 ( 1 , 1 ) . . . 1 ( 1 , ) . . . . . . . . . 1 ( , 1 ) . . . ( , )        .(6) Each output (band, in our case) can be seen as linear combination of functions defined by the same covariance ( , ). The rank of the matrix defines the number of component in the linear combination. Therefore, an ICM is generated by the products of two covariance functions: one that models the dependence between the outputs and one that models the input dependence (Álvarez et al. 2012). The restriction = 0 for ≠ describes the case with independent data for bands and . ICM models can be, however, computationally demanding. In the high rank case, the number of hyper-parameters to be learned is (in our case) larger than 20, making the computation of the likelihood computationally expensive. Therefore, for computational purposes, and also because the rather high degree of correlation and superposi-BF − BF − dB db 64 45 Table 4. Bayes factors (expressed as dB) computed for the light curves in all the four available bands considered together for a periodic kernel over the best-fit stationary one. tion among the four available bands makes high rank coregionalized matrices unnecessary, we applied the simplest model with the lowest rank, thus modeling our four bands simultaneously with either the AE or the RQ covariance function. As expected, once the kernel hyper-parameters are learned using the whole dataset, the RQ kernel outperforms the AE, essentially because the former is better suited to model correlation for long lags. The introduction of a periodic component, now with the same period for all bands, is highly favoured by the computed Bayes factors (Table 4), the probability in favour of the periodic model is about 4.2 . The PDF for the period exhibits a strong peak (Fig. 4) and yields a period of = 1525 +42 −33 days. As mentioned before, adopting a higher-rank formulation quickly makes the (Monte Carlo) computation of the likelihood (and the Evidence) impractical due to the large number of parameters. Popular and, for simple inferences, effective alternatives are available as the already introduced BIC, which is a proxy of the actual Bayes factor, i.e. log Evidence ≈ (− /2) (see, e.g. Kass & Raftery 1995;Raftery 1999, for pros and cons of the BIC for model inference). Else, more advanced techniques for approximate Evidence computation that scale efficiently with the number of parameters have also been developed (e.g. variational inference, Opper & Archambeau 2009;Blei et al. 2017;Cherief-Abdellatif 2019). Finally, as discussed in Sect. 3, another by-product of our analysis is an estimate of the PSD of the data. Applying again here a rank one ICM model with the best-fit periodic covariance function, we can derive a PDF directly from the GP regression (Fig. 2). Given that the PSD can be computed for any needed sampling rate, it can be used for complex inferences or even fit as, e.g., in Vaughan (2010) with any physically-driven model. More advanced GP based algorithms could also be applied, if needed, to derive the PSD with more general assumptions about the covariance of the data (e.g. de Wolff et al. 2021). DISCUSSION The analyses carried out in Sect. 4 show that a covariance function including a periodic component is supported by the data with a probability higher than 99% for the and bands. These improvements are of interest, although somehow less significant compared to the results reported by Liao et al. (2021). Results obtained through different techniques should be carefully analysed since their direct comparison can be occasionally misleading due to their different assumptions. In addition, we also stress that here we report probabilities supporting a periodic kernel with "any period" within the adopted prior, since we are considering the period as a nuisance parameter and thus integrating it out from our posterior. With essentially single-peaked unimodal distributions it makes little difference, but this does not need to be generally the case (see also, e.g. Gregory & Loredo 1992, for a similar discussion). The availability of multiple bands allows us to carry out the analysis making use of all the information provided by the dataset, and the probability supporting a periodic kernel given the data becomes higher, corresponding to an improvement with respect the purely stationary channel-independent solution of about 4.2 . The best-fit period, = 1525 +42 −33 days, is shorter than those reported in Liao et al. (2021) and those obtained by our LS analysis (Sect. 2), although still roughly consistent given the uncertainties. This difference might indicate that the period is not stable during the monitoring, since LS analysis is essentially a change of basis, and data are simply described by the frequency content. A GP regression models a covariance function and the level of correlation for short and long lags affects how the best period describing the data is obtained. Shorter periods than those identified by a LS analysis were reported in Liao et al. (2021) too (see their Table 2). In any case, the source considered in Liao et al. (2021) and in this study was singled out by a sample of 625 analysed objects (Chen et al. 2020). This implies the reported probabilities have to be corrected for the so-called "look elsewhere effect" (Bayer & Seljak 2020), i.e., the probability to find a positive result across multiple (independent) trials. In our case the ∼ 4.2 probability supporting the periodic kernel becomes, after the multiple trial correction (van der Klis 1989), ∼ 2.4 . This is still an interesting figure, yet far from being conclusive, implying that the periodicity proposed for J0252 can be just a chance result is still a plausible interpretation. We mention in passing that the multiple trial correction (see a discussion in Vaughan 2013) is often neglected since it is occasionally left unrecorded or badly described how large is the original sample of studied sources before devoting specific attention to the most interesting cases. Unfortunately, it is a factor that can easily dominate the final false alert probability for a proposed period. CONCLUSIONS In this paper we have collected data for an AGN with a long multiband optical monitoring, SDSS J025214.67-002813.7. This source was singled out by Liao et al. (2021) from a larger sample of objects and, after an extensive analysis, they proposed a possible periodicity of ∼ 4.4 years. The available data cover a long period, a couple of decades, but are affected by a very irregular sampling and long gaps. We re-analyzed these data modeling the light curves by Gaussian processes, i.e. Bayesian nonparametric models for regression of time series . We analysed the single band data and by means of multioutput GP models the whole dataset at once. We can confirm that a periodic component slightly longer than ∼ 4 years improves the description of the data in all bands, and is also supported by the full analysis of the whole dataset. Nevertheless, given that the considered source was originally identified in a large sample of several hundreds sources, after a correction due to the sample size the false alert probability of the periodicity does not appear to be low enough to rule out the possibility this is just a chance result. Figure 1 . 4 .Figure 2 . 142photometry for J0252 fromLiao et al. (2021). Only points with 1 magnitude error lower than 0.3 mag are plotted for clarity. The photometry covers about 20 years although with seasonal gaps and interruptions and with an irregular sampling. Superposed to the four bands we also plot the best fit multi-band GP model and the 1 uncertainty, as discussed in Sect. Lomb-Scargle periodograms for the bands of J0252. The colors lines are the actual periodograms while the black line is the PSD computed sampling at the maximum of the posterior distribution of the the best-fit GP model (see Sect. 4). The vertical lines indicate the maximum of the periodograms and the one year period. and also integrated out the hyperparameters from the posterior density by a Markov Chain Monte Carlo (MCMC, Hogg & Foreman-Mackey 2018) based on the "paralleltempering ensemble" algorithm (Foreman-Mackey et al. 2013; Goggans & Chi 2004; Figure 3 .Figure 4 . 34DCF for the bands of J0252. For clarity, only points with DCF errors lower than 0.4 are shown. The blue dashed line is the best-fit periodic covariance function for each band, and the cyan lines are 50 randomly chosen samples extracted from the posterior distribution of parameters. PDF for the periods computed marginalizing the posterior density distributions for the bands of J0252 and for the analysis with all the available bands together. The dashed vertical line identifies the period proposed inLiao et al. (2021). PDF for each band are artificially shifted for clarity by the amount reported in the legend. The most probable periods for the bands are 1511 +103 −109 , 1317 +323 −183 , 1570 +220 −133 , and 1373 +243 −215 days, respectively. The most probable period for the analysis including all the available bands is = 1525 +42 −33 days. Table 1. Prior information adopted for for analyses described in Sect. 4. Priors are properly normalized for the computation of the Bayes factors.Hyper-parameter prior ln Uniform [-20, 20] ln Uniform [-20, 20] ln Uniform [-10, 10] ln Uniform [ln(100), ln(3000)] Table 2 ) 2reflecting Table 2 ) 2, although derived following a different approach. Covino et al. MNRAS 000, 1-9 (2021) Note that the analogous formula reported in Covino et al.(2020)is shown with a typo. We thank the anonymous referee for having pointed it out.MNRAS 000, 1-9 (2021) http://www.python.org 3 http://www.numpy.org 4 https://www.scipy.org 5 http://www.astropy.org 6 https://github.com/willvousden/ptemcee 7 https://george.readthedocs.io/en/latest/ 8 https://gpflow.readthedocs.io/en/master/index.html 9 https://github.com/GAMES-UChile/mogptk 10 https://www.astroml.org/ 11 https://www.matplotlib.org 12 https://corner.readthedocs.io/en/latest/ ACKNOWLEDGEMENTSWe thank the anonymous referee for her/his valuable and accurate comments. 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[ "Native geometry and the dynamics of protein folding", "Native geometry and the dynamics of protein folding" ]	[ "P F N Faisca \nCFTC\nAv. Prof. Gama Pinto 21649-003Lisboa CodexPortugal\n", "M M Telo Da Gama \nCFTC\nAv. Prof. Gama Pinto 21649-003Lisboa CodexPortugal\n" ]	[ "CFTC\nAv. Prof. Gama Pinto 21649-003Lisboa CodexPortugal", "CFTC\nAv. Prof. Gama Pinto 21649-003Lisboa CodexPortugal" ]	[]	In this paper we investigate the role of native geometry on the kinetics of protein folding based on simple lattice models and Monte Carlo simulations. Results obtained within the scope of the Miyazawa-Jernigan indicate the existence of two dynamical folding regimes depending on the protein chain length. For chains larger than 80 amino acids the folding performance is sensitive to the native state's conformation. Smaller chains, with less than 80 amino acids, fold via twostate kinetics and exhibit a significant correlation between the contact order parameter and the logarithmic folding times. In particular, chains with N=48 amino acids were found to belong to two broad classes of folding, characterized by different cooperativity, depending on the contact order parameter. Preliminary results based on the Gō model show that the effect of long range contact interaction strength in the folding kinetics is largely dependent on the native state's geometry.	10.1016/j.bpc.2004.12.022	[ "https://arxiv.org/pdf/q-bio/0411044v1.pdf" ]	992,152	q-bio/0411044	8975f029c65399a44f1a36753ee73fb14a5f5964	Native geometry and the dynamics of protein folding 26 Nov 2004 P F N Faisca CFTC Av. Prof. Gama Pinto 21649-003Lisboa CodexPortugal M M Telo Da Gama CFTC Av. Prof. Gama Pinto 21649-003Lisboa CodexPortugal Native geometry and the dynamics of protein folding 26 Nov 2004(Dated: January 4, 2014)numbers: 8715Cc; 9115Ty Keywords: protein foldinglattice modelscontact orderlong-range contactskineticscooperativity In this paper we investigate the role of native geometry on the kinetics of protein folding based on simple lattice models and Monte Carlo simulations. Results obtained within the scope of the Miyazawa-Jernigan indicate the existence of two dynamical folding regimes depending on the protein chain length. For chains larger than 80 amino acids the folding performance is sensitive to the native state's conformation. Smaller chains, with less than 80 amino acids, fold via twostate kinetics and exhibit a significant correlation between the contact order parameter and the logarithmic folding times. In particular, chains with N=48 amino acids were found to belong to two broad classes of folding, characterized by different cooperativity, depending on the contact order parameter. Preliminary results based on the Gō model show that the effect of long range contact interaction strength in the folding kinetics is largely dependent on the native state's geometry. INTRODUCTION It is well known that most small (from ∼ 50-120 amino acids), single domain proteins fold via a two-state (single exponential) kinetics, without observable folding intermediates and with a single transition state associated with one major free energy barrier separating the native state from the unfolded conformations [1,2,3]. For this reason small protein molecules are particularly well suited for investigating the correlations between folding times and the native state equilibrium properties, a major challenge for those working in protein research. The energy landscape theory predicts that the landscape's rugedeness plays a fundamental role in the folding kinetics of model proteins: The existence of local energy minima, that act as kinetic traps, is responsible for the overall slow and, under some conditions (as the temperature is lowered towards the glass transition temperature), glassy dynamics [4]. On the other hand, rapid folding is associated with the existence of a smooth, funnel-shaped energy landscape [5]. In Refs. [6,7] Plaxco et. al. and Gillespie and Plaxco have provided experimental evidence that the folding energy landscape of single domain proteins is extremly smooth even at considerably low temperatures. Therefore differences in the landscape's 'topography' cannot account for the vast range of folding rates as observed in real proteins [8,9]. However, a strong correlation (r=0.94) was found between the so-called contact order parameter, CO, and the experimentally observed folding rates in a set of 24 non-homologous single domain proteins [10]. The CO measures the average sequence separation of contacting residue pairs in the native structure relative to the chain length of the protein CO = 1 LN N i,j ∆ i,j \|i − j\|,(1) where ∆ i,j = 1 if residues i and j are in contact and is 0 otherwise; N is the total number of contacts and L is the protein chain length. The empirical observation that the CO correlates well with the folding rates of single domain proteins, exhibiting smooth energy landscapes, strongly suggests a geometry-dependent kinetics for such two-state folders. The connection between the CO and the dominant range of residue interactions brings back an old, well-debated issue in the protein folding literature, that of the role of local (i.e. close in space and in sequence) and long range (i.e. close in space but distant along the sequence) inter-residue interactions in the folding dynamics. Several results appear to agree on the idea that long range (LR) contacts play an active role in stabilizing the native fold 2 [11,12,13,14]. In what regards the folding kinetics, results reported in Refs. [11,13,15,16] suggest that local contacts increase the folding speed, relative to LR contacts, while results in Ref. [12] suggest an opposite trend. In Ref. [17] Gromiha and Selvaraj have analysed explicitly the contribution of LR contacts in determining the folding rates of 23 (out of the 24) two-state folders studied by Plaxco et al [10]. These authors proposed the so-called long range order (LRO) parameter, measuring the total number of long range contacts relative to the protein chain length, as an alternative way of quantifying the native structure geometry. In fact, the LRO parameter correlates as well as the CO with the folding rates of the twostate folders analysed in Ref. [10]. The majority of protein folding theory is based, not only on results for real proteins such as those outlined above, but also on a vast number of findings obtained within the scope of simple lattice models and Monte Carlo (MC) simulations. Although lattice models do not encompass the full complexity of real proteins they are non trivial and capture fundamental aspects of the protein folding kinetics [18]. In the present study we investigate through Monte Carlo folding simulations of simple lattice models, such as the Miyazawa-Jernigan model and the Gō model, the dependence of two-state folding kinetics on the native state geometry. LATTICE MODELS In a lattice model the protein is reduced to its backbone structure: amino acids are represented by beads of uniform size, occupying the lattice vertices, and the peptide bond, that covalently connects amino acids along the polypeptide chain, is represented by sticks, with uniform length, corresponding to the lattice spacing. We model proteins as threedimensional, self-avoiding chains of N beads. To mimick amino acid interactions we use either the Miyazawa-Jernigan model or the Gō model. H({σ i }, { r i }) = N i>j ǫ(σ i , σ j )∆( r i − r j ),(2) where {σ i } represents an amino acid sequence, and σ i stands for the chemical identity of bead i. The contact function ∆ is 1 if beads i and j are in contact but not covalently linked and is 0 otherwise. The interaction parameters ǫ are taken from the 20 × 20 MJ matrix, derived from the distribution of contacts of native proteins [19]. H({ r i }) = N i>j B ij ∆( r i − r j ),(3) where the contact function ∆( r i − r j ), is unity only if beads i and j form a non-covalent native contact and is zero otherwise. Since the Gō model ignores the protein sequence chemical composition the interaction energy parameter is B ij = −ǫ. SIMULATION DETAILS Our folding simulations follow the standard MC Metropolis algorithm [20] and, in order to mimick protein movement, we use the kink-jump move set, including corner flips, end and null moves as well as crankshafts [21]. Each MC run starts from a randomly generated unfolded conformation (typically with less than 10 native contacts) and the folding dynamics is traced by following the evolution of the fraction of native contacts, Q = q/Q max , where Q max is the total number of native contacts and q is the number of native contacts at each MC step. The folding time t, is taken as the first passage time (FPT), that is, the number of MC steps corresponding to Q = 1.0. The folding dynamics is studied at the so-called optimal folding temperature, the temperature that minimizes the folding time as measured by the mean FPT. The sequences studied within the context of the MJ model were prepared by using the design method developed by Shakhnovich and Gutin (SG) [22] based on random heteropolymer theory and simulated annealing techniques. All targets studied are maximally compact structures found by homopolymer relaxation. NUMERICAL RESULTS Evidence for two folding regimes in protein folding and thirty SG sequences were prepared according to the method described in Ref. [22]. P f old (t), the probability of the chain having visited its target after time t, was computed as the fraction of (150) simulation runs, which ended at time t. Two distinct folding regimes were identified depending on the chain length. We name the regime observed for N < 80 the first regime while that corresponding to N ≥ 80 is the second regime. We have investigated the contribution of each target to the folding probability curve and found that for N ≥ 80 the folding performance is sensitive to target conformation, with some targets being more foldable than others as shown in Figure 1(b) for N=100. For N < 80 targets are equally foldable since all folding probability curves are consistent with asymptotic values P f old → 1. In order to investigate if kinetic relaxation in the first regime is well described by a single exponential law we have calculated the dependence of ln(1 − P f old ) on the time coordinate t. Remember that in a two-state process the reactant concentration (the equivalent in our simulations to the fraction of unfolded chains) is proportional to exp −t/τ where τ is the socalled relaxation time. Therefore, if first regime kinetics is single exponential ln(1 − P f old ) vs. t should be a straight line with slope=−1/τ . Results reported in Figure 2 show that single-exponential folding is indeed a very good approximation for the folding kinetics of small lattice-polymer proteins. 5 Contact order and the lattice-polymer model kinetics In a recent study [24] we analysed the folding kinetics of ≈ 5000 SG sequences and 100 target conformations distributed over the chain lengths N = 36, 48, 54, 64 and 80. Targets were selected in order to cover the observed range of CO (≈ 0.12 < CO < 0.26). Results reported in Ref. [24] show a significant correlation (r = 0.70 − 0.79) between increasing CO and longer logarithmic folding times. In Ref. The contact map is a N × N matrix with entries C ij = 1 if beads i and j are in contact (but not covalently linked) and are zero otherwise. Figure 4 shows the contact maps of targets T1 (CO=0.126), T2 (CO=0.189) and T3 (CO=0.259) respectively. One could argue that high-CO targets are associated with longer logarithmic folding times because they have predominantly LR contacts which, given the local nature of the move set used to simulate protein movement, eventually take a longer time to form. Let the contact time t 0 be the mean FPT of a given contact averaged over 100 MC runs. The longest contact time (ln t 0 = 12.24) observed for target T3 is two orders of magnitude shorter than T3's folding time (ln t = 17.59) and the sum of all contact times is ln( 57 i=1 t i 0 ) = 15.51, much lower than the observed folding time. Thus, the fact that T3 and other high-CO structures have predominantly LR contacts cannot justify, per se, their higher folding times. The contact map provides a straightforward way to compute the frequency ω ij = t ij /t with which a native contact occurs in a MC run, t ij being the number of MC steps corresponding to C ij = 1 and t the folding time. For each target studied we computed the mean frequency of each native contact < ω ij > averaged over 100 simulation runs, and re-averaged < ω ij > over the number of native contacts in each interval of backbone distance (we measure backbone distance in units of backbone spacing). We have found that while for the low-CO targets the backbone frequency decreases monotonically with increasing backbone distance, for the intermediate and high-CO targets such dependence is clearly nonmonotonic. that we have ruled out, is that of a negative correlation between the frequency and the energy of a contact; Could the most stable contacts be the most frequent ones? We found modest correlation coefficients r = 0.63 and r = 0.65 for targets T1 and T3 respectively and therefore we conclude that the observed behaviour is not energy driven. In Table I we show the dependence of the contact time, averaged over contacts in each interval of backbone distance, on the backbone distance for model targets T1 and T3. Since the average contact times, over a given range, are similar for these extreme model structures, the differences in the frequencies reported in Figure 5 must necessarilly distinguish different cooperative behaviors. Results outlined above suggest that two broad classes of folding mechanisms exist for small MJ lattice polymer protein chains. What distinguishes these two classes is the presence, or absence, of a monotonic decrease of contact frequency with increasing contact range that is related to different types of cooperative behaviour. The monotonic decrease of contact frequency with increasing backbone distance is a specific trait of low-CO structures. In this case folding is also less cooperative and is driven by backbone distance: Local contacts form first while LR contacts form progressively later as contact range increases. Contact order, long-range contacts and protein folding kinetics in the Gō model The energy landscapes of Gō-type polymers are considerably smooth because in the Gō model the only favourable interactions are those present in the native state. Therefore such models are adequate for investigating the dependence of protein folding kinetics on target geometry. In this section we investigate the contribution of LR and local interactions to the folding kinetics of targets T1, T2 and T3 (Figure 4) in the following way: The total energy of the native structure is kept constant but the relative contributions of LR and local interactions are varied over a broad range. With the above costraint the energy of a conformation is given by H({ r i }, σ) = C LR (σ)H LR ({ r i }) + C L (σ)H L ({ r i }),(4) where C LR (σ) = σ/[(1 −σ)Q L + σ(1 −Q L )] and C L (σ) = (1 −σ)/[(1 −σ)Q L + σ(1 −Q L )]; Q L is the fraction of local native contacts and H LR (L) is given by equation 3. The parameter σ varies from 0 (only local contacts contribute to the total energy) to 1 (only LR contacts contribute to the total energy). The constraint of fixed native state energy is enforced to rule out differences in the folding dynamics driven by the stability of the native state. Preliminary results reported in Figure 6 show the dependence of the logarithmic folding time, averaged over 100 simulations runs, on the parameter σ for the three native geometries. For σ < 0.15 we have not observed folding of the target T3 and no folding was observed for the target T2 if σ < 0.10. The behaviour exhibited by target T3 is easily explained: since approximately 80 percent of T3's native contacts are LR there is little competition between LR and local contacts. Moreover, such competition is not significantly enhanced when one varies σ towards unity. However, the effect of decreasing σ is equivalent to that of 'switching off' the LR contacts, that is, to force a structure to fold with only approximatly 20 per cent of its total native contacts resulting in longer folding times and for σ < 0.15 folding failure. More intriguing are the results obtained for the low and intermediate-CO target structures, T1 and T2 respectively. The curves are qualitatively similar (with a minimum at σ > 0.5) but closer inspection reveals an important difference, namely: for σ < 0.5 the dependence of the folding time on σ is much stronger for the intermediate-CO target, T2. Indeed, in this 8 case one observes a remarkable three-order of magnitude dispersion of logarithmic folding times, ranging from log(t) = 5.62 (for σ = 0.65) to log(t) = 8.50 (for σ = 0.10). We stress, however, that for both targets the kinetics is more sensitive (in the sense that the folding rate decreases more rapidly) to lowering σ: LR contacts appear therefore to have a crucial/vital role, by comparison with local contacts, in determining the folding rates of small Gō-type lattice polymers and this effect depends on target geometry. CONCLUSIONS AND FINAL REMARKS By using different target structures in MC simulations of protein folding we have identified two distinct folding regimes depending on the chain length. In close agreement with experimental observations we found a first regime that describes well the folding of small protein molecules and whose kinetics is single exponential. Folding of protein chains with more than 80 amino acids, on the other hand, belongs to a dynamical regime that appears to be target sensitive with some targets being more highly foldable than others. In this case we ascribe folding failure to existing kinetic traps but we have not been able to carry out our simulations for long enough times in order to observe escape and succesfull folding. Because the additive MJ lattice polymer model fails to exhibit the remarkable dispersion of folding rates observed in real proteins one should interpret the results for the dependence of folding times on contact order parameter with caution. However, our results strongly suggest that the geometry driven cooperativity is rather robust and this implies an increase in folding times for increasing cooperativity. We have analysed the role of LR contacts in the folding kinetics of small Gō-type lattice polymers and found a considerably strong dependence on target geometry. In particular, we have found that targets with a similar fraction of LR contacts (that is, targets with similar LRO parameter) and different contact order exhibit considerably different folding rates when LR contacts are destabilized energetically with respect to local contacts. We are currently investigating this issue and results will be published elsewhere (in preparation). This result may provide a clue to understanding the increadible dispersion of folding rates exhibited by real two-state folders: one can expect to observe longer folding times if the distribution of contact energies in real proteins is such that local contacts are, on average, more stable than LR contacts for specific native folds. sequences were designed per target. P f old , the probability of the chain having visited its target after time t, was computed as the fraction of simulation runs that ended in time t. (b) Separate contribution of each of the 100 bead long targets for the dependence of P f old (t) on log(t) [23]. There are 23 LR contacts in structure T1, 21 in structure T2 and 44 in structure T3. 24 Figure 6. Dependence of the logarithmic folding time log 10 t on the parameter sigma. The parameter σ varies from 0 (only local contacts contribute to the total energy) to 1 (only LR contacts contribute to the total energy). Miyazawa-Jernigan (MJ) model the energy of a conformation defined by the set of bead coordinates { r i } is given by the contact Hamiltonian Figure 1 ( 1a) shows the dependence on time t, of the folding probability P f old (t), for chain lengths N = 27, 36, 48, 54, 64, 80, 100. Five target structures were considered per chain length [ 25 ] 25Jewett et al. found a similar corelation (r = 0.75) for a 27-mer lattice polymer modeled by a modified Gō-type potential. In a recent study, Kaya and Chan[26] studied a modified Gō type model, with specific manybody interactions, and found folding rates that are very well correlated (r = 0.91) with the CO and span a range that is two orders of magnitude larger than that of the corresponding Gō models with additive contact energies. These results support the empirical relation found between the contact order and the kinetics of two-state folders.Contact order and structural changes towards the native fold in the Miyazawa-Jernigan model In order to investigate if native geometry as measured by the CO promotes, or does not promote, different folding processes, eventually leading to different folding rates, we have analysed the dynamics of 900 SG sequences with chain length N = 48, distributed over nine target structures with low (0.126, 0.127, 0.135), intermediate (0.163, 0.173, 0.189) and high (0.241, 0.254, 0.259) contact order. The averaged trained sequence energy shows very little dispersion ranging from -25.11±0.03 to -26.16±0.02. Within this target set the folding time and the contact order correlate well (r = 0.82) although the dispersion of folding times is small as reported in Figure 3. Figure 5 5illustrates this behaviour for model structures T1, T2 and T3 elements of the low, intermediate and high-CO target sets respectively. A possible explanation for this behavior, Figure 1(a); P.N.F. Faisca, Biophysical Chemistry Figure 3 ;Figure 6 ; 36PP.N.F. Faisca, Biophysical Chemistry Figure 1 . 1Dependence of the folding probability, P f old , on log(t). (a) For each of the chain lengths N = 27, 36, 48, 64, 80 and 100 five target structures were considered and 30 Figure 2 . 2Evidence for single exponential folding kinetics for chain length N = 48. The correlation coeficient between the logarithmic fraction of unfolded chains and 'reaction' time is r ≈ 0.97 for target T4 and r ≈ 0.99 for the remaining targets. Figure 3 . 3Dependence of the logarithmic folding times, ln e t, on the contact order parameter (r ≈ 0.82). Figure 4 . 4Contact maps of targets T1 (a), T2 (b) and T3 (c). Each square represents a native contact. We divide the 57 native contacts into two classes: LR contacts are represented by filled squares and correspond to contacts between beads for which the backbone separation is 10 or more backbone units. Local contacts are represented by white squares. Figure 5 . 5The backbone frequency, < ω \|i−j\| >, as a function of the backbone separation for the low-CO, high-CO and intermediate-CO target. The backbone frequency is the mean value of < ω > averaged over the number of contacts in each interval of backbone separation. The Gō model GōIn the Gō[11] model only native contacts, i.e. contacts that are present in the native state, contribute to the energy of a conformation defined by { r i }. In this case the contactHamiltonian is How do small protein molecules fold?. S E Jackson, Folding Des. 3S. E. Jackson, How do small protein molecules fold?, Folding Des. 3 (1998) 81-91 Contact order, transition state placement and the refolding rates of single domain proteins. K W Plaxco, K T Simmons, D Baker, J. Mol. Biol. 277K. W. Plaxco, K. T. Simmons and D. Baker, Contact order, transition state placement and the refolding rates of single domain proteins, J. Mol. Biol. 277 (1998) 985-994 Simple two-state protein folding kinetics requires near-levinthal thermodynamic cooperativity. H Kaya, H S Chan, Proteins. 52H. Kaya and H. S. 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Ball, Folding and form: Insights from lattice simulations, Phys. Rev. E 69 (2004) 051917 Table I, 38, 43[ T 1 8.14 ± 0.12 10.67 ± 0.21 11.47 ± 0.07 11.69 ± 0.13 11.39 ± 0.07 - 11.58 ± 0.10 - T 3 7.66 ± 0.10 11.14 ± 0.18 10.88 ± 0.10 11.60 ± 0.06 12.06 ± 0.05 12.24 ± 0.08 11.82 ± 0.05 11.61 ± 0.05The averaged contact time. 2838ln e < t 0 >, as a function of the backbone separation [27] Target backbone distance [3, 8[ [8, 13[ [13, 18[ [18, 23TABLE I: The averaged contact time, ln e < t 0 >, as a function of the backbone separation [27] Target backbone distance [3, 8[ [8, 13[ [13, 18[ [18, 23[ [23, 28[ [28, 33[ [33, 38[ [38, 43[ T 1 8.14 ± 0.12 10.67 ± 0.21 11.47 ± 0.07 11.69 ± 0.13 11.39 ± 0.07 - 11.58 ± 0.10 - T 3 7.66 ± 0.10 11.14 ± 0.18 10.88 ± 0.10 11.60 ± 0.06 12.06 ± 0.05 12.24 ± 0.08 11.82 ± 0.05 11.61 ± 0.05	[]
[ "A spherical Hopfield model", "A spherical Hopfield model" ]	[ "D Bollé \nInstituut voor Theoretische Fysica\nKatholieke Universiteit Leuven\nCelestijnenlaan 200 DB-3001LeuvenBelgium\n", "Th M Nieuwenhuizen \nInstitute for Theoretical Physics\nValckenierstraat 651018 XEAmsterdamThe Netherlands\n", "I Pérez Castillo \nInstituut voor Theoretische Fysica\nKatholieke Universiteit Leuven\nCelestijnenlaan 200 DB-3001LeuvenBelgium\n", "T Verbeiren \nInstituut voor Theoretische Fysica\nKatholieke Universiteit Leuven\nCelestijnenlaan 200 DB-3001LeuvenBelgium\n" ]	[ "Instituut voor Theoretische Fysica\nKatholieke Universiteit Leuven\nCelestijnenlaan 200 DB-3001LeuvenBelgium", "Institute for Theoretical Physics\nValckenierstraat 651018 XEAmsterdamThe Netherlands", "Instituut voor Theoretische Fysica\nKatholieke Universiteit Leuven\nCelestijnenlaan 200 DB-3001LeuvenBelgium", "Instituut voor Theoretische Fysica\nKatholieke Universiteit Leuven\nCelestijnenlaan 200 DB-3001LeuvenBelgium" ]	[]	We introduce a spherical Hopfield-type neural network involving neurons and patterns that are continuous variables. We study both the thermodynamics and dynamics of this model. In order to have a retrieval phase a quartic term is added to the Hamiltonian. The thermodynamics of the model is exactly solvable and the results are replica symmetric. A Langevin dynamics leads to a closed set of equations for the order parameters and effective correlation and response function typical for neural networks. The stationary limit corresponds to the thermodynamic results. Numerical calculations illustrate our findings.	10.1088/0305-4470/36/41/002	[ "https://arxiv.org/pdf/cond-mat/0305316v2.pdf" ]	17,105,768	cond-mat/0305316	1e0644cbdcf626c6fad9dd87019274e5d51cbb13	A spherical Hopfield model 15 May 2003 February 2, 2008 D Bollé Instituut voor Theoretische Fysica Katholieke Universiteit Leuven Celestijnenlaan 200 DB-3001LeuvenBelgium Th M Nieuwenhuizen Institute for Theoretical Physics Valckenierstraat 651018 XEAmsterdamThe Netherlands I Pérez Castillo Instituut voor Theoretische Fysica Katholieke Universiteit Leuven Celestijnenlaan 200 DB-3001LeuvenBelgium T Verbeiren Instituut voor Theoretische Fysica Katholieke Universiteit Leuven Celestijnenlaan 200 DB-3001LeuvenBelgium A spherical Hopfield model 15 May 2003 February 2, 2008 We introduce a spherical Hopfield-type neural network involving neurons and patterns that are continuous variables. We study both the thermodynamics and dynamics of this model. In order to have a retrieval phase a quartic term is added to the Hamiltonian. The thermodynamics of the model is exactly solvable and the results are replica symmetric. A Langevin dynamics leads to a closed set of equations for the order parameters and effective correlation and response function typical for neural networks. The stationary limit corresponds to the thermodynamic results. Numerical calculations illustrate our findings. Introduction In general, the introduction of continuous versions of discrete spin models in statistical mechanics has always been welcomed. Since the work of Kac and Berlin [1] the spherical model has lead to a better understanding of a lot of basic phenomena due to the fact that they simplify the mathematical calculations considerably. In spin glass research there is no doubt that an important role has been played by the spherical p-spin spin glass model [2] [3]. Since then it has been quite remarkable to see how a mean-field model with spherical spins resembles the results of realistic structural glasses (see, e.g., the many references in [4]). In neural network theory the Hopfield model is by now a well known classic found in many textbooks (e.g., [5], [6]) to study storage and retrieval in associative memory problems. Strangely enough, a spherical version of it has not yet been studied in the literature. One of the reasons might be that by relaxing the neurons from Isingtype to spherical-type variables the retrieval phase is absent, i.e. one of the most important properties of this model is lost. In this work we show that this problem has an easy and elegant solution. By introducing a quartic potential term in the Hopfield Hamiltonian such that the neurons stay spherical the retrieval phase is recovered. In this way we obtain a simplified version of the Hopfield model with an exactly solvable thermodynamics leading to marginally replica symmetric results. A phase diagram is obtained with an explicit solution for the spin-glass retrieval 'transition' line, showing no re-entrance. Moreover, the region of global stability for the retrieval solutions is larger than the corresponding region in the standard Hopfield model. Next, we study the relaxational Langevin dynamics of this model leading to a closed set of equations for the order parameters and effective correlation and response functions, typical for neural networks. These equations are similar to those of other models with continuous spins (see, e.g, [7] and references therein). We discuss the evolution of the overlap order parameter in the retrieval phase. The stationary limit of the dynamics is found to correspond to the thermodynamic results. The numerical solution of the dynamics illustrates this behaviour. The rest of the paper is organized as follows. In section 2 we introduce the model. Section 3 contains the thermodynamical properties with the temperature-capacity phase diagram. In section 4 we study the dynamics showing the evolution of the retrieval overlap. Finally, section 5 summarizes our conclusions. A spherical Hopfield model The Hopfield model [8] is defined through the following mean-field Ising-type Hamiltonian H({σ}) = − 1 2 N i =j=1 J ij σ i σ j ,(1) where the couplings J ij are related with the information one wants to store in the network through the Hebbian rule J ij = 1 N p µ=1 ξ µ i ξ µ j ,(2) with p = αN, where α is the loading capacity of the network. In this work, the p patterns ξ µ i are choosen to be a collection of continuous indepnedent identical random variables (i.i.d.r.v.) with respect to i and µ and drawn from a gaussian distribution P (ξ µ i ) = 1 √ 2π exp − (ξ µ i ) 2 2(3) and the neurons (spins) are also taken to be continuous and to satisfy the spherical constraint N i=1 σ 2 i = N .(4) As we will show explicitly below this setup of the spherical Hopfield model does not allow for a retrieval phase. Therefore, we add the following term to the Hamiltonian (1) − u 0 4 i,j,k,l J ijkl σ i σ j σ k σ l , J ijkl = 1 N 3 p µ=1 ξ µ i ξ µ j ξ µ k ξ µ l ,(5) which is a quartic term in the order parameter m characterizing the retrieval phase m µ = 1 N N i=1 ξ µ i σ i .(6) The introduction of this quartic term turns out to contribute macroscopically to the condensed part of the free energy but it only leads to sub-extensive contributions to the noise produced by the non-condensed patterns and to a sub-extensive contribution coming from the diagonal terms, as we now discuss in the following section. Thermodynamic and retrieval properties. We apply the standard replica technique [9] in order to study the thermodynamical properties of the model defined above. Starting from the Hamiltonian H(σ) = − 1 2 i =j J ij σ i σ j − u 0 4 i,j,k,l J ijkl σ i σ j σ k σ l(7) and assuming, for simplicity, one condensed pattern, the replicated free energy per site becomes βf = lim N →∞ lim n→0 −1 Nn ∞ −∞ n α=1 dm α ∞ −∞ α =β dq αβ i∞ −i∞ n α=1 du α 4πi i∞ −i∞ α =β d q αβ 4πi/N e −N g(8) with g = − β 2 n α=1 m 2 α − u 0 β 4 n α=1 m 4 α + nαβ 2 + 1 2 n α,β=1 m α (q −1 ) αβ m β − αn ln 2π + α 2 ln det(1I − βq) − n 2 ln 2π + 1 2 ln det Q − 1 2 n α,β=1 Q αβ q αβ .(9) In this expression m 1 α = 1 N N i=1 ξ 1 i σ α i , q αβ = 1 N N i=1 σ α i σ β i , α = β(10) are the retrieval overlap, respectively the neuron (spin) overlap order parameters, q αβ are conjugate variables and Q αβ = u α δ αβ − q αβ (1 − δ αβ ) .(11) Within a K-th order Parisi replica breaking scheme we assume that the spin overlap matrix has the following (ultrametric) structure q αβ = q i , if I(α/m i ) = I(β/m i ) and I(α/m i+1 ) = I(β/m i+1 )(12) with {q i } i=0,...,K a set of real numbers and {m i } i=1,...,K a set of integers such that m i+1 /m i is an integer (m 0 = n,m K+1 = 1), and we introduce the inverse of the Parisi function x(q) = n + K i=0 (m i+1 − m i )Θ(q − q i ) .(13) In the limit K → ∞, q 0 → 0 and q K → q M such that the free energy per site reads in the limit n → 0, N → ∞ βf [x(q), m] = − β 2 m 2 − u 0 β 4 m 4 + αβ 2 − α + 1 2 ln 2π − 1 2 + α 2 ln 1 − β(1 − q M ) − β q M 0 dq 1 − β 1 q x(q ′ )dq ′ − 1 2 q M 0 dq 1 q x(q ′ )dq ′ + ln(1 − q M ) − m 2 1 0 x(q)dq ,(14) where f has to be extremized with respect to x(q) and m. Taking, in general, x(q) to be a piece-wise continuous function it is straightforward to show following [2,10] that the solution for x(q) reads x(q) = Θ(q − q M )(15) and, hence, the replica solution is exact. The free energy then reads βf = extr m,q − β 2 m 2 − u 0 β 4 m 4 + αβ 2 − α + 1 2 ln 2π − 1 2 + α 2 ln[1 − β(1 − q)] − βq 1 − β(1 − q) − 1 2 ln[1 − q] + q − m 2 1 − q(16) with q M = q, the Edwards-Anderson orderparameter and m 1 = m. In agreement with this, one can show by investigating the stability against replica symmetry breaking fluctuations that the replicon eigenvalue is zero and, hence, the replica symmetric result is marginally stable. The saddle-point equations then become simple algebraic equations for q and m m u 0 m 2 − 1 − χ χ = 0 , q − m 2 χ 2 = αq (1 − χ) 2(17) with χ = β(1 − q) the susceptibility. We immediately remark that when we remove the quartic interaction by setting u 0 = 0 there exist no solution with m = 0 and, hence, there is no retrieval possible. Henceforth, we take u 0 = 1. The corresponding T − α phase diagram is shown in Fig. 1. Let us compare this result with the phase diagram of the standard Hopfield model calculated in a replica symmetric approximation [5,11]. Again we have three phases. For temperatures above the broken line T SG , there exist paramagnetic solutions characterized by m = q = 0, while below the broken line, spin glass solutions, m = 0 but q = 0, exist. The transition between the paramagnetic and the spin glass phase is continuous and the line T SG separating the two phases is easily computed T SG (α) = 1 + √ α(18) as in the standard Hopfield model. Next, below the thin full line T R , (locally stable) retrieval solutions, m = 0 and q = 0, appear. The transition between the spinglass and the retrieval phase is, however, a discontinous transition. Using then the information that the values of both the order parameters q and m jump when passing this line T R , it is possible to find an analytical expression for the latter. Writing the equations (17) as two polynomials in q and m, a jump means that complex roots of these two polynomials become real at the same time for all values of α, T obtained when crossing T R . After some algebra this leads to α R−SG (β) = 2 81β 5 8(β − 1) 2 β 2 (7β − 16) + sgn(∆) Σ 3 \|∆\| + sgn(∆) 3 \|∆\| (19) with ∆ = 81 √ 3(β − 1) 9/2 β 17/2 (\|125β − 128\|) 3/2 − ∆ 2 ∆ 2 = (β − 1) 5 β 6 {−2097152 + β(4849664 + β[−3459072 + 625β(1088 + 25β)])} Σ = (β − 1) 3 β 4 {−16384 + 5β[6144 + β(−2976 + 125β)]} .(20) For T = 0 this line T R leads to the critical capacity α(T = 0) = 4/27 = 0.14814815, as seen in Figure 1. Below the thick full line, T c , the retrieval states become globally stable. At this point we remark that, compared with the standard Hopfield model, there is no reentrance behaviour in the transitions since the spherical model is (marginally) replica symmetric. Furthermore, the region of global stability for the retrieval solutions in the spherical model is much larger. Macroscopic Dynamics. In this section we study the dynamics of the model. Since the neurons are continuous, we assume the relaxational dynamics to be given by the set of Langevin equations ∂σ i (t) ∂t = −µ(t)σ i (t) − δH[σ] δσ i (t) + η i (t) + θ i (t)(21) where the first term on the rhs controls the fluctuations of the neurons, η i (t) is gaussian noise with as first two moments η i (t) = 0 , η i (t)η j (t ′ ) = 2T δ ij δ(t − t ′ )(22) and θ i (t) is an external perturbation field. We discuss this dynamics using the generating functional approach [12,13]. In a straightforward way the following generating functional is introduced Z[ψ] = D[σ, σ] exp N i=1 dtψ i (t)iσ i (t) + A[σ, σ](23) with the action A[σ, σ] given by A[σ, σ] = N i=1 dt − T σ 2 i (t) + i σ i (t) ∂σ i (t) ∂t + µ(t)σ i (t) − θ i (t) + δH[σ] δσ i (t)(24) where the ψ i (t) are the generating fields, σ i (t) are conjugate variables and D[σ, σ] is the measure in path space. From this expression all physical quantities of interest can be computed by taking derivatives with respect to the generating and the external perturbation fields. Assuming that the initial neuron configuration is correlated with only one pattern, i.e. the condensed one, and averaging over the disorder we arrive, after some algebra, at the following effective single site dynamics in the thermodynamic limit ∂σ(t) ∂t = −µ(t)σ(t)+ m(t)+u 0 m 3 (t) ξ 1 +θ(t)+α dt ′ [(1I−G) −1 G](t, t ′ )σ(t ′ )+φ(t) (25) where φ is a colored noise with mean value and variance φ(t) ⋆ = 0 , φ(t)φ(t ′ ) ⋆ = 2T δ(t − t ′ ) + α[1I − G] −1 C[1I − G † ] −1 (t, t ′ ) . (26) In this expression m(t) is the retrieval overlap m(t) = ξ 1 σ(t) ⋆ ξ 1(27) and the matrices C and G are the dynamical order parameters of the problem, i.e. the correlation and response functions C(t, t ′ ) = σ(t)σ(t ′ ) ⋆ ξ 1 , G(t, t ′ ) = ∂ ∂θ(t ′ ) σ(t) ⋆ ξ 1(28) with · · · ξ 1 the average over the condensed pattern and · · · ⋆ the average over the effective noise φ. At this point we remark that in the course of the calculation we see explicitly that the quartic term in the Hamiltonian gives no extensive noise contribution. Taking the perturbation field θ(t) to zero, using the causality properties of the response function and the spherical constraint, C(t, t) = 1, it is possible to write down a closed set of equations for the macroscopic observables specifying the dynamics ∂ ∂t + µ(t) m(t) = m(t) + u 0 m 3 (t) + α t −∞ dt ′ R(t, t ′ )m(t ′ ) ∂ ∂t + µ(t) G(t, t ′ ) = δ(t − t ′ ) + α t t ′ dt 1 R(t, t 1 )G(t 1 , t ′ ) ∂ ∂t + µ(t) C(t, t ′ ) = 2T G(t ′ , t) + α t ′ −∞ dt 1 S(t, t 1 )G(t ′ , t 1 ) + m(t) + u 0 m 3 (t) m(t ′ ) + α t −∞ dt 1 R(t, t 1 )C(t ′ , t 1 )(29) where we have defined the effective correlation function, S(t, t ′ ), and response function, R(t, t ′ ), as S(t, t ′ ) = [1I − G] −1 C[1I − G † ] −1 (t, t ′ ) , R(t, t ′ ) = [(1I − G) −1 G](t, t ′ ) .(30) In order to obtain the stationary state from the equations (29) we assume that, close to equilibrium the order parameters become time translation invariant meaning that the one-time quantities become time independent and the two-time quantities satisfy C(t, t ′ ) = C(t − t ′ ), G(t, t ′ ) = G(t − t ′ ) (and similarly for R and S). Time translation invariance holds when the system is ergodic. Then the correlation and response functions are related through the fluctuation-dissipation theorem (FDT) [3], [4] β∂ τ C(τ ) = G(−τ ) − G(τ ) , β∂ τ S(τ ) = R(−τ ) − R(τ )(31) with τ = t−t ′ , the initial time −∞ and ∂/∂τ denoted by ∂ τ . With these assumptions we can write the evolution equation for the correlation function in the following way ∂ τ + µ − αβ[1 − S(τ )] C(τ ) + αβ τ 0 dt ′ S(τ − t ′ ) − S(τ ) ∂ t ′ C(t ′ ) = m + u 0 m 3 m + α ∞ 0 dt ′ R(t ′ + τ )C(t ′ ) + S(t ′ + τ )G(t ′ ) .(32) With the conditions C(0) = 1 and β∂ τ C(τ ) τ =0 = −1 we then have for τ → ∞ m + u 0 m 3 m = C(∞) β[1 − C(∞)] − αβS(∞)[1 − C(∞)] .(33) Finally, using the evolution equation for m and lim τ →∞ C(τ ) = q , lim τ →∞ S(τ ) = q [1 − β(1 − q)] 2(34) it is straightforward to arrive at the equilibrium saddle-point equations (17). At this point we remark that following [3] by starting from the evolution equation for the correlation function written as (∂ τ + µ − αβ[1 − S(τ )]) β[1 − C(τ )] − αβ 2 τ 0 dt ′ S(τ − t ′ ) − S(τ ) ∂ t ′ C(t ′ ) = 1 (35) and using the fact that the dynamics is purely relaxational, and hence, that ∂ t C(t) ≤ 0, ∂ t S(t) ≤ 0, we can derive the following condition for ergodicity 1 β[1 − C(τ )] + αβ − µ − µS(τ ) ≥ 0 .(36) This inequality seems to indicate that our system is ergodic in the sense of [3]. In fig. 2 we show some plots of the dynamics (29) obtained by discretizing the equations. Specifically, the overlap order parameter is shown for several values of the loading capacity at temperature zero with m(0) = 1. The behaviour for nonzero temperature is qualitatively the same. We clearly see that below the critical capacity (figure on the left) the system evolves to the stationary solution, while above the critical capacity (figure on the right) m(t) drops to zero. Conclusions In this work we have presented the spherical version of the Hopfield model. In order to have a retrieval phase a quartic interaction has been introduced which does not destroy the spherical character of the model. The thermodynamic phase diagram is qualitatively the same as the one for the standard Hopfield model, except that there is no re-entrance because the system is shown to be (marginally) replica symmetric. Furthermore, the region of global stability for the retrieval solutions is larger. A closed set of equations is obtained for the Langevin dynamics and the stationary limit is shown to correspond to the thermodynamic results. A numerical calculation of the evolution of the retrieval overlap order parameter illustrates these findings. Figure 1 : 1T − α phase diagram for the spherical Hopfield model. Full (dashed) lines indicate discontinuous (continuous) transitions: T SG describes the spin glass transition and T R (19)-(20) indicates the border of existense of the retrieval solutions, T c denotes the thermodynamic transition below which the retrieval states are global minima of the free energy. Figure 2 : 2The overlap order parameter m(t) as a function of time at temperature T = 0 for several values of the capacity: α = 0.041, 0112, 0127 from top to bottom on the left and α = 0150, 0160, 0180 from top to bottom on the right. The horizontal dashed lines correspond to the stationary value taken from theory. AcknowledgmentsWe thank A.C.C. Coolen and R. Kühn for informative discussions. This work has been supported in part by the Fund of Scientific Research, Flanders-Belgium. . T H Berlin, M Kac, Phys. Rev. 86821T.H. Berlin and M. Kac, Phys. Rev. 86, 821 (1952). . A Crisanti, H J Sommers, Z. Phys. B. 87341A. Crisanti and H.J. Sommers, Z. Phys. B 87, 341 (1992). . A Crisanti, H Horner, H J Sommers, Z. Phys. B. 92271A. Crisanti, H. Horner and H. J. Sommers, Z. Phys. B 92, 271 (1993). Violation of the fluctuation-dissipation theorem in glassy systems: basic notions and the numerical evidence. A Crisanti, F Ritort, cond-mat/0212490J. Phys. A (Math. Gen. A. Crisanti and F. Ritort,Violation of the fluctuation-dissipation theorem in glassy systems: basic notions and the numerical evidence, to be published in J. Phys. A (Math. Gen.), cond-mat/0212490. J Hertz, A Krogh, R G Palmer, Introduction to the Theory of Neural Computation. Redwood CityAddison-WesleyJ. Hertz, A. Krogh and R. G. Palmer, Introduction to the Theory of Neural Computation (Addison-Wesley, Redwood City 1995). B Müller, J Reinhardt, M T Strickland, Neural Networks: An Introduction. BerlinSpringerB. Müller, J. Reinhardt and M.T. Strickland, Neural Networks: An Introduc- tion (Springer, Berlin, 1991). Cugliandolo Lecture notes in Slow Relaxation and non equilibrium dynamics in condensed matter Les Houches Session 77. Leticia F , ; J-L Barrat, J Dalibard, Kurchan, cond-mat/0210312Leticia F. Cugliandolo Lecture notes in Slow Relaxation and non equilibrium dynamics in condensed matter Les Houches Session 77 July 2002, J-L Barrat, J Dalibard, J Kurchan, M V Feigel'man eds. cond-mat/0210312. J J Hopfield, Proc. Nat. Acad. Sci. USA. Nat. Acad. Sci. USA792554J.J. Hopfield, Proc. Nat. Acad. Sci. USA 79, 2554 (1982). M Mézard, G Parisi, M A Virasoro, Spin Glass Theory and Beyond (Singapore. World ScientificM. Mézard, G. Parisi and M.A. Virasoro, Spin Glass Theory and Beyond (Sin- gapore, World Scientific, 1987). . . M Th, Nieuwenhuizen, Phys. Rev. Lett. 744289Th.M. Nieuwenhuizen, Phys. Rev. Lett. 74, 4289 (1995). . D Amit, H Gutfreund, H Sompolinsky, Annals of Physics. 17330D. Amit, H. Gutfreund and H. Sompolinsky, Annals of Physics 173, 30 (1987). . P C Martin, E D Siggia, H A Rose, Phys. Rev. A. 8423P. C. Martin, E. D. Siggia and H. A. Rose, Phys. Rev. A 8, 423 (1973). . C De Dominicis, Phys. Rev. B. 184913C. De Dominicis, Phys. Rev. B 18, 4913 (1978)	[]

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